L.E.J.Brouwer Centenary Symposium Proceedings

THE L. E. J. BROUVVER CENTENARY SYMPOSIUM Proceedings of the Conference held in Noordwijkerhout, 8- 13 June, 198 1

Edited by

A. S. TROELSTRA Universiteit van Amsterdam Amsterdam. The Netherlands

and

D. VAN DALEN Rijksuniversiteit Utrecht Utrecht, The Netherlands

I9b2

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD

@

NORTH-HOLLAND PUBLISHING COMPANY, 1982

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PREFACE

From time to time a singularly gifted man with a purpose leaves his mark on the development of mathematics. In our century one of those men was Luitzen Egbertus Jan Brouwer. He made a fundamental contribution to topology which became part of the mainstream of mathematics almost immediately. This was different in the case of his second contribution, involving a reappraisal of mathematics, logic and language, and the relationship between them. Although Kronecker and the French semi-intuitionists, E. Bore1 in particular, had already advocated various forms of constructivism, Brouwer's radical approach was a novelty. At a time that Formalism in mathematics, and Neo-Positivism in the sciences and philosophy were the leading fashion, Brouwer presented his view of mathematics as a mental activity of man -with rather unpopular consequences for the r81e of language and logic. Brouwer's programme -the rebuilding of mathematics according to intuitionistic principles- was laid down in a series of papers, already foreshadowed by his doctoral thesis. Now, a hundred years after Brouwer's birth, his achievements have found recognition and ideas from intuitionism have established themselves firmly in logic and the foundations of mathematics. During the last few decades, intuitionism and constructivism have increasingly been brought into relation with independent disciplines such as recursion theory, proof theory and category theory. This development started already in the thirties; the late Arend Heyting's best-known contribution to intuitionism, to wit the formalization of intuitionistic logic and mathematics, were soon followed by proof-theoretic researches (Gentzen, GEdel) and the introduction of topological and algebraical semantics (Stone, Tarski). The study of realizability, initiated by Kleene, and functional interpretations, such as for example GEdel's Dialectica interpretation, linked the study of intuitionistic formalisms with recursion theory. The first objects of metamathematical study were intuitionistic logic and arithmetic, but in the sixties the investigation of the intuitionistic continuum was undertaken by, among others, Kleene, Kreisel, Myhill and D.Scott. . In recent years a good deal of attention has also been given to the development of formalisms that could do for constructive and intuitionistic mathematics what axiomatic set theory did for classical mathematics: intuitionistic set theory, theories of operators and classes, and type theories. In their turn these formalisms have led to many metamathematical studies. In the model theory of intuitionistic theories we have lately witnessed the advance of categorical logic and topos theory -initially created by Lawere- which

vi has provided a common generalization of the earlier semantics of Beth and Kripke, and topological semantics. Categorical logic, and in particular topos theory is now studied and used by people with a variety of backgrounds; it provides a unifying and flexible framework for many metamathematical researches. We only mention sheaf models over Grothendieck topoi and realizability toposes as examples. Thus methods ultimately deriving from algebra and algebraic geometry are introduced in the study of intuitionism. Alongside intuitionism various other schools of constructive mathematics, at least partly deriving their initial impetus from intuitionism, albeit with a somewhat different outlook, have developed: constructivism in the style of Markov, and E . Bishop's form of constructivism. Last, but not least; it should not be forgotten that also in "ordinary" mathematics there are problems and results directly connected with the search for constructive solutions; see for example the contributions of C. Delzell, G . Kreisel and A. MacIntyre in this volume. The preceding brief and incomplete sketch i s enough to illustrate that there is a wide range of activities connected with intuitionism and constructive mathematics, and so it seemed a fitting tribute to the founding father at his centenary t o bring together the rich diversity of constructivists into a memorial symposium. The opening address of the conference was presented by his excellency the minister of education, dr. A . Pais. The organizing committee consisted of A.S. Troelstra(chairman), D. van Dalen (secretary), K. Koymans(treasurer), M. Euwe and G. Renardel de Lavalette. Max Euwe, himself a student and friend of Brouwer, has made crucial contributions towards the organization of the congress; we lament the l o s s that his death in November last year has brought u s . The conference, held under the auspices of the Royal Dutch Academy of Sciences and the Wiskundig Genootschap (Dutch Mathematical Association) was most generously supported by VOLMAC Automation at the occasion of its third lustrum; VOLMAC's contribution made the organization o f the symposium possible. We are also grateful for financial support by AMEV, and for a contribution of Shell Nederland towards the social activities. Valuable assistance has been given by North-Holland Publishing Company. The organizing committee wishes to express its gratitude to the many individuals who have contributed towards the success of the meeting, the participants, the staff of the Conference Centre De Leeuwenhorst, and in particular Doke van Dalen who ran the office. The Teyler Foundation has kindly opened its museum to u s , and the City of Amsterdam has commemorated Brouwer's Centenary at a reception at the Historical Museum. These Proceedings contain most of the invited talks, some o f the contributed talks, and in addition some contributed papers by persons who were unable to attend the meeting. The editors.

THE LEJ. BROUWER CENTENARY SYmOSIuM AS. TkoeLFlra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

1

THE TYPE THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY: CHOICE PRINCIPLES

P e t e r Aczel Department of Mathematics Manchester University England

In an e a r l i e r paper I gave an i n t e r p r e t a t i o n of a system CZF of c o n s t r u c t i v e s e t theory w i t h i n an extension of Martin-Lgf's types.

i n t u i t i o n i s t i c theory of

I n t h i s paper some a d d i t i o n a l axioms, each

a consequence of the axiom of choice, a r e shown t o hold i n t h e i n t e r p r e t a t i o n .

The mathematical

deductions a r e presented i n an informal, but I hope rigorous s t y l e . INTRODUCTION

The axiom of choice does n o t have an unambiguous s t a t u s i n constructive mathematics.

the one hand i t i s s a i d t o be an immediate consequence of t h e

On

Any proof of

constructive i n t e r p r e t a t i o n of t h e q u a n t i f i e r s . Vx cA3y E B F(x,y)

must y i e l d a function

f

This i s c e r t a i n l y t h e case i n Martin-Lb;f's

E

A

+

B

such t h a t 'Vx c A F ( x , f ( x ) ) .

i n t u i t i o n i s t i c theory of types.

On

t h e o t h e r hand, from t h e very e a r l i e s t days, t h e axiom of choice has been c r i t i c i s e d as an excessively non-constructive p r i n c i p l e even f o r c l a s s i c a l s e t theory.

Moreover, i n more r e c e n t y e a r s , i t has been observed t h a t t h e f u l l axiom

of choice cannot be added t o systems of c o n s t r u c t i v e set theory without y i e l d i n g constructively unacceptable cases o f excluded middle (see e.g. Diaconescu C19751). In Myhill C19751 a system o f c o n s t r u c t i v e s e t theory was put forward as a s u i t a b l e s e t t i n g f o r t h e s t y l e of c o n s t r u c t i v e mathematics pursued by Bishop and h i s school ( s e e Bishop C19671 and Bridges C19791 and a l s o t h i s proceedings). y l h i l l argued informally t h a t t h e axiom constructively acceptable. CZF

DC

of dependent choices was

Aczel C19781 contains an i n t e r p r e t a t i o n o f a system

of c o n s t r u c t i v e s e t theory t h a t i s c l o s e l y r e l a t e d t o Myhill's system.

claimed t h e r e , without proof, t h a t

w i l l b e shown i n 15 of t h i s paper.

DC

was t r u e i n t h e i n t e r p r e t a t i o n .

I

This

I n 1 7 of Aczel C19781 I a l s o put forward an

axiom c a l l e d t h e p r e s e n t a t i o n axiom, which I thought t o be a p l a u s i b l e a d d i t i o n t o constructive s e t theory. interpretation.

But t h e r e I was unable t o v e r i f y i t s t r u t h i n t h e

A fundamental aim of t h i s paper i s t o show t h a t t h e p r e s e n t a t i o n

P.H.G. ACZEL

2

axiom i s indeed t r u e i n the i n t e r p r e t a t i o n .

In f a c t a strengthening,

IICI-PA,

of the p r e s e n t a t i o n axiom w i l l be v e r i f i e d i n 57. The i n t e r p r e t a t i o n of

given i n Aczel [1978] was c a r r i e d out w i t h i n a

CZF

n a t u r a l extension of Martin-Lb;f's Martin-Lof

C19751.

i n t u i t i o n i s t i c theory of types a s presented i n

Type theory i s intended t o be a fundamental conceptual

framework f o r t h e b a s i c notions of c o n s t r u c t i v e mathematics.

For t h i s reason I

beliege t h a t t h e i n t e r p r e t a t i o n of c o n s t r u c t i v e s e t theory i n type theory can l a y claim t o give a good c o n s t r u c t i v e meaning t o t h e s e t t h e o r e t i c a l notions.

In

view of t h i s i t i s very n a t u r a l t o explore f u l l y which s e t t h e o r e t i c a l axioms a r e true i n the interpretation.

To do t h i s I have found i t convenient t o develop a

f l e x i b l e informal s t y l e f o r p r e s e n t i n g deductions i n type theory.

Moreover, i f

type theory i s t o be a p r a c t i c a l v e h i c l e f o r t h e p r e s e n t a t i o n of mathematical deductions of c o n s t r u c t i v e mathematics i t s e l f , such an informal s t y l e w i l l be essential.

So I have t r i e d t o w r i t e t h i s paper r e l y i n g only on the informal

d e s c r i p t i o n of t h e type t h e o r e t i c notions given i n 5 1 .

There i s the danger t h a t

I may be c r i t i c i s e d f o r lack of r i g o u r , as i t may not be always transparent t o

the reader how each s t e p could b e formalised.

But my primary aim has been t o

present t h e mathematical r e s u l t s s t r i p p e d of excessive formalism.

Of course,

while w r i t i n g t h i s paper, I have had i n mind a formal language f o r type theory such a s t h a t presented i n Martin-Lgf C19791, and t h e reader may f i n d i t h e l p f u l t o r e f e r t o t h a t paper both f o r t h e l i s t of r u l e s of t h e formal language and f o r the explanations of fundamental notions.

Martin-Lof

119751 and 119791 give some

information concerning t h e pre-history of the framework of type theory used here. The reader should a l s o r e f e r t o Aczel C19781 f o r a more extensive discussion of than w i l l be given here.

CZF

intended t o be self-contained.

Apart from these t o p i c s t h e present paper i s I n view of t h i s , t h e paper s t a r t s with an informal

d e s c r i p t i o n of type theory i n 5 1 , and i n 52 and 53 goes on t o present t h e i n t e r p r e t a t i o n of

CZF

i n t h e framework described i n 5 1 .

o t h e r s e c t i o n s the discussion i n 54 c o n s t r u c t i v e s e t theory.

In c o n t r a s t t o t h e

takes place i n an informal framework f o r

That d i s c u s s i o n i s concerned with formulating t h e axioms

t h a t w i l l be v e r i f i e d i n t h e remaining s e c t i o n s .

Dependent choices i s v e r i f i e d

i n 55, and t h i s s e c t i o n a l s o introduces some e s s e n t i a l i d e a s t o be used i n t h e l a s t two s e c t i o n s .

57 respectively.

The axioms

IIZI-AC

and

IICI-PA

w i l l be v e r i f i e d i n 56

and

I n t h e last s e c t i o n i t turns out necessary t o use a method not

contemplated i n Aczel C19781 o r a v a i l a b l e i n t h e formal languages of Martin-LGf C19751 and C19791, although t h e p o s s i b i l i t y of the method i s mentioned i n Martin-LSf

C19751.

This method involves making e x p l i c i t the conception of t h e

type U of small types as a type i n d u c t i v e l y s p e c i f i e d by a s p e c i f i c l i s t of r u l e s f o r forming s m a l l types. The method i s t o allow d e f i n i t i o n s of functions on

U

by a t r a n s f i n i t e recursion following t h e inductive generation of the small

Constructive set theory

types.

3

The method i s used t o c o n s t r u c t a s u i t a b l e r e p r e s e n t a t i o n of each small

type a s an element of t h e type

of i t e r a t i v e s e t s .

V

A s pointed out i n Aczel

C19781 such a r e p r e s e n t a t i o n i s what i s needed t o v e r i f y t h e p r e s e n t a t i o n axiom. A s with any a c t i v e research program t h e type t h e o r e t i c approach t o c o n s t r u c t i v e

mathematics has been under steady development over t h e years and the publications I have r e f e r r e d t o only represent s t a g e s i n t h a t development.

There have been

some s i g n i f i c a n t changes between t h e s t a g e s represented by Martin-Lgf Aczel 119781 and the more recent s t a g e s represented by Martin-LGf paper.

A minor n o t a t i o n a l change has been t h a t t h e symbols

interchanged t h e i r meaning. i n Martin-Lgf the type

and

V

have

A more s i g n i f i c a n t change has been t h e introduction

C19791 of t h e new form of type

(Wx

A)BCxI.

E

of i t e r a t i v e s e t s can be defined simply a s

V

U

119751 and

C19791 and t h i s

With t h i s new form

(Wx

E

U)x.

But i n t h i s

paper the new form w i l l not b e used, and i t s e f f e c t on t h e i n t e r p r e t a t i o n of constructive s e t theory w i l l b e l e f t t o another paper. above d e f i n i t i o n of

I have used t h e n o t a t i o n

V

f o r the i t e r a t i v e s e t s y d o l i s e d by

I

{b(x)

x

E

I n keeping with the

sup(A,b)

or

(supxcA)b(x)

A1 i n Aczel C19781.

Perhaps t h e In

most s i g n i f i c a n t change has been t h a t involving t h e treatment of e q u i l i t y .

Martin-Lgf 119751 t h e r e i s one r e l a t i o n of d e f i n i t i o n a l e q u a l i t y which i s defined t o be t h e equivalence r e l a t i o n which i s generated by the p r i n c i p l e s t h a t a definiendum i s always d e f i n i t i o n a l l y equal t o i t s d e f i n i e n s and t h a t d e f i n i t i o n a l e q u a l i t y is preserved under s u b s t i t u t i o n .

I n t h e p r e s e n t version of type theory

each notion of object must c a r r y with i t an e q u a l i t y r e l a t i o n which i s not So t h e r e i s an

n e c e s s a r i l y t o be understood a s a d e f i n i t i o n a l e q u a l i t y r e l a t i o n . e q u a l i t y r e l a t i o n f o r types, and a l s o f o r each type A.

f o r elements of type

A,

a r e l a t i o n of e q u a l i t y

Martin-L6f w r i t e s a = b c A

to express the judgement t h a t

a

and

b

example i f

f

and

g

A.

a r e equal elements of type

form of type has i t s own c r i t e r i a f o r when such a judgement holds. a r e elements of t h e type

A

-t

B

Each

So f o r

of functions'from

A

toB,

f = g E A + B means t h a t

f

and

g

a r e e x t e n s i o n a l l y equal, i . e .

I n p r a c t i c e when a judgement the context t h a t

a

and

b

a

=

b

E

A

f ( x ) = g(x)

E

B

for

x E A.

i s made i t i s usually e i t h e r c l e a r from

a r e elements of type

A

o r e l s e i t does not matter

exactly what t h e type of a and b i s . For t h i s reason I propose t o follow t h e s t a n d a r d convention of w r i t i n g simply

a = b.

Whenever two expressions a r e d e f i n i t i o n a l l y

equal, and i n a given context one of them r e f e r s t o an o b j e c t of some s o r t then the o t h e r expression w i l l r e f e r t o an equal object of t h a t s o r t .

For t h i s reason

it i s s a f e t o follow t h e standard convention, when making d e f i n i t i o n s , cf simply

4

P.H.G. ACZEL

writing equalities.

This w i l l b e done here.

The mathematical r e s u l t s i n t h i s paper were obtained while preparing a s e r i e s of t a l k s on t h e type t h e o r e t i c i n t e r p r e t a t i o n of c o n s t r u c t i v e s e t theory given i n Munich i n October 1980.

I am g r a t e f u l t o Prof. Schwichtenberg f o r h i s i n v i t a t i o n

which was t h e spur t o a f r e s h look a t t h e topic. The contents of t h i s paper do n o t e x a c t l y r e f l e c t t h e t o p i c of my t a l k a t the I f e l t unable t o w r i t e on t h a t t o p i c before having completed t h i s

conference.

paper, and I am g r a t e f u l t o t h e e d i t o r s f o r accepting t h i s s u b s t i t u t e . 5 1 . AN INFORMAL DESCRIPTION OF TYPE THEORY SOME NOTATION

If

...,

bCxl,

v a r i a b l e s then

i s an expression and

xnl

..., xn)

(xl,

bCxl,

xl,

..., xnl

..., x

i s a non-repeating l i s t of

w i l l denote t h e n-place function

f

having d e f i n i n g equation f(xl,

So whenever

bCal,

..., an]

... , xn)

... , xnl.

= bCxl,

r e f e r s t o an o b j e c t of some s o r t then

f(al,

..., an)

will r e f e r t o an equal object of t h a t s o r t and we can w r i t e f(al,

..., an] ..., x i n

..., an)

..., an].

= bCal,

i s t h e r e s u l t of simultaneously s u b s t i t u t i n g

Here bCal, for xl,

bCx,,

..., x n l ,

ais

.**’

making s u i t a b l e changes i n t h e bound

v a r i a b l e s when necessary.

If

c

i s t h e ordered p a i r

components

a

and

b

( a , b)

then

p(c)

and

q(c)

respectively.

TYPES AND THEIR ELEMENTS

1.1.

The fundamental notions of type theory a r e type

and i f

A

i s a type element of A.

If

a

i s an element of t h e type

A

then we s h a l l w r i t e a

E

A.

w i l l be t h e two

an

5

Constructive set theory

1.2.

We s t a r t our survey of the forms of type by considering the f a m i l i a r forms A-+B

A

-+

B

i s t h e type of functions

A

X

B

i s t h e type of p a i r s

A x B

(x)bCxl

( a , b)

N.

such t h a t

such t h a t

a

bCxl A

E

E

and

B

b

for

x

B.

N

E

and

A,

E

i s the

type of n a t u r a l numbers and i s i n d u c t i v e l y s p e c i f i e d using t h e r u l e s n e N s(n) E N

OeN Associated with

N

is t h e noethod of d e f i n i t i o n by recursion over

example, given a type

a

C,

and

C

E

f

N x C

E

-+

C

For

N.

we may define

h e N

by

C

-+

recursion so t h a t h(0) = a h(s(n)) We s h a l l w r i t e

a

on

and

1.3.

f

f(n, h(n))

=

R(n, a , f )

for

n

for

N.

E

i f we wish t o make t h e dependence of

h(n)

h(n)

explicit.

The notions w e have introduced so f a r already s u f f i c e f o r t h e type But h e r e w e

s t r u c t u r e of Godel's p r i m i t i v e r e c u r s i v e functions of f i n i t e types.

wish t o have a r i c h e r notion of type and i n p a r t i c u l a r we wish t o have types t h a t For t h i s reason i t i s u s e f u l t o have

a r e beyond the l e v e l of t h e f i n i t e types. types whose elements a r e themselves types.

While a type of a l l types i s

unreasonable i t i s s e n s i b l e t o have a type

U

s u i t a b l e e x t e n t t h e notion of type i t s e l f .

Reflecting on t h e forms of type we

have considered so f a r leads us t o have A, B

U

E

and a l s o t o have

N

E

A

+

U

t h a t a r e b u i l t up out of

a r e l a r g e types.

by recursion functions such as

B E U

and

A x B

E

U

whenever

Note t h a t we do not wish t o have

U.

i s n a t u r a l t o c a l l types i n U

of types t h a t r e f l e c t s t o a

small types and types such as F

E

+

N

N

Using the type

U

E

U

It

U.

U

and

U

one may form

-+

U

where

+ U

F(0) = N F ( s ( n ) ) = F(n) F

for

i s an example of a family of types.

is a function 1.4.

F

t h a t a s s i g n s a type

n

A.

If

be w r i t t e n

(Cx

E

II(A, F)

(nx

E

A)BCxl

and

i s t h e type of functions

and t h e d i s j o i n t union

C(A, F)

N.

In general a family of types over a type A F(a) t o each

We now c o n s i d e r t h e forms of type

family of types over t h e type

E

II(A, F) F

is

A)BCxl

(x)bCxl

and

(x)BCxl

a

E

A.

E(A, F)

respectively. such t h a t

i s t h e type of p a i r s

where

is a

F

then t h e s e types w i l l a l s o bCxl

The Cartesian product E

( a , b)

F(x)

for

such t h a t

x

A,

E

a

E

A

P.H.G. ACZEL

6 b

and

F(a).

E

forms when

The forms

A

B

+

i s t h e function

F

and

A x B

(x)B

a r e t h e s p e c i a l c a s e s of t h e s e new

having as c o n s t a n t value t h e type

A

+

B = (nx

E

A)B,

A

x

B = (Cx

E

A)B.

L e t u s consider more c l o s e l y t h e elements of t h e new forms of type.

f

E

II(A, F)

then

p(c)

then E

q(c)

A,

by r e c u r s i o n on considered. where a

E

N

F(0)

f

for

F(p(c))

x

and

E

c

A

f = (x)f(x).

(p(c), q(c)).

=

If

If c

(nz

E

h

N + C

E

was d e f i n e d from

a

and

C

E

h

We may u s e t h e same e q u a t i o n s t o d e f i n e F)) F ( s ( p ( z ) ) ) ,

X(N,

E

where

F

t o r e f l e c t them i n t h e type

E

TI(N,

TI(A, F)

and

L(A, F)

t h e l a s t two having s p e c i a l i n s t a n c e s

1.6.

and

No

I(A, a , b)

It has an element

c

E

If

1.7.

Z(A, a, b ) .

If

and

A

of t h e two types. f o r each

b

E

F

has t h e

A.

A -+ B

N, U, TI(A, F)

and

A x B.

B.

and

Our p r e s e n t I ( A , a , b),

V.

i s a type provided t h a t

r

provided

a = b.

a

So

and

b

c = r

a r e elements of t h e type

and

a = b

A.

whenever

The s i g n i f i c a n c e of t h i s form w i l l perhaps become c l e a r e r l a t e r .

i s s m a l l then so i s

A

from

we should remember

survey of forms of type w i l l b e completed by c o n s i d e r i n g t h e forms A + B,

F)

So we have t h a t b o t h t h e s e types a r e small

U.

So f a r we have considered t h e p r i m i t i v e forms of type

C(A, F ) ,

x C + C

i s a family of types

i s a family o f small types over t h e s m a l l type

F

N

f E

C.

Having introduced t h e new forms o f type

whenever

C(A, F)

E

The method of d e f i n i t i o n

The e a r l i e r formulation i s now simply t h e s p e c i a l c a s e when

N.

constant value

1.5.

and

i.e.

a p p l i e s i n a s l i g h t l y m r e general c o n t e x t than t h a t p r e v i o u s l y

Previously

and

F(x)

E

E

w a s a type.

C

over

f(x)

B,

I(A, a, b).

a r e types then

A + B

I t has an element

i(a)

B

If

f

(TIx

E

i s a family of types over

A

E

A) C ( i ( x ) )

+B

i s a t y p e c a l l e d t h e d i s j o i n t union f o r each and

g

E

a

A

E

(IIy

then we have a f u n c t i o n

E

and an element

B) C ( j ( y ) ) h

E

j(b)

where

TI(A + B , C)

C

defined

by cases s o t h a t

h(c) and

h(i(a)) = f(a)

for

a E A

h ( j ( b ) ) = g(b)

for

b E B

w i l l a l s o be w r i t t e n

g

explicit.

If

A

D(c, f , g) and

B

i f w e wish t o make i t s dependence on

a r e small types then s o i s

A + B.

f

I

Constructive set theory

1.8. Ro

No

i s t h e empty type.

n(No, C).

E

For

Whenever

...

k = 1, 2 , 3 ,

is a family o f types over

C

t h e k-element

type

No

then

can be d e f i n e d as

Nk

follows N 1 = I(N, 0, 0 ) N2 = N1 + N1 Ng

=

+ N1

N,

...

etc. The type

If h

al

E

Nk

..., ak

C(lk),

n(Nk, C )

E

has elements

E

lk,

..., kk

C(kk)

where

where

i s a family of types over

C

Nk

then

can be d e f i n e d s o t h a t

h ( k ) = ak. k I f we wish t o make t h e dependence of write

\(c,

1.9.

..., %)

al,

for

h(c).

F i n a l l y we come t o t h e type

V

h(c)

on

Each

al,

\

...,

ak

e x p l i c i t then we

can be defined a s follows

of ( i t e r a t i v e ) s e t s .

This type is

inductively specified via the rule

A E U b c A + V sup(A, b) E V We s h a l l a l s o w r i t e No

E

U

and

Ro

s h a l l abbreviate

E

No

9.

(supx +

V

E

A)b[xl

for

sup(A, ( x ) b [ x l ) .

w e c e r t a i n l y have t h e s e t

More g e n e r a l l y , given s e t s

al,

Note t h a t because

sup(No, R o ) ,

... , g,

which we

we may form t h e s e t

P.H.G. ACZEL

8 (supx

k % (x, a,,

N )

E

..., a,)

which w e s h a l l abbreviate

{a,,

..., ak}.

In t h i s

way the h e r e d i t a r i l y f i n i t e s e t s can be represented a s elements of t h e type

i s t h e following

kssociated with t h e r u l e i n d u c t i v e l y s p e c i f y i n g t h e type

V

method of d e f i n i t i o n by t r a n s f i n i t e recursion on

d

d(A, b , e )

function such t h a t e

E

(nx

h

E

n(C, V)

for

A

E

A)C(b(x)),

E

If

for a l l

A

E

i s a family of types over

C

i s a three place b

U,

b

E

A

+

+ V

When we wish t o make t h e dependence of

V.

e x p l i c i t i t w i l l be w r i t t e n

T(c, d).

Note t h a t

V

and

h(c)

d

on

must be considered a l a r g e

type a s t h e r u l e used i n s p e c i f y i n g i t makes e x p l i c i t reference t o

1.10.

A

E

then we have

V,

defined so t h a t

and

U

where

C(sup(A, b ) )

E

V.

V.

Let us review what we have s a i d concerning t h e type

elements a r e themselves types, c a l l e d the smll types.

U.

It i s a type whose

U.

The following schemes

express our r u l e s f o r forming s m a l l types

N0eU

N E U

FEA-+U II(A, F) E U

A E U F E A + U Z(A, F) E U

A E U B E U A + B c U

A E U a , b E A I(A, a , b) E A

A E U

*

In t h i s paper we s h a l l need t o consider t h e s e r u l e s as giving an i n d u c t i v e s p e c i f i c a t i o n of t h e type

This means t h a t we have t h e following method of

U.

d e f i n i t i o n by t r a n s f i n i t e recursion on t h e method allows us t o form h and

a, b

E

E

II(U,

If

U.

C)

C

i s a family of types over

such t h a t f o r

A, B

E

U,

F E A

-+

U

U

A

In t h e s e equations

dN0

E

C(N ),

place functions such t h a t i f

A, B

E E

C(N)

U, F

E

and

A

+ U

d,, and

dZ, d+

a, b

E

and

dI

A

then

a r e four

Constructive set theory

for

c

E

C(A),

d

E

C(B)

and

e E (nx

E

A)C(F(x)).

PROPOSITIONS AS TYPES

LOGICAL NOTION

TYPE THEORETIC EXPLICATION

proposition

type

proof of A

element of A

A is true

A has a n element

A 3 B

A-+B

A & B

A X B

A V B

A + B

A E B

(A + B) x (B + A)

1

NO A + N0 I(A, a , b)

i A a =A b

FUNDAMENTAL THEOREM.

(vx

E

A)B[xl

(nx

E A)BCxl

(3x

E

A)BCxl

(Ex

E

A)B[xl

For every i n s t a n c e i n type t h e o r y of a n a t u r a l deduction

rule for i n t u i t i o n i s t i c p r e d i c a t e c a l c u l u s with e q u a l i t y , i f t h e premises a r e t r u e then s o i s t h e conclusion. Rather than give a d e t a i l e d formulation and proof of t h i s r e s u l t I s h a l l j u s t

9

10

P.H.G. ACZEL

examine a s e l e c t i o n of the r u l e s .

IMPLICATION INTRODUCTION

If x

E

B

i s t r u e on the assumption t h a t

A

so that

(x)bCxl

A

E

-f

B

is t r u e then t h e r e i s

A

bCxl

E

for

8

and hence t h e conclusion i s t r u e .

IMPLICATION ELIMINATION ( i .e. m d u s poneus) A 2 B B

I f the premises a r e t r u e then t h e r e a r e

A

f E A

and

+ B

a

so t h a t

A

E

f(a)

E

B

and hence t h e conclusion i s t r u e . UNIVERSAL QUANTIFICATION INTRODUCTION

Tx (Vx If

is true for

BCxl

(x)bCxl

E

(nx

E

x

A)BCxl

E

A

E

A1

BCxl E A)BCxl

then t h e r e i s

bCx1

E

B[xl

for

x

E

A

so t h a t

and hence t h e conclusion i s t r u e .

EQUALITY ELIMINATION a =A b

B[al

BCbl I f the f i r s t premise i s t r u e t h e r e i s some element of a = b

so that

Bra]

=

Cbl.

an element and hence s o does 1.13.

I ( A , a , b)

and hence

I f t h e second premise i s a l s o t r u e then

BCbl

BCal

ias

so t h a t t h e conclusion i s t r u e .

I n a d d i t i o n t o the purely l o g i c a l p r i n c i p l e s considered i n t h e fundamental

theorem t h e r e a r e a number o f o t h e r p r i n c i p l e s t h a t can be j u s t i f i e d i n type theory and w i l l be needed i n l a t e r s e c t i o n s of t h i s paper.

We l i s t them below

with an i n d i c a t i o n of why they a r e c o r r e c t . 1.14.

If

F

is a family of propositions ( i . e . types) over t h e type A i t i s

natural to c a l l

F

a species over

A.

11

Constructive set theory

N-INDUCTION.

For every s p e c i e s

F

over

N

V-INDUCTION.

For every s p e c i e s

F

over

V

(VA

U)(Vb

E

A

E

-f

V)[(Vx E A)F(b(x)) E V)F(a)

F(sup(A, b ) ) ]

3

(Va

U-INDUCTION.

For every s p e c i e s

F

F(NO)

F(N)

Oc

,

Qz

QTI

(VA where

U

over

E

@I

@+

U)F(A)

is

0,

is like

(VA

E

QII

with

U)(VB E A

C

(VA

-f

U)[F(A) & (Vx

replacing

E

U)(VB

E

TI

and

A)F(b(x))

E

Q+

U)[F(A) & F(B)

3

and

3

F(Il(A,B))l,

are

F(A+B)I

and

These p r i n c i p l e s a r e j u s t i f i e d by s u i t a b l e d e f i n i t i o n s by r e c u r s i o n . N-induction, i f t h e premises a r e t r u e then t h e r e a r e b E (TIn

E

N)(F(n)

+

F(s(n))

and t h e conclusion i s t r u e . V-induction then

1.15.

If

B

Similarly i f

d

THE AXIOM OF CHOICE (AC)

F(0)

Thus f o r

and

i s an element of t h e premise of

(a)T(a, (A, b, e ) d ( A ) ( b ) ( e ) )

F(x, y )

E

so that

i s a family of types over t h e type

f u n c t i o n such t h a t

a

is a proposition f o r

i s a n element of t h e conclusion. A

and

X E

A

F

i s a two p l a c e

and

y

E

B(x)

then

12

P.H.G. ACZEL

THE DEPENDENT CHOICES AXIOM (DC)

If

f

is an element of the premise of

h = (x)q(f(x)).

Then

g

E

II(A, B)

i s an element of the conclusion. If

AC

and So

Then

g

Now l e t

C

E

a

+

E

C,

A

where and

b

is

C

B(a).

E

X(A, B),

then l e t (Vx

E

E

g = (x)p(f(x))

A)F(x, g ( x ) )

and

so that

(g, h)

i s correct.

AC

i s an element of t h e premise of

f

h

DC

then l e t

and

k

E

(nu

C)F(p(u), p ( g ( u ) ) ) .

E

Then by recursion over

e

we may d e f i n e

N

E

N

+

so t h a t

e ( 0 ) = (a, b) e(s(n)) = g(e(n))

i s an element of

(32

E

N

+

A)G(a, z ) .

words we have t h e conclusion of 1.16. f, g

EXTENSIONALITY. E

II(A, B)

then

If

B

n

E

N.

Thus given an element of

has an element we have found an element of

B(a)

for

(32

E

A

such t h a t

N + A)G(a, z ) .

In o t h e r

DC.

i s a family of types over the type A

and

C

Constructive set theory

If c is an element of the premise then c(a)

E

13

I(B(a),

f(a),

g(a))

for a

E

A

that f(a) = g(a) for a E A and hence f = g. As f = g the type I(II(A, B), f, g) has the element r and hence the conclusion is true.

so

1.17. Let F be a species over the type C. X-EXISTENCE. then

If C is X(A, B)

(32

where B is a family of types over the type A

( 3 E A)(%

C)F(z)

E

B(x))F((x,

E

y))

+-EXISTENCE. If C is A + B where A and B are types then

%-EXISTENCE.

If C is Nk (32

where in case k =

0

for E

k =

C)F(z)

I

0,

1,

F(lk) v

the right hand side is

...

then

... v

F(kk),

1.

In each of these equivalences the implication from right to left involves a simple direct application of the existence introduction rule. For the other direction let us just consider +-existence. If a E A then

where D is the right hand side of the +-existence equivalence. Hence (IIx

E

A) (F(i(x))

has an element

+

so

D)

has an element.

Similarly (Ily

that using definition by cases

element. So (Vz E A + B)(F(z) (32 E A + B)F(z) 3 D is true.

3

D)

E

B)(F(j(y))

(IIz E A + B)(F(z)

+

D)

+

D)

also has an

is true, and using intuitionistic logic

A WARNING

Of fundamental importance in understanding type theory is an awareness of the distinction between the notions of judgement and proposition. The distinction is critical and attempts to avoid it are liable to lead to confusion. Nevertheless, from the practical point of view it seems convenient to leave the distinction implicit inour informal deductions. Before doing so it may be worthwhile to give the distinction our explicit attention.

P.H.G. ACZEL

14

Examples of judgements a r e N i s a type, 0

E

N,

O = O E N ,

s(x)

E

for

N

x

E

N.

Examples of propositions a r e

Note t h a t each proposition false i f

A + No

when f a l s e .

A

i s a type, which i s t r u e i f i t has an element and Such a p r o p o s i t i o n is a meaningful o b j e c t even

has an element.

On t h e o t h e r hand judgements a r e n e c e s s a r i l y c o r r e c t a s such, and an

i n c o r r e c t l y formed judgement i s meaningless. Martin-LSf's

formal language has a system of f i n i t a r y r u l e s f o r deriving

judgements.

This i s i n c o n t r a s t t o t h e standard formal systems (e.g.

or

CZF) which involve f i n i t a r y r u l e s f o r deriving propositions.

for

HA

Nevertheless,

as we have seen, the l o g i c a l notions a r e represented i n type theory and t h e standard r u l e s f o r d e r i v i n g propositions a r e a l s o represented i n type theory. So, conceptually, t h e r e a r e two d i s t i n c t l e v e l s of d e r i v a t i o n .

There is t h e

fundamental l e v e l where judgements a r e derived, and t h e r e i s t h e secondary l e v e l concerned with t h e d e r i v a t i o n of propositions.

In our informal p r e s e n t a t i o n t h e

d i s t i n c t i o n between these l e v e l s w i l l n o t always be e x p l i c i t .

For example, we

have the notion of ( i t e r a t i v e ) s e t . and i n l a t e r s e c t i o n s we s h a l l make a l o t of use of t h e notion of an i n j e c t i v e l y presented set. a set is simply t h e judgement

a

E

V.

a is

Now the statement t h a t

On t h e o t h e r hand i f

a

E

V

then t h e

a i s i n j e c t i v e l y presented i s a p r o p o s i t i o n , which may be f a l s e .

statement t h a t

In following t h e arguments i n t h i s paper, which a r e presented i n a combination of the English language and symbolic expressions, i t i s necessary t o be aware of t h i s distinction. levels.

I n c e r t a i n cases t h e r e i s an easy i n t e r p l a y between t h e d i f f e r e n t

For example i f

the proposition

a

and

( a =A b ) ; i . e .

a c o r r e c t l y formed j u d g e w n t . proposition t h a t t h e type formed.

a = b A

i s true.

I(A, a , b)

b

a r e elements o f t h e type

t h e type

I ( A , a , b).

I f i t i s then

r

E

A

I ( A , a , b)

Conversely i f we know t h a t

then we may form

a = b

Now

a =A b

has an element, then the judgement

E

A

may not b e

s o t h a t the

is true, i.e.

a = b

E

A

can be

Constructive set theory

15

82. V AS A TYPE OF EXTENSIONAL SETS In t h i s s e c t i o n we d e f i n e t h e s p e c i e s on

of e x t e n s i o n a l e q u a l i t y , and

V x V

e x t e n s i o n a l membership, and show t h a t v a r i o u s set t h e o r e t i c a l p r o p e r t i e s hold. axiom system

CZF

An

f o r c o n s t r u c t i v e s e t theory w i l l be formulated i n t h e n e x t

s e c t i o n and t h e work i n t h i s s e c t i o n shows t h e t r u t h of t h o s e axioms.

In this

s e c t i o n and t h e n e x t we a r e r e t r a c i n g the ground of 8 5 4 - 6 of Aczel 119781, but now within t h e informal s e t t i n g of t h e previous s e c t i o n .

2.1.

THEOREM.

a

E

a

= b.

Moreover

PROOF. for

Define

a

E

a

a

= b.

E

E

2.2.

U

E

V + V

and

b

V + (Zx

a

a

E

and

U

b

A

E

a

a

E

U

A

E

A

-f

where

E

and

U

A

+

V.

b

r

E

whenever

g(a) = a

To show t h a t

a

a for

E

This

2 . 3 . THEOREM. V

V

E

then

a

+

a

= A

to

V

and

so t h a t

V

and

If

F

far all

a

E

then

V

Define

=

sup(A, b)

a

E

I(V, g(sup(A, b ) ) , sup(A, b ) )

V

a V a=A

for

+

and

g = (x)sup(x, x ) . f o r some

we need t o argue for a l l

V , T(a, (x, y , z ) r )

E

A

E

U

and

I ( V , g(a), a)

V.

i s a proposition f o r

E

g(a) = a

Hence, by t r a n s f i n i t e r e c u r s i o n on

THEOREM.

+

by t r a n s f i n i t e r e c u r s i o n on

= q ( T ( a ) ) . Then c l e a r l y

and by t h e above

g(a) =

BCxl

a, 6

E

-

a

and

V.

E

U)(x + V)

E

= sup(A, b)

W e know t h a t

A + V.

PROOF.

to

E

A

where

for

I t remains t o prove t h e f i n a l p a r t of t h e theorem. g

so t h a t If

- -

= sup(~~ a ),

= p ( ~ ( a ) ) and

as follows. b

= sup(A, b)

a

U

E

b c A + V

Also, i f

V.

Then

T

and

A E U

Now l e t

A

a

such t h a t i f

V

a

There a r e one-place f u n c t i o n s a s s i g n i n g

x

E

then d e f i n e

V

i s a s p e c i e s over

then

V

s an immediate a p p l i c a t i o n of V-induction from 1 . 2 . There i s a s p e c i e s on such t h a t

v

x

v

a s s i g n i n g a small p r o p o s i t i o n

(a,: 6 )

P.H.G.ACZEL

16

6) = CVx

(a

a3y

E

E

y)l&"dy

B(x

Define three-place functions

PROOF.

u + (V

-+

U),

E

u3y

E

v w(x)(y),

E

v3x

E

u w(x)(y),

w

E

u

(V

+

U).

-+

Gi(u, v, w) so t h a t

It follows t h a t

T(a, G)

T(sup(A, b), Now d e f i n e

(a

6)

E

-+

U

G) = G(A,

6) = T(a, G)(B)

(a

for

6 , f ) = vy

E

B3x

E

for

a(x L y),

s o t h a t we get t h e d e s i r e d r e s u l t . 2.4.

LEMMA.

For

(i)

a L a,

(ii)

a

6

a , 6, y

3

6 I a,

(iii) a L 6 8 B

PROOF.

V

E

(i) For a

L y 3

E

V

a

y.

a

E

u

for

a, 6 then

U, v

u

E

E

U, z

E

and

E

V

u

-+

V

U, b

E

and

and

V

b, (u)T(b(u), G))

and s i m i l a r l y

Gp(a,

V + U

E

f = (u)T(&(u), G)

and i f

U

V

E

v, w) & G2(u, v, w)).

is a small type f o r

G(u, z, w)

s o that

Gg

v, w) = Vx

i = 1, 2

E

and

G1, Gp

G(u, z, w) = ( v ) ( G l ( u ,

Then f o r

63x E a ( x f y)].

G 2 ( u , v, w) = Vy

G1(u,

w

E

E

V.

for

A

Then f o r

E

a,

B

E

V,

A

-+

V.

'(K

x)a

3

x p

3

KA

c

(x

X)(D

3 XA)

P.H.G. ACZEL

18

2.5. DEFINITION.

For

a, 8 E V

let

(x

8 = (3y

E

a 58 Then b o t h

a

A species

F

E

8 over

and

V

a

58

2.6.

THEOREM.

If

F

V)(F(x)

E

THEOREM.

For

a, 8, y

8).

E

E

V

a, 8

E

V.

if

3

(Vy

E

V)(y

i s an e x t e n s i o n a l s p e c i e s o v e r

PROOF.

2.7.

B ) ( a L y)

a r e small propositions for

i s extensional (Vx

E

= (Vx E a ) ( x

x

3

V

F(y))).

then f o r

a

E

V

Constructive set theory

- 5 -c a E vx

Similarly,

E

V(x

5

E

x

3

E

19

a).

Hence, by 2 . 3 and 2 . 4 ( i i )

( a ~ B & 5 ~ a )

a = B Z E

vx

E

E

V

=

V(x E a

x

5).

E

2.8. THEOREM. (i)

Unordered P a i r s .

a, 6

If

then t h e r e i s

y

E

V

such t h a t f o r a l l

n E V

n E y - ( n = a v T l = 5 ) .

(ii)

Union.

If

a E V

then t h e r e i s

I- E

( i i i ) Small-Separation. that for a l l

n

E

If

E

V

y z 3x

E

a

(n

and

F

E

V

=

E

a(F(x) &

V

LY E

E

y

3x

using t h e N - e x i s t e n c e p r i n c i p l e from 1 . 1 7 . k Let

a E V.

Then

y

E

V

where

n

E

V

y

E

E X).

+

U

then there i s

V

n

(ii)

such t h a t f o r a l l

y

n

x).

V

such

20

P.H.G. ACZEL

(iii)

y

2.9.

a

Let

that

E

E

and

V

where

V

THEOREM.

F

V

E

y = (supz

If

Then

U.

E

A)a(p(z)).

i s a s p e c i e s on

F

A

-f

V

x

where

U,

E

Now i f

let

V

A

(Cx

=

E

so

a)F(G(x)),

q E V

F'

be t h e s p e c i e s on

V x V

given by F'(x,

Then ( i )

(ii)

y) = 'du E x 3v

a, 6

If

Strong C o l l e c t i o n .

(Vx

a

If

a)(3y

E

-a

such t h a t

V

E

y F(u, v) & Vv

E

( i i i ) Subset C o l l e c t i o n .

V) F ( x , y)

E

a, 6

If

-6

y 3u

E

x F(u, v).

then

then

V

E

=

E

36

6).

V F'(a,

E

then there i s

V

E

3

y

E

n o t depending on

V,

such t h a t

F,

VX

E

Vy

E

(Y 3y 6 3x

(ii)

Let

Hence by let

a, 6

( i ) Let

PROOF.

6).

(iii)

Let

a=3

such t h a t

V

3

F(a(x),

$(y))

so that

E

a F ( x , y).

So

F ' ( a , 6).

a

E

V

such t h a t

Vx

i n 1.15 t h e r e i s

AC

6 = sup(;,

F'(a,

E

E

b).

a, 6

E V

So

6 E V,

and d e f i n e

Vx

E

a 3y

b

E

a

-

Vx

and

a 3y

E

such t h a t

and Vx

E

E

a F(&(x),

5 F ( x , y).

V F ( x , y).

E

a+V = 6

E

Then

Vx

E

a F(a(x),

8(x)).

Then

Similarly

Vx

E

(1. 3y

E

V F ( & ( x ) , y).

F(a(x), b(x)). i(x))

Then

s o t h a t by ( i )

Constructive set theory

It i s easy t o see t h a t vx E a 3y

E

B F(x, y).

there i s

f

E

6 = (supx and

vx

2.10.

a

-t

!

y

Then Then

F(a(x), 6(x))

a

+ N1)D(x,

for

a

E

V , a'

(iii) for

a

E

V

PROOF.

+

( i ) That

3y

Vx

E

(x F(G(x), s ( f ( x ) ) ) .

E

4

n

i n 1.15

AC

so that

6 E y.

Also

a

a

and f o r (I E V

E

define

V

and f o r

n

V

E

V

E

n

For

4 :a x

Hence by Let

F ' ( a , 6).

Then (i)

and f o r

E

y(f)

6

= sup(No, Ro)

(y)a).

Now assume t h a t

!F ( & ( x ) , E(y)).

E

and

V

is clear.

V

E

a,

V

E

a

so t h a t by ( i )

Recall t h a t

THEOREM.

= (supx E

(ii)

6

F.

E

Vx

such t h a t

a)s(f(x)):

E

E

and i s independent of

V

E

21

E

E

V

No)(n

I Ro(x))

- 1

by No-existence i n 1.17. If

(ii)

a

and

a'

E

V.

E

a V

If

V

E

TI

then

y

for E

V

E

N1

a

E

U

so t h a t

so t h a t

D(x,

then, using 1.17

a a,

+ N1

E

(y)a)

U. E

Also V

for

&(x) x E

a

E

V

for

+ N1,

x

E

Hence

=

6

P.H.G. ACZEL

22 (iv)

a, B

For

V

E

6'

a'

J)

a E B' & B

J)

(a

8

E

V

E

by ( i i )

a',

8)

a

&

(8

E

a

V

8 &a)

2 a L B .

The l a s t s t e p

uses

THEOREM.

For

n E N

for

n

2.11.

Then A h )

V

E

(a

E

5 & B

E

a)

3

which can be proved using 2 . 2 .

I

let

so t h a t

c N,

w E V

where

w = ( s u p x ~N)A(x),

and

E

w

,

( i i ) (Wa

E

w)(a'

0

(i)

E

w),

( i i i ) for every s p e c i e s

PROOF.

As

and

$ E V

a'

F

V

for

for

n

E

V

on

a E V

i t follows t h a t

A(n)

E

V

for

n

E

N.

(Ah))'

E

w

for

n

E

N

As N E U , W E V .

(Ah))'

( i i ) As and hence (iii) (Wn

E

A(s(n))

(Va

Assume

E

N)(F(A(n))

3

(Vx

and

F(4)

(Vn E N) F ( A ( n ) ) ,

E

N,

i t follows t h a t

w) (a' E w ) .

E

(Vx

E

3

F(x')).

Then

F(A(0))

and

so t h a t by N-induction from 1.14 w e g e t

F(A(s(n))))

i.e.

w)(F(x)

w)

F(x).

13. THE CZF AXIOM SYSTEM

3.1.

In t h i s s e c t i o n I review t h e language and axioms o f t h e system CZF of

c o n s t r u c t i v e set theory. s e c t i o n show t h a t t h e type model of

It w i l l then be clear t h a t t h e r e s u l t s of the previous V , with extensional e q u a l i t y and membership, i s a

CZF.

The language o f

CZF

i s e s s e n t i a l l y a standard one for s e t theory.

As t h e

23

Constructive set theory

underlying l o g i c i s t o be i n t u i t i o n i s t i c , a l l t h e l o g i c a l o p e r a t i o n s (vx

E

V)

(3x

(3x

E

V)

r a t h e r than t h e more customary

$

$J

5

and

E

V)

w i l l be t r e a t e d as p r i m i t i v e .

w i l l abbreviate x

$

3

(4

and

I

3

(Vx)

and

& ($I

$J)

Note t h a t I use (3x). $)

3

take as primitive t h e r e s t r i c t e d q u a n t i f i e r s

is r e s t r i c t e d

(Vx

E

y)

As usual

respectively.

w i l l be t r e a t e d a s p r i m i t i v e atomic formulae.

y

I & v

(Vx

and

16

3

V)

E

and

and

x

Both

E

(3x

E

y).

i f i t has been b u i l t up without using t h e q u a n t i f i e r s

A formula

(Vx

E

V)

V).

and

(3x

CZF

i s axiomatised u s i n g a s t a n d a r d a x i o m a t i s a t i o n of i n t u i t i o n i s t i c p r e d i c a t e

logic.

3.2.

E

The remaining axioms a r e p r e s e n t e d below.

STRUCTURAL AXIOMS

R e s t r i c t e d Q u a n t i f i e r axioms

f o r e v e r y formula $ [ X I ,

vx

E

VD

[XI

3

where

v z E V(x

is

O

k

z

3

4 Cz1)l.

E x t e n s i o n a l i t y axioms

S e t Induction

Vx

E

V(Vy

f o r every formula

E

x $ Cyl

$[XI.

3 . 3 . SET EXISTENCE AXIOMS

3

$ [XI)

3

vx

E

V$[xl,

y

In addition I s h a l l

24

P.H.G. ACZEL

Union Va

E

V 3y

V Vn

E

(n

E

V (n

E

y I

3

E

a

E

V

E

y

3x

E

a ($[XI & rl

x)).

E

R e s t r i c t e d Separation Va

E

V 3y

V Vn

E

(rl

x))

f o r every r e s t r i c t e d formula $ Cxl. Strong Collect ion Va

E

v

(VX €

f o r every formula

Vx

E

a 3y

a 3y

E

$Cx, y l ,

v $ cx,

yl

where

$‘[a, 21

E

z $Cx, y l & Vy

E

v

E

3

32

z 3x

E

E

v

$“a, 21)

is a $[x, y l .

Subset Collect ion Va

E

v

VB

f o r every formula

3y

E

v

Ocx, y l

Vu

E

v

(VX

E

a 3y

E

B $Cx, y l

’ 32

€

y

$“a,, 21)

( t h a t may contain f r e e occurrences of t h e v a r i a b l e

u).

Infinity

3y where

v

E

(3x

succ(x, y)

E

y Vy

If

Nx,,

x 1

&‘VX E y

3y

E

y succ(x, y)

is

X € Y & V U E X

3.4.

E

(uoy) & V u E y ( u E X V U ~ y )

..., xnl

i s a formula, a l l of whose f r e e v a r i a b l e s have been

displayed then by i n t e r p r e t i n g t h e formula i n type theory i t y i e l d s an n-place function a s s i g n i n g a proposition of elements of t h e type proposition OCx,,

$[al,

..., xnl

is

..., a

V.

OCa,,

.,., an]

Note t h a t when

t o each n-tuple

..., xnl

QCx,,

1 w i l l always be small.

valid i f

the proposition

(Vx,

al,

..., a

i s r e s t r i c t e d the

I say t h a t t h e formula E

V)

... (Vxn

E

V) OCx,,

..., xnl

i s true.

The language and a x i o m t h a t w e have given f o r

CZF have been chosen s o t h a t t h e

following r e s u l t has as d i r e c t a proof as possible.

25

Constructive set theory

THEOREM.

PROOF.

Every theorem of

i s valid.

CZF

The c o r r e c t n e s s of i n t u i t i o n i s t i c p r e d i c a t e l o g i c h a s a l r e a d y been

discussed i n 8 1 .

The v a l i d i t y of t h e s t r u c t u r a l axioms follows from theorems

2.6, 2 . 7 and 2.2.

2.11.

For t h e s e t e x i s t e n c e axioms use theorems 2.8, 2.9,

2.10 and

The d e t a i l s a r e l e f t t o t h e reader.

3.5. REMARKS.

I n t h e r e s t r i c t e d q u a n t i f i e r axioms t h e formula

the s p e c i e s over

d e f i n e d by t h e formula

V

induction on t h e way t h a t t h e formula

$[XI

$[XI

expresses t h a t

0

is e x t e n s i o n a l .

A routine

i s b u i l t up s u f f i c e s t o prove 0 It follows t h a t

using only t h e r e s t r i c t e d q u a n t i f i e r and e x t e n s i o n a l i t y axioms.

can be dropped from t h e r e s t r i c t e d q u a n t i f i e r axioms without

the assumption

0

altering

s o t h a t t h e r e s t r i c t e d q u a n t i f i e r s can be t r e a t e d i n t h e s t a n d a r d

way.

CZF,

Because o f t h i s t h e r e s t r i c t e d s e p a r a t i o n axiom can be rephrased i n t h e more

f a m i l i a r form u s i n g

(0 E

a & $Cql)

3x

i n s t e a d of

E

a ($[XI &

x).

0

14. CHOICE PRINCIPLES FOR CONSTRUCTIVE SET THEORY

4.1.

I n t h i s s e c t i o n we s h a l l work e x c l u s i v e l y i n an informal framework f o r

CZF.

The s t a n d a r d conventions and n o t a t i o n s of c l a s s i c a l s e t theory w i l l be used. ordered p a i r s a r e d e f i n e d as usual.

A, B

A

The C a r t e s i a n product

x

B

So

of t h e s e t s

can b e shown t o e x i s t a s t h e s e t

using replacement (which i s a consequence of s t r o n g c o l l e c t i o n ) and t h e union axiom.

More g e n e r a l l y we can d e f i n e t h e d i s j o i n t union

of s e t s indexed by t h e s e t

A + B

The d i s j o i n t union

A s usual a r e l a t i o n such t h a t

A function

< x , y> f

<x, y > , <x, z> domain

A

E

R

A

A, B

of s e t s

i s a s e t of ordered p a i r s whose domain i s t h e s e t of y

such t h a t

i s a s i n g l e valued r e l a t i o n , i . e . one where E

f.

f

of a family

i s a f u n c t i o n from A

and range a s u b s e t of

of s e t s indexed by t h e s e t A.

B

i s defined t o he t h e s e t

and whose range is t h e s e t of

R

Z(A, B)

as t h e s e t

B.

to

B

if

y f

z

<x, y,

L

x R.

whenever

is a f u n c t i o n with

Above we have used t h e n o t i o n of a fami1.y

B i s simply a f u n c t i o n w i t h domain

A.

B

P.H.G. ACZEL

26 4.2.

I N D U C T I V E DEFINITIONS

We s h a l l use t h e informal n c t i o n of c l a s s a s i n c l a s s i c a l s e t theory. class

@

< a , A>

E

THEOREM.

the class

is

X

@-closed i f

A

implies

X

a

X

E

For any

f o r every p a i r

The following r e s u l t i s u s e f u l .

0.

For any c l a s s

t h e r e i s a s m a l l e s t @-closed c l a s s

@

I(@),

called the

c l a s s of @-generated s e t s .

Call a relation

PROOF SKETCH.



such t h a t

E

g

i f whenever

<x, y>

E

g

there i s a s e t

C a l l a s e t @-generated i f i t i s i n t h e range of some good r e l a t i o n .

I(@)

the class s e t s , where

of @-generated s e t s i s @-closed l e t

< a , A>

A 3g

E

i s good & 3x

(g

By s t r o n g c o l l e c t i o n t h e r e is a s e t

and

L UG u {<;,

-

<x, a >

-

E

g

so t h a t

a g

G

To s e e t h a t

be a s e t of @-generated

V (<x, y>

E

E

8)).

o f good s e t s such t h a t

x L {x

a > ] where

i s a @-closed c l a s s and

A

Then

E 0.

Vy

Now i f

A

and

Q

I

3y

E

i s @-generated.

V<x, y >

E

UG}

i s good

then

Thus I(@) is @-closed.

Now i f

i s good then an easy proof by s e t induction on

X

will

x

show t h a t <x, y>

I(@)

so that

E

g

=

y

E

x,

5 X.

4 . 3 . THE SET OF NATURAL NUMBERS

This may be c h a r a c t e r i s e d a s t h e unique s e t

where

4

i s t h e empty s e t a n d

The e x i s t e n c e of such a s e t

w

y'

w

such t h a t f o r every s e t

is t h e set of

z

such t h a t

z

E

y

x

V

z

y.

follows from t h e axiom of i n f i n i t y u s i n g r e s t r i c t e d

Constructive set theory

separation.

21

Its uniqueness can be proved by s e t induction, a s can the scheme of

mathematical induction f o r

Our f i r s t choice p r i n c i p l e i s the following.

w.

4 . 4 . DEPENDENT CHOICES (DC)

f o r a l l formulae

e[x]

and

$Ex, y l .

OCa, y l

Here

expresses t h e conjunction

of t h e following statements. (i)

y is a function with domain

(ii)

<$, a> E y .

(iii) sets

(

E

y &



y)

E

w.

($[x, y l & e [ x l )

3

for a l l

n

E

w

and a l l

x, y.

The following s p e c i a l i s e d form of

DC

may be reduced t o a s i n g l e axiom using

strong c o l l e c t i o n . LIMITED DEPENDENT CHOICES (LDC)

f o r a l l formulae x

E

4.5.

B

replaces

$[x, y l ,

where

i s defined a s before except t h a t

OCa,yl

eCxl.

For t h e next choice p r i n c i p l e we need t h e

DEFINITION.

is a

The s e t

A

function with domain

A.

base

i f every r e l a t i o n With domain

A

extends a

The following p r i n c i p l e was introduced i n Aczel C19781. PRESENTATION A X I O M (PA).

set

A

Every s e t has a p r e s e n t a t i o n , where a p r e s e n t a t i o n of a

i s a f u n c t i o n w i t h range

A

I n Aczel [19781 i t was observed t h a t countable choice) expresses t h a t

4.6.

w

whose domain i s a base.

PA

3

LDC

3

CC,

where

CC

( t h e axiom o f

i s a base.

The remainder of t h i s s e c t i o n w i l l be taken up with t h e formulation of a

strengthening of

PA.

F i r s t r e c a l l from Aczel L19781 t h a t Myhill's exponentiation

axiom i s a consequence of subset c o l l e c t i o n .

So f o r any s e t s

A, B

we may form

P.H.G. ACZEL

28

the s e t

BA

of f u n c t i o n s from II(A, B)

C a r t e s i a n product

More g e n e r a l l y we may form a s a s e t t h e

of any family

{f

{z E

B.

B

of s e t s indexed by a s e t

A.

In fact

A}

for

is the s e t

II(A, B)

where

to

A

I

C = U{B(x)

{$I

I

a

DEFINITION.

Let

is

(i) w A

i s the s e t



such t h a t e i t h e r

D

and

D

I

L {A} u {B(x)

x

h

A,

{A}

f o r some s e t

and some

A

a , b E A.

IIXI-closed i f

X,

E

in

E

X

for a l l families

B

of s e t s i n

X

indexed

X,

and ( i i i ) I ( a , b)

Note t h a t any

C L C(A, B ) ,

and

( i i ) II(A, B), C(A, B) by a s e t

I(a, b)

$,

or

o f s e t s indexed by

X

B(X))I

E

In the next d e f i n i t i o n

D

II(A, B)

o r ( i i i ) C L I ( a , b) Then a c l a s s

A (f(X)

be t h e c l a s s of p a i r s and

w L

B

some family

IICI

C

or ( i i ) C

A}.

E

E

X !k

so t h a t

b}

(i)

x

cA I

E

E

X

for all

IIXI-closed c l a s s

a, b

E

A

A

and every s e t

in

X.

X

i s e a s i l y seen to have t h e following a d d i t i o n a l

X

whenever

closure properties ( i v ) $, (v)

4.7.

{$}

E

X

BA, A x B , A + B

E

A, B

E

X.

We can now f o r m l a t e our f i n a l choice p r i n c i p l e s .

THE IICI-AXIOM OF CHOICE (IICI-AC) Every IICI-generated s e t i s a base.

THE IICI-PRESENTATION AXIOM (IICI-PA). Every

IICI-generated s e t i s a base and every s e t has a

IIXI-presentation,

i.e.

t h e r e is a f u n c t i o n whose range i s t h e s e t and whose domain i s a IICI-generated base.

Note t h a t that

IICI-PA

IIZI-AC

i s a s i m l t a n e o u s s t r e n g t h e n i n g of

i m p l i e s t h a t s e t s such a s

t h a t t h e system HAw + AC + e x t

w, ww, w ( ww)

IICI-AC

,

...

and

PA.

a r e bases.

Also n o t e It follows

of f i n i t e type a r i t h m e t i c has a s t a n d a r d

Constructive set theorj CZF + IICI-AC.

interpretation in

4.8.

29

We end t h e s e c t i o n with an i n t e r e s t i n g c h a r a c t e r i s a t i o n of t h e notion of

base when assuming

IICI-PA.

A s e t i s a base i f and only i f i t i s i n one-one

(assuming IICI-PA).

THEOREM.

correspondence with a IIXI-generated s e t . Clearly every s e t i n one-one correspondence with a base i s i t s e l f a base.

PROOF. So using

By

As

B'.

B

B

whose domain i s a IICI-generated

g

from

b

b

{x

I

g(f(x)) e

B'

for a l l

with t h e s e t

Z(B'

b

E

XI,

, F)

B'

X(B'.

Hence

g

B

to

B'

such t h a t

i s a one-one correspondence between

B

and

and t h i s l a t t e r s e t is c l e a r l y in one-one correspondence where

i s IIXI-generated,

ilZI-generated.

f

Then

B.

F(x) As

be a base.

B

with range

f(g(b)) E

For t h e o t h e r d i r e c t i o n l e t

i s a b a s e t h e r e is a f u n c t i o n

there i s a function

IICI-PA

set

we g e t one d i r e c t i o n .

IIXI-AC

B

so

I ( g ( f ( x ) ) , x)

is

F(x)

for

f o r each

x

x

E

E

B'

B'. so t h a t

i s also

F)

Z(B',

i s i n one-one correspondence with the IICI-generated s e t

F).

55. DEPENDENT CHOICES The aim of t h i s s e c t i o n i s t o prove t h e v a l i d i t y of each instance of the set t h e o r e t i c a l axiom scheme of dependent choices formulated i n 4 . 4 .

Note t h a t a

type t h e o r e t i c a l v e r s i o n was proved i n 1.15 and i n order t o apply t h a t version t o the i n t e r p r e t a t i o n of s e t theory we s h a l l need t h e notion of an i n j e c t i v e l y presented s e t .

Although t h i s i s n o t an e x t e n s i o n a l n o t i o n of set theory i t t u r n s

out t o b e c l o s e l y r e l a t e d t o t h e extensional n o t i o n of a base and w i l l play an e s s e n t i a l r o l e i n t h i s and t h e l a t e r s e c t i o n s .

W e s h a l l continue t o work

informally i n type theory following t h e s t y l e of 52.

We s h a l l need the standard

s e t t h e o r e t i c a l n o t i o n s , described i n 5 4 , a s i n t e r p r e t e d i n type theory. example t h e i t e r a t i v e set

a

E

V

is a relation i f

In our work with r e l a t i o n s t h e following function w i l l be useful. such t h a t

-

-

a = B

then l e t

So f o r

If

a, 3 I E V

P.H.G. ACZEL

30

a

Note t h a t

x

-

E

a.

&(x), g(x)

and

U

E

Hence

S(a,

5)

V

with

V

E

for

x

S(a,

5)

E

a,

so t h a t



V

E

for

V.

E

5.1. LEMMA. (i)

a, 5

If

(ii)

5

E

a, y

If

-

a = is

then

Let

PROOF.

Let

such t h a t

V

E

-

a = 6,

with

V

(i)

E

i s a r e l a t i o n with domain

a

and

5.

range

a

5

-

A = a.

5

such t h a t

V

E

then, f o r some

a

y.

and l e t

V

E

5)

S(a,

i s a r e l a t i o n w i t h domain

y

y = S ( a , 5).

and l e t

= A

Then, by choosing

x = y = u , Vu

E

A 3x

E

A 3y

E

A (v(u) g ) ,

Vu

E

y 3x

E

a 3y

E

5 (u

so that

Hence

y

is a r e l a t i o n whose domain i s a s u b s e t of 8.

s u b s e t of

a 3u

E

E

y 3y

5 (u

E

y

(ii)

y

Let

V

= sup(A, b)

5.2. DEFINITION.

y,

a

E

we g e t

and hence

a.

be a r e l a t i o n w i t h domain

By t h e AC i n 1.15 t h e r e i s

5

and whose range is a

y

has domain

a.

B.

has range E

a

L <x, y>),

i s a s u b s e t of t h e domain of

a

Similarly

that i f

u = y = x,

A l s o , by choosing

Vx so t h a t

<x, y>).

b

then

E

A

+

6

E

V

Then

Vx

;= a

such t h a t

E

A 32

and

Vx

i s i n j e c t i v e l y presented i f f o r a l l

V

&(xl)

&(x2)

3

E

V (<(u(x), z> E y)

V x e A (<&(x), b ( x ) >

such t h a t

V

X I =_

E

y),

S(a, 6 ) ( x

E

xl, x2

.

So E

y).

E

a

xp.

a

5 . 3 . LEMMA. (i)

If

5

Let

be i n j e c t i v e l y p r e s e n t e d .

a E V

V, such t h a t

E

-

-

6 = a

and

6

E

V

then f o r a l l

( E S(a, 6))

i s a f u n c t i o n w i t h domain

If

y

E

V

then

f o r some

6

E

V

such t h a t ..;

(ii)

y

:(6

=

a.

x

E

a

B(x)). a

i f and only i f

Y

S(a,

8)

31

Constructive set theory

PROOF. (i)

f3

E

If y

E

Let

(ii)

a

Let

S(a, B )

c y. B

Also, f o r

so t h a t

As

-

6

such t h a t

a,

E

if

=

B,,

6,

S(a,

6,)

a.

Let

a.

5 S(a,

8,)

B,

and

-

E

b'x

E E

-

a

such t h a t

V

a

S(a, 8 )

and

B

=

a

E

a

and

is a r e l a t i o n

For t h e converse l e t

y

S(a, 6 )

L

i s a r e l a t i o n with domain

y

then, by ( i ) ,

-

a.

;(y)

L i(x).

a.

Hence

Then

( c S ( a , 6 , ) ) (il(X)

L

i2(x)).

interchanged, g i v i n g t h e d e s i r e d r e s u l t .

be i n j e c t i v e l y p r e s e n t e d and l e t

V

Then by 5.1 ( i i )

S(a, 8)

S(a, 8 ) y'

f o r some

i s a f u n c t i o n w i t h domain

car

y

6

Hence

E E

V

be a r e l a t i o n w i t h

V

such t h a t

a

i s a base.

The n e x t r e s u l t gives some examples of i n j e c t i v e l y p r e s e n t e d s e t s . e x t e n s i v e range of examples w i l l be a v a i l a b l e i n t h e n e x t s e c t i o n .

5.5. LEMMA.

-

a.

B,= B,=

E vx

Then

then i t i s c e r t a i n l y a r e l a t i o n

B

y.

E y

a.

E

Every i n j e c t i v e l y p r e s e n t e d s e t i s a base.

THEOREM.

domain

x

a

By 5.1 ( i )

such t h a t

V

E

6,

Similarly with

PROOF.

and

S(a, 8 )

-

i s a f u n c t i o n w i t h domain

y

5.3 (ii)

V

E

i s a f u n c t i o n w i t h domain

y

i t follows t h a t

a

x, y

( i i i ) Let

5.4.

u, 6

s o t h a t by 5 . 1 ( i i ) t h e r e i s

V

E

=

i s a f u n c t i o n with domain

V

with domain where

such t h a t

V

a

with domain

be i n j e c t i v e l y presented.

V

E

The following are i n j e c t i v e l y presented.

(i)

$,

(ii)

{ a } f o r any

(iii)

W.

a

E

V,

- -

8 = a:

A more

By

P.H.G. ACZEL

32 PROOF.

Q1

( i ) Recall t h a t so t h a t for

(Vz E N O ) l

( s u p z ~NO)RO(z). But, by

s

all

zl, z2

No

E

i n 1.17,

N,,-existence

any p r o p o s i t i o n i s t r u e , and i n

particular RO(z2) 3 z 1 -No -

R0(z1)

0

So

z

2'

i s i n j e c t i v e l y presented.

(ii)

{a} = ( s u p z ~N 1 ) R 1 ( z ,

Recall t h a t

so t h a t

z 1 =N1 z2

hence

R1(zl,

For

a).

a) L R 1 ( z 2 ,

a)

zl, z 2

z1

Nl,

E

and

= z2

is

Hence { a )

z1 =N1 z 2 .

3

i n je c t i v e l y presented. ( i i i ) Recall t h a t for

n

w

(supz

=

E

N)A(z)

where

n, m

We must show t h a t f o r

N.

E

A(m)

A(n)

E

A(0)

m

By 5 . 4 and 5.5 ( i i i ) axiom.

Let

Then f o r each

V x V

a

Let

a

E

i n 1.15 t h e r e is

w,

E

rl = sup(N, c ) .

<$, a> <XI,

N

-+

Then

a. E

As

6.

y> E 6.

by 5 . 3 ( i ) , and

F

<$, a>

A c(n)


and

V

E

x

such t h a t

n

c(0) = a

7

=

z,

and N

x

B, y

c(s(n)).

a r e e x t e n s i o n a l , we g e t

i(O)> E V

IA h ) ,

n

and f o r

a

so that

6

g(a)

such t h a t x'

E

is a

5, y

E

V

DC

N

6 = S(w,

so t h a t i f

<;(O),

c(O)>

E

6

and a l l

w

E

Then by t h e type t h e o r e t i c a l v e r s i o n of

with

x E w y

6

i s i n j e c t i v e l y presented,

w

Then f o r some

6

there is

and f o r every

B(a).

rl E V

Finally, let

6

F b e an

V (B(y) & F(x, y ) ) ) .

E

B(a) E

and l e t

V

e x t e n s i o n a l i n each argument) such t h a t

such t h a t

V

and by 5.3 ( i i ) , as

V

domain

B

c

3y

3

G$, a>

such t h a t

V

(i.e.

such t h a t

V

E

V (B(x)

E

f u n c t i o n with domain

E

( i v ) a r e needed.

be an e x t e n s i o n a l s p e c i e s over

B

Vx

6

and w i t h i n

N

is a base, and we have proved CC, t h e countable choice

w

e x t e n s i o n a l s p e c i e s over

Let

E

But we have t h e following s t r o n g e r r e s u l t .

5 . 6 . THEOREM.

PROOF.

n

f i r s t on

A t t h e key s t e p s 2.10 ( i i i ) and 2.10

N.

E

A(s(n) = ( A h ) ) '

and

n =N m.

3

This can be shown by a r o u t i n e double N-induction, t h a t on

Q

=

N

q)

then

is a f u n c t i o n w i t h i t follows t h a t <x, B> e 6

A(s(n))

and

and hence,

Hence, by (*) and t h e ' a s s u m p t i o n t h a t

B(5) & F(8, y ) .

Constructive set theory

33

I f w e apply t h i s r e s u l t t o the extensional p r e d i c a t e s defined by formulae i n t h e language of s e t theory we get the main r e s u l t of t h i s s e c t i o n .

5.7. THEOREM.

The s e t t h e o r e t i c a l

DC

i s valid.

16. THE IIXI-AXIOM OF CHOICE The aim of t h i s s e c t i o n i s t o show t h a t every IIXI-generated s e t i s a base.

We do

t h i s using t h e following notion.

6.1. DEFINITION.

B

presented

E

a

E

V

i s a s t r o n g base

if

6

a

f o r some i n j e c t i v e l y

V.

Note t h a t t h i s i s an extensional v e r s i o n of the notion of an i n j e c t i v e l y presented It will be convenient t o extend our use of t h e terminology associated with

set.

c l a s s e s t o such e x t e n s i o n a l notions, even though they may n o t be definable i n the purely s e t - t h e o r e t i c a l language.

By t h e r e s u l t s i n 12 the axiom schemes of

CZF

As the n o t i o n of base i s extensional i t

s t i l l apply t o t h i s wider context.

follows from 5 . 4 t h a t every s t r o n g base i s a base.

Hence t h e r e s u l t t h a t we a r e

aiming a t w i l l be a consequence of t h e r e s u l t t h a t every IIXI-generated s e t i s a strong base, and f o r t h i s i t s u f f i c e s t o show t h a t t h e c l a s s of s t r o n g bases is IIXI-closed.

We s h a l l need t o foruiulate e x p l i c i t i n t e n s i o n a l versions of the set

theoretical IIZI

operations.

X(a, 6)

In t h e following d e f i n i t i o n s we s h a l l d e f i n e a, 6

E

-

-

a = 6.

such t h a t

V

E

II(a, 6) E V

and

V

for

i s i n j e c t i v e l y presented

a

We then show t h a t when

then t h e s e a r e t h e d i s j o i n t union and Cartesian product, r e s p e c t i v e l y , of the family of sets

Let

a, 6

E

B) indexed by t h e s e t a.

S(a,

i = is.

with

V

6.2. DEFINITION OF Z(a, C

E

U, where

i(p(z))

E

V

C = (Zx E

,

q(z)

E

B).

As

a

E

U and

a) m. For

8(p(z))

so t h a t

z

E

f(z)

B(x)

E

E

V

for

U

C, p(z) E

u

x

a

E

so t h a t

where

i t follows t h a t a(p(z)) E V

f(z) = (e(p(z)))"(q(z)).

Now d e f i n e

6 . 3 . DEFINITION OF n(a, 6). z

E

D.

For

x

E

a,

Note t h a t

& ( x ) E V, B(x)

E

V

D

E

and

U,

where

z(x)

E

and

D = (IIx

B(x)

E

so that

a)

i(x).

Let

P.H.G. ACZEL

34

6.4.

-

-

Let

THEOREM.

a

E

be i n j e c t i v e l y presented and l e t

V

E

such t h a t

V

E

E = a. (i)

E(a, E)

is i n j e c t i v e l y presented f o r a l l

(ii) If

B(x)

a r e i n j e c t i v e l y presented.

By 5.3 ( i i ) S(a, 6)

indexed by t h e s e t (i)

Let

rl

i s t h e Cartesian product of t h e

S(a, 6).

n(a, 8)

PROOF.

n(a, 6 )

i s t h e d i s j o i n t union and

family of s e t s

E

a

then both

i s a function with domain

Z(a,

8)

and

a, i . e . a family of s e t s

a. Then

V.

E

x

S(a, 8) & 3y

E

a 36

E

V (<x, 6 >

:3x

E

a

36

E

V (<(Y(x), 6>

5

3x

E

cr

36

E

v

E

3x

E

a

3y

E

! 32 E

c (n

:n

Z(a,

E

S(a, E)

i s i n t h e d i s j o i n t union of

q

:3x

(6

h

E

E

S(a,

$ ( x ) & 3y

E

E

(n

6

E) & 3y

E

6

L

<x, y > ) )

(n

<(Y(x), y > ) )

6(ll 2 < a x ) , y>))

i ( x ) ( n A .&x) , Y>) <(Y(p(z)), f ( z ) > )

8).

The t h i r d of t h e above :uses 5.3 ( i ) and t h e f i f t h uses t h e Z-existence p r i n c i p l e of 1 . 1 7 .

For the Cartesian product f i r s t n o t e t h a t by 5.3 ( i i )

with domain

a i f and only i f 3y

Hence

rl

v (7

E

=

E

&

n

S(a, 6 ) ) .

i s i n t h e Cartesian product of 3y

where, i f

is a f u n c t i o n

rl

y

E

V

with

F(a, B, y) = ‘dx

E

(7 7 = E, V

E

a 36

Using 5.3 ( i ) twice, t h e

E

AC

=

&

n

V (<x, 6 >

E

S(a,

8)

S(a,

Y) 8. F ( a , 8, Y)),

S(a,

6) & 3y

i f and only i f

E

6 ( < x , y>

from 1.15 and 5.3 ( i i ) we get t h a t

E

S(a,

y))).

35

Constructive set theory

I t follows t h a t E

D (n

(ii)

Let

32

y = C(a, 8 )

show t h a t

7(zl)

i s i n the Cartesian product of

II

S(a, g(z))),

z1

y(z2)

=c z 2 .

and hence i f and only i f and l e t

Let

zl,

z2

xi = p(zi)

i f and only i f

n

8).

E n(a,

q(zl)

such t h a t

C

E

S ( a , 8)

and

yi = q ( z i )

for

;(z2).

i = 1, 2.

We must

As

i t follows t h a t

< a x 1 ) , (8(X1))"(Yl)>

<&x,),

(E(xz))-(y2)>

so t h a t a(xl) L G(x2) (8(xp) 1 -(YZ).

(B(Xl))-(yl) As

i s i n j e c t i v e l y presented

a

x1 =_ x p a

(i(xl))-(yl) g(xl)

As z1

let

y = n(a, 0)

that

and l e t Note t h a t

z1 = D_'2'

5 y(z,),

(*)

VXl

a 3x2 a

then

then

x1

=_

;(zl)

Hence

;(z2).

Now

We must show

= ( s u p x ~a ) < k ( x ) , ( g ( z i ) ) - ( x ) > .

s o that i f also

x2,

..a

(B(xl))-(z1(xl))

z l ( x l ) =B(xl) z 2 ( x l )

* ;(XI).

i s i n j e c t i v e l y presented.

such t h a t

D

E

B

( < a x l ) , (g(zl))-(xl)> L <%xz), (g(z2))--(Xp)~).

E

a ( x l ) L a(x2)

Z(a, 8)

= S ( a , g(zi))

?(zi)

-

v(zl) E

z2

__

y1 =B yp where Thus

z2.

zl,

Hence, as

But i f

(i(xl))-(Yz).

5

i s i n j e c t i v e l y presented

=c ( x l , y l ) =c (xz, y2) =c

and hence

(E(xl)) ( z 2 ( x l ) ) , where

B(xl) =

i(xl).

and a s

XI)

g(zl))-(xl)

(g(z2))-(xp)

is i n j e c t i v e l y presented,

Hence i t follows from (*) t h a t

so t h a t

which y i e l d s

z 1 =D z 2

using t h e e x t e n s i o n a l i t y p r i n c i p l e 1. 16.

The following r e s u l t , needed i n 17, can be proved along the l i n e s of the previous result.

6.5.

LEMMA.

For

a,

a + B = ( s u p z a (I +

E

V, a + E

i) D(z,

E

where

V

(x)<$, & ( x ) > , (y)<{$}, g ( y ) > ) ,

and (i)

a + f3

(ii) if

a

i s t h e d i s j o i n t union of and

E

a

and

8,

a r e i n j e c t i v e l y presented then s o i s

a + E.

P.H.G. ACZEL

36

The next r e s u l t w i l l be needed f o r t h e next theorem and a l s o i n 57.

6.6. LEMMA.

For

A

and

U

E

a, b

A,

E

a , b)

?(A,

i ( A , a , b) = ( s u p 2

E

where

V

E

I ( A , a, b))g,

and (i)

i ( A , a , b)

(ii)

if

PROOF. (i)

a

E

y

=

I ( A , a , b)

As

Let

i s i n j e c t i v e l y presented,

E

0

and

U

I ( A , a , b)

We must show t h a t z1

-

i s i n j e c t i v e l y presented and

V

and l e t

V

E

i t follows t h a t

z l , z2

But, as

z1 =_ z 2 .

then f o r a l l

a = A

zl,

z2

E

?(A,

a, b)

I(A,

E

E

V

Y a

Hence f o r

E

V

n

E

6 . 7 . THEOREM.

7((z2).

so t h a t

= z2

i s i n j e c t i v e l y presented and

a, b

E

A =

a

V

The c l a s s of s t r o n g bases i s

IICI-closed.

0 i s i n j e c t i v e l y presented by 5.5 and hence i s a s t r o n g base.

PROOF.

V

;(zl)

z1 = r

I ( A , a, b),

E

Y

=_ z 2 .

( i i ) As

a , b)

such t h a t

n

Let

6

We wish t o show t h a t t h e

be a family of s t r o n g bases indexed by a strong base.

Choose

Cartesian product and d i s j o i n t union of t h e family a r e a l s o s t r o n g bases.

a

E

such t h a t

V

is i n j e c t i v e l y presented.

a

that

i s a function with domain

6

presented there i s

y f

E

such t h a t

i+V

E

< & ( X I ,f ( x ) >

V

E

6.

a.

As

S(a,

<&(x), y>

5) 5 6

5

and

II(a, 8)

E

V

x

E

and

i.

and d i s j o i n t union of

y E V

B,,

=

x

x

-

E

t h e r e i s an i n j e c t i v e l y

a

By the type t h e o r e t i c

6.

a

Without loss we may assume

a.

E

i,

5 = sup(;,

S ( a , 8)

By 5.3 ( i i )

Z(u, 5) S(a,

AC of 1.15

i s i n j e c t i v e l y presented and

f(x)

where

i s a f u n c t i o n with domain

6

and

product and d i s j o i n t union of a s t r o n g base and

E

such t h a t , f o r each Then

i n j e c t i v e l y presented f o r

By 6 . 4 both

For each

f),

and

a

8 ) , i.e. of

Hence t h e Cartesian product

6.

a r e s t r o n g bases. It remains t o show t h a t i f E

rl E

such t h a t V

S(a, 5)

w e m u s t have

a r e i n j e c t i v e l y presented and a r e t h e Cartesian

6,

such t h a t f o r a l l

is

i(x)

is a f u n c t i o n with domain

6

V

V

E

B,,

5,

E

a

a

E

V

is

then t h e r e i s ' a s t r o n g base

6.

31

Constructive set theory

Without loss w e may assume t h a t where

a.

-

E

a

6. = a ( a i )

is i n j e c t i v e l y p r e s e n t e d and t h a t

a

By t h e lemma i t s u f f i c e s t o l e t

..-

y = I(a, al, a2).

The ILXI-axiom of choice i s v a l i d .

6.8. THEOREM.

PROOF.

i = 1, 2.

for

By t h e previous theorem every IIXI-generated s e t i s a s t r o n g base and hence

i s a base. 57. THE IICI-PRESENTATION AXIOM The main a i m of t h i s f i n a l s e c t i o n i s t o show t h a t every s e t has a IIZI-presentation. This i s done by coding each s m a l l t y p e set

T(A)

set

S(T(;),

= A.

such t h a t

a s a n i n j e c t i v e l y presented HCI-generated

A

It then e a s i l y follows t h a t f o r each s e t

i s a f u n c t i o n with range

a)

a

a, t h e

and domain t h e ILCI-generated

set

T(;).

The s e c t i o n ends w i t h a r e s u l t g i v i n g s e v e r a l c h a r a c t e r i s a t i o n s of t h e n o t i o n of a base.

7.1.

THEOREM.

and

a, b

E

There is a f u n c t i o n

T E

U

-t

V

such t h a t f o r

A, B

E

U,

F

E

A

+

A T ( N ~ )=

0

r(N) = w T(II(A, F ) ) = ~'I(T(A),( s u p x ~A)T(F(x))) T(C(A, F)) = C(T(A), ( s u p x ~A)T(F(x))) T(A + B) = T(A) + T ( B ) T ( I ( A , a , b ) ) = I(A, a , b ) . Moreover

-

T(A) = A

for a l l

Note t h a t t h e symbols

II, C

A

E

and

U. +

equations r e f e r t o t h e o p e r a t i o n s on

PROOF.

Define

T(A) = sup(A, u(A))

defined by t r a n s f i n i t e r e c u r s i o n over

t h a t occur on t h e r i g h t hand s i d e of t h e s e introduced i n 56.

V

for U

A

E

U where

so t h a t

u

E

(I[A E U)(A

+

V)

is

U

38

P.H.G. ACZEL o(I(A, a , b ) ) = ( z ) $

for

A, B

E

U,

F

A

E

+

and

U

a, b

E

4 , w , II,

follow using t h e d e f i n t i o n s of

be i n c o r r e c t t o use t h e equations f o r recursion over Z(a,

B)

7.2.

THEOREM.

The required equations f o r

f

+,

C,

6

a,

For a l l

A

the s e t

U

E

T(A)

by t r a n s f i n i t e

T

II(a, B )

-

without the assumption t h a t

V

E

E

5.

a =

now

T

Note t h a t i t would

given i n 16.

t o d i r e c t l y define

T

That would only work i f we had t h a t

U.

for

V

E

A.

and

V

i s i n j e c t i v e l y presented and

IIZI-

generated. PROOF.

A

Call

No

and

good and t h a t

F

A

E

a, b

7.3. THEOREM.

Let

E

E

V

B and range

a.

x

E

A.

A, B

That

are

U

II(A, F)

i s good and by 6.6

A + B

E

and

I(A, a , b)

A.

For each

a

i s good f o r

F(x) By 6.5

a

E

B

and l e t

the s e t

V

B

A s , by 7.2,

i s a IIXI-presentation of

S ( T ( ~ ) ,a )

B

E

V,

S ( 6 , a)

E

V

Then

= T(;).

So by 5.1 ( i ) and 5.3 ( i i )

presented.

F i r s t note t h a t

As induction hypothesis assume t h a t

such t h a t

U

-+

i s i n j e c t i v e l y presented and IICI-generated.

T(A)

i s good by U-induction (see 1 . 1 4 ) .

U

E

a r e good follows from 6.4.

i s good f o r

PROOF.

A

a r e good by 5.5.

N

C(A, F)

if

U

E

We show t h a t every

=

a

a.

i s injectively

and

i s a f u n c t i o n with domain

i s IICI-generated t h e r e s u l t follows.

As a consequence of t h i s theorem and 6.8 w e have t h e main r e s u l t . 7.4. THEOREM.

The IICI-presentation axiom i s v a l i d .

We next give a type t h e o r e t i c c h a r a c t e r i s a t i o n of t h e notion of a IIXI-generated set. 7.5.

THEOREM.

t h e set

T(A)

A s e t i s IIXI-generated i f and only i f i t i s e x t e n s i o n a l l y equal t o f o r some

A

E

U.

The implications from r i g h t t o l e f t follows from 7.2.

PROOF.

implication, c a l l a s e t U-generated some

A

Clearly

U.

E

w

For t h e converse for

i f i t i s e x t e n s i o n a l l y equal t o T(A)

It s u f f i c e s t o show t h a t t h e c l a s s of U-generated s e t s is IICI-closed.

i s U-generated because

w = T(N).

Now suppose t h a t

of U-generated s e t s indexed by a U-generated s e t . function with domain

Vx

Then f o r some

So

T(A). E

T(A) 3y

E

U (<x, r ( y ) > e 6),

6

V

E

A

E

U

i s a family 6

is a

39

Constructive set theory

and hence .vx

E

A 3~

E

u

( < ( T ( A ) ) - ( x ) , T ( Y ) >E 6 ) .

from 1 . 1 5 t h e r e i s

AC

By t h e type t h e o r e t i c

VX

E

A (<(T(A))-(x), T(F(x))>

F

E

A

+

such t h a t

U

6).

E

It follows t h a t S(T(A), ( s u p x ~A)T(F(x)))

6 , and hence i s e x t e n s i o n a l l y e q u a l t o

is a s u b s e t of

T(A).

with domain

6

union of

a r e t h e U-generated s e t s

6

Finally let

~ ( l l ( A ,F ) )

a, 6

be U-generated and l e t

V

E

i s U-generated i t i s e x t e n s i o n a l l y equal t o a, b

A

E

Then

y

E

a

T ( A ) ( ~ ) and

V,

and f o r

~l E

z

(ll L

Il

E

6,

as both a r e f u n c t i o n s

From t h i s i t follows t h a t t h e C a r t e s i a n product and d i s j o i n t

y

6

L

T(A)(b).

and E V

T(A)

T(C(A, F))

f o r some

A

respectively.

a, 6

such t h a t E

E

As

6.

6

So f o r some

U.

y = T ( I ( A , a , b ) ) = f ( A , a, b ) .

Let

V

$ & a

s o t h a t the U-generated s e t

5)

L

I ( a , B) of 4 . 5 .

is t h e s e t

y

We end with t h e promised c h a r a c t e r i s a t i o n s of t h e n o t i o n of a base.

7 . 6 . THEOREM.

For

t h e following a r e e q u i v a l e n t

a E V

(i)

a i s i n one-one correspondence with some IIXI-generated s e t .

(ii)

a i s i n one-one correspondence with some s t r o n g base.

( i i i ) a i s a s t r o n g base. (iv)

a i s a base.

PROOF.

We show t h a t ( i )

6.5.

For

base

6.

(ii)

3

a.

and range

$

(ii)

let

a

3

(iii)

y

E

V

6

L

y l , y2

a.

(i).

3

(i)

3

(ii)

follows from

such t h a t

6

i s i n j e c t i v e l y presented.

Hence, as

y

By our

i s a one-one f u n c t i o n w i t h domain

y

y 2 S(5, 6)

i s one-one and

B

f o r some

6

E

V

5 such

i s injectively

E

8(y1)

8(y2)

3

IB(y,)

’ Y1

6

(iv)

By 5.1 and 5 . 3 i t follows t h a t and

=

presented, i f

so t h a t

3

be i n one-one correspondence with t h e s t r o n g

Without loss we may assume t h a t

assumption t h e r e i s

that

3

(iii)

B(y2)

=- Y 2 9

B i s i n j e c t i v e l y p r e s e n t e d and hence

is a consequence of 5.4.

a

i s a s t r o n g base.

By 7 . 4 we may u s e 4.6 t o g e t

(iv)

3

(i).

(iii)

3

(iv)

40

P.H.G. ACZEL

REFERENCES Aczel, P. C19781 The type t h e o r e t i c i n t e r p r e t a t i o n of c o n s t r u c t i v e s e t theory, i n Logic Colloquium ' 7 7 , eds. A . Macintyre, L. Pacholski and J. P a r i s , North Holland. Bishop, E . C19671 Foundations of Constructive Analysis, McGraw H i l l . Bridges, D.S. C19791 Constructive Functional Analysis, Research Notes i n Mathematics, Vo1.28, Pitman. Diaconescu, R. C19751 Axiom of choice and complementation. Proc. h e r . Math. SOC., Vol .5 1, pp .176-178. Martin-Lgf, P. C19751 An i n t u i t i o n i s t i c theory of types: P r e d i c a t i v e p a r t , i n Logic Colloquium '73, eds. H.E. Rose and J . C . Sheperdson, North Holland, pp. 73-118. Martin-LGf, P. C19791 Constructive Mathematics and Computer Programing. To appear i n t h e Proceedings of the 6 t h I n t e r n a t i o n a l Congress f o r Logic, Methodology and Philosophy of Science at Hannover i n 1979. Myhill, J. C19751 Constructive S e t Theory, J.S.L.,

V01.40, pp.347-382.

THE L.E.J. BROUWER CENTENARY SYMPOSIOM A.S. Troelstra and D. van Dalen (editors) 0North-HoNandPublishingCompany, 1982

41

RECENT PROGRESS IN CONSTRUCTIVE APPROXIMATION THEORY Douglas S . Bridges University College at Buckingham, Buckingham MK18 lEG, England.

Approximation theory can be regarded as a good testing-ground for constructive mathematics. Recent work on a general aspect of approximation theory, and on the particular case of best Chebyshev approximation over a compact interval, shows that constructive techniques can handle this material successfully, in such a way that new classical results and estimates are obtained as by-products of the constructive development.

1 If constructive mathematics is to gain widespread acceptance, it will have to satisfy the mathematical community on several issues quite apart from those of philosophy and foundations. One of these issues is its ability to produce significant results in functional analysis and measure theory. Although there is much to be done in these fields - for example, almost nothing constructive exists in the very important area of operator algebra theory - much has already been accomplished, particularly by Bishop cl,Zl. Certainly, enough constructive analysis has been developed to invalidate the argument of those mathematicians who are prepared to dismiss constructive mathematics on the ground that its methods cannot lead to deep theorems.

I would like to suggest that there is another test which constructive mathematics must pass before the day arrives when it "will be the accepted norm" [l, p. XI. In order to justify the constructivists' claim that their mathematics is characterised by numerical meaning and computational method 11, pp. 2-31, they must produce a fruitful constructive development of numerical analysis and related subjects. In this article, I shall outline the first steps of such a development of approximation theory. All the work below will be in the constructive spirit of Errett Bishop C1,31. Thus all the proofs will be acceptable to the intuitionist, but there may be (in fact there are) places where the use of intuitionistic principles would simplify matters considerably.

2

The fundamental theorem of classical approximation theory states that: (*)

If X is a finite-dimensional linear subspace of a real normed space a E E, then there exists b E X such that

E, and if

Ila

-

bll = dist(a,X) = infflla - xII : x

E

X}

It is a remarkable fact that, although this theorem lies at the heart of a large and growing body of computational mathematics, all its proofs are nonconstructive. The constructive content of these proofs seems to be the locatedness of the finite dimensional subspace X; the further conclusion that b exists invariably depends on an application of the nonconstructive proposition that a continuous, real-valued function on a compact space attains its infimum.

D.S. BRIDGES

42

I n f a c t , theorem (*) i t s e l f appears t o be n o n c o n s t r u c t i v e , a s i s shown by t h e following Brouwerian counterexample, which was suggested t o me by Fred Richman. Let

a

be an a r b i t r a r y r e a l number, and l e t X be t h e one-dimensional subspace

R ( c o s a , s i n a ) of Et2, where each r e a l number t , d e f i n e

lR2 i s given t h e norm

$ ( t ) : 11(0,1)

=

-

(x,y)

+

max( 1x1, l y l ) .

For

t ( c o s a , sina)ll

m a x ( J t c o s a 1 , ) l- t s i n a l )

Suppose t h a t theorem ( * ) o b t a i n s ; so t h a t we can compute a r e a l number T with $(T) = i n f { $ ( t ) : t E El}. W e s h a l l show t h a t a t 0 or a 2 0 . C l e a r l y , we may assume t h a t l a ( < Tr/4, E i t h e r 0 < T or T < l / ( c o s a + \ s i n 4 ) . I n t h e former c a s e , i f a < 0 t h e n , a s a l i t t l e computation shows, T = - l / ( c o s a s i n a ) < 0 , a c o n t r a d i c t i o n ; hence a L 0 . I n t h e o t h e r c a s e , a > 0 e n t a i l s t h e c o n t r a d i c t i o n T = l / ( c o s a + s i n a ) , and so a 5 0. Is t h e r e a reasonable c o n s t r u c t i v e s u b s t i t u t e f o r t h e fundamental theorem of c l a s s i c a l approximation theory? I n o r d e r t o answer t h i s a f f i r m a t i v e l y , w e need some d e f i n i t i o n s . Let E be a m e t r i c s p a c e , X a l o c a t e d s u b s e t of E , and a an element of E. An element b of X is a best approximant of a i n X i f d ( a , b ) = d i s t ( a , X ) . I f each element of E has a b e s t approximant i n X , w e s a y t h a t X i s proximinal i n E . On t h e o t h e r hand, a has a t most one b e s t approximant i n X i f max(d(a,x),d(a,x'))

>

dist(a,X)

whenever x , x ' belong t o X and d ( x , x ' ) > 0. The subspace X is quasiproximinal i n E i f , t o each element of E t h a t has a t most one b e s t approximant i n X , t h e r e corresponds a (unique) b e s t approximant i n X . I t i s t r i v i a l t o prove c l a s s i c a l l y t h a t q u a s i p r o x i m i n a l i t y and p r o x i m i n a l i t y a r e e q u i v a l e n t p r o p e r t i e s . That they a r e u n l i k e l y t o be e q u i v a l e n t i n a c o n s t r u c t i v e framework i s shown by t h e above Brouwerian counterexample and t h e following theorem, which i s c l a s s i c a l l y e q u i v a l e n t t o ( * ) . THEOREM 1

A f i n i t e - d i m e n s i o n a l l i n e a r subspace X of a r e a l normed space E

quasiproximinal

& I E

C71.

&

0

The proof of t h i s theorem proceeds by i n d u c t i o n on t h e dimension of X , t h e one c o n s i d e r a b l e d i f f i c u l t y (which occurs i n t h e i n d u c t i o n s t e p ) being r e s o l v e d by extending t h e n e s t e d i n t e r v a l s argument used i n t h e c a s e dim X = 1. I t i s worth n o t i n g t h a t t h i s d i f f i c u l t y can be overcome much more e a s i l y by t h e i n t u i t i o n i s t , by an appeal t o h i s theorem t h a t a continuous mapping o f a compact m e t r i c space i n t o t h e p o s i t i v e r e a l l i n e has p o s i t i v e infimum. Theorem 1 s u g g e s t s t h a t t h e next s t e p i n c o n s t r u c t i v e approximation theory i s t o c o n s i d e r c l a s s i c a l s i t u a t i o n s where t h e b e s t approximant i s unique, and t o conv e r t t h e c l a s s i c a l proof of unique e x i s t e n c e i n t o a c o n s t r u c t i v e proof t h a t t h e r e is a t most one b e s t approximant of t h e element i n q u e s t i o n . I n t h e c a s e of Chebyshev approximation over C O , l l , where E = C [ O , l ] and X i s a Chebyshev subspace of C[O,ll, t h i s procedure was c a r r i e d o u t i n C5l (cf.C41). In t h a t c a s e , we can c o n s t r u c t a mapping P which a s s i g n s t o each a i n CC0,ll i t s unique b e s t approximant i n X. Now, c o n s i d e r a t i o n s f a m i l i a r to anyone working i n c o n s t r u c t i v e mathematics suggest t h a t t h e c o n s t r u c t i o n of P should embody a proof of i t s uniform c o n t i n u i t y on compact s u b s e t s of CC0,ll. As I s h a l l i n d i c a t e below, t h i s proof of c o n t i n u i t y of P can be obtained by c a r e f u l l y a n a l y s i n g t h e p r o o f s , and thereby s t r e n g t h e n i n g the results, in [5l.

43

Constructive approximation theory

As it happens, these stronger results lead to a proof of existence of P which does not require an.application of Theorem 1. I suspect that this phenomenon may be quite general, and that every time we examine carefully a classical situation where every element of a normed space has a unique best approximant in a finitedimensional subspace, we will be able to prove constructively both existence and continuity of the best approximation process without appeal to Theorem 1. (See also the remark following [4, 3.11.) However, Theorem 1 remains of interest as both a guide for action in particular cases, and a constructive substitute for the fundamental theorem of classical approximation theory.

3 Let us now look more closely at Chebyshev approximation over [0,11. Let $1, ..., $ be elements of Cc0,ll (taken with the supremum norm). Define N

and

where l l . l l z is the Euclidean norm on lRN. positive real line as follows:

~ ( a )=

infIl$l(x)I

=

Define mappings B,y of (0,1/N1 into the

for each a :

x

E

: 0 5 xl,

J

l
if N = 1,

~0,113

inf{ldetC$.(xi)ll min

(O,l/Nl,

E

IXi

-

Xjl

2

. . . , xN

1,

5

a3

if N t 2 ,

and

~ ( a )= min(ll+ll,

B ( ~ ) / Nf

(N-I)!II:=~(I

+ II$~II)).

In the case N t 2 , B(a) is well defined, as the mapping is continuous on the compact set

IdetC$j(xi)ll 5 1,

minlsi<ja

-

xjl

..., bN}

We say that

(g)

Ixi

{(xl,

(xl,

...,

x

N

..., xN) )

: 0 5

-f

xl,

...,

x

N

a). is a Chebyshev system if

6(a) > 0 for each a

E

(O,l/Nl.

Condition ( g ) is classically equivalent to finite linear combination of $1, ..., @ a simple Brouwerian counterexample to thys N = 2, $,(x) :1, $,(x) E a x , and it is

the well known Haar condition: every has at most N-1 zeroes in C0,ll. For equivalence, consider the case where not known if a = 0 or c1 f 0. Intui-

tionistically, (9is equivalent to the simpler condition: whenever xl, ..., xN are distinct points of C0,ll.

]detC$j(xi)l]

> 0

The best known example of a Chebyshev system is the set of polynomial functions 2

{l,x,x , sin 4nx,

.. . ,x?. ...,

Another example is the set cos h n x , sin 2nnx).

From now on, we shall assume that

I$,,

. . ., $N}

{l, cos 2nx, sin ZTx, COS 4nx,

i s a Chebyshev system, and we

shall denote by H the N-dimensional real linear subspace of Cr0,llspanned by . . . , I $ ~ } . Given a E C[O,l], we shall discuss the characterisation, existence and uniqueness of a best Chebyshev approximant of a in H: that is, an

D.S. BRIDGES

44 element b of H such that \\a- bn

=

=

dist(a,H)

-

inf{lta

c~=lci$ill : cl,

...,

CN E

m}

The classical characterisation of best Chebyshev approximants depends on the notion of "alternant". If a E C[O,l] and p E H, then an alternant of a and p is an where j E {0,11, 0 5 x1 x2 < . < XN+l 5 1 ordered pair (j,(xl,. . . ,x~+~)), and

..

Classically, a necessary and sufficient condition for b E H to be the best Chebyshev approximant of a is that there exists an alternant of a and b [12, Theorem 3-11. To see the nonconstructive nature of this characterisation, consider best Chebyshev approximation of a by constant polynomials. In this case, if m and M are respectively the infimum and supremum of a on [ O , l l , then dist(a,H) = &(M - m) and the best approximant of a in H is &(M + m). Were the above characterisation constructively valid, we would be able to find 5 in C0,ll with a(c) whence

a

- &(a + m)

=

&(M

-

would attain its supremum M at

m),

5.

To obtain a constructive characterisation of best Chebyshev approximants, we weaken the notion of "alternant" in the obvious manner. Let a E C[O,ll, P E H and E > 0. An E-alternant of a and p is an ordered pair (j,(xl,...,xN+l)), where j E {O,l}, 0 5 X1 < X2 < .. < xN + 1 5 1 and

> If also 0 < E < \la - pll and m p is an ordered pair (j,(xl, < x2m+4 = 1 ,

- pll

-

(1 5 k 5 N + 1 ) .

E

...,N-l},

0,

E

...,

Ila

x2m+4)),

an (m,E)-prealternant of a and where j E {O,l}, 0 = x1 < X2 < .

..

__ and

sup{((a

-

p)(x)J

:

x~~ s x

5

x2k+l1

<

la

-

(1 5 k 5 m+l).

The only reason (m,E)-prealternants are introduced is to facilitate the construction of E-alternants. In section 3 of 151, I extracted enough of the constructive essence from the proof of the classical characterisation given in Cl21, to ensure the existence of best Chebyshev approximants. However, on closer examination of their proofs, the tedious details of which we omit, we can obtain stronger versions of Lemmas 3.1 and 3 . 2 of C5l: LEMMA 1

Let I) be an element of H with iiI)ll

S

1

0 < c 5 inf{$(x)

: x

E

Constructive approximation theory

Let a

[O,ll}.

+

dist(a,H)

-

S 1/N.

0 <

E

< ilia

-

(m,E)-prealternant there exists ---

pll,

of

p

0

<

-

< Ila

E

pll.

m

...,

{O,

E

and p.

a

N-21,

of

for

(t > 0 ) and suppose that there exists an

Then either

an (m+l,E)-prealternant

2

0

&

R+ + R+

U:

ty(6(2t)/16Nl$l)

and p.

a

Ila - pll

Then either

of

and let 6 be a modulus of continuity

H,

E

Define

E

u(t) Let -

and

H

E

C[O,l],

E

6

p such that

p

or there exists a (0,E)-prealternant

Let a

L E W 2

a

C[O,l],

E

cE/4

45

Ila - PI1 2 dist(a,H)

+ U(E)

Or

0

a and p.

In connection with Lemma 1, I should point out that it is not by any means trivial to construct an element of H with positive infimum. To do so in C51, I first proved that the distance c in IRN from 0 to the convex hull A of {@(x) : x E [O,l]} is positive. Application of the Separation Theorem [l, Chapter 9, Theorem 31 then yields (al, , %) E RN such that

...

The required element of H is then

N

ai@i

We now arrive at two constructive substitutes for the classical characterisation of best Chebyshev approximants.

Let A

PROPOSITION 1 subset of H. -E

> 0

and

totally bounded subset of

B a bounded

CcO,ll,

a: R+ + such that if a E A, p dist(a,H) + a(~), then there exists an E-alternant

Then there is a mapping IIa

-

pll

S

E

B,

of

a

and p. Proof: As H is finite-dimensional, B is contained in a totally bounded subset of H; so that A-B is contained in a totally bounded subset of CC0,ll. By c 3 , Chapter 2 , 5.31, there is a common modulus of continuity 8 for the functions in A-B. Let @ be an element of H with II$II 5 1 and 0 < c Z inf{@(x) : x E [0,11). Define w: R+ + R+ by w(t) Let a E A , Ila - pll or

5

p E B, E > 0 and la - pll 5 dist(a,H) + LO(€). Ila - pll > ~ / 4 . In the former case, as

IIa - pll if

O S X

< x2 <

-

(t > 0 )

min(ct/4, ty(min(l/N,6(2t))/16NII$II))

E

<

< -na

-E/Z

... < xN+,

S

1

-

pll s I(a

then

- p)(x)

(O,(xl,

I

Either

(x

..., x ~ + ~ ) )is

E/2

>

CO,ll),

E

an E-alternant of

a and p. On the other hand, if f a - pll > ~ / 4then, by Lemmas 1 and 2 , there exists an (N-1,E)-prealternant (j,(tl, t2N+2)) of a and p. Choosing xk in [t2k-l,t2kl so that

...,

(-1lk-’(a we obtain an E-alternant

-

p)(xk) (j,(xl,

>

ila - PII-

..., xN+l))

(1

E/4 of

a

and P.

S

k

S

N+1),

0

D.S. BRIDGES

46

Let

THEOREM 2

a

and

C[O,ll

E

an

E-alternant

of

of

and b.

a

A

b E H.

b t o be a b e s t Chebyshev approximant

necessary

s u f f i c i e n t condition

is t h a t , f o r each

a

E

for

> 0, there exists

0

For t h e proof of Theorem 2 , s e e c4, 4.41. From P r o p o s i t i o n 1 w e can o b t a i n a v e r y u s e f u l lemma. A be5 t o t a l l y bounded s u b s e t o f

LEMMA 3 H.

and

Q: IR+

Then t h e r e i s a mapping max(lla

-

pll,Ila

-

911)

-f

IR+

dist(a,H) +

5

e

C[O,l],

B a bounded s u b s e t o f

such t h a t i f

a

E

A, p E B, q

a(€), then

Ilp

-

qll S E .

E

B,

E

>

0

Proof: As i n t h e proof of P r o p o s i t i o n 1, w e can f i n d a common modulus of c o n t i n u i t y 6 f o r t h e f u n c t i o n s i n A-B, w i t h 6 S 1 / N . With w: R+ + IR+ a s i n Propos i t i o n 1, d e f i n e mappings c , Q o f IR+ i n t o R+ a s f o l l o w s : f o r each E > 0 , ( II $11

NY (6( ~ 1 2))- ~ I I $11

c ( ~ )E

-

/y(6 ( € 1 2 ) )N+i-l

1)

and

: m i n ( ~ / 4 c ( 6 ( ~ / 2 ) )+, w ( E / ~ c ( ~ ( E / ~ ) ) ) ) .

Q(E)

Let

a E A, p

E

B, q C B

Ila

-

and

pll + Ila

-

-

IIp

911

2

911

>

€

> 0.

Ilp

-

911

>

As E,

we may assume t h a t IIa - qll > E/2. E i t h e r Ila - 911 > d i s t ( a , H ) or, by P r o p o s i t i o n 1, t h e r e i s an € / 4 ~ ( 6 ( ~ / 2 ) ) - a l t e r n a n n t ( j , ( x l , a and q . I n t h e l a t t e r c a s e , f o r 1 % k 5 N , /(a

Hence

x k+l

-

-

q)(xk+l)

-

(a

-

-

qll

-

E/~c(~(E/z)))

t

z(lla

- qll

-

~/4)

>

E/2.

1 S k 5 N.

-

min l_
Ilp

(as

c 2 1)

Were

>

9) (x,)

Lemmas 4 . 2 and 4 . 3 of [S] would e n t a i l

&~(E/4~(6(€/2))) x ~ + ~ )o)f

...,

z(lla

s)(xk)I

xk 2 6(~/2) f o r

+

-E/~(~(E/Z)),

-

< Ilp

911 5 E

-

911.

Hence t h e r e e x i s t s

k with

-

<

( - I ) ~ - J ( ~ q) (x,)

-E/zC(~(E/Z)).

For t h i s k w e have IIa

-

(-1lk-J(a

PII

-

p)(x,)

=

(-l)k-J(a

>

la

-

~ / 4 c ( 6 ( ~ / 2 ) )+

=

dist(a,H)

+ E/4c(6(€/2)).

-

911

q)(x,)

Thus i n b o t h p o s s i b l e c a s e s w e have

+

max(lla

(-l),-J(q

-

p)(xk)

~/2~(6(~/2))

- pll ,IIa - 911)

>

dist(a,H)

+

R(E).

47

Constructive approximation theory

0

The result follows immediately.

The existence and uniqueness of best Chebyshev approximants follows simply from Lemma 3 and Theorem 1. However, Lemma 3 also enables us to prove existence without reference to Theorem 1. To each

THEROEM 3

H.

whenever

Proof:

p

E

n: R+

Moreover, with

n(E)

a

-t

and

H

E

there corresponds a

C[O,ll

as in Lemma 3, we have

R+

Ilp

unique best approximant b &

-

bll >

E

Ila - pII

t

dist(a,H) +

> 0. lim [la - p 11 = dist(a,H). By n* being finite-dimensional, is complete,

Let (p ) be a sequence in H such that

Lemma 3, (p

is a Cauchy sequence.

)

there exists b in H with

As H,

b = lim n-m pn.

Clearly,

Ila - bll

rest of the theorem follows immediately from Lemma 3.

=

dist(a,H).

The

0

In view of Theorem 3, there is a well defined mapping P - the Chebyshev projection of C[O,ll onto H which carries each element of C[O,l] to its best Chebyshev approximant in H.

The Chebyshev

THEOREM 4

projection *uniformly

continuous on each totally

bounded subset of C[O,ll. ___-and choose R > 0 s o that Proof: Let A be a totally bounded subset of C[O,l], IIall I R for each a in A . With B E {p E H : llpll I ZR), let the mapping n be as in Lemma 3. Given E > 0, let a,a' be points of A with IIP(a) - P(a')ll > E. Then IIP(a)ll

and so P(a)

B.

E

-

I

Ila

P(a)II

5

Ilall + Ilall

+ IIall

dist(a,H)

+ IIall

ZR,

I

Similarly, P(a')

IIa - P(a')ll

=

-

B.

E

=

max(lla

2

dist(a,H) +

By Lemma 3,

p(a)ll,Ila

-

p(a')ll)

a(€),

and so IIa

-

a'II t t

IIa - P(a')ll

- Ila' - P(a')ll

+

dist(a,H)

Q(E)

- dist(a',H).

Likewise, Ila'

-

all t

dist(a' ,H) +

n(E)

- dist(a,H).

Thus IIa

-

all1 2

It follows that

IlP(x)

Idist(a,H)

-

P(x')ll

-

I E

dist(a' ,H) whenever

I

+

Q(E),

x,x'

belong to A and

IIx

- x'I

0 For points in the metric complement of H , the uniqueness property in Theorem 3 can be strengthened considerably.

<

-

D.S. BRIDGES

48

THEOREM 5

&a

=A

{dist(a,H)

: a

a - P(a)

(a

E E

A}. A).

totally bounded subset of C[O,l] such that

Let Let

0 < d E inf

6 be a common modulus of continuity for the functions 0 < CI

min(l/N,6(d)),

5

c 5 N

and let

-2

(y(~~)/lI$ll)~~+~.

Then Ila whenever

a

A

E

-

pll

p

E

2

dist(a,H) + clip

-

P(a)II

0

H.

The essence of the proof of Theorem 5 is found in [ 5 , 4.61. Under the conditions of Theorem 5, the argument of [ 9 , p. 82, Theorem] shows that IIP(a)

-

P(a')l

-1 2c la

5

-

a'l

whenever a,a' belong to A. Thus the Chebyshev projection is uniformly locally Lipschitzian on totally bounded subsets of C[O,ll that are bounded away from H. This observation can be used to produce another proof of Theorem 4 L8, Theorem 21. Although the continuity and Lipschitz properties of P are well known classically L111, our constructive treatment enables us to write down explicit expressions for moduli of continuity and Lipschitz constants. The new information embodied in these expressions is just the kind that classical approximation theorists want, but frequently cannot obtain.

4

The practical computation of best Chebyshev approximants involves a process of discretisation in which we approximate P(a) by best approximants of a over finite subsets of [O,ll. Let X = {x , C[O,ll defiAe

...,

0 i x1 < x2 <

where

E

Ila

-

bllX

=

of

approximant

inf{lla

< x ~ 5+ 1.~ For each f in

: 1 5 k 5 N+1)

sup{lf(xk)I

best Chebyshev

An element b of H is a

...

-

p'lx : p

a

E

C[O,ll Over X if

HI

E

The existence and characterisation of such b are much easier to establish than in the case of approximation over the whole interval C 0 , l l .

Let a

THEOREM 6

E

C[O,ll, and let

(bl,...,bN+l)

be the unique solution of

the equations k bi$i(xk) + (-1) bN+l Then

N

b !Xizl biOi

= IbN+l

I.

=

is a best Chebyshev approximant

Moreover,, Ila

-

pllx > II a

-

bllX

whenever

H.

A

tion that b be the best Chebyshev approximant

of

THEOREM 7

~ e at

E

CLO,ll, and let

---- j (0,l) such

b

E

__-

there exists

(1 5 k i N+1)

a(xk)

E

(-l)k-j(a

-

of

a

p

H

E

necessary a

Over X

over X, and la Ilp

- bllX

sufficient

bl

> 0.

X

0

--

is that for each

E

> 0

that b)(xk)

>

Ila

-

bllX

-

E

(1

< k

5

N+l).

0

49

Constructive approximation theory

For the proofs of these theorems see 15, section 51. From our point of view, it seems natural to look at best approximation over alternants: THEOREM 8 alternant

Let a - E C[O,ll, p (j,(xl,

approximant ~f

E

and E > and p,

H

..., X~+~))f x

a

Ixl,

a

0. Suppose that there exists an

0 <

E

E-

-

and let p be the best Chebyshev

..., x ~ + ~ } .hen

Under the conditions of Theorem 8 , if continuity for a - p with 6 5 1/N,

E-

Ilp - p ,.l,lx 5

< ilia - pll

NE

0

C5, 5.31.

and 6 is a modulus of

then

so that, by Theorem 8 and C5, 4.31,

-

IIP- PII s In particular, if p = P(a)

2

N y(6(11a

then

p,

- PII))-'II$II~ +

P(a)

as

E

+

0.

It is possible that this observation may lead to an efficient algorithm for computing P(a). However, there is an efficient algorithm - the Remes algorithm in common use already. Now it might be suspected that the classical proof of convergence of the Remes algorithm is essentially a constructive proof of existence of the best Chebyshev approximant. There are at least two reasons why this is not the case. First, the convergence proof assumes the existence of the best approximant from the outset; and, secondly, the proof itself is highly nonconstructive. In fact, the classical convergence proof is really a proof of the impossibility of nonconvergence: it does not furnish an explicit estimate of the rate of convergence [9, Chapter 31. It seems to me that this is a typical situation where constructive mathematics may be able to provide results of immediate appeal to the classical mathematician. I would certainly expect a numerical analyst to be interested in explicit convergence rates for an algorithm, especially when he only knows that the algorithm cannot fail to converge. In any case, a constructive proof of convergence of the Remes algorithm has been given and, of necessity, embodies explicit estimates of the rate of convergence [61. Of course, these estimates are not at all efficient ones, as they cover even the most intractable cases of Chebyshev approximation. (It would be interesting to see whether, as I believe would be the case, a close inspection of the constructive proof of convergence of the algorithm would produce efficient estimates in commonly occurring special cases.) There are two further constructive remarks to be made about the Remes algorithm. The first is that what I have called the "Remes algorithm" in C6l is not quite the standard Remes algorithm, as it has been modified to make all its procedures fully constructive in Bishop's sense. From a classical point of view, my algorithm boils down to the standard one in practice. (See also [lo] in this context.) My second remark here is that the constructive analysis of the Remes algorithm required an interesting result on approximate interpolation which appears to be new even in the classical framework.

Let a

THEOREM 9 such that,

if

E

>

0

E

C[O,ll.

Then there exist mappings

and if xl,

..., xN+l

6,w

of

R+

into

are -points of C0,ll with 0 -

<

R+

D.S. BRIDGES

50 minliksN ( x ~ + x ~,) 0,

then there exists p

< 6(E),

such that maxlsk
-

P)(x,)\

2 E

E

C6, 2.21

H with modulus of continuity

U

It is easy to find an element p of H which interpolates the value of a exactly at Theorem 9 says that this interpolation can be performed with maximum error E on a set of N+1 distinct points, provided that this set is close to degenerating into an N-point one. Moreover, the interpolating function can be chosen to have modulus of continuity 0 , independent of E .

N distinct points of C 0 , l l .

5 The work I have outlined ahove is but a first step in constructive approximation theory. Perhaps the next steps will be to look at rational approximation and at approximation in the mean.

In any event, I believe that a constructive approach to approximation theory and numerical analysis will continue to shed light on the subject, and to produce theorems which are new even classically. Moreover, I suspect that there may be many areas of numerical analysis where constructive techniques will provide hitherto unavailable, explicit estimates which, by subsequent more careful analysis, will yield efficient estimates in important practical cases.

REFERENCES

El1 Bishop, E., Foundations of Constructive Analysis (McGraw-Hill, New York, 1967).

c21 Bishop, E. and Cheng, H., Constructive Measure Theory, Amer. Math. SOC. Memoir No. 116 (1972). C3l Bridges, D . S . , Constructive Functional Analysis, (Pitman, London, 1979).

[41 Bridges, D . S . , On the Foundations of Best Approximation Theory, J. Approx. Theory 28 (1980), 273-292.

[51 Bridges, D.S., A Constructive Development of Chebyshev Approximation Theory, J . Approx. Theory 30 (1980), 99-120.

C6l Bridges, D . S . , A Constructive Analysis of the Remes Algorithm, J . Approx. Theory 32 (1981), 257-270.

[71 Bridges, D . S . , A Constructive Proximinality Property of Finite-dimensional Linear Subspaces, Rocky Mountain J . , to appear. C8l Bridges, D . S . , Lipschitz Constants and Moduli of Continuity for the Chebyshev Projection, Proc. Amer. Math. SOC., to appear.

C9l Cheney, E.W., Introduction to Approximation Theory (McGraw-Hill, New York, 1966). [lo] Dunham, C.B., The Weakened First Algorithm of Remer, J . Approx. Theory 31 (1981), 97-98.

[111 Henry, M.S. and Schmidt, D . , Continuity Theorems for the Product Approximation Operator, in: Law, A.G. and Sahney, B.N. (eds), Theory of Approximation with applications (Academic Press, New York, 1976). [121 Rice, J . R . , The Approximation of Functions, V o l . I (Addison-Wesley, Reading Mass., 1964).

THE L.EJ. BROUWER CENTENARY SYIctpOSIUM A S . TbwLFtra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

51

FORMULAS OF ONE PROPOSITIONAL VARIABLE IN INTUITIONISTIC ARITHMETIC Dick H . J .

de Jongh

Mathematisch Instituut University of Amsterdam Roetersstraat 15 1018 WB Amsterdam

The present paper circulated originally as Report 73-03, Department of Mathematics, University of Amsterdam. A brief account of its contents is given in Troelstra (1973, 3.1.14-18 and 5.3.20). Since the paper has been referred to repeatedly in the literature, the editors thought it worthwhile to publish the paper in the present volume. The text is essentially that of the report; some references to later developments, which I owe to A.S. Troelstra and A . Visser have been inserted between square brackets r I, and the bibliography has been updated.

I. TMRUPUCTION It is known (cf. Nishimura 1963, de Jongh 1968) that in the intuitionistic propositional calculus IC, the following sequence is an enumeration of representatives of each of the equivalence classes of formulas in one propositional variable P :

In this article we will study formulas in arithmetic arising from the substitution of a number - theoretic formula A

for P

in the formulas of this sequence. I owe

much to A . S . Troelstra for many stimulating conversations on this and related subjects. Also, I have to thank N. Goodman for his patience in listening to my explanations and for correcting some of my mistakes. We will consider intuitionistic arithmetic HA Troelstra 1971, writing

k for derivability in HA

as formalised e.g. in

.

(In that paper, as in others,

the number - theoretic language is extended to include symbols for all primitive recursive functions. One of the main purposes of this is to make all

c-

formulas

equivalent to purely existential formulas. If one does not insist to be able to reduce a sequence of existential quantifiers to a single one, then it follows from Matijasevic 1971 that the introduction of these symbols is not necessary for this purpose. However, it is slightly more convenient to keep following this practice.) From de Jongh (M) it follows that (assuming the consistency of in the following)

for each

i 5 0

there are number

-

HA as we will

theoretic sentences A , B

52

D.H.J. DE JONGH

If

such that

fi(A),

.

Hgi(B)

In the first four sections we will investigate in

more detail the behaviour of number theory with respect to the sequence of formulas in one propositional variable. In Section 5 we will solve a problem G. Kreisel suggested to the author concerning the possibility of the enumeration by an arithmetical formula of the sequences arising by the substitution of numbertheoretic formulas in the above mentioned sequence. In de Jongh (M) it was noted that the sentences A form in the sense that

Iffi(A)

and

Ifg.(A)

for each

where

T

such that

I-gi+l(A) but not

I-fi+](A)

but neither

kfi(A)

nor

is i 2 0,

T-predicate. One of the questions investigated here is, for which

there are formulas A or

. In particular,

i t 0

v VylT(x,x,y)l

this was proved to be the case for Vx[lVylT(x,x,y) Kleene's

constructed there are uni-

kgi(A),

. A convenient formulation is made possible by the f(P1,...,Pn) is a formula of IC (in such formulas we

l-fi(A)

following definition. If

always assume all propositional variables displayed) and A,, ...,An are formulas of HA , then we say that exactly Ef(A ] , . . . ,A ) , if I-f(AI, ...,A ) and, for each propositional formula g(P,, +

g(P,

HA

,...,P ) . In de Jongh

such that exactly

I-gi(A)

Vx[lVylT(x,x,y) are

+

A

.

V

IkIC

iff

f(Pl,

...,P

)+

.

such that exactly

respectively. Since we assume the consistency of HA

I-fi(A)

or

there is no formula

I-A A 1 A . Moreover, by the results of de Jongh (M)

such that exactly

k A

+

I-g(AI, ...,An)

,

)

there exists a sentence o r formula A

fi,gi A

I-A

...,P

(M) it was shown that there are sentences A,B of B The main question to be studied here is for which

VylT(x,x,y)l

is an example of a formula A

such that exactly

It will be shown here that the only other cases with a positive answer

fo, go, g ] , g2

(for sentences) and in addition

f, and

f2

(for formulas).

Examples for those cases can easily be constructed. Of course, fo and are trivial cases, since exactly E T A . Let IfG

and

IflG

istic logic

,

iff

k A and exactly

Il0-sentence independent of HA

G be a

HA ) . For this G

k fo(A)

kgl(G v 1 G )

and

I-g2(G)

,

since by assumption

s o , by the disjunction property of HA IfG

1-11G

+

G,

I-ll(G

V

1G)

and

hLll(G

V

1G)

V

+

1G (G

and by intuitionV

be nice to be able to give some characterisation of sentences A I-gl(A)

or

I-g,(A)

then clearly exactly

.

If we define H(x) I-f,(H(x))

. For

as

rx=O

the case

fl

A

I=Ol

exactly

kfi(A)

for any

i

t 1

.

V

[x#O

1G)

.

It would

for which exactly A

[G v lGll,

a similar construction is pos-

sible. Immediately obvious, by the disjunction property of sentence A

80 iff

(e.g. a Rosser-sentence for

I

, exactly

I-go(A)

HA

, is

that for no

The main result implies that, even though there are infinitely many non-equivalent formulas of one propositional variable,only a finite number of them can really be distinguished when it comes to provability in first order arithmetic. The corresponding problem for formulas of two o r more propositional variables is open and seems rather difficult. We do not even dare to raise a conjecture in this respect. Also left open is the question whether more formulas of one propositional

53

Formulas of one propositional variable can be distinguished by provability in arithmetic of formulas in which

variable

they occur as subformulas. (It isn't sure that this can be formulated in a satisfactory way.)

2. NUN-COMTRUCTI U E RESULTS We will now first give an (at least at first sight) non-constructive proof for the main result for sentences based on methods due to Smorynski (1973). Thm 2.1. If

HA

f(P)

I-

such that

Proof. -

is an unprovable monadic formula of f(A)

,

then

FllA

I-11A

or

+

IC and A

a sentence of

A.

Our notations will be based on Smorynski 1973 and Fitting 1970. Assume

If12+ A . Then there are Kripke-models K 0 = (K0,<,1kD+,0) and such that K II-HA and K l l - H A , KOI-7A, K l I I - 1 7 A , K 1 V A . 1 0 Let us define Without loss of generality it can be assumed that KO n K1 = K2 = {qeK, I q II- A} We now 'extend KO and K , to the Kripke-model K = (K,<,lk,D) : VllA

K

and

=(Fl,<,l-gl)

.

.

by taking and all

K

p.

X

a set disjoint from and KO U S1 U {pi} 22 (with distinct) and with < as indicated in the picture above and according

=

to the clause that, for each p

-D(pi)

= w

KII-HA. q

E

El,

E

K 1 , p2 5 p

iff

p

E lC2

and, for each

i 2 2

,

(the standard interpretation of the natural numbers). By Smorynski 1973, K , q IF 1 A

One easily sees that, for each

q

q k-

q It- A

1lA + A

iff

p2

5

q

and

E

V

lA

q

iff

iff

q

E

E

s,

K2 o r

q IF 1 1 A iff q

E

KO.

By

D.H.J. DE JONGH

54

induction it follows then that, for each iff

pi+2

and

5 q

K lh' fi(A),

q If

K I# gi(A)

fi+2(A)

iff p.

for each

i 2 0

& and each i

q

E

2

q

.

or

0, q 11- gi+2(A) implies that

2

. This

pi+l 5 q

0

For the simplest cases in the recursive hierarchy the theorem can be strengthX0 2-sentences show the full complexity.

ened, but already the Thm 2.2. a) If A FllA

+

a

II -sentence of HA

is a

1

,

then

I-A

or

I-lA

or exactly

A.

.

0-sentence of HA , then I-A or k l A or exactly \-A + A I0 c) If A i s a II -sentence of HA , then \ - A or I-1A or exactly I-11A + A 2 or exactly k A + A . d) There are X 0-sentences A,B of HA such that exactly I-llA and I-llB + B , 2 Proof. a) is trivial by intuitionistic predicate calculus. b ) follows from some results by Kreisel (cf. Kreisel 1958, Kreisel 1959 and Troelstra 1971): if C(x) b) If A

is

C

is prime and

I-l13xC(x)

kll3xC(x)

3xC(x)

sult, if

then

F l l A , then

I-A

dent of HA to be

,

,

+

. Let

VxC(x)

by classical logic, so ative formula, but

I-3xC(x)

I-l3xC(x) ,

d) For

be a

. If

v C(y)l

3xVy[lC(x)

then

and, if

or B

C(x)

[-3xC(x)

.

is prime and

c) Again by Kreisel's re-

Z i s Peano-arithmetic, then

I-l13xVyClC(x)

V

v C(y)l

Is3xVy[lC(x)

0 1

Il -sentence indepen-

one can just take a

0 II -sentence independent of 1

,

C(y)l

,

HA , then take A I-,3xVyrlC(x)

v C(y)l

since that is equivalent to a neg-

since, for no

n

,

CVyrlC(n) v C(y)l

.

Obviously a), b ) and c) can be strengthened in appropriate ways to generalize to formulas with free variables.

0

[Remark. Leivant 1980, Theorem 3 strengthens Thm 2.1 above, and is established by proof-theoretical methods.]

3. S O M E FORMALlZATlONS OF KLEENE'S SLASH As the formulas gi(P) try to use Kleene's notion

are disjunctions under implication one is tempted to

I

of Kleene 1962 in an attempt to improve on Thm 1 . 1 .

Indeed one obtains in this way an alternative proof of Thm I . I . ,

but it still con-

tains an application of Markov's principle and does not apply to formulas with free variables. (However, this method will give an easy proof for the analogous theorem for IC

itself, as, by the way, will the method of the previous section.)

By formalizing Kleene's

I

in HA

one can reduce this application of Markov's

principle to an application of Markov's rule for HA

(for terminology see

Troelstra 1971). The first formalization given here does not fulfill this purpose, but it is the easiest one and of independent interest. It is also investigated in Troelstra 1973.

Formulas of one propositional variable of HA

Define for each pair of formulas E,A on the length of A i)

E L A

ii) iii)

E E

1. (AAB)

1. (AVB)

1. (A+B) E 1. VxA(x) E L 3xA(x)

iv) v)

E

vi)

55

a formula ELA

by induction

as follows:

,

if A

A

E

++

E + A

++

E

f-f

[E

A

A

rE-fAlI

V [E

++

rE

1. A

A

rE+AlI

+

++

VxlE

I A(x)l

++

3xCE

.1 A(x)

If in case (v)

x

for some variable y

1. A

is prime,

1. B

, rE+BI]

A

,

,

, [E+A(x)ll

A

occurs free in E

not occurring in E

It is then obvious that E J A free in E or A

1. B

E J B

,

take instead ELVXA(X)

or A(x)

++

VyrEJA(y)l

. Proceed similarly in case vi).

contains exactly those variables free which occur

.

We can now proceed in a way exactly analogous to Kleene 1 9 6 2 . L e m a 3 . I. For any formula A (a) (b)

1. 1=0

+

CE+Al

I- E L 1 = 0

+

ELA.

k E

,

of HA

. k E L 1=0

Proof. By definition

induction on the length of A

t+

[E+l=Ol

.

,

if

x

is not free in E

I- [t/xl[ELA(x)l

tf

Proof. First note that

t

E

So (a) is trivial and (b) follows by

of HA

Lemma 3 . 2 . For any two formulas E,A ~in A

.

0

,

and any term

t

A(t)

.

is free for x

Lemma 3 . 3 . For any two formulas E,A in A

(a)

tVx[EJA(x)l

(b)

I-ELA(t)

-f

-f

,

in E

IA

.

(If necessary we change some

If E

of HA

,

if x

.

0

is not free in E

and

t

then

EJA(t)

3xCELA(x)l

0

Proof. Immediate from Lemma 3 . 2 . Thm 3 . 4 .

free for x

then

bound variables.) Then proceed by induction on the length of A

is free for x

of HA

and A

are formulas of HA

and

E k A , then E L E k E L A .

Proof. Without l o s s of generality we can assume that the proof o f - A

from E

does

~

not contain bound variables which are free in E length of the proof of A

from E

. The proof

. Now proceed by

induction on the

is essentially the same as the proof

in Kleene 1 9 6 2 , except that we use Spector's axiomatisation of HA (Spector 1962). Note however that Kleene unnecessarily uses the excluded middle by distinguishing between the cases where the sequence

r

(corresponding to our E ) is consistent

D.H.J. DE JONGH

56 or not. For axioms A

k ELA.

we obtain in fact

0

[The following considerations give some extra insight in the character of

ELA. be obtained by simultaneously inserting E

Let AE mulas of A

+

in front of all subfor-

(if necessary with renaming bound variables so as to prevent variables becoming bound by the insertion). Let

free in E

IQC be intuitionistic predicate

logic. Then we have Lemma. -

E

(i)

IQC I-E

(ii) (iii)

IQC I- E L A

[At+ A 1 ,

+

IQC I-AE

+

CE

+

A1

,

.

++

Proof. (i) is proved by formula induction. (ii) follows from E k E + [A t+ A 1 * k E + [AE + A1 * C A E + CE + A1 (iii) is gain roved by a for-

.

0

mula induction.

We can then establish (which is slightly easier), that E k A

*

EEI-AE ; this is

equivalent to Thm 3 . 4 . Let

%

be obtained by simultaneously inserting "E v"

formulas of A

by A Visser, (1981, 6 . 3 . 1 ) , ATE

+

++

in front of all sub-

(again with renaming of bound variables if necessary).As observed for theories with decidable prime formulae

E LA.

is the mapping introduced in Friedman 1978 for establishing closure under

Markov's rule. Semantically, the mappings A

+

AE

and A

+

AE

correspond to the

construction of a so-called open, respectively closed quotient (see e.g. FourmanScott 1979, 2.18). Visser's observation simplifies the treatment of ping A

+ AE

EL

slightly, but it should be noted that

and hence of the map-

1.

can also be applied to certain theories with not all prime formulae decidable; see e.g. Troelstra 1975.1

Thm 3.5. a) If and

x

k E

-f

B

V

,

not free in E

C

,

1E

kE

Corollary 3 . 6 . a) If

kE

+

3xB(x)

,

x

+

B

V

We have to prove

sitional calculus.

0

k x

[EK]

. b)

If

k E

+

3xB(x)

kE

IE

and

,

then

k E

E

,

I- IE+Bl V CE+Cl k 3x[E+B(x)]

then

.

.

0

Proof. Immediate from Thm 3.5. -

Proof.

and

C

not free in E

Lemma 3.7. For any formula A

V

I- 3x[E+B(x)l.

0

Proof. Immediate from Thm 3 . 4 . -

b) If

E I-[E+B]

then E

then E

of

IA

HA , k7A L1A A

C1A+AI

+

.

[lA+1=01, which follows by propo-

,

Formulas of one propositional variable

Corollary 3.8. 1-3.

b) If

a) If

+

1-1A

,

3xB(x)

+

then

B

,

C

V

I-3x[lA

then

I-[lA+BI

.

[7A+CI

V

.

B(x)l

+

0

Proof. Immediate from Lemma 3.7 and Corollary 3.6. -

Aslightlyweaker form of this result was proved in Troelstra 1971. Actually a stronger form of the result was already announced in Kreisel's review of Harrop 1960 (Kreisel 1963). We can prove Kreisel's result too with the aid of the presently following Lemma. of HA

Definition. A formula A each occurrence of

v

or

has Harrop's property (we write

, if

FX(A))

is in the antecedent of an implication sign in A

3

(see Harrop 1960). Lemma 3.9.

is a formula of HA

If A

RH(A) , then

such that

Proof. By induction on the length of A -

.

(2) A

implies RH(B)

is B

C

A

.

duction hypothesis logic (4) A

I-llC (5) A

I-llB

I-llrB

A

C] +B

is

C

.

+

B

+

is VxB(x)

Thm 3.10. kA

A

Then

C

+

.

B

and

kllC

(3) A

3xB(x)

I-llVxa(x) and

V

If A C

+

C

implies N(C)

RH(A)

. In that case

(a) If RH(A) +

B

is

C. In intuitionistic logic

and again by logic

and

In that case RH(A)

(I)

+

I-A

B

+

v

.

so

.

by the in-

So

\-[A+BI

V

+.

tB+Cl

.

B(x)

rA-tcI

.

( 6 ) If

RH(A)

0

Proof. Immediate from Lemma 3.9 and Corollary 3.8. Unfortunately we have been able to prove

+

I-llB(x)

0

(6) is like ( 3 ) .

C , then

I-llrB+CI

So

.

, then I-3x[A+B(x)l

.

RH(C)

rB+llCI,

implies RH(B(x))

VxB(x)

+

and

and therefore by intuitionistic

. Then M(A) is impossible. . So by induction hypothesis

kll[B+C]

RH(A)

1-73.+ A .

is prime, then it is trivial.

only for formulas E ,with

I-ELE

Harrops property (or equivalent to such formulas), even though an equivalence of

-

these properties seems highly unlikely. [From the observations above we can see

I

that this is nevertheless the case:

E E

a trivial formula induction), hence

E

so

E

t+

1%

Ex

,

hence E L E

+

(1%

+

under the assumption of E

E)

LE

(by

,

is equivalent to aHarropformula under this assumption. Note that we relied

on the decidability of prime

formulas here. We owe this remark to A. Visser.1

Also, no converse to Corollary 3.6 like Corollary 2.11 of Kleene 1962 is provable. It is not in general true that E

1E

does follow from Kleene's Cor. 2.11).

implies that

(although the converse

I-ELE

A counterexample will be conscructed later

in this section. But first we will discuss a slightly more complicated formalization of Kleene's Kleene's

I .

Let Prf(x,'A1)

represent in a natural way (cf. Kreisel

.

-

LQvy 1968)

Subst(yl, ...,yn, rAi- ) stand for the gadelnumber of the formula resulting from the substitution of the

the fact that x

is the gadelnumber of a proof of A

Let

D.H.J. DE JONGH

58

q,...,%

numerals

for the free variables of A

in order of their occurrence.

(It will be too cumbersome to actually strictly adhere to this.) We now write Prf(x,yl

,...,yn,‘A1)

for Prf(x, Subst(yl,. ..,Y,,~A~)),

for 3xPrf (x,yI,. . . ,yn,rA1)

(or ThmCAl))

and

Thm(yl,.

.

again define E R A by induction on the length of A t+

Thm(rE+Al)

++

EpB

++

[E p B

t+

EpB

,

if

A

A

A

Thm(rE+B1)l

E R (VxA(x))

cf

VxCE,pA(x)l

E,p (3xA(x))

++

3x[EpA(x)

A

[E p C

V

Thm(rE+B1)

(v)

-f

if E 1- A , then

(11)

if A

(111)

ri I-Thm( A )

.

Thm(‘E+A(x)’)I

Lemma 3.11. (a) (b)

Thm(rA+B1)

+

Thm(‘E+Ai)

+

EpA

Proof. By remark 111 above

!-A +

+

Thm(rE+A1)

Thm(‘B1).

\-Thm(‘E

-f

of HA

1=0-’)

+

Lemma 3.12.

For each formula A(x,yl

,...,yn)

t

,

(a) follows immediately

.

of HA

0

,

if

is free for x

t

contains at most the variables x,yl,...,yn , then yn, ‘A(x,yI ynF) * Thm(x,yl, yn, ‘A(t,YI,

...,

,...,

,...,

Proof. It is sufficient to prove that, if y or

,

Thm(‘E+Al);

from this and (b) by induction on the length of A

A and t kThm(t,yl

can also assume that,

I-Thrn(‘E+A1)

For each pair of formulas E,A

I- E p 1=0 I- E p I = O

1. , while we

of HA ,

is prime, then A

Thm(‘E+C1)l

A

E ,p C

We again obtain Lemmas as in the case of

(I)

kThm(y,yl,.

(2)

k=y

+

in

...,yn)1 )

is a variable not occurring in A

-

then

(1 1

.

is prime,

EpC

A

(vi)

for all formulas E,A,B

.. ,Y,,~A’)

. .,yn, ‘A(x,yI.. .ynP ) Thm(y,yl,. . . ,yn, ‘A(y ,...,yn, ‘A(y,yI ,...,ynP)) Thm(y .Y ..,yn, ‘A(t

(Thm(y,yl

.

,yl.. ynP )

,

and

tf

,y1,.

..,yn)’ ) .

For ( I ) we just have to assume that !-Subst(y,yl,...,~,, rA(x,~I,...,~nF) = Subst(y,yI,...,yn, ‘A(Y,Y~,...,Y~F) For (2) we note that kt=y + [A(y,yI, ...,y, ) * A(t,yI, ...,y n)l, and we apply remarks I1 and 111. L e m a 3.13. free for x Proof. -

0

For each pair of formulas E,A in A

and

x

By induction on the length of A

Lemma 3.14.

of HA

is not free in E

,

,

then

if

t

is a term of HA

k[t/xl[EpA(x)l

with the aid of L e m a 3.12.

For each pair’.offormulas E,A

of

HA

,

if

t

f-t

,

EpA(t)

D

is free for x

in A

.

Formulas of one propositionalvariable and

x

(a)

I-Vx[EpA(x)l

(b)

I-EpA(t)

,

is not free in E

+

then

EgA(t)

-f

3xCEgA(x)l

0

Proof. Immediate from Lemma 3 . 1 3 .

of HA , if E

Thm 3 . 1 5 . For each pair of formulas E,A

Proof.

59

I-

By induction on the length of the deduction of A

A , then E p E I- E p A

. The proof

from E

is

exactly like the proof of Thm 3.4 except for the use of the remarks I and 11. We will just execute it for the schema of induction. Assume ( I ) EkA(O), (2) EI-A(x)

( 6 ) \-Thm('E+A(x?)

+

Thm('E+A(x)l)

+

Thm(rE+A(x)l)

A

Thm('E+A(x'P)

( 7 ) klO/xIThm('E+A(x?) A

and (4) EeECEg(A(x)

( 3 ) EpEkEpA(0)

A(x'),

-f

obtain (5) E2EI-EpA(x)

rx'/xl[EpA(x)

.

remark I and Lemma 3 . 1 2 , (8) EpEkEga(x)

kE p E

,

then

U

kE+A

or

I-E+B

,

then

I-E

+

A(2)

of

HA , if

I-E

-+

A

B

V

. ,

(b) For each pair of closed formulas E kE Q E

A

by ( 6 ) , ( 7 ) and Lemma 3 . 1 3 and

Thm(%+A(x)l)l

A

Corollary 3 . 1 6 . a) For each pair of closed formulas E,A and

. Then we

A(x'))

+

by (4),

Thm(rE-+A(x')l)

by (2) and remark 111,

by ( I ) ,

finally EpEI-EgAfx)

+

3xA(x)

of HA

,

if

1-E

-+

3xA(x)

and

.

for some n

0

__ Proof. Follows quickly from Thm 3.15.

such that E I E , We have not been able t o find an example to show that

We will close this section with an example of a sentence E but not I-EpE

kELE and

or

I-EpE

I-ELE

.

are not equivalent although we certainly don't believe in

their equivalence. Let A(x,y)

be a prime formula representing the graph of a gen-

. We

eral recursive function which is not provably recursive in HA

.

VxjyA(x,y)

E 1 E means, for each m

there is an n

take E

to be

such that

Vx3yA(x,y) l-A(~n,n) and this obviously holds. (Actually we have shown by this that the set of all

D

such that D

r.e. so that in effect we have shown the non-equivalence of ~ D Q D ,

ID

but we will continue with this particular example.) If

Church's rule, f o r some n But then n is A(x,y)

D

,

I-VxC{n}(x)+

A

rVx3yA(x,y)

-+

ID

is not

with

I-DLD

or

I-ELE , then by

A(x, {n}(x))ll

would be a GEdelnumber of the recursive function the graph of which

,

so

pVx[{~$(x)+l

means Vx3y(Thm(x,y, tradiction as

and we have a contradiction. So

%x3yA(x,y)

kE L E

-+

A(x,y?

) ; I-E g E

IfELE. E Q E

easily leads t o a similar con-

.

[A. Visser showed us the way to find an example demonstrating the non-equivalence of

I-E Q E

A

Thm( )'A'

+

and

I-E

L E . Let

. We will .show

A be a 1-E p E

0 1

II -Rosser sentence, and take E but not

I-E E

.

ERE

becomes

to be

60

D.H.J. DE JONGH [(A+Thm('Al))

A

( ' m h T

A+Thm('Al))

+

A')

+

r(A-+Thm('A'))

~Thrn(~Al)l

.

is a II -sentence, the second conjunct of the antecedent is equivalent

Since A in HA

QAI 0

I

, which

(by intuitionistic logic) to)'A'(mhT Thm('Al)

using the fact that

3x Thm('(A+ThmrA1))

equivalent to

x , ' J 3x P r f (

)

+

.

is 3xPrf(x,'A1)

Again

one obtains that the consequent is

3xPrf(x,'A1)

I

P'r f) (x, 'A'

. From the fact that

)

I- 3x Thm( 'Prf )l)'A',x(

it follows that

I-

3xPrf(x,'A7) whence

I

1A]

) 1 A '

A

-+

Since P r f (x, 5 ' )

%'))

((A+Thm(

A'x)

A

is +

A)

1Thm( 'A1)1 .

(A+Thm('Al))

+

A

+

5'))

3x((A+Thm(

+

is decidable, this implies that A

o r , equivalently, A + Thm(rA1)

. Now

v 1Thm('A1)

.

Prf(x,%'))

+

3x Prf (x,%

assume

I-ELE

)

V

lThm('i?

. Then by

)

the fact

II$- and the disjunction properties of HA under implication,

is

that A

ELE

Prf(x,'A1?)

this is equivalent (in HA ) to

VxA'x Vx((A+Thm('A'))

)'A'(mhT I-A +

,

k A + lThm('A1)

or

Rosser sentence (suppose that

neither of which is the case if A

,

so

we would find

not to be the case). If on the other hand follow that

I-A

+

is a

, then, A being a Rosser sentence,

I-A +)'A'(mhT

[-A + (Thm('A1)+Thm('X1))

also

+

I-ERE. On the other hand

c (A+Thm( If A

3xThm('(A+Thm('A1)

I-A

+

which is known

\-A + 1 C o n

,

1Thm('A1)

it would immediately

Con, which is also not the case). So indeed

IfELEI.

4 . MAIN RESULTS

The second formalization of

1

in Section 3 enables us to prove Thm 2 . 1

constructively, but not yet to treat the case of formulas with free variables. For that purpose we also have to consider derivability in HA restricted to formulas of a definite complexity n

with derivations

. Remember

that in intuitionistic

systems for the complexity of a formula we do not only count quantifiers but also impiication-signs Let

Thm,

and

(cf. Kreisel - L6vy 1968 and Section 5 of this article). Prfn

represent the counterparts of

with complexity S n , and let us write

\knA f o r

Thm

and

Prf

"there is a proof f o r

for proofs A

in

HA with complexity S n " . Next we define, f o r each E,A of HA , and each by induction on the length of A

n,E%A ERA,

by replacing in the definition of

E everywhere by % and Thm by Thm,

then leads to Thm 4 . 1 . For each pair of formulas E,A if

knE+A

Proof. We -

,

of HA:

then E % E k E % A .

only have to check that we can assume

.A

similar proof as for Thm 3.16

Formulasof one propositionalvariable

61

k- E-tA , then kThm-(rE-tA1)

(a)

if

(b)

I-Thmn;'rE+J )

[ A + B ] ' )

Thmn(rE

A

,

Thmn( 'Ed)

+

++ Thmn(rE + A A . (c) kThmn(rE-+k?) A Thmn(rE+$) These are rather trivial. e.g. (b) follows from the fact that, if

2)

(I)

knE+A

(3)

I-,E+E

(4)

I-n

(5)

knE

En

and (2)

E

CA+Bl

+

,

then

,

E

+

A

E

A

from ( 3 ) and ( I ) ,

A

A

+

B

from (2) and

from ( 4 ) and (5).

knE+B

Note however that this is dependent on the particular axiomatization for HA

U

(Spector's) used.

Corollary 4.2. For each pair of closed formulas E,3xA(x) if

kn E

Proof. -

+

,

3xA(x)

then E&Ekn

,

then

Proof. Assume

of HA

,

.

0

Obvious.

Thm 4 . 3 . For each formula A

kf(A)

for some n

E%A(rn)

k1X

,

kf(A)

contains only y

C1-h

V

of HA

and

f(P)

of

IC

,

if

yIc f(P)

and

.

A1

+

IfICf(P).

Without loss of generality we can assume that A

free and that for some i t 0

.

(*) l-gi+2(A) v gi+l(A) The theorem will be proved by induction on

i

. If

i = 0

, it

is trivial. Nowlet

us first assume i > I. Then gi+2(A) is gi+l(A) gi-l(A) v gi-2(A) (A) is gi(A) + fi-l(A) By Church's rule for HA , for some n gi+ I bVy3zT(g,y,z) and +

.

kCJZCT(Z,y,Z)

A

UZ=OI

+

gi+2(A)I

A

C3ZTT(g9y.Z)

A

UZZOI

+

, and

gi+l(A)I.

Therefore k[T(:,y,z)

Uz=O

A

+

[gi+l(A)

+

gi-l(A) A

gi-2(A)11

V

[T(g,y,z)

A

A

Us0

So, for some m , (1)

(2)

km gi+l(A) km T(n,y,z)

+

A

CT(g,y,z)

Uz*O

+

A

Uz=O

gi+l(A)

+

gi-l(A)

V

gi-2(A)1

.

Moreover, if we take m h 2 (3) (4)

I- ~(g,y,z) A uz=o T(;,y,z)

A

UzZO

-+

~hm~(r~(g,y,z)A u z = o l )

,

+

Thmm(rT(n,y,z)

.

A

From ( 2 ) and ( 4 ) we obtain

(5)

k T(n_,y,z)

A

Uz*O

r

-+

Thm( gi+](A)')

.

UztOl)

,

+

gi+l(A)l

D.H.J. DE JONGH

62

I-Ic

From

* gj(P)l

[gi+l(P)

tf

-

gj(P)

I- Thmm(rgi+l(A) * g j ( A ? )

(7)

,

j 5 i

for

Thmm( rgi-l

(A)1)

.

v Thmm( rgi-2(A)1)

i t follows that

T h m m ( r Ig . ( A p ) ,

Now by a p p l y i n g ( 7 ) and ( 4 ) t o ( 6 ) we o b t a i n +

,

I- T ( n , y , z )

1Thmm(‘gi(A)’)

Thus, s i n c e

t,,

i .

j

for

gi-*(A)

+

giM

Uz=o

A

*

,

Then by (8) and (5) and a g a i n p r o p o s i t i o n a l c a l c u l u s

Therefore V

gi(A)l)

k l l T h m m ( r g i + l (A)

k

and f i n a l l y

flection principle for

gi+l(A)

,

Thm,

duced i n t h e a s s u m p t i o n (*)

i

Now j u s t t h e c a s e g2(A)

+

g0(N

If

A

if

knf(A)

[A. HA

V

f,,W

is a sentence,

,

then

gi(A)l)

V

=

i

by

A

V

i s a f o r m u l a we u s e a r e -

T r o e l s t r a 1973, 1 . 5 . 6 ( a ) l and w e h a v e re-

see e . g .

1

[Because

gi(A)

V

I- Thmm(rgi+l(A)

and by Markov’s r u l e

.

1

r e m a i n s . But e x c e p t f o r t h e f a c t t h a t

t h e p r o o f r e m a i n s e x a c t l y t h e same.

g3(A)

is

0

t h e n t h i s p r o o f shows

t n l l A

or

I-,llA

+

A.

Visser o b s e r v e d t h a t t h i s c a n i n t u r n b e f o r m a l i z e d so a s t o y i e l d

I- Thm(rf(Ajl)

+

Thm(?lA

V

11A-tA)’)l

5. A PROBLEM OF G. KREISEL I n a l e t t e r t o t h e a u t h o r o f May 23, 1 9 7 0 , G . K r e i s e l

suggested e s s e n t i a l l y

t h e f o l l o w i n g problem. Does t h e r e e x i s t , f o r e a c h f o r m u l a

A

w i t h one a d d i t i o n a l f r e e v a r i a b l e

n

x

such t h a t , f o r each

of

,

HA, a f o r m u l a

I-B(n)

t+

B

gn(A).

We w i l l p r o v e h e r e t h a t n o t e v e n f o r e v e r y s e n t e n c e d o e s such a f o r m u l a e x i s t and i n f a c t t h a t no s u c h f o r m u l a e x i s t s f o r any words f o r any f o r m u l a

A

A

such t h a t e x a c t l y t A + A

,

i n other

f o r which no p r o p o s i t i o n a l f o r m u l a i s p r o v a b l e e x c e p t

when it i s p r o v a b l e a l r e a d y i n

.

IC

a n i n f i n i t e number o f n o n - e q u i v a l e n t

So t h i n g s go wrong a s soon a s t h e r e r e a l l y are f o r m u l a s t h a t c a n b e o b t a i n e d from

A

with

p r o p o s i t i o n a l c o n n e c t i v e s . Kreisel s u g g e s t e d t h i s problem b e c a u s e i t s n e g a t i v e sol u t i o n shows v e r y c l e a r l y ’ w h y i n

HA

i t i s n e c e s s a r y t o i n c l u d e t h e number o f im-

Formulas of one propositionalvariable

63

plications when assigning a degree of complexity to a formula (cf. Kreisel L6vy 1968). Thm 5.1. If, for a sentence A

HA , exactly

of

I-A

A , then for no formula B

+

of HA , for each n , I-gn(A) ++ B(n) . Proof. We form the same Kripke-models KO, K , and K as in the proof of Thm 2.,1. It was proved there that, for each q E K and each 1 2 3 , qll-gi(A) iff pi 5 q . Let us say that a sentence C has degree d

in the Kripke-model

the smallest natural number with the property that, for each pkll-C.

K

,

if

d

is

j,k t d, p.II-C iff J

For our purpose it will be sufficient to prove that, for any formula

...,xn)

C(xl,

the degrees of the sentences C(g,

will be proved by induction on the length of

,...,%)

C

in K

are bounded. It

that there is a number nC which

is such a bound.

(i)

If

C

is a prime formula, then nC = 0 , since K

in HA (ii)

If

C

is A

(iii)

If

C

is A v B

, ,

(iv)

If

C

is A

, then for each q

qll-A

+

is a model for HA

and

all prime formulas are decidable.

B

A

+

B B

then we take nC

=

again take n C

max(nA,nB).

iff, for each

r

2

q

,

=

max(nA,nB).

K ,

E

rll-A , then rlkB.

if

One sees easily from the picture in the proof of Thm 2.1 that this implies (v)

that we can take nC = max(n A ,nB ) + 1 . C is V x D , then, since the domains of all the p.

If

the same we can take nC (vi)

If

C

is

=

.

max(%,2)

3 x D , then we can take n

=

C

n D ’

for

i

t

2

are

’

[Remark. A stronger result, obtained by proof-theoretical methods, is found in Theorem I of Leivant 1981.1

BIBLIOGRAPHY Fitting, M.C. (1970),

I n t u i t i o n i s t i c logic, model theory and forcing. North-Holland

Publ.Co., Amsterdam 1970. Fourman, M.E. and D.S. Scott (1979), C.Mulvey, D.S.Scott(editors),

Sheaves and logic in: M.E.Fouman, Applications of sheaves. Springer-Verlag,

Berlin, 302-401. Friedman, H. (1978),

Classically and i n t u i t i o n i s t i c a l l y provably recursive func-

tions, in: G.H.MCller, D.S.Scott(editors),

Higher set theory, Springer-

Verlag, Berlin, 21-27. Harrop, R . A .

(1960),

Concerning f o m l a s of the types A

intuitionistic formal systems.

JSL

25, p.27-32.

+

B

V

C, A

+

(Ex)Bx

in

D.H.J. DE JONGH

64

Jongh, D.H.J. de (19681, Investigations on the i n t u i t i o n i s t i c propositional calculus. Ph.D. Thesis, University of Wisconsin, Madison. (Univ.Microfilms 69-894)

---------

(M), The maAmality o f the i n t u i t i o n i s t i c predicate calculus with respect t o Heyting’s arithmetic. Unpublished.

Kleene, S.C. (1962), Disjunction and existence under implication i n elementary int u i t i o n i s t i c formalisms. JSL 27, p. 11-18. Kreisel, G. (1958), The non-derivability of T x A ( x ) + 3 x l A ( x ) , A primitive in i n t u i t i o n i s t i c f o m a l systems. (Abstract), 3SL 23, p.456-457.

-------

-------

RefZection principle f o r subsystems of Heyting’s arithmetic

(1959), (Abstract),

JSL 24, p.322.

(1963),

Review of Harrop 1960. Zbl.Math. 98, p.242-243. Reflection principles and t h e i r use f o r establish-

Kreisel, G. and A. LQvy (1968),

ing the complexity of a3ciom systems. Z.Math.Logik Grundlag Math. 14, p.97-142. Leivant, D. (1980), Innocuous s u b s t i t u t i o n s .

-------

JSL 45, p.363-368.

Implicational complexity in i n t u i t i o n i s t i c arithmetic. JSL 46,

(1981),

p.240-248. MatijaceviP, Yu.V. ( 1 9 7 1 ) , . Diophantine representation of recursively enumerable predicates, in: J.E. Fenstad(editor), Proceedings of the second Scandinavian logic symposium. North-Holland Publ.Co., Amsterdam, pp. On formulas of one propositional variable in i n t u i t i o n i s t i c

Nishimura, I. (1960),

propositional calcuLus.

JSL 25, p.327-331.

Smorynski, C.A. (1973), Investigations of i n t u i t i o n i s t i c formal systems by means of f i i p k e models. Ph.D.Thesis, University of Illinois at Chicago Circle.

-------

(1973A),

Spector, C. (1962),

Applications of Kripke models. In Troelstra(l973).

p.324-391.

Provably recursive functionals of analysis, a consistency

proof of analys,is by an extension of principles formulated in current in-

t u i t i o n i s t i c mathematics, in: J.C.E. Dekker(editor), Proceedings of the symposia in pure mathematics V. American Math.Soc., Providence, RI, p. 1-27.

Troelstra, A.S. (1971),

Notions of r e a z i z a b i l i t y f o r i n t u i t i o n i s t i c arithmetic i n

a l l f i n i t e types, in: J.E. Fenstad(editor), Proceedings of the second Scandinavian logic symposium. North-Holland Publ.Co., Amsterdam, p.369-425.

-------

-------

(1973)(editor),

Metamathematical investigation of intuitionistic

arithmetic and analysis. Springer Verlag, Berlin. (1975), Markov )s principle and Markov ) s rule f o r theories of choice sequences, in: J. Diller, G.H. Mcller(editors), Proof theory symposium Kiel 1974. Springer Verlag, Berlin, p.370-383.

Visser, A. (1981).

Aspects of diagonalization and provability. Akademisch Proef-

schrift, Rijksuniversiteit Utrecht.

THE L.E.J. BROUWER CENTENARY SYWOSIlM A S . Troelstra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

65

COATINDOUS SUMS OF SQUARES OF F O R M

Charles N. Delzell Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA

We give a continuous representation of positive semidefinite (psd) n-ary quadratic forms over an ordered field as sums of (almost n!e) nonnegatively-weighted squares of linear forms. This answers a question of Kreisel, who noticed in 1980 that (already for n = 2) the usual "completion-of-square" process gives a discontinuous representation. For n = 2 J.F. Adam has recently reduced the required number of continuous summands to 2, but only over Euclidean ordered fields. We also show that any universal representation of psd quartic forms as sums of squares of quadratic forms must be discontinuous at (X2 + Y2)'. Contents:

0. Introduction 1. Continuous Versions of Some Classical Results (a) Lagrange's and Siegel's Theorems on Sums of Squares (b) The "Weak" Hilbert Nullstellensatz (c) Hilbert's 17th Problem 2 . Sums of Squares of Forms

3. Statement of Continuity Results 4 . Significance of Continuity (a) Geometric Significance (b) The Case of Recursive Ordered Ground Fields (c) The Case of IR 5. The Proof for Quadratics 6 . The Proof for Ternary Quartics

0. Introduction We construct (5.1) a representation of psd (positive semidefinite) quadratic forms in X1, Xn over an ordered field K as sums of (almost n!e) nonnegat ively-weighted squares of linear forms such that each summand, regarded as a function of the Xi and their coefficients, is rational with integer coefficients and (extensible to a function which is) continuous with respect t o the usual interval topology on K, hence also with respect to finer "computational" topologies on enrichments of IR by certain kinds of representations. This answers a question of Kreisel, who noticed in 1980 that (alread for n = 2) the usual SOS (sum of squares) representation of aX2 + 2bXY + CYJ (writing (X,Y) for (Xl,X2); a > 0, c 0, ac - b2 > 0) as

...,

(0.1)

a-'[

(ax +

+ dY2], o r

Research supported in part by NSF grant No. MCS8102744. 1980 Mathematics Subject Classification: Primary 03355. 10C05, 10C10, 10505, 54C05; Secondary 10C04, 54H13.

66

C.N. DELZELL + (bX + CY)']

c-l[dX2

(0.1')

(d = a c - b 2 ) , o b t a i n e d by " c o m p l e t i n g t h e s q u a r e , " i s d i s c o n t i n u o u s n e a r t h e ray ( a , b , c ) = (O,O,c) ( c > 0 ) , s i n c e a s a + 0 , b 2 / 4 a v a r i e s between 0 a n d c. Our p r o o f o f 5 . 1 c o n s i s t s , f o r n = 2 , o f t a k i n g t h e convex combinat i o n s (by a / s and c / s , where s = a + c ) o f 0 . 1 and 0 . 1 ' t o g e t t h e c o n t i n uous r e p r e s e n t a t i o n

+ by)' + dY2 + dX2 + (bX + cY)'].

(ax

s-'[

(0.2)

W e a l s o show ( 6 . 1 ) t h a t any map o f psd t e r n a r y q u a r t i c forms i n t o e q u i v a l e n t SOS o f q u a d r a t i c forms must have a jump d i s c o n t i n u i t y a t ( X 2 + Y2)' ( H i l b e r t [1888] gave t h e f i r s t d i s c o n t i n u o u s map of psd t e r n a r y q u a r t i c s i n t o e q u i v a l e n t SOS o f q u a d r a t i c s ) .

1. C o n t i n u o u s V e r s i o n s of Some Other C l a s s i c a l R e s u l t s S i n c e t h e e a r l y s i x t i e s Kreisel h a s asked whether "continuous" v e r s i o n s of v a r i o u s c l a s s i c a l r e s u l t s a r e p o s s i b l e . L e t u s l i s t some o f t h e a n s w e r s : ( a ) L a g r a n g e ' s and S i e g e l ' s Theorems on Sums o f S q u a r e s I n t h e c a s e o f L a g r a n g e ' s theorem, H e i l b r o n n [1964] c o n s t r u c t e d i n t e g r a l functions f l , f 2 , f 3 , f 4 which s a t i s f y ( w r i t i n g m = 4 ) (1.1) and s u c h t h a t

&fi(z)

I n [Delzell, follows: let F

m

such t h a t

9 . for

0

<

z

E

Q.

i n p r e p a r a t i o n ( c ) ] w e g e n e r a l i z e H e i l b r o n n ' s Theorem as be a c o u n t a b l e s u b f i e l d of C . For m > 1 write Fm f o r

t h e s e t o f "sums of fixed

E

m

squares" i n

we c o n s t r u c t +cfi(z)

E

m F

F, and

F

m

integral functions for a l l

z

E

for fl,

u,,

...,fm

Fm.

B e n f o r each

s a t i s f y i n g 1.1 and

It i s i n t e r e s t i n g t o combine t h i s

Fm.

g e n e r a l i z a t i o n w i t h S i e g e l ' s Theorem t h a t i f K i s an a l g e b r a i c number f i e l d (hence i d e n t i f i a b l e with a s u b f i e l d of a: o r even t h e a l g e b r a i c numbers @ , 'Ihe r e s u l t i s t h e t h e n K 4 = &,, t h e s e t o f t o t a l l y p o s i t i v e e l e m e n t s o f K. c o n s t r u c t i o n of f u n c t i o n s gl , g 2 , g 3 , g 4 which a r e a n a l y t i c i n t h e complex p l a n e m i n u s t h e n e g a t i v e r e a l a x i s , which s a t i s f y Z ? = l g i ( z l 2 = z , and such t h a t gi(z) E K f o r each z E & . ( b ) The "Weak" H i l b e r t N u l l s t e l l e n s a t z F o r t h e weak H i l b e r t N u l l s t e l l e n s a t z , t h e n e g a t i v e answer t u r n e d o u t t o b e e a s y t o prove: t h e two p o l y n o m i a l s f(X) = a and g(X) = bX + 1 i n C[X] h a v e no common z e r o i n C1 for ( a , b ) E S = I ( a , b ) E C21a + 0 o r b = 01; b u t i f we a p p l y t h e N u l l s t e l l e n s a t z t o w r i t e 1 : p(X)f(X) + q(X)g(X) ( p , q E C [ X ] ) , t h e n t h e c o e f f i c i e n t s o f p and q , a s f u n c t i o n s on S , must v a r y d i s c o n t i n u o u s l y n e a r a = b = 0. ( c ) H i l b e r t ' s 1 7 t h Problem

I n t h e c a s e o f H i l b e r t ' s 17th problem, l e t K b e a n o r d e r e d f i e l d ( e . g . = { x , E Klx > 01, c o n t a i n e d i n a r e a l c l o s e d o r d e r e x t e n s i o n f i e l d (e.g. R), with R+ = { x E Rlx > 01. L e t X = (Xo X,) be indetermi-

Q) w i t h R

nates, let

x = (xo,

...,x n )

,...,

E

Rn+l,

let

f

E

Z[C;X]

b e t h e g e n e r a l form i n

X

Continuous sums of squares

of degree

d

with c o e f f i c i e n t s

C = (Cj),

1< j

61

C

cnAd).

For f i x e d

say that f(c;X) i s p o s i t i v e s e m i d e f i n i t e over R in X i f f ( c ; x ) > 0. W e s h a l l abbreviate "positive semidefinite" "psd."

as

Pnd = { c then Delzell functions

([19801

p i : Pnd

R

E

( n:d)

and

i s psd o v e r

[ i n preparation

R+

+

If(c;X)

f(c;X) = Xpi(c)ri(c;X)2,

w i t h e a c h smmand

multaneously i n

c

and

x

and r a t i o n a l i n

X.

Unfortunately, the

pi

pirf o u s ( r e l a t i v e t o t h e u s u a l i n t e r v a l t o p o l o g y on

for

(c;x)

and

E

in

X};

( a ) ] ) constructed

r i : Pnd x Rn+'

and

R

+

a f i n i t e set of

satisfying,

R

c , we

v x E Rn+l, Let

t l c E Pnd,

a f u n c t i o n which i s c o n t i n u R) (and " s e m i - a l g e b r a i c " )

Pnd x Rn+',

and w i t h

ri

si-

homogeneous

need n o t t a k e p o i n t s w i t h c o o r d i n a t e s i n

ri

is r e a l closed; i f

K i s a c o u n t a b l e suba s a b o v e c a n b e found which do t a k e p o i n t s

K t o elements of K, unless K f i e l d o f R, t h e n pi and r i

w i t h c o o r d i n a t e s i n K t o e l e m e n t s of K; w e l o s e o n l y s e m i - a l g e b r a i c i t y , n o t constructivity [Delzell, i n preparation (b)]. I f , furthermore, every p o s i t i v e e l e m e n t o f K i s a sum o f a bounded number o f s q u a r e s i n K, o r e q u i v a l e n t l y , i f K i s u n i q u e l y o r d e r a b l e w i t h f i n i t e P y t h a g o r a s number ( e . g . i f K = Q o r i f K i s E u c l i d e a n ) , t h e n by ( a ) we may a b s o r b t h e p i i n t o t h e r i . I am g r a t e f u l t o P r o f e s s o r Kreisel f o r many c o n v e r s a t i o n s on a l l t h e topi c s d e a l t with i n t h i s paper.

2 . Sums of Squares of Forms

In t h i s p a p e r we c o n s i d e r c o n t i n u o u s v e r s i o n s o f a n o t h e r c l a s s i c a l result:

for

d

even

and

(2.1)

d

< 2

Pnd

or

a s above, t h e c o n d i t i o n

n

<

1

or

(n,d) = ( 2 , 4 )

imp1 i e s

(2.2)

Vc E Pnd t h e r e e x i s t f i n i t e l y many g i E R[X] ( n e c e s s a r i l y homogeneous o f d e g r e e d / 2 ) such t h a t f ( c ; X ) = Cgi(X)2.

[

For d < 2 t h i s g o e s back t o Lagrange; f o r n < 1 t h i s w a s well-known i n 1888 ( a c c o r d i n g t o H i l b e r t ) , and p u b l i s h e d by Landau [ 1 9 0 3 ] ; f o r (n,d) = ( 2 , 4 ) H i l b e r t ' s o r i g i n a l p r o o f 118881 was f o l l o w e d by a more e l e m e n t a r y p r o o f by C h o i a n d Lam [ 1 9 7 7 ( b ) ] . When c E Pnd h a s c o o r d i n a t e s i n K C R i t i s gi E KIX] S R [ X ] a s i n 2 . 2 , b u t t h i s i s i m p o s s i b l e a l r e a d y n = 0 if K h a s p o s i t i v e e l e m e n t s which are n o t SOS ( e . g . i f K = Instead, t h e proper formulation of t h i s d e s i r e is: Q(fi)). n a t u r a l t o seek when

(2.2')

vc [and

E

PndnK gi

E

(,Ad,

K[X]

,

t h e r e e x i s t f i n i t e l y many

such t h a t

pi

E

K+

f ( c ; X ) = Zpigi(X)'.

Note: if e v e r y p o s i t i v e e l e m e n t of K i s a SOS ( e q u i v a l e n t l y , i f K u n i q u e l y o r d e r a b l e ) , t h e n t h e pi i n 2.2' c a n be absorbed i n t o t h e gi. c e r t a i n l y have t h a t c o n d i t i o n

is We

C.N. DELZELL

68

d < 2 or

(2.1')

n < l

implies 2 . 2 ' ; indeed, the case d < 2 is classical, and the case n < 1 may be proved, e.g., by combining (a) Henkin's refinement [1960] of Artin's solun+d tion to Hilbert's 17th problem, that for c E PndflIc( there exist pi E K+ and hi E K(X) such that f(c;X) = Zpihi(X)', with (b) the CasselsPfister Theorem [Cassels 1964, .Pfister 1967, Gerstein 19731 that for f E FLY] (F any field and Y an indeterminate), if f = Zpihf (pi E F and hi E F(Y)), then there exist gi E F[Y] such that f = Zpig?,. I do not know whether 2.1 implies 2 . 2 ' ; the case (n,d) = (2,4) seems problematic. However, once we impose continuity requirements (§3), we shall have (Theorem 6.1) that for (n,d) = (2,4), the topological versions of both 2 . 2 and 2 . 2 ' are false.

I,

3 . Statement of Continuity Results

Let us formulate continuous versions 3.1 and 3.1' of 2 . 2 and 2 . 2 ' respectively. Continuity is understood to be with respect to the usual interval topology on R. We write h = d/2, g E Z[D;X] for the general form of degree h in X with indeterminate coefficients D = (D.), 1 < j < e, where e = J

Khh) *

!

m e r e exist finitely many functions ai: Pnd (3.1) Zg(ai(c);X)', with each summand g(ai(c);x)' for (c;x) E pnd x R"+'.

+

Re such that f(c;x) = continuous in c and x

!

There exist finitely many functions pi: Pnd + R+ and ai: Pnd + Re which take points with coordinates in K to points with coordinates (3.1') in K, such that f(c;X) = Zpi(c)g(ai(c);X)2, with each summand pi(c)g(ai(c);xP continuous for (c;x) E pnd x ~ n + l . Obviously 3.1' implies 3.1. The following Proposition shows that the continuity of each summand is equivalent to the continuity g(ai(c);x)' in 3.l'for (c;x) E Pnd x Rn+' of the ai. Proposition: Let g h be as above, and let a: A + Re & a function from a topologix space A i z o T h x f f i c i e n t space of g. & continuous for ( ~ ; X I E A x a & continuous if and only if g(a(c);x) Rn+'. Proof: lhe "only if" direction is trivial. For the "if" direction, we use induction on

n.

For

n = 1, write

a =

(ao,. ..,ah), g(a(c);Xo ,X1) = Z"o ai(c)XiX:-i. For the (h+l)-dimensional R-vector space spanned by the indeterminates {Di10 < i < h}, we have another ,Dh;l,i) 10 C i < h}; the linear independence of B is a basis B = {g(Do,. consequence of the non-vanishing of the Vandermonde determinant

..

69

Continuous sums of squares I1

So t h e r e a r e

uij E R

Substituting

a

for

1

1

... 1 I

such t h a t f o r

< i < h,

0

D, w e s e e t h a t t h e c o n t i n u i t y o f

g

implies t h a t o f

a.

Now s u p p o s e t h e " i f " d i r e c t i o n o f t h e P r o p o s i t i o n h a s been e s t a b l i s h e d f o r n-1; t o e s t a b l i s h i t f o r n, suppose t h a t g ( a ( c ) ; x ) i s c o n t i n u o u s , and e x p a n d g i n powers o f ,X, w i t h c o e f f i c i e n t s forms i n Xo,...,Xn-l. By t h e r e s u l t for x

R";

n = 1, t h e s e c o e f f i c i e n t s a r e c o n t i n u o u s f o r

by t h e i n d u c t i v e h y p o t h e s i s ,

a

is continuous.

3 . 1 ' d o e s n o t imply t h e c o n t i n u i t y o f t h e

pi

( C ; X ~ , . . . , X ~ -L~ A)

Q. E . D .

and

ai

separately.

We now s t a t e t h e two main r e s u l t s of t h i s p a p e r , which appeared i n my d i s s e r t a t i o n [ D e l z e l l 19801: F i r s t , t h e p o s i t i v e r e s u l t ( 5 . 1 ) is t h a t f o r d < 2, 3 . 1 ' h o l d s ; i n f a c t , t h e pi and t h e ai are ( t h e u n i q u e ' c o n t i n u o u s ex-

t e n s i o n s o f ) r a t i o n a l f u n c t i o n s of c with i n t e g e r c o e f f i c i e n t s . n e g a t i v e r e s u m ( 6 . 1 ) i s t h a t f o r ( n , d ) = ( 2 , 4 ) , 3.1 i s f a l s e . (d > 2 ) , I d o n o t know w h e t h e r e i t h e r 3.1 or 3 . 1 ' h o l d s .

Second, t h e For n = 1

The c o n t i n u i t y q u e s t i o n d o e s n o t a r i s e f o r v a l u e s o f n and d n o t s a t i s f y i n g 2.1, s i n c e f o r such (n,d) 2.2 i s f a l s e , as shown f i r s t by H i l b e r t [1888]. Work o f E l l i s o n (1968, u n p u b l i s h e d ) , Motzkin [1967, p. 2171, and R . M. Robinson [1973] toward s i m p l e r a n d / o r more e x p l i c i t p r o p f s o f t h i s f a c t , c u l m i n a t e d i n Choi and Lam's [ 1 9 7 7 ( a ) , ( b ) ] c o n s t r u c t i o n o f t h e psd forms

+ Y2Z4 +

X2Y4 w4

Z2X4

-

3X2Y2Z2

+ X2Y2 + Y222 + 22x2

-

and

LXYZW,

which t h e y e a s i l y showed t o b e n o t r e p r e s e n t a b l e a s SOS o f ( r e a l ) f o r m s .

4. The Significance of Continaity ( a ) Geometric S i g n i f i c a n c e I n h i s book (18991 on t h e f o u n d a t i o n s o f geometry, H i l b e r t p o i n t e d o u t t h e s i g n i f i c a n c e o f r e p r e s e n t i n g ( n o t o n l y q u a d r a t i c and q u a r t i c ) p o s i t i v e d e f i n i t e forms a s sums o f s q u a r e s o f r a t i o n a l f u n c t i o n s , f o r s o l v i n g problems i n p l a n e g e o m e t r i c a l c o n s t r u c t i o n by means o f o n l y a s t r a i g h t e d g e and gauge ( a n example o f a gauge i s a compass whose u s e i s r e s t r i c t e d t o t h e l a y i n g o f f o f d i s t a n c e s on a s t r a i g h t l i n e ) . I n view o f t h i s g e o m e t r i c o r i g i n o f H i l b e r t ' s 1 7 t h p r o b l e m , s t r e s s e d i n h i s own p r e s e n t a t i o n o f [ 1 9 0 0 ] , i t seems n a t u r a l t o impose t o p o l o g i c a l c o n d i t i o n s , o f t h e k i n d f a m i l i a r from Hadamard's p r i n c i p l e (for well-posed problems: s o l u t i o n s h a v e t o be c o n t i n u o u s i n t h e p a r a m e t e r s , t a c i t l y , f o r t h e t o p o l o g y and c h o i c e o f p a r a m e t e r s a p p r o p r i a t e t o the situation).

C.N. DELZELL

70

( b ) The Case of R e c u r s i v e Ordered Ground F i e l d s L e t K and R be r e c u r s i v e o r d e r e d f i e l d s , i . e . o r d e r e d f i e l d s s u c h as and t h e r e a l a l g e b r a i c numbers, w i t h a numbering o f t h e i r u n d e r l y i n g s e t s f o r which t h e f i e l d o p e r a t i o n s and t h e o r d e r ( h e n c e a l s o t h e e q u a l i t y ) r e l a t i o n s a r e recursive. Then r e c u r s i v e v e r s i o n s o f b o t h A r t i n ' s s o l u t i o n t o

Q

H i l b e r t ' s 17th problem and t h e r e s u l t s t h a t 2 . 1 i m p l i e s 2 . 2 , and 2.1' i m p l i e s 2.2', were known. More p r e c i s e l y , r e c u r s i v e maps from t h e c o e f f i c i e n t s (E For A r t i n ' s theorem, A. P n d ) i n t o s u i t a b l e r e p r e s e n t a t i o n s had b e e n found. Robinson [1963] used model t h e o r y and K r e i s e l [1960] used p r o o f t h e o r y ; f o r 2 . 1 ' i m p l i e s 2.2', t h e c a s e d = 2 simply amounts t o d i a g o n a l i z i n g a s y m m e t r i c m a t r i x o v e r K ; t h e c a s e n = 1, f o r 2.2, amounts t o f a c t o r i n g a b i n a r y form o v e r a r e a l c l o s e d f i e l d and u s i n g t h e 2-square i d e n t i t y , and f o r 2 . 2 ' , t h e c o n s t r u c t i o n g i v e n above i n 12 u s i n g ( a ) and ( b ) p r o v i d e s a r e c u r s i v e representation; finally, i n the case ( n , d ) = ( 2 , 4 ) , Choi and Lam's e l e m e n t a r y c o n s t r u c t i o n o f 2.2 i s a l r e a d y r e c u r s i v e . ( c ) The Case o f

R

However, when K = R = R a l l o f t h e s e c o n s t r u c t i o n s (and a f o r t i o r i t h e i r p r o o f s ) become i n a d e q u a t e from an i n t u i t i o n i s t i c p o i n t of v i e w , s i n c e t h e o r d e r r e l a t i o n (and e v e n t h e e q u a l i t y r e l a t i o n ) i s u n d e c i d a b l e , w h e t h e r t h e elements of R a r e p r e s e n t e d a s , s a y , d e c i m a l s , Dedekind c u t s o f r a t i o n a l s , o r p a i r s ((rn),IJ) o f a Cauchy s e q u e n c e o f r a t i o n a l s and a "modulus o f convergence"

function

IJ

satisfying

vk

>

0,

vn,m

> u ( k ) , [ lrn-rm\< l / k ] .

Our c o n t i n u o u s v e r s i o n s o f A r t i n ' s theorem ( § l ( c ) ) and o f (d = 2 i m p l i e s 2 . 2 ' ) f i l l i n t h i s g a p , and n o t o n l y f o r K = R = R , b u t more g e n e r a l l y whenever R c o n t a i n s a dense r e c u r s i v e s u b f i e l d F: t h e n i f t h e c o e f f i c i e n t s o f a psd form a r e g i v e n by a p p r o x i m a t i o n s from F, w e c a n compute a p p r o x i m a t i o n s o f t h e SOS r e p r e s e n t a t i o n t o c o m p a r a b l e a c c u r a c y ; t h e p r o o f s a l s o a r e i n t u i t i o n i s t i c . On t h e o t h e r hand, o u r d i s c o n t i n u i t y r e s u l t f o r (n,d) = (2,4) does n o t n e c e s s a r i l y imply t h a t 2.2 c a n n o t be proved i n t u i t i o n i s t i c a l l y f o r (n,d) = (2,4),even when R = R . 'Ihe r e a s o n i s t h a t t h i s d i s c o n t i n u i t y r e s u l t i s w i t h r e s p e c t t o t h e u s u a l i n t e r v a l t o p o l o g y on R , and n o t w i t h r e s p e c t t o f i n e r " c o m p u t a t i o n a l " t o p o l o g i e s on v a r i o u s "enrichments" o f R by s p e c i f i c r e p r e s e n t a t i o n s , say o s c i l l a t i n g b i n a r y expansions with t h e corresponding Baire s p a c e o r "weak"

t o p o l o g y on

y,

o r Cauchy s e q u e n c e s o f r a t i o n a l s w i t h t h e to-

&.

While any f : R + R conp o l o g y i n h e r i t e d from t h e p r o d u c t t o p o l o g y on t i n u o u s w i t h r e s p e c t t o t h e u s u a l topology i s o b v i o u s l y c o n t i n u o u s w i t h res p e c t t o t h e c o m p u t a t i o n a l t o p o l o g y on Cauchy s e q u e n c e s , t h e c o n v e r s e i s d i s proved by e a s y examples; i n f a c t , f u n c t i o n s c o n t i n u o u s f o r t h i s c o m p u t a t i o n a l t o p o l o g y need n o t r e s p e c t e q u i v a l e n c e between Cauchy s e q u e n c e s a t a l l . Brouwer had o b s e r v e d t h a t e v e r y c o n s t r u c t i v e l y d e f i n e d f u n c t i o n from R t o P (whose e l e m e n t s h e d e s c r i b e d by " f r e e c h o i c e sequences" u o u s r e l a t i v e t o t h e p r o d u c t t o p o l o g y on

value

sn

h a s been established, then

o f information about

(rn).

f

&,

for i f

(rn)

E

and a

c a n h a v e used o n l y a f i n i t e amount

'Ihus i f 3.1 c a n n o t b e proved f o r

with respect t o t h i s particular proved i n t u i t i o n i s t i c a l l y .

e) i s c o n t i n -

f ( ( r n ) ) = (sn),

computational

topology,

then

(n,d) = (2,4)

it cannot be

5 . The Proof for Quadratics Let let

C = (C. .) IJ

be i n d e t e r m i n a t e s ,

c = ( c i j ) E Sym(n+l,R),

for

0

< i , j < n,

with

t h e t o p o l o g i c a l (R-vector)

Cij

= Cji,

and

s p a c e o f symmetric

71

Continuous sums of squares

( n + l ) x ( n + l ) m a t r i c e s o v e r R. i n X = (X,,.. ,Xn) and write PA,? = { c

1J

R("+l)'

be t h e g e n e r a l q u a d r a t i c form

I

1

i s psd o v e r

Sym(n+l,R)lZc;jXiXj

E

i s an i s o m o r p h i c image i n

PA,2

Z i j C..X.X.

Let

..

of

N(n) = ( n + l ) !

Pn,2.

k=O

R

Set

&.

0 < < n, such t h a t

Theorem 5.1: For f i x e d n and f o r c o n s t r u c t r a t i o n a l f u n c t i o n s pk, a,cC)

XI;

in

1 < k < N ( n ) , we can

(5.2) throughout

Rn+l,

x

and

> 0,

pk(c)

for

1 < k < N(n)

,

p k ( c ) ( ~ a .akE(c)xa)'

continuous

(c;x)

B e f o r e g o i n g t h r o u g h t h e p r o o f , i t i s i n s t r u c t i v e t o c o n s i d e r t h e simp l e s t case, n = 1 , and v e r i f y c o n t i n u i t y of 0.2. The o n l y p o i n t i n P1,* where t h o s e c o e f f i c i e n t s c o u l d be d i s c o n t i n u o u s i s where s = 0 , h e n c e a = c = b = O ( t h e t r i v i a l f o r m ) , and even h e r e e a c h c o e f f i c i e n t e x t e n d s c o n t i n u o u s l y (by 0 ) : namely, t h e i n e q u a l i t i e s 0 < a 2 / s < a , 0 < c 2 / s < c , and 0 < max{d/s,b2/s} < a c / s < a show t h a t t h e c o e f f i c i e n t s a2/s, c2/s, d/s, b2/s, and h e n c e a l s o 2 a b / s and Lbc/s must approach 0 a s a and c d o . Proof o f 5.1: For

n

I n d u c t i o n on

>

0

n.

For

ccijxixj (% ,...,Xa.-l,Xa.+l

=

so have t h a t

(5.3)

(cij)

p a

p1 = Coo

a10 = 1.

and

we make u s e o f t h e r e p r e s e n t a t i o n c o n s t r u c t e d f o r

t o construct the representation for

where

n = 0, t a k e

€

PA,2

0, and b o t h

P h , 2 , a s follows.

+ 2Xa.Zifa.

= CLa.X$

ciaxi

,...,X,,),

and where

i f f , for

0 <

qa.(Xi)

For

8

< n

P,',-1,2 write

+ qa.(Xi),

qa.(Xi) =

< n,

Z

i+a..j

da.(Xi) = cfia.qa.(X;)

and

<

0

C..X.X

-

ij 1

j.

W e a'-

(Ei+a

3.

a r e psd i n L

Writing +

0

t r c = Ecii

forces

cii

+

0.

5 . 3 implies that i f

c

varies within

Complete t h e s q u a r e w i t h r e s p e c t t o

then

tr c

Xa.:

EC We now form t h e convex c o m b i n a t i o n , by

(5.4)

E c i j x i X j = ( t r C)-'

a. =O [ ( E y = o

We s h a l l n e e d t h e i n e q u a l i t y for given

i

and

C,,/tr

c:j

C , o f t h e above,

Ciaxi)'

< c 11 ..c..

J~

+

II:

de(xk)].

in

j, t h e inequality clearly holds for

for a l l

If c

E

PA,2.

n

< 1, t h e n If

n

>

1,

C.N. DELZELL

12 i

then given

qa(O i s psd f o r

c

<

j , we can pick

II

d i f f e r e n t from

,...,0,Xi.O ,...,o,xj,o ,...,0) E

PAd

11 1

lCijCIIrnl

+ 2 c . .x.x + c . . x ? IJ 1 j JJ 1

tr c

tr c

ctj

< ciicjj

in

PA,*.

t h e second i s j u s t i f i e d by symmetry,

c.1 1. c j j

C t j

<-

O<-

j; since

and

(by 5 . 3 ) , w e s t i l l c o n c l u d e t h a t

T h i s p r o v e s t h e t h i r d i n e q u a l i t y below; i.e. w e may assume t h a t lcIIml < I c i j l :

(5.5)

= c..x?

i

<

cii + c

< cii. jj

T h e c o e f f i c i e n t s o f t h e f i r s t s q u a r e summand o f 5 . 4 a r e o f t h e form C. lhe o n l y p o i n t i n PA,2 where t h e s e c o e f f i c i e n t s c o u l d b e d i s -

CillCja/tr

t r c = 0 , hence

c o n t i n u o u s is where

c i i = 0.

5.5 now i m p l i e s t h a t e v e n n e a r

t r c = 0 , t h e s e c o e f f i c i e n t s e x t e n d c o n t i n u o u s l y , by 0. S i n c e t h e f i r s t summand o f 5.4 v a r i e s c o n t i n u o u s l y , so d o e s t h e psd quad r a t i c form D = d I I ( X i ) / t r C . By t h e c o n s t r u c t i o n f o r PA-l,2 w e have D r e p r e s e n t e d a s a w e i g h t e d s m o f s q u a r e s of l i n e a r forms whose c o e f f i c i e n t s are c o n t i n u o u s f u n c t i o n s o f D ' s c o e f f i c i e n t s , h e n c e of C . The l a s t t h i n g t o do i s t o c o u n t t h e number o f s q u a r e s .

For

D

we h a v e

+ 11 = (n+l)[n!Z,?=O l / k ! ]

=

o b t a i n e d by d i a g o n a l i z i n g a symmetric m a t r i x , i s d i s c o n t i n u o u s , i t h a s o n l y + 1 summands, i n c o n t r a s t t o 5.1. T h i s s u g g e s t s t h e

n

squares ( i n N(n)

Xi).

l h u s 5.4 r e q u i r e s

s q u a r e s , a s claimed.

( n + l ) "(n-1)

Q . E . D.

While t h e c l a s s i c a l SOS r e p r e s e n t a t i o n

Question:

For d = 2, i s c o n t i n u i t y ( e i t h e r 3.1 o r 3 . 1 ' ) p o s s i b l e w i t h o n l y n + 1 summands, e v e n f o r n = l ?

P r o f e s s o r J. F. Adams r e c e n t l y s u p p l i e d me w i t h a p a r t i a l answer: n = 1, 3.1 i s t r u e w i t h o n l y n + 1 = 2 summands: (5.6)

ax2 + 2bXY + cY2 =

((a&)X

C o n t i n u i t y i s e a s y t o check: c a n v a n i s h i s where e.g.,

+ by)'

for

+ (bX + ( c + a ) Y ) '

s + 2 6 t h e only point i n

s = 0, hence

Pi,2

where t h e d e n o m i n a t o r

a = c = b = d = 0 ; even h e r e , we h a v e ,

which show t h a t t h e c o e f f i c i e n t s o f t h e f i r s t s q u a r e summand e x t e n d c o n t i n u o u s l y , by 0. U n f o r t u n a t e l y , t h e u s e o f s q u a r e r o o t s makes t h i s r e p r e s e n t a t i o n

Continuous sums of squares

u n s u i t a b l e f o r p r o v i n g 3.1', e x c e p t when p o s i t i v e element of K has a s q u a r e r o o t . Write

D =

fi.

K

i s Euclidean,

d e f i n e s a q u a d r a t i c mapping from by ( a , b , c , D ) - s p a c e ( a n o t h e r R4) a = a2 + y2, and t h a t t h i s mappin t h e s u b s p a c e a2 + f3'

+ c

i.e.

when e v e r y

Adams' key o b s e r v a t i o n was t h a t t h e r e l a t i o n

ax2 + 2bXY + cY2 = ( a x + BY)'

a

13

the

b = aB + y6,

+ (YX + 6Y)'

(u,B,y,b)-space c = 6'

+ 6',

(an

D = a6

to

R4)

the

- BY,

S3 + S 2 , from r e s t r i c t s t o t h e c l a s s i c a l Hopf f i b e r i n g , y 2 + 6' = 1 (which i s an S 3 ) t o t h e s u b s p a c e

+

(7)'

+ b2 + D2 = 1 / 4

= 1,

(which i s an

S2).

T h i s r e s t r i c t i o n makes no e s s e n t i a l d i f f e r e n c e , s i n c e t h e problem i s homogeneous. Now t h e Hopf f i b e r i n g h a s no s e c t i o n c o n t i n u o u s o v e r t h e whole o f S2, b u t w e o n l y need a s e c t i o n c o n t i n u o u s o v e r t h e h a l f - s p a c e D > 0 , f o r example, The c l a s s i c a l c o m p l e t i o n - o f - s q u a r e p r o c e s s , 0 . 1 , and such s e c t i o n s do e x i s t . amounts t o a s e c t i o n , b u t as w e p o i n t e d o u t i n 50, i t h a s a s i n g u l a r i t y o n t h e ray ( a , b , c , D ) = (O,O,c,O), w h i l e i t i s v a l i d and c o n t i n u o u s everywhere else. So w e c a n a p p l y an automorphism t o t h e f i b e r i n g so as t o move t h i s s i n g u l a r i t y or ( a , O , a , a ) . w h e r e i t c a n c a u s e no harm, f o r example t o t h e r a y ( a , O , a , - a ) 5 . 6 c a n b e o b t a i n e d more e l e m e n t a r i l y by s e e k i n g t h e s e c t i o n w i t h B = y. This means we want a =a2

+ B2,

b = B(a + 6 ) ,

c = B2 + 6',

D = a6

- B2,

which g i v e s a + c + 2 D = ( a + 6 ) ' ,

a + D = a ( a + 6 ) ,

b=B(a+G),

c + D = 6 ( a + 6 ) ,

which amounts t o 5 . 6 . 6.

The Proof for Ternary Quartic6

Write (X,Y,Z) for (% ,X,,X2), l e t f(C;X,Y,Z) E Z[C;X,Y,Z] g e n e r a l q u a r t i c form i n X , Y , Z with c o e f f i c i e n t s C = (Cijk), i,j,k Let

i+j+k = 4.

g(D;X,Y,Z)

>

be t h e 0 and

b e t h e g e n e r a l q u a d r a t i c form w i t h c o e f f i c i e n t s

D = ( D i j k ) , i , j , k > 0 and i + j + k = 2 , and c o e f f i c i e n t s p a c e R6 ((2;2) = 6 ) . F o r f u n c t i o n s a': P 2 , 4 + R6 ( 1 < 1 < n ) , write a t j k f o r t h e i,j,k-component of

a'.

Write t h e v e c t o r o f f u n c t i o n s

i,j,k

w r i t e t h e vector of

Write

aijk(s)aijk(t)

IIaijk(t)

11'

for

Theorem 6 . 1 :

V c

(a',

i,j,k-components

f o r t h e "dot

product"

...,am)

1 (aijk,

a , and f o r f i x e d

Zg a i j k ( s ) a t j , ( t )

as

aijk*

and w r i t e

aijk(t)aijk(t). T h e r e do n o t e x i s t f u n c t i o n s

a':

E P2,4,

(6.2) (6.3)

as

...,amr. J. k )

a

8200

and

a

continuous.

P2,4

+

R6

such t h a t

C.N. DELZELL

14 Proof:

We show by c o n t r a d i c t i o n t h a t f o r some

c,

a jump d i s c o n t i n u i t y a t where A p p r o a c h ? a l o n g two p a t h s a f(a(t);X,Y,Z)

= (X2

+ Y2

0 , either

aoo2(a(t))

5

(6.5)

lla200(a(t))

(6.6)

\Iao2,(a(t))

0 ao2,

or

has

c

i s d e f i n e d by f(E;X,Y,Z) = (X2 + Y 2 ) 2 . and 8 i n P 2 , 4 : d e f i n e a: R + P2,4 by

+ tXZ)2,

and

B:

R

+

P2,4

X2(X + t Z I 2 + 2X2Y2 + ? . E q u a t i n g c o e f f i c i e n t s ( e v e n w i t h o u t assuming c o n t i n u i t y ) t h a t (6.4)

a:00

by

f(B(t);X,Y,Z)

=

i n 6.2 we s e e immediately

0,

11 11

1,

1

1,

hence (6.7) (6.8)

aoll(a(t))

0 , and

\ \ a l o l ( a ( t ) ) 11

=

t.

Replace a w i t h 8 i n 6.4 t h r o u g h 6 . 8 , and c a l l t h e r e s u l t i n g f o r m u l a s 6 . 4 ' t h r o u g h 6 . 8 ' ; 6 . 4 ' t h r o u g h 6 . 8 ' a l s o f o l l o w from 6 . 2 . W e d e a l f i r s t w i t h a. By a l p l ( a ( t ) ! a 2 0 p ( a ( t ) ) and 6 . 5 , t h e Cauchy-Schwarz ("CBS") i n e q u a l i t y i m p l i e s

(6.9)

ta2,,(a(t))

By alol(a(t))ao2,(a(t)) implies

5

(6.10)

tao2,(a(t))

t

t

Z

combined w i t h 6.8

= alol(a(t)).

combined w i t h 6.8 and 6 . 6 ,

t h e CBS i n e q u a l i t y

= aloL(a(t)).

t f 0, a,,,(a(t)) = a O 2 , ( a ( t ) ) , so i f a2,, and 6.9 and 6.10 g i v e , f o r ao2, a r e c o n t i n u o u s a t = a(O), t h e n = ao20(C)1 Thus, by 6.5 or 6.6, a 2 , , ( ~ ) a o 2 , ( ~ )f 0 , h e n c e vc i n a neighborhood o f c , a 2 0 0 ( c ) a 0 2 0 ( c ) f 0.

c

We d e a l f i n a l l y with 8. BY alo,(B(t))a2,,(~(t)) 6 . 8 ' and 6 . 5 ' , t h e CBS i n e q u a l i t y i m p l i e s

(6.11)

tap,

,(B

( t ) ) = al

, (6

(t ) )

= t

combined w i t h

.

But 6 . 7 ' i m p l i e s a 1 0 1 ( 8 ( t ) ) a 0 2 0 ( B ( t ) ) : 0 , so by 6.11, t a , o o ( B ( t ) ) a , 2 0 ( B ( t ) ) D i v i d e by t and l e t t + 0 t o c o n t r a d i c t t h e p r e v i o u s p a r a g r a p h . Q. E . D.

I 0.

Bibliography C a s s e l s , J.W.S., On t h e r e p r e s e n t a t i o n of s q u a r e s , Acta k i t h . 9 ( 1 9 6 4 ) , 79-82.

rational

functions

as

suns

of

Choi, M . D . , and Lam, T.-Y., An o l d q u e s t i o n of H i l b e r t , Proc. Conf. on Quad r a t i c Forms--1976 (Queen' s P a p e r s on P u r e and A p p l i e d 6.M -i-.h.at Orzech, e d . ) , Queen's Univ., K i n g s t o n , O n t a r i o , 1 9 7 7 ) , 385-405. E x t r e m a l p o s i t i v e s e m i d e f i n i t e forms, Math. Ann. 231 ( 1 9 7 7 ) , 1-18. D e l z e l l , C . N . , A c o n s t r u c t i v e , c o n t i n u o u s s o l u t i o n t o H i l b e r t ' s 1 7 t h problem, and o t h e r r e s u l t s i n s e m i - a l g e b r a i c g e o m e t r y , Ph.D. d i s s e r t a t i o n , Stanf o r d Univ., 1980 (Univ. M i c r o f i l m s I n t e r n a t i o n a l , Order No. 8024640);

75

Continuous sums of squares

c f . a l s o D i s s e r t a t i o n A b s t r a c t s I n t e r n a t i o n a l 41, no. 5, 1980.

A c o n t i n u o u s , c o n s t r u c t i v e s o l u t i o n t o H i l b e r t ' s 17th problem, i n preparation (a). (See a l s o a p r e l i m i n a r y a b s t r a c t i n Abstracts 2(1) ( J a n . 1 9 8 1 ) , #783-12-28.) On H i l b e r t ' s (b).

1 7 t h p r o b l e m o v e r c o u n t a b l e s u b f i e l d s of

R, i n p r e p a r a t i o n

A n a l y t i c v e r s i o n o f S i e g e l ' s theorem on sums of s q u a r e s , i n p r e p a r a t i o n ( c ) ; p r e l i m i n a r y a b s t r a c t i n AMS A b s t r a c t s , J a n . 1982, #792-12-269. G e r s t e i n , L . J . , New p r o o f o f a theorem o f C a s s e l s - P f i s t e r , S O C . 4 1 ( 1 ) ( 1 9 7 3 ) , 327-8. -

P r o c . Amer. Math.

H e i l b r o n n , H . , On t h e r e p r e s e n t a t i o n o f a r a t i o n a l as a sum o f f o u r s q u a r e s by means o f r e g u l a r f u n c t i o n s , J . London Math. SOC. 39 ( 1 9 6 4 ) , 72-6. Henkin, L . , Sums o f s q u a r e s , Summaries o f T a l k s P r e s e n t e d a t t h e Summer I n s t i t u t e o f Symbolic Logic i n 1957 a t C o 3 U n i v e r s i t y T G t E D e f e n s e A n a l y s e s , P r i n c e t o n , 1 9 6 0 ) , 284-91. H i l b e r t , D . , itber d i e D a r s t e l l u n g d e f i n i t e r Formen a l s Summe von Formen-quadraten, Math. Ann. 3 2 ( 1 8 8 8 ) , 3 4 2 - 5 0 ; see a l s o Ges. Abh. 2 ( S p r i n g e r , B e r l i n , 19331, 154-61. Grundlagen d e r Geometrie (Teubner, 1 8 9 9 ) ; t r a n s l . by E.J. Townsend (Open C o u r t P u b l i s h i n g Co., La S a l l e , I L , 1 9 0 2 ) ; t r a n s l . by L. Unger from t h e t e n t h German e d i t i o n (Open C o u r t , 1 9 7 1 ) . Mathematische Probleme, G o t t i n g e r N a c h r i c h t e n ( 1 9 0 0 ) , 253-97, and Archiv d e r Mathematik und P h y s i k 3d ser. 1 ( 1 9 0 1 ) , 44-53, 213-37. Transl. b y M.W. Newson, B u l l . Amer. Math. SOC. 8 (19021, 437-79; r e p r i n t e d i n M a t h e m a t i c a l Developments A r i s i n g from H i l b e r t Problems (F. B r a r d e r , e d . ) , P r o c . Symp. i n Pure Math. 28, ( h e r . Math. Soc., P r o v i d e n c e , 1 9 7 6 ) , 1-34. K r e i s e l , G . , Sums o f s q u a r e s , Summaries of T a l k s P r e s e n t e d a t t h e Summer Ins t i t u t e f o r Symbolic Logic i n 1957 a t C o r n e l l U n i v e r s i t y ( I n s t i t u t e for ~Defense A n a l y s e s , P r i n c e t o n , 1 9 m 3 1 3 - 2 0 . Landau, E., h e r d i e D a r s t e l l u n g Math.Ann. 57 ( 1 9 0 3 ) , 53-64. Motzkin, O.,

definiter

binzrer

Formen d u r c h Q u a d r a t e ,

T.S., 'lhe a r i t h m e t i c - g e o m e t r i c i n e q u a l i t y , I n e q u a l i t i e s 1, S h i s a , e d . (Academic P r e s s , New York, 19671, 204-24.

P f i s t e r , A . , D a r s t e l l u n g d e f i n i t e r F u n k t i o n e n a l s Summe von Q u a d r a t e n , I n v e n t . Math. 4 ( 1 9 6 7 ) , 229-37. Robinson, A . , I n t r o d u c t i o n t o Model Theory and t o t h e Metamathematics o f Algeb r a (Nor th-Ho 11and P u b ~ s h i n g C o . - , s t e r d ~ , ~ 63 1

.

Robinson, R.M., Some d e f i n i t e p o l y n o m i a l s which are n o t sums o f s q u a r e s o f r e a l p o l y n o m i a l s , Notices Amer. Math. Soc. 16 ( 1 9 6 9 ) , 554; S e l e c t e d Q u e s t i o n s i n A l g e b r a and Logic (Vol. d e d i c a t e d t o t h e memory of A . I . M a l ' c e v ) , I z d a t . "Nauka." S i b i r s k O t d e l N o v o s i b i r s k ( 1 9 7 3 ) , 264-82; o r Acad. S c i . USSR. (E4912647). S i e g e l , C.L., D a r s t e l l u n g t o t a l p o s i t i v e r Zahlen d u r c h Q u a d r a t e , -Math. Z e i t . 11 ( 1 9 2 1 ) , 246-75; Ges. Abh. 1 ( 1 9 6 6 ) , 47-76. -

THE L.E.J. B R O W E R CENl'ENAR Y SyMpoSIlM A.S. lbelptra and D. van Dalen (editors) 0North-Holland Atblishing Company,1982

MONOTONE INDUCTIVE DEFINITIONS

Solomon Fef erman Department of Mathematics Stanford University Stanford, C a l i f o r n i a U.S.A.

The main r e s u l t of t h i s paper i s t h a t ""D:I of

i s a conservative extension

where t h e f i r s t of t h e s e t h e o r i e s expresses t h e i t e r a t i o n

ID,(W)',

of inductive d e f i n i t i o n s by successive monotonic operators while

IDa(W)'

( w ~I)a~.

I n both cases

1981 t h a t

Buchholz,

, which

i s a n i n t u i t i o n i s t i c theory f o r a - i t e r a t e d accessi-

b i l i t y inductive d e f i n i t i o n s .

(and

1981 t h a t

nE )

I n a d d i t i o n f o r limit

o r d i n a l number c l a s s e s

ID<

a (@)i

Qp(@< a).

, an

it follows from t h e work

i n t u i t i o n i s t i c theory of t h e constructive

However, t h e exact r e l a t i o n s h i p between

t h e s e t h e o r i e s on t h e i n t u i t i o n i s t i c s i d e s i . e . of or

CY

i s a conservative extension f o r negative a r i t h m e t i c

ID?;'

sentences of

even with

It follows from t h e work of

i s a conservative extension f o r negative a r i t h m e t i c

ID?'

sentences of

of Sieg,

,

rA(X) = (xlA(x,X))

a i s an e f f e c t i v e l y presented o r d i n a l

indicates t h a t the logic i s classical.

'c'

times

i s a theory of t h e i n d u c t i v e l y generated r e c . codes of well-founded

tree classes and

Q

IDfs'i

IDn:J1

'

with

or

( i t e r a t e d p o s i t i v e inductive d e f i n i t i o n s ) i s s t i l l

There a r e o t h e r i n t e r e s t i n g open questions concerning i t e r a t e d monotone i n -

open.

ductive d e f i n i t i o n s which w i l l be r a i s e d a t t h e conclusion of t h e paper. Most of t h e s e t h e o r i e s of i t e r a t e d inductive d e f i n i t i o n s were f i r s t formul a t e d o r informally suggested i n Kreisel, 1963, and t h e question of t h e i r exact r e l a t i o n s h i p s was r a i s e d t h e r e ( c f . a l s o Kreisel,

1965 8 2 . 6 f o r t h e case a = l ) .

The main questions concerning t h e r e l a t i o n s between 'D:I IDa(@),

'
ID,(W)

and

i n p l a c e of

IDaCC

a

and t h e subtheories

i n c l a s s i c a l o r i n t u i t i o n i s t i c l o g i c , as w e l l as with

' a 'f o r l i m i t a, have been s e t t l e d by t h e work of Buchholz,

78

S . FEFERMAN

Pohlers and Sieg; f o r t h e most r e c e n t e x p o s i t i o n c f . Buchholz, Feferman, Pohlers,

1981 [B,F,P,S]. ')

Sieg

Thus only t h e question of

remained open.

ID?

Sur-

p r i s i n g l y , it turned out p o s s i b l e t o hancUe t h i s i n a completely s t r a i g h t f o r w a r d way, as will be shown i n t h e following. Examples of

Iesfi are

which a r e not immediately r e d u c i b l e t o

.

ID,(Q)'

found by applying t h e double-negation t r a n s l a t i o n t o , say,

The question

on t h e f a c e of it r e q u i r e s sub-

of c o n s t r u c t i v e j u s t i f i c a t i o n of even s c r i p t i o n t o impredicative p r i n c i p l e s which

-

it could be argued

t o accept t h e f u l l i n t u i t i o n i s t i c t h e o r y of s p e c i e s ( c f . K r e i s e l ,

IDrnji

-

would l e a d one

1968, 5 5 )

. The

i s justified indi-

p r o o f - t h e o r e t i c reductions obtained here show t h a t

r e c t l y by much more r e s t r i c t e d evidence, j u s t concerning p a r t i c u l a r g e n e t i c a l l y However, t h e status of monotone i n d u c t i v e d e f i n i t i o n i n a more

presented s p e c i e s .

general c o n s t r u c t i v e set-ting i s s t i l l open (as will b e explained a t t h e end). For s i m p l i c i t y , t h e r e s u l t s will be formulated i n d e t a i l and proved f o r

a: < w.

Extension t o g e n e r a l 0 poses no problem.

U n t i l otherwise i n d i c a t e d ,

a l l t h e o r i e s t o be considered w i l l be based on c l a s s i c a l l o g i c .

proof i s t o i n t e r p r e t IDn(W).

ID?

in

IDL2)(W),

Standard arguments show t h a t

Our method of

a weak 2nd-order extension o f

IDE)(W)

i s a conservative extension of

IDn(W) *

It i s assumed i n t h e following t h a t t h e reader i s familiar with Feferman

1970

(especially

$0 1-2)

and has had a look a t t h e volume [B,F,P,S].

Chapters

r e f e r r e d t o i n t h a t volume a r e l i s t e d s e p a r a t e l y by t h e i r authors i n t h e b i b l i ography below. The n o t a t i o n f o r t r e e s i s much as i n Feferman here i n p l a c e of

'T

'.

Briefly:

I s ' ,

's

' , ...

1

1970, except

slz

s.

,

Further,

'f'

s (slC

'g'

, . ..

s)

holds i f

s

1

'W'

range over sequence numbers,

s = ( ( ~ ) ~ , . . . , ( s ) ~ ~ , ( ~ S* ) - s1 ~ ) i s concatenation,

empty sequence,

t h a t we u s e

(

)

i s t h e number of t h e

i s an i n i t i a l (proper) segment of

range over number-theoretic f'unctions;

f(n)

is the

Monotone inductive definitions

,..., f ( n - 1 ) ) .

sequence number numbers.

e

(f(0)

codes a t r e e r e c u r s i v e i n

condition a s ( i ) Tot(e,Y).

5 S[(elY(s1) =O])

(SIVSl

, 'Y',...,

'X'

Y

range over s e t s of n a t u r a l

(elY

if

79

i s t o t a l ; we abbreviate t h i s

The s e t of nodes of t h e t r e e coded by

(which i s empty i f

founded i f (ii) V f 3y [ e l Y ( i ' ( y ) )

f

0.

) ) f 0).

[ely((

e

is

The t r e e i s well-

i s d e f i n e d t o be t h e c o l l e c t i o n of

Wy

e

s a t i s f y i n g t h e s e two conditions ( i ) , ( i i ) . e, e '

determine t h e same t r e e ( r e l . t o Y ) i f

i s written

e

e'

. w'

i s closed under

(e [ s )

r s' =

Wy

e

Given

e, s

(e')'(s)];

, effectively

Y i n e, i . e . ( e l S ) ( s ' )

which codes t h e t r e e below s

e r s

.

Vs[[e)'(s)

2

this

associate

Y ( e l ( s Y s ' ) . Then

r (sYs').

i s equally w e l l defined i n d u c t i v e l y as t h e l e a s t s e t

X

satisfying the

c l o s u r e conditions:

f

( i ) Tot(e,Y)

A

(elY(( ) )

(ii) Tot(e,Y)

A

Vx[e r ( x ) E X I

0

--t

e E X

+ e

E X.

(Actually it i s s u f f i c i e n t i n ( i i ) t o assume t h a t by i n d u c t i o n t h a t n

< u) . Let

Thus n

W1

2

e

t

E Wo

for

B(x) = B ( x , .

Wy

+ Tot(e,Y).) We t a k e

1. The f i r s t - o r d e r language

Wk

(PRA)

and

=

Wn+l

n'

for

= W

L(l)(W)

has t h e symbols of p r i m i t i v e

t o g e t h e r with unary p r e d i c a t e c o n s t a n t s

as s e t c o n s t a n t s and w r i t e

t = t . The axioms of

..

W

) ) 1 ; one t h e n proves

i s t h e s e t of (codes o f ) r e c u r s i v e well-founded t r e e s .

recursive arithmetic We t r e a t t h e

8

(elY((

i s any forrmiLa of

I D (W)

t

Wk

Wk(t);

also write

a r e as follows, where

k=l,. . , n

Lp)(w):

6

for

...,W .

W1,

.

and

S. FEFERMAN

80

The second-order language

Lr)(W)

simply by adjoining s e t variables

X,Y,Z,.

Write

Y

X

for

Y

Vx[x E X + x

a r e defined t o be s e t s X f , g,.

..

E

1

satisfying

i s an extension of

..

LF)(W)

obtained

and t h e membership r e l a t i o n

and X = Y

for X

Vx 8!y(x,y)

X

E

5Y

.

A

Y

5X

.

E

.

FUnctions

Then f i n c t i o n variables

a r e taken by convention t o range over such sets, and functional notation

f ( x ) , ?(x) ,Dp)(W)

i s explained accordingly.

The system

by replacing t h e induction schemes 2"

axioms 11 and

Iv

IDf)(W)

, 4"

i s obtained from

by second-order induction

(also called r e s t r i c t e d 2nd order induction) and by adding a

predicative comprehension axiom scheme V

.

That i s , t h e axioms of

IDf)(W)

are

...,n-1):

(with

k = 1,

1. Axioms of

PFVi

11. VX[OEXA V X ( ~ E X - - ~ X ' E+X V)X ( X c X ) ]

(

111.

=

) 3'

above

V X ( Vz[Tot(x, Wk-l)

IV.

vz[Tot(x, wk-l)

A

3X Vx[x

V.

A (Z)Wk-l((

E

XH

B(x)],

A vX(z

)) f 0

r (x) x )

f o r each formula

--f

XI

Z E

z

xI

+

B(x) =B(x,

wk 5 x I

... )

of

. Lf)(W)

with-

out bound s e t variables.

A s usual, t h e formulas

admitted i n the scheme V a r e said t o be i n

- when it i s necessary t o show dependence on t h e Wk's - i n I I ~( wl,. ..,wn) - form) an3 v i s denoted (n: - CA). We a l s o write I X \ B ( ~ )f )o r

:l

- form

B

the set

( or

X

asserted t o exist i n

hridently Theorem 1. ----------

IDn(w)

IDf)(W)

5

V, which i s unique up t o = of s e t s .

IDF)(w).

i s a conservative extension of

IDn(W).

81

Monotone inductive definitions

Proof. We describe t w o proofs. m= (M,. . . )

of

(Model-theoretic).

This i s standard.

h(2)

can be expanded t o a model

IDn(W)

preting t h e s e t variables t o range over a l l subsets of

,..., ij)) wh.ere

( x ~ B ( xil ,

ml,

. . .,mi

B(x, y1

(Proof-theoretic).

(nz - CA)

mrther,

LP)(W)

and

We here follow t h e method of

i s enlarged t o a system i n which every

IDf)(W)

formula becomes equivalent t o a

+-operator.

by i n t e r -

definable i n t h e form

i s a formula of

This i s more-or-less standard.

Feferman, Sieg 1981, $ 2 . 1 - 2 . 2 .

:I'i

M

IDf)(W)

.

M

a r e elements of

,..., y,)

of

Any model

(quantifier-free formula) by use of the

QF

i s replaced by use o f an abstraction operator.

I n the expanded language individual terms

t, s e t terms

T

and

QF-formulas

C

are generated as follows:

(i) (ii)

Every indiv. ( s e t ) varia'ble i s an indiv. ( s e t ) term. The indiv. terms a r e closed under t h e prim. rec. operations.

(iii) t

(iv) (v)

G

formula.

is a

T

IJ. x C

i s an indiv. term.

( x ~ c ) i s a s e t term.

( v i ) The

formulas a r e closed under

QF

7

and A .

t

Formulas a r e then obtained from t h e atomic formulas A

together with

V

E

T

by closing under

applied t o both indiv. and s e t variables.

D=VXl

i n purely universal form i f

...

... Vxm C

Vxl

A formula

where

C

is

7 ,

D

is

QF.

In t h e extended language t h e +-axioms a r e a l l universal closures of statements of t h e form

(Id for

c(x, C(x,

... )

in

...

The

QF.

+ c ( + x c ( x , ... ) ,

... )

QF-abstraction axioms a r e a l l universal closures of

statements of t h e form y

(W-M) where again

C(x,

.. . )

G

(XI c(x,

i S

in

... ) I

oc(y,

... ) ,

QF. With each -:II

formula

B

i s associated a

82

S. FEFERMAN

QF-formula

with

B*

B

(p)

H

B*.

We w r i t e

i s meant t h e system obtained from I D f ) ( W ) and replacing t h e scheme V p-axioms Clearly

IDF)(W)

5 U -IDF)(W).

Lp)(W)

then

U

by

Now i f

of s u b s t i t u t i o n i n s t a n c e s

(QF-AA)

F

of

U

.

B

H B*

, so

- IDf)(W)

B

where

B

i s a closed

t h a t t h e r e i s a f i n i t e number

such t h a t

IDn(W)

*

It follows by Herbrand’s Theorem

1981, p.103)

that finally

I -1V

a r e i n p u r e l y universal form.

nay be i n t e r p r e t e d back i n t h e f i r s t - o r d e r theory preserving

*

I - I V by

and f i n a l l y by adding t h e

IDf)(W)

- ID(2)(W) k B*

( i n t h e form given i n Feferrnan, Sieg

By U-ID;*)(W)

by r e p l a c i n g Axioms

U - IDf)(W)

Thus t h e axioms of

(p).

f o r m l a of

- CA)

(TI:

(TXB)*=TXB*.

1B.

MF

I D (W)

t B* .

F

and

B*

(cf. loc. cit.p.111)

T h i s completes t h e d e s c r i p t i o n

of t h e second proof.

It i s now shown t h a t t h e t h e o r i e s I D f ) ( W )

Here it i s convenient t o i d e n t i f y

ductive d e f i n i t i o n . Arithmetic) and

IDo(W)

with i t s conservative extension :

IDf)(W)

with

(nz-CA)

PA (Peano with re-

W.

s t r i c t e d i n d u c t i o n on

Theorem 2. Suppose -----__--

a r e closed under monotonic i n -

A(x,X)

is a

IIz(W,,

...,Wn-l)

formula containing a t most ‘x’,

‘ X ’ as f r e e v a r i a b l e s , and suppose f h r t h e r t h a t

k Vx,X,Y[A(x,X)hX 5 Y

IDf!(W) Then

IDf)(W)

-1A(x,Y)].

proves t h e e x i s t e n c e of a unique s e t

PA s a t i s f y i n g

( i ) clos(PA) and (ii) WIClosA(X)

where Proof. -

3

PA 5 X I ,

Clos ( X ) = Px[A(x,X)

A

-1

x E X

1.

This w i l l b e c a r r i e d o u t i n two lemmas, which w i l l show t h a t t h e standard

d e f i n i t i o n of

PA from above,

x E PA

H

VX[Cl0SA(X)

actually serves t o define a set i n

-1

x E

xI

IDf)(W).

This i s done by formalizing t h e

Monotone inductive definitions usual recursion t h e o r e t i c argument t h a t in

83

i’It i n W

PA i s

n- 1

and hence recursive

Wn.

Every formula of t h e form

--_-___ 1. Lemma ______-

form i s equivalent i n recursive i n

Proof.

Put

IDE!(W)

S

Wn-l

for

7B(x,X)

j

where

B

is i n

t o a formula of t h e form

If(W,,

. ..,Wn-l)-

Vf 3yR(x,?(y)) with R

‘n-1. i n prenex

recursive i n

with

VXB(x,X)

< n).

which e x i s t by

W

n- 1

n.f.

(which i s possible since each W .

J

i s recursive i n

Use t h e Skolem functions

(I:

-CA)

for

..

i = 1,. ,k. Then take

It follows by usual r e c u r s i o n - t h e o r e t i c simplifications t h a t YXB(x,X)

c)

c . )

f o r suitable Lemma 2.

-------

f o r which

Sl, R

Suppose

VfoVfl

vf

3y R(x,

3yl

... 3yk S 1

(Y))

recursive i n W n-1 R(x,s)

IDf)(W)

... Vfk

i s recursive i n

Then we can f i n d prim. rec. n

Wn-l.

proves

Vf 3y R(x,?(y))

H

n(x) E Wn

.

S. FEFERMAN

84 Proof.

i s defined t o b e t h e G h e l number of t h e t r e e of

n(x)

-)

R(x,

unse-

cured sequences, i . e . :

The proof of t h e d e s i r e d equivalence i s i n two s t e p s .

(2 1

5 R(x,

Yf

F(Y))

i s proved by c o n t r a d i c t i o n . 'n-1 (el s)

we have

induction on e

r(

y

.

) f Wn

Bs(dh(s) = y

e rs

I f it i s t r u e f o r

e E ?(y) Wn

shows

n(x)

Hence

4

s

5X

C

=

(v)

c--1

means

f

5s

R(x,S1).

suppose

Tot(z, Wn-l)

r

s A

s=?(dh(s)).

suppose

2 3

z r (v)

n(x)

=

n(x) r s

rs

f

s

++ (v) , and

1

f

(again with

0

t h e n it would

R(x, f ( y ) ) .

This

Now t o prove

Fix

f

s

f .

* (Y)

and p u t

IV f o r W n . z

n(x)

3

(v) Let

c f

G

X

1

rs

First, if then also

+z E X

SO

z

.

To show codes t h e

( ~ ( X ) ) ~ " - ~ ( Sf ) 0

3y R(x, ? ( y ) )

v = f ( y ) where

,

1,

X=(zlB(z,f)].

we c l e a r l y have

s C f

C

+ 3y R(x, ? ( y ) )

s C f

We show Vv[Z

with

Vy

by

Vy(e/i'(y))f Wn.

f

(e?-'(?(y)) y, so

k=O

y + 1 since then

Hence

Wn).

proving ( 2 ) .

) ) f 0 and

Thus i f

.

It will be proved by

This holds f o r

W,).

for all

we apply t h e i n d u c t i o n axiom

3sl

.

Wn

is total

i n IIZ-form definecl b y

B(z,f)

V s [ z z n(x)

4

( e )'-1

e r ( s + ( v ) ) . Now d e f i n e

R(x, ? ( y ) ) ,

1

empty t r e e , i . e . {zIwn-'((

so

f

e

f(y)=pv((er?(y)) t(v)

+ 3f Vy

Wn

B(z,f)

where

r

Suppose

Since

t h e n it i s t r u e f o r

y

'n-1 {el (f(y)) = 0

consider t h e property

(41

A

e=c(x).

were a code of t h e empty t r e e , i . e .

.

be i n

Wn

(er s)

r e c u r s i v e l y by:

II:-CA)

If

s.

and

Wn

0

x and l e t

Fix

t o t a l f o r every

that

Kv((e [ s ) I ( v ) f Wn)

n(x)

--t

First

.

Next

To o b t a i n t h e conclusion y=dh(s).

3y R(X, ? ( y ) )

Then

by hypothesis.

We

Monotone inductive definitions can now conclude W

C

n-

(5 1

VZ

Taking

z =n(x)

n(x)

by induction, so

X

[

85

Z

+B(Z,f)].

E Wn

r(

)

it follows t h a t

z

8

W + 3 y R ( x , ? ( y ) ) , which f i n a l l y n

proves ( 3 ) , and hence Lemma 2 . To complete t h e proof of t h e theorem, take Lemmas 1 and 2 , VXB(x,X) PA= (x(VXB(x,X ) )

exists.

trX[ClOSA(X) 4 PA 5 X I B(x,X)

f o r each

H

X

.

.

ID?

f o r suitable prim. rec. n .

X ( X )E Wn

It i s trivial t h a t

Thus suppose

by

x E X

5X

ClosA(X), q.e.d.

Finally,

PA i s immediate by ( i ) and ( i i ) .

i s a f i r s t - o r d e r theory obtained by i t e r a t i n g at each stage inductive

shown t o be monotonic i n t h e preceding stage. and

we prove

by (ii) so A(x,X)

d e f i n i t i o n with respect t o a l l closure conditions given by A(x,X)

IDn(W)

By

Hence

To show x E PA

ClosA(X). Then PA

Hence

1.

PA s a t i s f i e s t h e condition ( i i )

Now f o r (i)suppose A(x,PA).

by t h e monotonicity hypothesis. unicity of

B(x,X) = [ C l o s A ( X ) + x E X

mon I D < u) i s t h e union of t h e

ID

IDmn

i s t h e union of the

< ,(W) for

n

which a r e

<

u).

The following i s

now a corollary of Theorems 1 and 2. Theorem 3 . ( i ) ”D:I _----____

5

IDf)(W),

IDY, 5

i s a conservative extension of

(ii) ID?

extension of

D :I:

(W)

.

I D ~ ( W ) and : :DI

i s a conservative

ID <w(w).

-

Remark.A proof of Theorem 3 i i ) can be obtained which follows more closely t h e argumerits of Feferman, weaker

CA

1970 using a

variant 2nd-order intermediary

but stronger induction.

IDn(2)(W) with

For t h e idea c f . 1 o c . c i t . Theorem 2.1.3.

How-

ever, t h e burden of t h e recursion work then faUs on formalizing Kleene’s basis theorem i n

IDn(W).

To r e l a t e our r e s u l t s t o those already known involving i n t u i t i o n i s t i c systems, we now restore t h e superscripts ‘ c ’ , ‘ i ’ t o distinguish use of c l a s s i c a l from int u i t i o n i s t i c logic.

S. FEFERMAN

86

___-----_ Co_ro_ll_aL..Y* i s conservative over

IDacc, i

metic formulas, and t h e proof-theoretic ordinals of

ID^"'^

( i ) (Buchholz

1981)

f o r negative a r i t h -

ID^(@)^

and

are

t h e same. (ii) (Sieg 1981)

i s conservative over

f o r negative a r i t h -

I D < w (@)I

l$ formulas.

metic and

By ( i ) ,the proof-theoretic ordinals of

and

IDm
ID

(@)i

<w

are also the

same.

i s conservative over I D n ( @ ) i

Questions -________ 1. The f i r s t question i s whether Sieg 1981 has shown t h a t f o r limit

i s conservative over I D <

Q:

IIZ(CJ~) i n which

f o r a l l sentences i n

p a r t i c u l a r , t h e provable statements r e l a t i o n between

Ql

aE

has only positive occurrences. a r e t h e same.

Q1

an3

~

(@)i In

However, t h e exact

appears t o be open even f o r

Q:

<w

.

Given our present s t a t e of knowleclge, one may ask t h e same question as above with IDPos,i n

IDaCC,i n

or

i n place of

ID,(Q)~,

and of course inquire a s t o p a r t i a l

conservation r e s u l t s as well. 2.

The next question has t o do with closure ordinals.

c l a s s i c a l l y one defines :P

The closure ordinal of

PA

5

f o r ordinals

i s the least

Given monotonic

A(x,X),

by:

4)

5

with P(*) A

.

=

These were

f i r s t studied i n a systematic way by Spector 1961, who showed t h a t for A monotonic and arithmetical (or even

1

il1)

t h e closure ordinal of

One c a n ' t improve t h i s bound even f o r A and t h e closure or3inal monotonic w+1,

A

E

but not

established i n

5 w.

E

II; , but for A

E

PA i s Cy

5

we have

wik . PA E C y

Kreisel1965 p . l 3 8 ( f t n . ) remarked t h a t f o r c e r t a i n

C,"

,

w

as i n t h e c l a s s i c a l case.

t h e closure orclinal can be established constructively t o be

for

Kreisel has asked what bound can be

restricted t o

CE operators.

It i s not

c l e a r how t o f o m l a t e t h i s question precisely, since t h e d e f i n i t i o n by recursion

.

87

Monotone inductive definitions

on o r d i n a l s r e q u i r e d t o i n t r o d u c e t h e

pi')

i s not d i r e c t l y a v a i l a b l e i n t h e B u t one f e e l s t h e r e should be some way

t h e o r i e s which have been considered here. t o d e a l with c l o s u r e o r d i n a l s formally. o f monotonic i n d u c t i v e d e f i n i t i o n s

3.

What can be s a i d about c l o s u r e o r d i n a l s more generally?

in IDPOS,i n

1~:~'

The preceding suggests c o n s i d e r a t i o n of more extensive t h e o r i e s i n which clo-

sure o r d i n a l s can be described d i r e c t l y . I D (B)

by a p r e d i c a t e

1

given

Consider f o r example an extension o f

which i s thought of as expressing

P(a,x)

X E

P i 1 a 1 )for

A(x,X), with t h e axiom:

Va(a

It (A/G1)

Then p u t

E

4[ P ( a , x )

O1

P "(x) = 3a P ( a , x ) . A

containing

P

+

t r X ( ~ ( x ,P;)


a A(x,?P(b, y ) ) ] ] . 1

3b

f--f

I n d u c t i o n on

Bl i s extended t o apply t o formulas

Can it be proved ( c l a s s i c a l l y o r i n t u i t i o n i s t i c a l l y ) t h a t

as w e l l .

P,V(X) 1

f o r ( i )A

positive i n

X,

(ii)A

Note t h a t one can prove T x ( A ( x , $ B ( y ) ) + B ( y ) ) ] +ilxVa(a What i s t h e s t r e n g t h of t h e t h e o r y

E

monotonic i n

X?

O1 A P ( a , x ) + B ( x ) )

.

S i m i l a r questions can be

IDl(@) + It(A/Bl) ?

One should a l s o consider i n t h i s connection

r a i s e d f o r extensions of ID,(@).

j u s t what p r i n c i p l e s a r e needed t o formalize t h e work of Spector 1961.

4.

To i n Feferman, 1979 provides a f l e x i b l e broad

The language of t h e t h e o r y

framework w i t h i n which t o consider g e n e r a l c o n s t r u c t i v e operations on c l a s s e s . Thus, for example, c o n s t r u c t i o n of h i e r a r c h i e s r e c u r s i o n theorem which g i v e s

(Gala

archy o f number c l a s s e s b E

B1 , a

hierarchy

fa

(Q,bJb c B a )

for suitable

411

Q1)

E

,

(ii)

m a (fx t%

Write Mon(f)

Y)

I

n [ ~Y

(or

+ fx

, etc.

fY

f o r t h e conjunction of

ment expressing t h a t

f

f; i n p a r t i c u l a r a h i e r -

( l o c . c i t . p.190) f

.

The most g e n e r a l mono-

such t h a t

trx c CI By E c d ( f x

5

i s immediate by t h e

i s obtained i n t h i s way, then f o r each

t o n i c operations on c l a s s e s a r e given by (i)

( P ( " ) l a cO1)

y)),

I

I. ( i ), (ii). Let

3 L F P ( f ) be t h e state-

has a l e a s t fixed-point, BX [ f X C X A W ( f Y

5Y

+ X ZY)]

.

S. FEFERMAN

88

What i s t h e strength of

To+Vf[Mon(f) + 3 L F P ( f ) ] ?

t i o n f o r t h e subsystems o f

To

considered i n Feferman,

We may ask t h e same ques-

1979, e.g.

IG f'

with

.

It has been proved i n Feferman, Sieg 1981 t h a t t h e l a t t e r theory ( i n i n t u i t i o n i s t i c l o g i c ) i s equivalent i n strength t o t h e c l a s s i c a l system

51- A C , while i s equiva-

i & e r and Pohlers ( t o appear), J&er ( t o appear) proved t h a t full T l e n t i n strength t o

4 - AC + BI .

my i n t e r p r e t a t i o n of

To

in

I have t r i e d , but did not succeeed, t o extend

$ - AC + B I

t o include t h e statement

Vf [ Mon(f) + 3 L F P ( f ) The theory

To

+ Vf[Mon(f)

+ PLF P ( f ) ]

1.

includes a l l constructive formulations

of i t e r a t i o n of monotone inductive d e f i n i t i o n of which I am a m r e , while (in i t s

To

IG axiom) i s based squarely on t h e general i t e r a t i o n of a c c e s s i b i l i t y

inductive definitions.

Thus it would be of great i n t e r e s t f o r t h e present sub-

j e c t t o s e t t l e t h e relationship between these theories.

Footnotes. 1) Research supported by NSF Grant.

2)

A l l these conservation r e s u l t s a r e established by f i n i t i s t i c reduction procedures.

3 ) The proof-theoretic ordinals of c e r t a i n useful weaker theories evaluated i n Feferman ( t o appear). cluded i n [ B , F, P, s ] ,

4)

qy<

PA,q )

.

were

Those r e s u l t s came too l a t e t o be i n -

though some of t h e same methods were applied.

The other usual way of defining approximations i s as (xlA(x,

A

ID,

The r e l a t i o n i s t h a t

closure ordinal i s t h e l e a s t

5

with

P A,'

P A,<

-

= A '

-

P A,%

('+'I ,

,,YgpA,q

'

and t h e

89

Monotone inductive definitions Bibliography - r u l e , i n [B,F,P,S](1981) (Ch.IV), 188-233. IJ. +1 W. Buchholz, S. Feferman, W. Pohlers and W. Sieg, I t e r a t e d Inductive Defin i t i o n s and Subsystems of Analysis: Recent Proof-Theoretical Studies, Lecture Notes i n Mathematics 897, Springer-Verlag (1981). [B,F,P,s]. W. Buchholz, The Ci

S . Feferman, Formal t h e o r i e s f o r t r a n s f i n i t e i t e r a t i o n s of generalized i n d u c t i v e d e f i n i t i o n s and some subsystems of a n a l y s i s , i n I n t u i t i o n i s m and Proof Theory (eds. Kino, Myhill, Vesley), North-Holland, (1970) 303-325. S. Feferman, Constructive t h e o r i e s of f u n c t i o n s and c l a s s e s , i n Logic Colloquium 78 (eds. Boffa, van Dalen, McAloon), North-Holland (1979) 159224. S . Feferman, I t e r a t e d fixed-point t h e o r i e s : a p p l i c a t i o n s t o Hancock’s conj e c t u r e , ( t o appear i n Proc. P a t r a s symposium).

S. Feferman and W. Sieg, P r o o f - t h e o r e t i c equivalences between c l a s s i c a l and c o n s t r u c t i v e t h e o r i e s f o r a n a l y s i s , i n [B,F,P,S] (1981) (Ch. I1 ), 78-142. G. JLger, A well-ordering proof f o r Feferman’s t h e o r y 2;ger

To

,

( t o appear).

and W. Pohlers, A b i s s i b l e Proof Theory ( t o appear).

G. Kreisel, Generalized i n d u c t i v e d e f i n i t i o n s , i n Stanford Report on t h e Foundations of Analysis (mimeographed), Stanford (1963), Ch. 111.

G. K r e i s e l , Mathematical l o g i c , i n Lectures on Modern Mathematics Vol. I11 (ed. Saaty), Wiley (1965), 95-195. G. K r e i s e l , Functions, o r d i n a l s , species, i n Logic, Methodology and Philosophy of Science I11 (eds. van Rootselaar, S t a a l ) , North-Holland 145- 159.

(1968)

W. Sieg, I n d u c t i v e d e f i n i t i o n s , c o n s t r u c t i v e o r d i n a l s and normal d e r i v a t i o n s , in’[B,F,P,S] (1981) (Ch. I11 ) 143-187. C . Spector, I n d u c t i v e l y defined s e t s of n a t u r a l numbers i n I n f i n i t i s t i c Methods, Pergamon (1961)97-102.

THE LB.J. B R O W E R CENTENARY SYMPOSILW A.S. Troelstra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

91

NOTIONS OF CHOICE SEQUENCE Michael P. Fourman Department of Mathematics Columbia University New York, New York 10027 U.S.A.

We use sheaf models to undertake a constructive analysis of the effects of admitting non-constructive choice sequences to mathematics. 01 PREAMBLE

“A choice sequence is an infinite sequence of

natural numbers whose terms are generated in succession; inthe process of generating them, free choices may play a part. At one extreme, the selection of each term may be totally determined in advance by some effective rule: a sequence generated by such a rule is a lawlike sequence. At the other extreme, we have a sequence the selection of each term of which is totally unrestricted: these are the lawless sequences. In between are those choice sequences the selection of whose terms is partially restricted in advance, but not completely deterDummett (Elements, p. 415) mined. ” The general notion of choice sequence allows that future choices need not be entirely free: they may be subjected to effective restrictions laid down at any stage. For example, we may impose the condition that the finite initial segments of our sequence should belong to some subtree S of the tree of all finite sequences, or that they should be generated by applying some function to the results of some other generating process. Different notions of choice sequence arise from differing types of restriction. Following Troelstra ( C S ) and Dummett, we shall consider in turn various different notions. Our analysis starts, not from a conception of a particular notion of choice sequence but rather, from a particular conception of the data which may be available at some stage. This then determines a notion of choice sequence. Our aim is to provide a formalization for various notions of data which will allow the informal but rigorous discussions found in Troelstra and Dummett to be replaced by calculations. For those who understand the jargon, we say immediately that sites codify particular conceptionsof data. Formally, a type of data is a site; a category equipped with a Grothendieck topology. The forcing definition for these sites formalizes the type of conceptual analysis described by Troelstra. We give the basic definitions below. For the general theory we refer the reader to Makkai & Reyes (1977) and Kock (1981). 1.1 Representation of data by a site. A category is a collection of states (objects) and arrows representing incoming data:

92

M.P. FOURMAN

f

Note that, incoming data may transport one from one state to another backwards along the arrow, or may leave one in the same state. In a given state only certain items of data may be received. Data may be received cumulatively (provided the associated states match up) so we have a partial operation *, obviously associative. The identities required by the definition of a category correspond to giving no information < > : V + V. We normally call the arrows morphisms. By introspection, we may recognize a truth whose verification seemingly requires more data bv observina that. no matter which of an exhaustive coilection of possibilities fo; extra data transpires, the verification will occur. We call such exhaustive collections of possibilities for future data covering families. Formally we require that these satisfy the axioms for a Grothendieck pretopology: 1) I < > ) covers U for each state U, f: Vf + U I fEK 1 covers U, and e: W -t U 2) If K = then I g 1 e*g factors through some feK } covers U. These are clearly valid for the intuitive notion to hand. 1.2 Remarks. Our models are similar to the familiar Beth and Kripke models. Formally, the forcing definition for sites is that for Beth models with bars replaced by abstract covers, and passage to a later stage, 5 , replaced by arrows representing incoming data, +. We make a distinction between a constructive explanation of meaning, given in terms of a notion of proof, and an intuitionistic explanation of meaning which we shall give in terms of data. Unlike Dummett, (Elements p. 403) we do not view these as rival accounts. We assume a basic conception of the mathematics of lawlike or constructive objects. (We discuss later the minimum demands we make on this metatheory - which may be classical.) We introduce various notions of data and representations for non-contructive objects based on these notions. Our analysis of the meaning of predicates involving non-constructive parameters leads us to the justification of various intuitionistic principles; it does not affect the mathematics of lawlike objects. Technically the theory of non-constructive objects is a conservative extension of our metatheory. Our-project is not novel: Beth models were introduced to formalize iust such an explanation of meanins. Our use of catesories in place of posets arises from a basic philosophical difference. We aim; not to qive a model of the activities of a sinqle idealized mathematician, but-rather, to analyse the objective mathematical truths which may be justified on the basis of a particular conception of data. Thus, for us, the information to hand at a particular time is not, in general, part of the state, but part of the representation of a particular nonconstructive object.

Notions of choice sequence

93

Two extreme examples of categories are posets and monoids. In a poset, there is, for each pair of objects , at most one morphism p5q from p to q. Posets represent a totally subjective conception of data, which identifies the state with the information to hand. A monoid is a category with only one object, only the morphisms and the (total) operation of composition matters. We use them to represent objectively conceptions of data for which we can recognize that future possibilities for data are independent of the information to hand. Some types of restriction on future data compel us to consider separate states to represent the differing data which is acceptable. In these cases an objective viewpoint amounts to being able to juxtapose two states given independently to give a single process. Formally, given two states A and B, there is a state A X B accessible from A and B by morphisms TA nB A -A X BB which are covers and for each pair e: CA and f: DB there is a unique e x f making the diagram commute A

k

f A

1 C-B

X

3B

C

tx D

e x f

b

I D

Note that, A x B is not a categorical product since we have no pairing in general. The requirement that the “projection“ maps cover says that, we can always introduce a new process independent of that under consideration. 1.3 Definitions. A spread is a subtree S c N4N with every branch N, we say ~ E S infinite: if aES then a*ns.S for some nsN. For a:N+ iff Va. ( m a 4 aES). If S and T are spreads, a neighbourhood function F: S + T is a monotone function such that for each nsN, the set of nodes aaS such that Lth (F(a))zn is an inductive bar of S. Given a s s and F: ST we define F(a) by asa + F(a) E F(a). Spreads represent subsets of 7 8 , neighbourhood functions represent continuous functions. Composition of neighbourhood functions gives the composition of the associated functions. Some functions are canonically represented: in particular, the open inclusion corresponding to a finite sequence e is represented canonically by Xa.e*a (and in many other ways by merely deferring the information); more generally, if f is an open map it has a canonical representation F(a) = A { b I aEf-’(b) 1 . Countable dependent choice and a suitable form of Bar Induction imply that every continuous function has a neighbourhood function. We write B for the universal spread of all sequences. 12 TYPES

OF DATA

The examples which follow should make clearer the translation from the informally rigorous description of a notion of choice sequence to the appropriate site.

M.P. FOURMAN

94

2.1 Open Data. The simplest data we shall consider, open data, consists of finite sequences a of natural numbers. We can construct a sequence a from such data in many ways, the simplest of which is to consider the information thus far received as an initial segment aEa. We have decided in advance that, having received the information a we will treat subsequent data (another finite sequence, b) in a particular way: we concatenate a*b. Thus there are various states which in this example may be identified with the information to hand. Incoming information takes us from one state to another.

Abstractly, we have a category whose objects are the states and whose morphisms represent finite amounts of information. In our present example this structure is represented abstractly as the tree of finite sequences or, more concretely, as the category of basic opens of Baire space and open inclusions between them. Since, to recognize that aEa*n for some nEN it suffices to recognize that aca, we must let { a*n I nEN 3 cover a for each finite sequence a. These covers generate the open cover topology for formal Baire space (see Fourman & Grayson (this volume) ) . Our analysis has merely served to reconstruct the Scott (1968) - Moschovakis (1973) model. From a more objective view of mathematics the distinction between various states seems unjustified: it portrays the activity of a particular idealized mathematician rather than the mathematics which results from reflection on the general nature of such activity. A more satisfactory model (from this point of view) is given by the monoid of finite sequences. We can picture this concretely as the monoid of neighbourhood functions canonically representing open inclusions. Again the appropriate topology is the open cover topology. This model corresponds to the liberation of the idealized mathematician: realizing her situation, she can transcend it and is free b?

.

-

This conception of open data allows that all possible data can be coded up in a single choice sequence. 2.2 Independent Open Data. We modify our model to consider not a single generating process but a potentially infinite collection. It is essential to distinguish this from a potentially infinite sequence of processes, which we could code as a single choice sequence. The force of this distinction is that, at any stage, the information we have is just the collection of initial segments to hand. They are not taken in any particular order.

The subjective states for this notion consist of finitely much information about finitely many sequences. Concretely we represent such a state by a basic open U c B n modulo any action by a permutation of n.

Notions of choice sequence

95

Actually it is more convenient to consider basic opens as states. At any stage we may introduce finitely many independent generating processes as well as obtaining more information about those already considered. We represent such information by a map

u-v which is induced by the projection corresponding to some injection n r+ m. The morphisms induced by permutations of n have the effect of identifying states which represent different orderings of the same collection. In addition to allowing open covers as before, we stipulate that the projection

u

ni

-

Bm-

T(U)

ni

Bn

is a cover. This reflects the possibility of adding finitely many independent processes to any discussion. In general, a family of morphisms covers iff the union of the images is an (open) cover of V. All the morphisms here are open maps and we shall later consider them as represented canonically by neighbourhood functions. Again, we have arrived at a well-known model: according to Hyland (personal communication), the forcing definition for this model corresponds to the Kreisel-Troelstra elimination of lawless sequences. more objective representation of this type of data is obtained by identifying states in which the same number of generating processes are considered. The site we use to represent this type of data has as objects, the various Bn, and as morphisms, compositions of projections Bm-Bn induced by n-m and open inclusions Bn-Bn induced by n finite sequences. Once more, all the morphisms are open maps canonically represented by neighbourhood functions and we use the topology in which a family of morphisms covers iff its images cover.

A

2.3 Lawless Data. This conception of*data was motivated by the following passage from Troelstra (CS p. 16) : "Suppose we have started two lawless sequences a and 0, alternately selecting values: a0, 00, al, 01, 1x2, 02, Now we may also regard this as a single process y , with y(2n) = an, y(2n+l) = On. However, we cannot regard cx,B,y as all being lawless within the same context: either we have to decide a and 0 to be lawless, and then y is a sequence which is not itself lawless .; or we consider y as lawless, inwhich case a,B are sequences (not lawless ones) constructed from y."

... .

.. .

..

This discussion cannot be expressed in the Kreisel-Troelstra formalisation of choice sequences. This is because, their notion of lawlessness is not an objective one. In our previous example, all the states are, in fact, remarkably similar: Bn is homeomorphic to B. Essentially, the differences between the states arise because we have chosen a coding B Bn in term of which we choose which maps to put in our category. We now take the point of view that different codings

MP. FOURMAN

96

simply reflect different ways of considering the same reality. Thus the controversy as to which of a, B and y are lawless, in our example above, arises from the difference in viewpoint formalized by the pairing B = B x B. We now consider an abstract view of the same kind of data, which is independent of such codings. We call this lawless data, it consists of those endomorphisms e: B + B which can be decomposed as "projections modulo some coding": e = j-1.r,i for some homeomorphisms i and j.

Bm& UI

Bn UI

u-v i\ B

4

A s usual, all our morphisms are open maps and we use the topology in which surjective families cover.

We shall see that lawless sequences for this conception of data behave more sociably than is traditional. For example, two views of the world may at some stage turn out to be the same so equality is not decidable. To formalize our discussion of this type of lawlessness, we shall introduce a notion of independence: basically, a and B are independently lawless iff y = < a , % >is lawless. Returning to Troelstra's example, a, B and y are lawless a and 4 are independent and y is independent of neither of them. 2.4 Spread Data. Here we attempt to formalize Brouwer's description of the generation of a free choice sequence.

...

" the freedom of proceeding, without being completely abolished, may at some time p, undergo some restriction, and later on further restrictions. " Brouwer (Cambridge p. 13)

The restrictions discussed by Brouwer demand that future choices belong to some spread. Spreads correspond to certain sublocales of B. We consider such sublocales F s Bn and morphisms between them induced by projections. We take as covers projections and open covers. This gives us (in this example) the topology in which a family covers iff the interiors of its images cover. This topology involves no new insights, many stronger topologies (more covers) are conceivable: It is certainly plausible that we might justify the conclusion that every member of a spread S belongs to one of the spreads Ti without showing that the interiors of the Ti cover S , by appealing to particular properties of S. This would be reflected in our models by adopting a stronger topology. What we will show is that it is consistent to assume that the only covers are those we have built into the definition of the topology. The main insights justified by this conception of data are the relativisation of V a 3 % choice and continuity for lawless a to lawless elements of some spread and the extension of Bar Induction to give induction over arbitrary spreads. Brouwer's conception of choice sequence has been criticizad for not

Notions of choice sequence

97

b e i n g c l o s e d u n d e r c o n t i n u o u s o p e r a t i o n s . The s p r e a d s w e have i n t r o duced a r e b l a n k o r naked s p r e a d s , which, f o r Brouwer, s i m p l y p r o v i d e a framework f o r t h e g e n e r a t i o n o f m a t h e m a t i c a l e n t i t i e s . By a t t a c h i n g In " f i g u r e s " t o some nodes of a s p r e a d S w e p r o d u c e new o b j e c t s . p a r t i c u l a r , any neighbourhood f u n c t i o n F: S + T p r o d u c e s f o r e a c h c h o i c e s e q u e n c e a€S a s e q u e n c e F ( ~ ) E T . The i n f i n i t e s e q u e n c e s gene r a t e d i n t h i s way a r e c l e a r l y c l o s e d u n d e r t h o s e c o n t i n u o u s o p e r a t i o n s which have neighbourhood f u n c t i o n s . We s h a l l s e e t h a t ( i n o u r models) a l l l a w l i k e f u n c t i o n s have neighbourhood f u n c t i o n s . F u r t h e r more u s i n g s u c h d r e s s e d s p r e a d s ( w i t h S E < S , F > i n t e r p r e t e d a s , f o r some ~ E S , 5 = F ( a ) ) , we s h a l l see t h a t an axiom of " s p r e a d d a t a " i s v a l i d f o r t h e s e sequences. 2 . 5 Continuous d a t a . W e s t a r t from B r o u w e r ' s 1933 d e s c r i p t i o n of a d r e s s e d s p r e a d a s r e p o r t e d by van Dalen (Cambridge p . 1 7 1 . Here A g e n e r a t e s a l a w l e s s sequence and B a p p l i e s t o it a neighbourhood f u n c t i o n t o o b t a i n a s e q u e n c e F ( a ) a s d e s c r i b e d e a r l i e r . W e modify t h i s p i c t u r e by no l o n g e r r e q u i r i n g t h a t A ' s sequence b e l a w l e s s : it may i n f a c t b e g e n e r a t e d a s a c o n t i n u o u s f u n c t i o n of some sequence g e n e r a t e d by X who, i n t u r n , r e f e r s t o Y , and s o on. W e r e q u i r e t h a t a l t h o u g h t h i s c h a i n of dependence may b e p o t e n t i a l l y i n f i n i t e , a l l t h a t B c a n be aware of a t any g i v e n s t a g e i s a f i n i t e c h a i n of dep e n d e n c i e s , r e s u l t i n g i n t h e knowledge t h a t B = r ( a ) f o r some n g e n e r a t e d by someone down t h e l i n e , and some neighbourhood f u n c t i o n

r.

We r e p r e s e n t s u c h d a t a by a neighbourhood f u n c t i o n r : S + T between s p r e a a s . Note t h a t , a l t h o u g h i n p r i n c i p l e w e s h o u l d want t o c o n s i d e r dependence on more t h a n one s e q u e n c e , s u c h d a t a r e d u c e s t o dependence We g i v e t h i s on a s i n g l e s e q u e n c e by means o f t h e p a i r i n g B x B c. B. c a t e g o r y t h e "open c o v e r t o p o l o g y " i n which t h e c a n o n i c a l r e p r e s e n t a t i v e s of a c o v e r i n g f a m i l y of open i n c l u s i o n s form a c o v e r . Of a l l our models w e b e l i e v e t h a t t h i s one b e s t r e p r e s e n t s t h e n o t i o n of c h o i c e s e q u e n c e . N e v e r t h e l e s s , w e d i s c u s s two v a r i a n t s . F i r s t l y , i f w e a r e concerned o n l y w i t h e x t e n s i o n a l p r o p e r t i e s , w e can use c o n t i n u o u s f u n c t i o n s i n p l a c e of neighbourhood f u n c t i o n s . Secondly, i f i n s t e a d of u s i n g a r b i t r a r y s p r e a d s w e c o n s i d e r t h e monoid of c o n t i n u o u s f u n c t i o n s B + B , w i t h t h e open c o v e r t o p o l o g y , w e o b t a i n a model f o r Kreisel and T r o e l s t r a ' s t h e o r y C S . ( T h i s was observed i n d e p e n d e n t l y by Moerdijk & van d e r Hoeven ( 1 9 8 1 ) , Grayson ( 1 9 8 1 ) and t h e a u t h o r ( 1 9 8 1 ) ) . The f o r c i n g d e f i n i t i o n f o r t h i s model c o r r e s p o n d s t o t h e e l i m i n a t i o n mapping f o r c h o i c e s e q u e n c e s of Kreisel and T r o e l s t r a ( 1 9 7 0 ) . I n t h e s e models w e v e r i f y f u l l V a 3 B c h o i c e and c o n t i n u i t y p r i n c i ples. The a d v a n t a g e of t h e e x t e n d e d model i n which w e a l l o w a r b i t r a r y s p r e a d s as domains i s t o j u s t i f y r e s t r i c t e d v e r s i o n s of t h e s e and e x t e n d e d Bar I n d u c t i o n a s f o r s p r e a d d a t a . 2.6. O t h e r t y p e s of d a t a . I n o u r p a p e r C o n t i n u o u s T r u t h (1982) we c o n s i d e r more g e n e r a l t y p e s of d a t a ; i n p a r t i c u l a r , d a t a r e p r e s e n t e d by c o n t i n u o u s maps between opens of Rn. W e a l s o g i v e a g e n e r a l t r e a t ment of t h e " e l i m i n a t i o n mappings" a s s o c i a t e d w i t h e a c h t y p e of d a t a and t h e r e l a t i o n s h i p s between v a r i o u s t y p e s of d a t a m e d i a t e d by geometric morphisms between t h e c o r r e s p o n d i n g t o p o i .

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M.P. FOURMAN

13 NON-CONSTRUCTIVE OBJECTS

We now embark on the analysis promised in 2.1. An understanding of a collection of objects is merely an understanding of what it is to be presented with such an object and of what it is to show that two such objects are equal. This does not automatically give rise to a determinate collection of predicaterrather we must introduce predicates by explicitly giving their meanings. Other predicates may, of course, be compounded from ones previously understood using the logical connectives. We suppose the meanings of statements involving lawlike parameters, quantification over lawlike objects and the meanings of the logical connectives applied to such statements, to be understood. Traditionally, an explanation is given in terms of an informal notion of construction (for example, Dummett (Elements p . 12ff.)). Our explaination of the meaning of statements involving non-constructive objects is independent of this (and, to a large extent, of its results), similar to it in form, and different from it in content. The meaning of a statement involving non-constructive objects is given in terms of a constructive understanding of which items of data justify a given assertion. 3.1 Non-constructive Objects. Our archetype is given by Brouwer's notion of a dressed spread: A partial function $ assigning lawlike objects to the nodes of some spread S. The idea is that any choice sequence a of the spread S generates successive approximations, $(a) for aca, to a non-constructive object $ ( a ) . Abstractlv, we assume that the constructive objects @(a) have a preorder, x < y if x contains "more information" than y, and that $ is monotone, a (I-

@.

Firstly, we have two basic properties of this notion of justification: if K is a cover of

u and ell- $for each ecK, then

if VII- @le then ell- $ where e: V

+ U

UII-

$

,

and @le is the result of restricting each parameter

,

Notions of choice sequence

99

Secondly, we give the meaning of basic (atomic) predicates. The basic statements we may make concerning non-constructive objects are few (two): intensional equality: if ale = Ole then el1-a-B finite information: if a(e(<>))z p then ell-acp For basic predicates involving only constructive parameters, if $ then ell-$. Finally, we explain the meanings of the logical connectives: A if Ull-$ and 1l-i then \ l - $ ~ I j l V if Uli-$ or /I-$ then /I-$ v Ijl 3

+ V

1

if Ull-$(a) then UII- sx.$(x) if for all e, if ell-$ then ell-$, then Ull-$ +$ if for all e: V +U and each 5 given at V we have Vll-$le(a), then U / I - VX.$(X) is never explicitly justified, although it may become s o at some nodes by virtue of the general clauses above (this eventuality does not arise in our present models).

We claim that this definition reflects the intended meaning of statements involving non-constructive parameters. It coincides with the standard forcing definition for sites. We also define quantification over independently generated free choice sequences: for e: V +B. if UxV/I-$lnl(el.rr2)then U(I-FclEe.4 if UxWII-$lnl( B ~ I T ~then ) UII- ?ase.$ (for any 8: W + B which factors through e) (In 9 3 . 4 we see how to regard e in general as a subset of 78 . ) 3 . 2 . 1 Lemma.

1) fll-$Ie iff e*fll-$ if ell-$ then e*f IF$ 3 ) ell-$ iff $, if $ has no non-constructive parameters or quantifiers. 2)

3 . 3 Basic Types. We have represented non-constructive objects abstractly as (local) sections of certain separated presheaves. We now see that this representation gives us the higher types defined formally in sheaf models. To save space we then treat these models more or less formally. We ask the reader to bear in mind that the models are intended to reflect the meaning of non-constructive mathematics as well as its formalism.

3.3.1 Lemma. If F, G: U T represent non-constructive objects at U then iff F is a neighourhood function UII- F E NN Ull- F = G iff F and G represent the same function -+

Where = is extensional equality. In any sheaf model the discrete spaces N and N < N are represented by constant presheaves. In our present models, the (set of points of) Baire space is represented by the separated presheaf of continuous

M.P. FOURMAN

100

Baire - valued functions (see CT). Thus the non-constructive sequences we have defined correspond extensionally to the abstract sequencesgivenby the higher-order logic in sheaf models, The presheaf IB (U) 2 (U,NGN) of neighbourhood functions represents Baire space intensionally. The representable presheaf 0 (U) 5 IB (U) represents the free choice sequences for the notion of data concerned. The idea is that nothing more is known of them than can be given directly by data. Non-constructive objects are represented extensionally as continuous maps to formal spaces (see Fourman & Grayson). Restriction is given by composition. The basic predicates are given by equality if ale = a l e then ell-a = B finite information if (aIe)-’(p) = T then ell-acp. For consideration of extensional properties it suffices to consider our sites extensionally as categories of continuous maps. 3.4 Lawlike Objects In general, we think of constant presheaves as representing lawlike objects. In most sheaf models there is no canonical way in which to define “the“ collection of lawlike elements of a given sheaf. In our present models we define the collection of lawlike elements of a sheaf X to be the subsheaf L(X)EX generated by global sections of X. We claim that L(X) is a constant sheaf: Suppose a and b are global sections if A //- a=b then AXB 11- a=b whence B I/- a=b (as projections are covers). (For monoid models of course the topology is irrelevant and we always have a notion of “lawlike“ given in this way.) Of course, the discrete spaces N and NCN are lawlike. We now give some other examples: 3.4.1. The lawlike elements of B a r e given by the constant functions in B (u), which are canonical representatives of the constant functions

.

3.4.2. The collection Kx of lawlike operations the Yoneda Lemma: r(K,) = X(B) with the action

0

-t

X is given by

F I ~

u 11- F(c.) 5 iff = 5 Note that these are given intensionally. However, every lawlike operation with non-constructive objects as values is given as a neighbourhood function and thus acts extensionally. 3.4.3. Lawlike data; Morphisms with codomain B may be viewed as subobjects of B as follows: For e: U -c B and 5 : V -+ B if 5 factors through e then V (I- Cce In each model tce is in fact definable: e may be viewed as an element Fe of KU and v (I- vg.(tce++ ~c.E;.F,(~) = 5). (Where g is the representable presheaf.) If u: u w B is monic then CcX(U) may be viewed as a lawlike function defined for all C ~ E Uby U IF F (ulB1 = 5 1 B .

101

Notions of choice sequence

14 P R O P E R T I E S OF THE MODELS All sheaf models provide interpretations of HAH. These ones have special properties. We view this as justifying certain intuitionistic principles on the basis of particular conceptions of data. We assume choice principles for lawlike objects, (which is constructively unexceptionable). This allows us to reduce existence on a cover of U to existence on U itself for lawlike objects (as every open cover of U has a disjoint clopen refinement). For continuous data, existence on a cover always reduces to existence on U, as covering families of monomorphismsgenerate the topology. Below, choice use x, €,ELto

5 , 5 range over non-constructive objects, a, 0 , y over free sequences, e, f over data a, b overB and n-.m over N. We y a s variables for lawlike objects in general, and write signify that €, is lawlike.

4.1 Choice Principles. We obtain countable lawlike choice with non-constructive parameters. ACN*

l k v A(

V

n 3 x.A(n,x)

-t

3 f Vn.A(n,f (n)))

.

From the remarks above, this is standard. For continuous data, the same proof gives 3 f Vn.A(n,f(n))). ACN&* /I-VA(Vn>€,.A(n,€,) It is perhaps surprising that for open data (for example) this is not justified. As 8 is representable, we obtain, for those models in yhich projections are covers, forms of V a 3 x choice: AC& 11- vAEL(VaEU.3X.A(a, X) -+3fEL.VaEu.A(a,f(x))) ACB* \k V A ( yclEU.3X.A(a,X) 3 y 3 f E L F U E U . A ( U , f ( U , y ) ) ) -+

+

For AC& , suppose A is lawlike and U Va3xA(a,x), then E X(B). This UxB IF V a3x.A(a,x) whence B It A(id,a) for some a re resents the required function as in 3 . 4 . For A m * if UxV TF 3 x.Al'iT1 (ulap,x) then UxV IF Alal (ul'rrl,a)for some a E X(UXV). Now introduce parameters 7 for U and view a as a function of these and a m as in 3 . 4 . Note that, for continuous data, and V coincide. For continuous data, the same proofs give V a 3 J 5 choice principles: AC06 IF VAEL( V ~ E U , L ~ ~ . A ( ~ , E ) 3 FEL V ~ E U . A ( ~ , F ( ~ ) ) ) AcB~* 11- VA(V ~EU.LIE.A(~,S) 3 FEL. 3 7 . v ~ E u . A ( ~ , F ( ~ , ~ ) ) ) -+

-+

Forthemonoid model for continuous data,in which we do not allow restrictions to arbitrary spreads, we obtain a stronger form: -+3fEL.Va.A(a, F(a))) ACbb**))-VA('VdE.A(a, 5 ) Suppose BII- va3E.A(a, €,) then BII- A(id, E ) , for some SEX(B) which represents the required function.

U In each case above, we may replace V a by V ai,...an and Y ~ E by. anLun and obtain the corresponding choice principles with Ea1Eu1, the same proofs. The same remark applies to our principles below.

...,

4.2

Continuity Principles.

Data. -

The first principle,we call Existence of

M.P. FOURMAN

102

Given aE M(U) with UlI-A(a), view a as an element of D to justify ED. We also obtain a stronger form for data in which independent generating processes may be introduced: ED* 1)- " _ i , , ,Up(A(Ui ,. ,Up) + 3 Ui,. ,Up

..

..

..

..

cliEUi hy'BiE~1,. ,BpEUp.A(Ei ,. . .'Bp))) up) where W is a basic open and the ui Suppose BPxU WxVIl-Ala(ul, are components of the inclusion. These Ui do the trick: WxWxVII-A(ul,. ,up) as the two possible orders for considering W are isomorphic. ( M

...,

..

:'he second general principle is that every lawlike functicn defined on 0 is given by a neighbourhood function. We have already see (3.4) that elements of B (B) represent lawlike functions. Ey 3.2.1.(3) these nay also be viewed as neighbourhood functions in the model. The application defined in 3.4 is just standard application of neighbourhood functions carried out internally. Thus, K

11-

VFEL: 8

+

B

.

"F

is given by a neighbourhood function."

The same remark holds for functions to other non-constructive domains and for those defined on subspaces of 63 given by monic data u. Note that the F above are a priori given intensionally, it is a consequence of K that they act extensionally. This remark is not deep, it shows how little intensional information we have taken into account. We finish this section with the remark that all objects are lawlike functions of a finite number of free choice parameters U I ~ V E ~ ~ ~ a1F ,..., E Lan.S . ~ = F(a1 an) (U is for Uniformization).

,...,

.

This enables us to make good an earlier promise. Dressed spread data holds in all our models as a consequence of the existence of data for free sequences because if 3?X,F.S = F ( E ) then for any A we have A(S)++A(F(a)).

4.3 Erouwer's Dogma of Bar Induction is also built into our models. We show that IF V K C N < N (K monotone A K inductive + (Va3nEK.aEn -+ < > EK)) Note that K is not required to be lawlike. Suppose that Ull- "K is a monotone, inductive bar". Consider IK = { I V(I- n KIV with VE @U) , n€NcN} IK is monotone and doubly inductive (i.e. IK is a closed crible in o(UxB)). Thus if IK covers UXE then > E IK and we are done.

~ ~persistence m, of forcing so Now UxB l l - V m ~ K l ~ ~ . nby M * = { < ~ , n >I Vxn 11- ZE ~ 1 A I~ T -1~ for E ~ some rnl covers UXB. But IK * ,C IK because Vxn 11- mE Klal iff V 11- mEK as projections cover and Vxn 11- n 2 e m iff n 7m. Thus we are done. Clearly, the same proof justifies bar induction over any spread which occurs as a state in our representation of data. 4 . 4 Equality. We begin by considering open data, which models the theory LS, lawless data and spread data, which give another kind of lawlessness. For all of these, extensional and intensional equality

103

Notions of choice sequence

-

coincide : ll-.'dct,R

( a =B

a:

a).

The major difference between them is that, for open data free sequences have decidable equality

11- V a , B ( a = R v l a = 0 ) Whereas for lawless and spread data, 11- 1 V a , B ( a = R v l a = B ) In these models we can represent the notion of non-constructive objects generated by independent processes. Given XEX(U) and YEY(V) let uxv 11- XlTl YITZ "x is independent of y". W with Then W 11- x $ y iff locally there is a cover p: UXV xjp = xlnl and y1p = yqliz for some X E X(U) and YE Y(V).

.

+

-f

For free sequences, independence is definable: 11- v a , B ( a $ B ++ l a = B ) for subjective open data, 11- v a,B ( a R ++ 3 y . y = < a , B > ) for lawless and spread data. Note that for other types of data such a predicate becomes trivial since a priori independent processes might later be found to be related.

z,

For these types of data, the quantifiers 3 satisfy the axioms and rules given by Troelstra ( C S , p. 35). They have the effect of quantifying over lawless sequences generated by processes independent of any under consideration 11- 1 a1 r r an( r * * tan)) For lawlike predicates we have

---

I~VAEL

(EE.A(CL)

+

-

++VE

($(z)+ A(G)))

Thus ED* coincides with the usual form of open data. We conclude our discussion of these variations on open data by listing some simple properties whose verification is left to the reader. Free sequences are dense. 11- v a 3 a.aEa ) sequences are very dense. (I-V'lVCi( 7 S = a + 5 ~ ~ 1Free For open and lawless data, we also have, Free sequences are not lawlike. 11- V a . l a € L For continuous data, extensional and intensional equality do not coincide 11- 3 c1,a ( C Y = B A B) every sequence is free 11- v 6 3 a . c : c1 4 . 5 Identification of Data. All the data we have used is given by neighbourhood functions. As we remarked in 4 . 2 , these just act internally as lawlike neighbourhood functions. Thus data is given in general as analytic data with a class of functions restricted appropriately for each notion of data. For open and lawless data, this reduces internally to open data. For spread data it is just spread

104

M.P. FOURMAN

data in the tradional sense. For continuous data it is analytic data. We know the internal and external characterization coincide by 3-2.1 (3)- For each type of data, we have the property of density D ik VeeD. 3 a.aee , triviaily, as c1 can be taken equal to e.

55

CONCLUDING REMARKS

5.1 Metatheory. We have been somewhat cavalier in our use of higher types. In particular, we have stated many principles in universally quantified form, using quantifiers which are constructively unacceptable. The corresponding schemata may be justified by our methods using a metatheory equivalent to IDB (CS p. 31). For a discussion of this see v.d. Hoeven & Moerdijk (1982). 5.2 The Other Kind of Lawlessness.

The model we presented at the Brouwer Symposium, we now view as ad hoc. We considered the monoid of local homeomorphisms of Baire space with the open cover topology, and picked out a domain of "lawless sequences" represented by the local projections. We now view the lawless data presented in 2.3 as a better representation of the notion we had in mind. The free sequences for this notion of data have all the properties we mentioned in our abstract.

5 . 3 Other Models. Once this project was well-advanced, the author

realized that few of the ideas here are really new. The elimination translations of Kreisel and Troelstra coincide with our models for LS (open data) and CS (continuous data) as remarked above. Thus, it cannot be said that there is anything more than a difference in viewpoint distinguishing our approach. More concretely, Dragalin (1974) uses essentially the same ideas as us to construct essentailly the same type of model. We hopethat the presentation of these ideas as sheaf models will at least aid progress by providing a mathematically apt setting fox comparing various notions of choice sequence. It is on the level of philosophical analysis that we hope to have provided something novel. For example, we claim that the verification of the axioms of CS in the monoid model for continuous data, provides an adequate conceptual basis for these axioms: acoherent notion of choice sequence and a verification of the axioms based on this notion. 5 . 4 Acknowledgements. Dana Scott's suggestion, made while discussing an early version of CT, that I consider lawless sequences as abstractly "generic" elements in sheaf models, led to the present paper. I am also grateful to Michael Dummett: both for his lectures which introduced me to intuitionism, and for helpful comments on an earlier draught of this paper. I hope his influence is discernible. I wish to thank Dirk van Dalen and Anne Troelstra for their lectures at Oxford while I was a student, their encouragement and hospitality during many visits to the Netherlands, and a most stimulating conference. Finally, I wish to thank the ZWO and the NSF for their financial support.

Notions of choice sequence

105

REFERENCES

Brouwer, L.E.J., (Cambridge), Brouwer's Cambridge Lectures on Intuitionism, van Dalen, D., ed.(Cambridge Univ. Press, Cambridge 1981). van Dalen, D., An interpretation of intuitionistic analysis, Ann. Math. Logic (1978) 1-43. Diaconescu, R., Change of base for toposes with generators, Journal of Pure and Applied Algebra 5 (1976) 191-218. Dummett, M., (Elements), Elements of Intuitionism (Oxford Univ. Press, Oxford, 1978). Dragalin, A.G., Constructive models for theories of intuitionistic choice sequences, in: Investigations on formalised languages and non-classical logics (Nauka, Moscow, 1974) 214252 (in Russian). Fourman, M.P., (1981) A model for the theory of choice sequences (CS) (abstract), Amer. Math. SOC. Abstracts, to appear. (1982) Continuous Truth, in preparation (abstract, Amer. Math. SOC. Abstracts 2. (1981) 282). - & Grayson, R.J., Formal Spaces, this volume. Grayson, R.J., A sheaf model for choice sequences, manuscript, Mtinster, 1981. van der Hoeven, G.F. & Moerdijk, I., Sheaf models for choice sequences, to appear. Johnstone, P.T., Topos Theory (Acad. Press, London,1977). Kock, A., Synthetic Differential Geometry, London Math. SOC. Lecture Notes =(Cambridge Univ. Press, Cambridge, 1981). Kreisel, G., Lawless sequences of natural numbers, Comp. Math. 2 (1968) 222-248. - & Troelstra, A., Formal systems for some branches of intuitionistic analysis, Annals of Math. Logic 1. (1970) 229-387. Makkai, M. & Reyes, G., First Order Categorical Logic, Lecture Notes in Math. 611 (Springer-Verlag,Heidelberg, 1977). Moschovakis, J.R., A topological interpretation of second order 26 (1970) 261-275, intuitionistic arithmetic, Comp. Math. Scott, D.S., Extending the topological interpretation to intuitionistic analysis, Comp. Math. 2 (1968) 194-210. Tierney, M., Forcing topologies and classifying topoi, in: Heller, A . & Tierney, M. eds., Algebra, Topology and Category Theory: a collection of papers in honor of Samuel Eilenberg (Acad. Press, New York, 1976) 211-219. Troelstra, A., (CS), Choice Sequences, A chapter of intuitionistic mathematics (Oxford Univ. Press, Oxford, 1977).

THE L.E.J. BROUWER CENTENARYSYWOSILM A S . Troelrtra and D. van Dalen (editors) 0 North-Holhnd Publishing Company,1982

107

FORMAL SPACES M.P. Fourman Dept. of Mathematics, Columbia University, New York, NY 1 0 0 2 7 , U.S.A. R.J. Grayson* Inst. fur mathematische Logik und Grundlagenforschung, EinsteinstraBe 64, 4400 Munster, West Germany Infinitary geometric propositional theories are axiomatisations of frames (complete Heyting algebras). Working in the dual category of locales gives rise to the notion of formal space of a theory. This formal space has enough points to distinguish its opens iff the theory is complete. Completeness of the theories of (formal) Baire space, Cantor space and real numbers is equivalent constructively to Bar Induction, the Fan Theorem and compactness of the unit interval. A proof of completeness for countable theories may be viewed geometrically as a covering theorem for separable locales by formal Baire space.

PREAMBLE This paper is an exercise in geometric logic. Treating intuitionistic logic algebraically leads to frames or complete Heyting algebras which are to intuitionistic logic as complete Boolean algebras are to classical logic. Formally "reversing the arrows" gives the dual category of locales, which are thought of as geometric objects or "generalised spaces". These arose from the observation that much of (classical) pointset topology may be expressed in terms of the frame of open subsets of a space, without mention of points. (The literature on locales is now considerable; Johnstone ( 1 9 8 2 ) gives a comprehensive bibliography.) Joyal has pointed out that locales may be viewed as spaces of models for certain propositional theories; these formal spaces are a simple example of the presentation-invariant view of theories as geometric objects which has long been espoused by Lawvere. Our primary concern, however, is with the interplay between presentations of a theory and properties of its formal space, especially with respect to the existence of points, that is, models of the theory. Locales are generalised spaces because they may fail to have points (to distinguish the open sets); from the logical point of view this happens precisely because consistent theories may not have enough (or any) models (even classically, as our logic is infinitary). Formal spaces which classically have enough points may fail to do so constructively. For example, the statement that formal Baire space has enough points is equivalent to a form of Bar Induction, BI. A standard proof that countably axiomatised theories in countable languages are complete (have enough models) may be viewed geometric-

*

Research Fellow of the Alexander von Humboldt-Foundation

M.P. FOURMAN, R.J. GRAYSON

108

ally as constructing an open cover of the corresponding countably presented formal space by a sublocale of formal Baire space. If this sublocale has enough points (a form of Bar Induction) then the theory is complete. From a logical viewpoint we prove various completeness theorems which rely oncertain induction principles. Geometrically we prove, constructively, localic versions of classical covering theorems of general topology. In the presence of induction principles which ensure that our fundamental spaces have enough points, we obtain constructive proofs of the classical theorems for topological spaces. In Fourman and Scott (1979) complete Heyting algebras are used naively as domains of truth-values for modelling intuitionistic higherorder logic. In the interpretation over the formal space of a theory we get a generic model for the theory, giving a systematic way of constructing models in contrast to the ad hoc methods of Fourman and Hyland (1979 )

.

This paper has had a long gestation. Our thoughts have been influenced by Martin Hyland and Dana Scott who have been very patient. The idea of viewing locales as geometric representations of theories is due to Andri? Joyal, whose influence on this paper has also been considerable.

5 1

THEORIES AND MODELS

We begin with an abstract version of propositional logic which corresponds to presentations of locales ( 2 . 3 ) . We shall show by examples how this subsumes and is covered by more traditional syntactic presentations. 1.1 Definitions. A propositional language P , g is a preordered set of formulae with conditional finite meets: if a finite set of formulae has a lower bound then it has a greatest lower bound, in particular, if ! I is J inhabited then it has greatest element T .

PSP

piqsr psr

PST

r r p and r s q rsPAq

preorder rules A-rules

In the A-rule (and always) we interpret mention of p A q strictly (as in Fourman & Scott (1977)), to imply that p A q exists. We say p and q are compatible iff p A q exists. The meaning of such a language will be given in terms of truth. We consider sequents $ * r where $ , T F P and @ i s finite. If m c P is some collection of "true" formulae, we say $ * r is valid in m, @kmr, iff $ G m implies there is some q E r Om. A sequent @ * r should thus be viewed as asserting that M $ implies V r (a finite conjunction of formulae implies an arbitrary disjunction). We shall often write sequents more legibly using these notional conjunctions and disjunctions explicitly. In particular we use 1, "false" , for the empty disjunction. A model of P,< is a subset m of P which forms a filter: in terms of

109

Formal spaces

va 1idity :

P bm q

if P < q

‘=m PI9

km

P Aq.

(The latter clause means: if p,q E m then p hq exists and is in m.) , transWe may also phrase this in terms of truth-values in ~ ( l ) the lation being given by: *€[[pll iff p E m

.

The conditions become

13 c “sll if P _
“P

Later we generalize this definition using other lattices of truthvalues. We give some examples of languages and sequents.

1.2 Examples. 1 ) Syntactic languages. Let P be a set of propositional letters. We take as P the collection of finite subsets of P (viewed as formal conjunctions) ordered by reverse inc1usion:psq iff p a q . This is the traditional notion of propositional language with conjunction. A model is simply an assignment of truth values for which [[p A q]] = Tiff [[PI] = T and “q]] =T, and “TI1 =T; it is of course determined once we know which propositional letters(sing1etons) are true. 2 ) Trees. L e t T be a tree (growing downwards), A model of T is a branch: a linearly ordered subsetl with a t b E m implies a E m , and T Em. This branch is infinite iff a i=,V{blb is an immediate successor of a} for each a € g . 3 ) Function spaces. If A and B are sets let P be the set of partial functions p: A- B with finite domain, ordered (again) by the principle of “more (information) is less (variation)”: p s q iff p a q . A model m of tp is a partial function F: A- B: Let p e m iff p c F . F is total iff TkmVI I b E B} for each a € A F is

F is

1-1 iff for all a,a’,b ,Fmv{TI a=a’) onto

iff

T bmV{la € A }

for each b EB.

The effect of the second axiom could be obtained by considering in place of P the collection P ‘ of 1-1 partial functions with finite domain. 4 ) The reals. Let [p = { (p,q)l p < q an: p,qEQ* = Q u { - m , + m } } ordered by (p,q) < (p’,q’)iff p’ < p and q s q I the poset of proper rational intervals. A model m of tp corresponds to a disjoint pair of sets L , U c Q l which are closed downwards, respectively,upwards,with

110

M.P. FOURMAN, R.J. GRAYSON

r E L iff (r,m) E m and r E U iff (-m,r) Em. These sets are open and inhabited iff (p,q) k,VI(p',q') Ip q E Q implies ~ E Q . Then Q , s is a propositional language. Models of Q,< are those models m of P such that for each P E P

-.

P

F,VITIPEQ1

.

.

We note that we need really only consider sequents a singleton since, for example, p,q km r iff, if p h q exists, then p h q We now axiomatise the logic of such sequents.

4J-r where $ i s

Fm r.

1.3 Definition. An entailment relation 11-is a collection of sequents closed under the following rules (writing p 11- r for "p * r is in II- " ) :

P 11-9

Pltr

if P _ < 4 if r c A

P Il-A

P ii-r P hq 1l-r hq p 11- r {q 11- A 1q

( localisation)

E

r}

(composition)

PIC A 1.4 Examples. 1 ) Grothendieck topologies. The axioms we have given are equivalent to those for a Grothendieck topology on P, so we will speak interchangeably of entailments and topologies: in topological language p1l-T is read as " r covers p". 2 ) Provability. A theory is any set of sequents. Any theory T generates a least entailment t~ containing T, obtained by closing T off under the rules. We define pkTr iff every entailment containing T contains p* r If p t T r w e may also say that p* r is provable in T. Every entailment I- arises in this way simply by setting T = I - , but we are interested in presentations or axiomatisations of entailments by "simple" theories (e.g. countable ones, for the completeness Theorem 3 . 1 1 ) . 3) Semantic entailment. A model m (of P) is a model of a theory T , written F T, iff each sequent in T i s valid in m. Semantic entailment k T it given by

.

p kT!? iff p km rfor every model m of T. This is an entailment containing T, so kTE; I = T ; the converse inclusion is discussed in 3 . 1

.

We now start on the path which leads to formal spaces. LetII- be some fixed entailment on P. 1.5 Definition. A subset K of P is closed (with respect toll-) iff

pII-K implies p EK.

Formal spaces 1.6

111

Lemma. For any rclP r = {PIpII-r} is the least closed set containing r -

r c r = F r A A = 7 n Z If p < q E

7

where r then p E

IPA~IPE~,~EA}. (We say is a crible.)

A A =

7.

Proof. The only nontriviality is for each q E T , p Aq1l-A A q s r A A, for entailments.

T h a z ? n a . If pll-r and pll-A then, SO

that pl1-r

A

A, applying the rules

1.7 Definition. A frame is a complete lattice (H,<,h,v,A,V) such that

P A V q i = V ( P A q i )* An H-valued model m of lp is an assignment of truth-values [[p]lmEH to the formulae subject to the conditions (compare 1 . 1 )

“PI] “TI1, “PI] sequent $ + r A [[p]lm< q Er PE$ each sequent of holds for every

A

v

5

“~11

if P 5 9

= T A “Sll
T is valid. We say that p forces r frame-valued model of T.

, p k T r r iff plCm.r

(Note: Frames are also called complete Heyting algebras, under which name a good deal of the (constructive) theory of frames and framevalued models is developed in Fourman and Scott ( 1 9 7 7 ) . We discuss applications of this theory in 3 . 8 - 9 . ) Now we want to relate forcing , I k T , to provability,l- T; one direction is immediate.

.

1.8 Lemma. (Soundness) If p I-TT then p IkTI‘. Proof. For any H-valued model m of T the relation pll- r is easily seen to be an entailment relation containing T; but IFTis the least such. For the converse we want to provide a universal (“generic”)framevalued model for T. 1.9 Lemma. The closed subsets of lp with respect to any entailment relation form a frame; in the case of kTthis is denoted O(T),called the formal opens of T.

Proof. Since arbitrary intersections of closed subsets are closed, these form a complete lattice, with KAM = K ~ M and VKi = -~ M hVKi = M n U Ki = M n U Ki = U (M nKi) = v(M AKi). Then

q.

1.10 Corollary. (Completeness) p h , r iff p1I-s. Proof. We write [p] for in O ( T ) , and define an O(T)-valued model m of T b y [[p]]m=[p]. Then pll- r iff [p]cr iff pkTr. This model is called the generic model over ?he formal opens of T . Before proving the universality of the model in 1.10, we need to discuss morphisms of frames, and locales.

F}

M.P. FOURMAN, R.J. GRAYSON

112

5

2

LOCALES AND FORMAL SPACES

We now introduce locales as ~eneralisedspaces, which are given by their frame of open subsets. Of special interest are presentations of a locale as the formal space of a theory, and the relationship between this and the actual space of models of the theory. The following example provides motivation for the definitions. 2.1 Exam le. If X is a topological-space, O(X) is a frame. If f :X--> Y ? s a continuous map then f : O(Y)-> O(X) preserves finite infs,T,A, and arbitrary sups,v. If Y is a sober space then f is completely determined by f-l. Definitions. A morphism of frames is a map f*:H-->K preserving T , h , V . The category of locales is the dual of the category of frames. That is, a locale X is given by giving a frame O(X) called the opens of X. A morphism of locales, called a continuous morphism f : X - > Y , is given by a morphism of frames, called its inverse image map, f*:O(Y)->O(X) For any topological space A we identify A with the locale whose opens are O(A) (Formally this gives a functor Top +Lot.) In particular each discrete space A gives a locale whose opens are arbitrary subsets of A. A point of a locale X is defined to be a continuous morphism t:*-> x that is a map t* : O ( x ) -->P ( 1 ) preserving T , A ,V. 2.2

.

.

2 . 3 Definition. For a theory T, the formal space M(T) ofmodels of T

is defined to be the locale whose opens are O(T) , the I-T-closed subsets of IP (Lemma 1.9); these are also called the formal opens of T.

2.4 Theorem. Points of M(T) are in 1-1 correspondence with models of T. If X is a locale, O(X)-valued models of T are in 1-1 correspon-

.

dence with continuous morphisms X-> M(T) Proof. Since models of T are just P(1)-valued models, the second assertion includes the first. Clearly every frage morphism f*:O(T)-> O(X) induces a model m by [[p$ = f ([PI). Conversely each model m induces a morphism f* by f (K) = V [[PI] m. PEK Viewed algebraically, theorem 2.4 says simply that we have a universal interpretation P- [

I

O(T)

satisfying T. We want to consider it geometrically: our identification of the points of M(T) as models explains the terminology "formal space"; we view an O(X)-valued model as a family of models continuously parametrised by X. 2.5 Definition. For any locale X the points of X form a topological space Pt(X) whose opens are given by u* = {tft*(u) = T 1 , for UEO(X).

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2.6 Examples. We refer back to the examples introduced in 1 . 2 . 1) Syntactic languages. Models of a language P are just subsets of

P; these are given the "finite information topology" with basic opens Ve = {ml e c m} , where e is a finite subset of P. By 2 . 4 the points of M(T) are just the models of T, so 2.5 topologises the space of models, denoted by Mod(T) ; the frame O(Mod(T))of open subsets corresponds to the semantic entailment relation, k T ( 1 . 4 (3))

-

2 ) Trees. The space of infinite paths, Path(T), in a tree T i s given

the topology with basis va = {a[a an initial segment of a}, for a E T . The corresponding formal space is denoted by P(T), with topology generated by "immediate successors cover"; we call this the inductive topology onlf. (See 3 . 5 for notation for special cases here.) 3 ) Function spaces. Here the topology on points is the usual product topology. 4) Reals. The points here are real numbers (presented as "Dedekind cuts" (see 3 . 5 - 7 1 , and their topology is the usual one with basis the rational open intervals. Thus our presentation via propositional theories gives various collections of points a natural topological structure. We shall see that, from a constructive point of view, the formal space is often better behaved than the space of points, and that even classically the former may contain more information. 2 . 7 Example. A pointless locale. For any sets A and B let the locale Surj(A,B) be the formal space of the theory of a total function from A onto B (Example 1 . 2 ( 3 ) ) . If A = N and B is uncountable, clearly Surj(A,B) can have no points. On the other hand Surj(A,B) is non-trivial in the sense that T # I in O(Surj(A,B)) (indeed, constructively, it may be shown to be both proper and open, in the sense of Lemma 2.12):for each partial function p: A 7 B, the set [PI = {qlqzp} is closed (for the entailment generated by the axioms)

In this connection we refer forward to 3 . 9 where it is shown that theories which classically have no models may have some constructively. The general relationship between a locale and its points is treated further in 83. Although locales may lack points, many topological notions may be generalised to them more or less straight-forwardly: 2.8 Definition. A locale X is compact iff every cover of T in O(X) has a finite subcover. An open U EO(X) is positive, Pos(U), iff every cover of U is inhabited (this is a constructively positive way of saying U # I). X is connected iff U v V = T and-Pos(U) and Pos (V) together imply Pos (U A V)

.

In what follows we shall take care to use constructively apt versions of such definitions. Morphisms. A continuous*morphism f:X+Y is (X+>Y) surjective (onto) iff f* is 1-1 injective iff f, is onto (XQY) open iff f- preserves A (arbitrary infs). For an open surjection we write f: X 4 Y . 2.9

I14

M.P. FOURMAN, R.J. GRAYSON

For any f:X-Y the map f* has a ri ht ad'oint f*, characterised by f*U -< v iff U 5 f*v, and given by * f V}. Similarly f is open iff f9 has a left ad'oint f, , with f!V
j:O(Y)-->O(Y) given by j U = f*f U (see Fourman and Scott 1 9 7 9 ) ) . Among these generalised subspaces are the open sublocales given by j U = (v+U) , and the closed ones, given by j U = ( VV U) for V E O(X); topologically these correspond to the subspace V and its complement. Note that a sublocale of a topological space need not be topological (i.e. have points) as our pointless example ( 2 . 7 ) shows. From the logical perspective, sublocales arise by considering the subspace of a formal space obtained by imposing additional axioms. For example, the formal closed unit interval [ O , l l results from the formal space of the reals ( 1 . 2 ( 4 ) ) by adding the axioms ( - m , O ) * 1 and ( I , m ) * l

.

2 . 1 1 Products. The product XxY of two locales is obtained by amalgamating the two theories: we take P = O(X) x O(Y) with the product ordering and axioms (U,V) * V (UIVi)for UEO(X) and V < VVi in O(Y) and (U,V) * V (Ui,V) for V E O(Y) and U sVUi in O(X) We leave the reader to check that this defines a product in the category of locales and continuous morphisms. If S,T are theories in languages P,Q the product M ( S ) x M ( T ) can also be more directly axiomatised, by

.

and so on. For any locale X there is just one continuous morphism from X to the one-point space, whose inverse map " : P ( l ) - + O ( X ) is given by PI-+V{TI*EP} = P We characterise (constructive) non-triviality of X in terms of this map.

*,

.

2 . 1 2 Lemma. The morphism X->*

is surjective iff Pos(T) open iff IUlPos(U) 1 is a basis for O ( X ) . If these conditions hold, X is called proper and open,respectively. Proof. By definition, Pos(T) gives (6 = T + * E p) So t j i s gives ( * Eq* Ep), whence is 1-1. Conversely, if T sVW, TcVCTI3u Em] = {*13UEU]"; s o , if is 1 - 1 , { * 1 3 U € X } = { * I , that is, 3 U E U .

"

.

"

Similarly, if V s V U , V g {*13U €%I"; but I*IPos(V)] = ( * I t / covers U of V. 3U E M ) is given by an A in P ( 1 ) , so that, if the morphism is open, V g C*IPos(V%" Then also VrV{VlPos(V)} = V {U i VlPos(U)}, that is, the positive elements form a basis.

.

Conversely we show that, if the positive elements form a basis,then u i $ iff POS u implies * E p, from which openness of is immediate. (Note also that then the left U}.) adjoint to is the map U-{*[Pos The hypothesis gives U 5 V I U l P o s (U) 1 , so, if Pos (U) implies * E p,

Formal spaces

115

U
For the case of the formal space M(T) of a theory T we write Con($) for Pos([@l), which is read as "6 is consistent", since it is a positive version of $kT1. Then M(T) is proper iff Con(T), that is, the theory is consistent. It is easy to see that, if 6 has a model (i.e. bm@) , 0 is consistent; the converse implication is discussed in 3 . 1 2 .

8 3

COVERINGS AND COMPLETENESS

We now start investigating the relationship beiween a locale X and its space of points Pt(X) ( 2 . 5 ) . The map U W U = {tl* E t*(U) 1 clearly gives an injection of locales Pt(X)->X. Definition. A locale X has enough points iff this morphism is surjective (and then, in fact, an isomorphism).

3.1

This condition is clearly equivalent to U* = V* implies u = V, which says that X has "enough points to distinguish opens". For the case of the formal space M(T) of a theory T, having enough points is precisely the equivalence between semantic entailment, b , and provability, I-T, that is, the completeness of T with respecx to its (P( 1 ) -valued) models. Relations between completeness of different theories will mostly be established by giving a "covering" (surjection) of one formal space by another, using the following lemma. 3.2 Lemma. If X covers Y (i.e. there is a continuous surjection from X to Y) and X has enough points, then so does Y.

Proof. Let f*: O(Y)+>O(X) and U * = V*, for q,VEO(Y). Then f*(u)* = f*(V)* (by composinq with f*), so f (U) = f*(V), so U by the assumptions,

=

V.

In general, completeness of a theory T can be read as an induction principle: any crible K such that p kTK (K covers the points of p), which is closedunderall restrictions of axioms of T, must contain p (see Postscript). We consider some special cases. Definition. L e t T be any tree. The principle of monotone bar induction for F , BI ('A) , is the implication

3.3

(1)

A

(2)

A

(3)->(4)

where ( 1 ) VaE K v b s a. bE K ( 2 ) Va(Vb immediate successor

( 3 ) Va E Path(G) (cEa+

3

(K is a crible) of a.bE K-aE K) (K is closed)

a € Kn a )

(K covers paths through c)

116

M.P. FOURMAN, R.J. GRAYSON

(4) c E K for c any node of g a n d K any subset o f T .

3.4 Theorem. The locale P(T) (2.6(2)) has enough points iff BI(m holds. Proof. P(T) has enough points iff any cover of all paths through c covers c in the inductive topology, iff BI(T) (see Postscript). If we now consider the special case of the tree A < N of finite sequences from some set A, the principle BI(AP (i.e. n g k implies F(k) P generates a (sequential) model m = {p[3n.F(n) sp} of T iff, for every axiom p *ref T , 3n.F(n) i p implies 3n 3 q E r . F(n) 5 q . 3.7 Examples. 1) Trees. Paths in a tree " A are, almost by definition, always sequential models. 2) Reals. A sequential model of the theory of reals is just a Cauchy real, that is, one for which there is a sequence of rationals converging to it. In some sheaf models (e.g. over the reals themselves), where AC-NN fails, there are many non-Cauchy reals (see Fourman and Hyland (1977), for example). On the other hand, the Completeness Theorem below shows that (under assumptions of bar induction) every countable theory has "enough sequential models". (Classically, of course, every model of a countable theory is sequential.) 3.8 Interpretations in Sheaves. In order to interpret all this in sheaves over a locale X (that is. in O(X)-valued models), we need first to generalise Definition I 17 and 'define a artial.0 ( X )-valued model of a theory T to be an assignement ~pll, :a in 1.7, but withinstead measures the out the condition "TI1 =T; the value "TI], "existence" of the m o d a , and may be written also as Em 11. In other words (as in 2.4) we are considering functions f*:O(T)->O(X) which preserve A and v but not necessarily T ; that is, in localic terms, partial continuous from X to M ( T ) with open domain. If X has

Formal spaces

117

enough points, these can be treated as partial functions from Pt(X) to Mod(T)

.

Now we want to interpret O(X)-valued models as the internal models of some internal theory, in sheaves over X,_namely the "constant" theory ?. For any set A the constant sheaf A is simply the set A with [[ a=b ]]=V{~la=b} lsee Fourman and Scott ( 1 9 7 7 ) ) ; for any propositional language P,s-,B also inherits its structure trivially, with [[ p - < q]]=V{Tlp " p ]Im, and the remaining conditions for a partial O(X)-valued model are easily checked.

Of course, in general, the theory $ may not at all be the same as that obtained by interpreting some definition of T in sheaves over X. When this is (always) s o , we call T (or rather, the definition) absolute. Sufficient for absoluteness is that P should be countable and that T should be arithmetically defined (in terms of some enumeration). A s examples we have the theories of Baire space, Cantor space and the reals; as a corollary of the above discussion we have the well-known representation of these spaces in topological models as sheaves of partial functions with open domain. Finally we can interpret the internal formal space M($)as the sheaf M(T)X, which is defined as the product locale ( 2 . 1 1 ) (XxM(T)) with the evaluation "W5W'Il =VIUlWuS.w y , where Wu = {(U'h U,V')I (U',V') EW]. 3.9 Applications of Sheaves. Such interpretations have an immediate application in the present context: Fourman and Hyland ( 1 9 7 7 ) have described sheaf models in which FT holds but not BI, and in which FT fails (one may also use recursive realisability), showing the constructive difference in strength of these induction principles. The comparative strength of the induction principles corresponding to other spaces (especially of the reals,R) is discussed in 4.6-11.

Another application, which we mention only briefly here, is to the construction of models of theories which classically have no models. The simplest example is the generic model (Lemma 1.10) over the formal space of the theory of a artial function from N NN Example 1 . 2 ( 3 ) ) ; this theory is not aEsolute, but it turns out that Nd is the constant sheaf in this interpretation, so that one gets a model of the right internal theory, and one has NN subcountable in this model. A more complex example is Joyal's theory of an ideal of rational intervals covering the reals, of measure less than 1 / 2 . Both these formal spaces are proper and open, in the sense of Lemma 2.12.

e

We now describe the construction of a "universal proof tree" for any countably axiomatised theory, in which infinite paths are models and inductive bars are proofs. In localic terms we are covering the formal space of the theory by the tree with its inductive topology. Then when the tree has enough points so does the formal space, that is, the theory is complete. 3.10 Definition. Let T be a theory, in a language with decidable<, with countably many sequents C Bi * TiliEN); for convenience we

MI'. FOURMAN, R.J. GRAYSON

118

assume that each sequent occurs infinitely often in this enumeration. We define a tree of finite sequences of formulae, the admissible sequences, inductively as follows: < > E T <@> E g

for every formula I$. If aEl' is of length i+l and B i * ri is the i-th sequent in T then E T if a(i) d B i a * B i and 0 E Ti and a * E a if a(i)

.

We view the admissible sequences as finite attempts at constructing a model of T. Note that if i < j then a(i) ta(j) for any admissible a. A sequence a of length i+l is said to secure @ iff a(i)i $, and to secure r iff it secures some @ in r . Clearly a(i) is the least formula secured by a; we denote it also by 0,. 3.11 Theorem. Every infinite path a through T g i v e s rise to a model m of T with @ E m iff 9 is secured by a(i) for some i EN.

A sequent @ * r is provable in T iff, for any node a securing 0, the collection of nodes below a securing r is an inductive bar of a.

If" is given the inductive topology (generated by covers by immediate successors - see 2 . 6 ) then there is an open surjection of locales from P ( T ) to M(T), taking each infinite path through 1p to the corresponding model of T. Proof. For any infinite path a the collection m of formulae secured by some a(i) clearly forms a filter. If Oi* ri is an axiom and 0 is secured by a(j) we may suppose that jJ:i las each axiom occursi infinitely often), so that T i is secured by a(i+l), by construction. Thus Bi kmri

.

Next suppose that K = Iblb < and b secures TI is an inductive bar of <$>, and consider K * = {blb 5 <$> and I$b I- TI: K * contains K, since, if b secures T, I$b5 )I for some J, E r, so $bl- r . K * is also inductively closed, since the immediate successors of anode b are either b *<@,> or b * < + b ~ $ >for every $ E r i , with 8 . * T i an axiom and $b s ei In the latter case, by assumption, @bA 41- 1' for each $ Eri; which together with $bl-ri gives $bl- r . Thus + > E K as K is an inductive bar; that is, 0l-r.

.

,

The converse is shown by an induction on proofs. If Bi* r l is an axiom and f3i is secured by a of length j we may assume j < 1, and then ri is secured by all extensions of a of length i+2; these form an inductive bar of a. Closure under the rules is clear since the inductive bars form a topology. Now define f:P('p) ->M(T) by f*(U) = {alt$,I Ef*U.

Formal spaces

119

3.12 Some Applications. By the theorem, if P ( T ) has enough points, so does M(T), that is, T is complete. This will always hold classically; constructively we need various forms of bar induction ( 3 . 3 1 ,

depending on the assumptions about T. Let T now be countably axiomatised in a countable language in such a way that an enumeration of Ti, for Bi e Ti the i-th sequent of T, is given uniformly in i (in order to avoid the use of countable choice). (a) Without further conditions g may be treated as an arbitrary aubtree of N

.

-

.

3.13 Remark. Our preoccupation with pen surjections serves to enable us to extend our completeness results to first-order theories. Let T be a countable geometric first-order theory and E its classifying topos (see Makkai + Reyes ( 1 9 7 7 ) ) . The Diaconescu cover of B gives an open surjection from the category of sheaves on a locale (Johnstone ( 1 9 8 0 ) for example). This locale is itself countably presented so we have an open surjection f:Sh(B) ->E. Since f* preserves first-order logic, this tells us that sheaf models over (formal) Baire space are complete for first-order consequences of countable geometric theories.

0

4

COMPACTNESS PROPERTIES

We now show that the formal space of the reals, R, has enough points iff the point space R , is locally compact (4.10) and that formal Cantor space, C, covers every compact regular separable locale ( 4 . 6 ) . This localic version of a classical covering theorem from general topology arises, from our point of view, because the topological

120

MI'. FOURMAN, R.J. GRAYSON

properties in question may be seen as describing a simple generating family of covers. Definitions. If X is a locale, U,VEO(X) then U is well-below V iff there is a W with U A W = I and W v V = T ; we write this U 7V. We say U is way-below V iff every ideal covering V contains U (i.e. if every cover of V finitely covers U), written U<
The locale X is locally com act iff V = V{UlU<
.

Lemma. In any locale X, the set {UlUeV} is an ideal as is {UlU 7V). The set {VlU 7V) is a filter. If U e V and W 7 U then W e U , so if X is compact 7 implies e . If X is regular, Q: implies 7. 4.2

4 . 3 Definition. A locale X has stable finite covers iff [VIU<
is a filter. This holds, for example, in any compact regular locale. The way-below relation is best looked at from the point of view of the Stone space. 4 . 4 Definition. If X is a locale, its Stone space P i s given by equipping O(X) with the finite cover topology: Ull- V U i iff for some

finite F G I we have U <

v Ui

in O(X)

.

iE I

The closed subsets of O(X)

iE F for this entailment are the ideals of O(X). We have a canonical inclusion f:X->X* with f* (K) = V K ; for the meanings of f, and f! see 2.9. 4.5 Lemma. We have U e V iff,for J an ideal in O(X),V sf*J implies f,U< J. The locale X is locally compact iff f is open (Joyal); in this case f l U = {VIVeU}. The locale X is compact, locally compact, with stable'finite covers (Johnstone ( 1 9 8 2 ) calls this compact; stably locally compact) iff f! is an inverse image map (i.e. a morphism of frames).

Proof. We deal only with the last assertion. As f l is a left adjoint it suffices to show it preserves T (compactness) ind A (stable finite covers). In the terminology of Fourman ( 1 9 8 2 ) this says that X is compact, stably locally compact iff it is an adjoint retract of its Stone space. To make our proof of the next theorem constructive requires some assumptions which we discuss after the proof. Theorem. Any compact, stably locally compact separable locale X is a continuous image of formal Cantor space, C. Hence the Fan Theorem implies that X has enough points. 4.6

Proof. A countable basis for X gives a countable basis for X*,so the Stone space, being generated by finite covers, is covered by C (3.12 (a)). As we have (Lemma 4.5) a retraction f a x we obtain a cover

c ->x.

4.7 Remarks. What we assume here is an explicit enumeration of a collection of finite covers generating x*, and that < is decidable for the basis used for this presentation. There are examples in which X is covered by C but is not stably locally compact: The unit interval, [ O , l l , with the usual topology except that 0 and 1 have only

Formal spaces

121

the neighbourhoods [O,l)and (0,11, respectively, is such an example. On the other hand, among regular locales it is precisely the compact separable ones that are continuous images of C. 4.8 Lemma. The formal reals R are locally compact and regular: formal [ O , 1 ]is compact regular. Proof. For rational intervals (p',q') 7 (p,q) iff p < p ' < q ' < q . Thus, by an axiom, (p,q) I-VC(p',q')I (p',qI).7 (p,q)} , so R is regular. Now we show that for bounded rational intervals, if U 7 V then U e V . Let K be a cover of a bounded interval V. Consider K* = {Wleach U < W is finitely covered by K}. Clearly K* is a covering crible of V. Using denseness of the rationaJs, U 7W1v W2 may be expressed as U = U1 where U1 z W l , a n dU2 <W2; so K is closed un>er finite unionvsy2hso, if V < W implies V E K * , then WE K*, as V < W implies 3V'. V Z V ' <W. Thus K* is closed underall restrictionsof the axioms for R, so V E R * , as K* covers V (see Postscript). This holds for any cover of V, s o ? implies << Now as [0,1] is a bounded closed sublocale of R(i.e. [O,l]cU for some U*: T ) , [0,1 1 is compact.

.

4.9 Corollary. The Fan Theorem implies that R has enough points. Proof. As C->>[O,I I, by 4.6 and 4.8, FT implies that [0,1 1 has enough points; R is covered by such closed intervals (all isomorphic to [O,ll). 4.10 Theorem. R and [0,1] have enough points iff IR is locally compact (equivalently Pt [ 0,1 I is compact)

.

Proof. Suppose OR is locally compact as a topological space. Let V*c VU; and let K = { W EO(R) I WsVUi}. As finite covers in R coincide with those in IR, each W Z V belongs to K. S o , as K is closed, V EK. The converse direction follows from 4.8, since R has enough points means that R and R are isomorphic. 4.11 Concluding Remarks. The picture of the comparative strength of completeness for various interesting theories is still far from complete: for example, even the case "R has enough points" implies "C has enough points" seems to be open still (does R have enough points in the model over K(R2) of Fourman and Hyland (1977)? ) In this connection, we conjecture that a localic version of the Hahn-Mazurkiewicz Theorem on continuous images of [O,ll should be provable, using now the "connected" Stone space ,'X in which the open subsets are those cribles J of O(X) closed under connected finite joins (compare 4.4): U E J and V EJ and Pos(U A V ) implies UV V E J.

.

Other interesting cases arise from products and exponents of locales. In relation to products we remark that, provided X has enough points, the locale YX has enough points in sheaves over X (3.8) iff the product (X x Y) has enough points (2.11) : this holds always if Y is locally compact. On the other hand, it is not known whether BI holds in sheaves over formal Baire space, when BI is not assumed externally The exponent Yx of locales may be constructed (with the compact-open topology) when X is.locally compact; this is best described by axiomatising the relationship U <
122

M.P. FOURMAN, R.J. GRAYSON

Hyland (private communication) has described a topological model in which RR has not enough points, thus showing that this principle is stronger than the Fan Theorem. On the other hand, the special case $ X I where $ i s Sierpinski space, always has enough points: these are exactly the opens of X. There are clearly many questions to be answered here and many directions for generalisation. We hope that, by describing some simple examples, we will have made this area more accessible and may encourage others to further research. REFERENCES Fourman,M.P.(1982): Continuous truth, to appear. Fourman,M.P. and Hyland,J.M.E. (1979): Sheaf models for analysis,% Applications of Sheaves, Springer Lecture Notes 753, 280-301. Fourman,M.P. and Scott,D.S. (1979): Sheaves and Logic, Applications of Sheaves, Springer Lecture Notes 753. Grayson,R.J. (1982): Constructive properties of frames, in preparation. Johnstone,P.T. (1980): Open maps of toposes, Manuscripta Mathematica, vol. 31, p. 214-247. Johnstone,P.T. (1982): Stone spaces, London Mathematical Society Lecture Notes, to appear. Joyal, A . and Tierney,M. (1982): Open maps and descent, to appear. Makkai,M. and Reyes,G.E. (1977): First-order categorical Logic, Springer Lecture Notes 611. Troelstra,A.S. (1980): Extended bar induction of type zero, in The Kleene Symposium, eds. Barwise, Keisler and Kunen, Studies in Logic, vol. 101, p.277, NorthHolland, 277-316. POSTSCRIPT. At various points (immediately before 3.3, in Theorem 3.4 and in Lemma 4.8) we have tacitly made use of the following lemma, which is proved by an induction on the definition of provability I-T: Lemma. If K is a crible of (P,() (that is, closed downwards) and is closed under all restrictions of axioms of a theory T (that is, for p = * T in T and any q, if r h q E K I then phq EX), then K is closed with respect to kT. Note that in the case of trees every restriction of an axiom is again an axiom, so that one need not refer explicitly to restrictions.

THE LBJ. BROUWER CENTENARY SYMPOSICM A S . Tmelstra and D. van Dalen (editorsJ 0North-HolIandPublishing Company, 1982

123

CONSTRUCTIVE MOUELS FOR SET THEORY WITH EXTENSIONALITY

'I

L e v Gohdeew, Miinchen

I NTROOUCT I ON There a r e two well-known methods t h a t can be used t o i n t e r p r e t J . M y h i l l s t y l e e x t e n s i o n a l C o n s t r u c t i v e Set T h e o r i e s w i t h i n subsystems o f a n a l y s i s ( [ F r

31 ),

and S. F e f e r m a n - s t y l e c o n s t r u c t i v e systems o f f u n c t i o n s and c l a s s e s ( [ B e Z ] ) . Both i n t e r p r e t a t i o n s use f a m i l i a r ideas o f a g r o w i n g w e l l - f o u n d e d u n i v e r s e o f s e t s w i t h ( s a y ) s t a n d a r d e x t e n s i o n a l e q u a l i t y and membership r e l a t i o n s . f e r e n t i n t e r p r e t a t i o n s were proposed i n [GZ]. Compared t o [Be21

Somewhat d i f -

, [Fr 3 1 ,

( a ) u n i v e r s e s o f s e t s from [G21 a r e i n no way r e q u i r e d t o be w e l l - f o u n d e d b u t ( b ) t h e i r e x t e n s i o n a l e q u a l i t y and membership r e l a t i o n s a r e b o t h d e f i n e d by elementary f o r m u l a s .

I n accordance w i t h N . S h a n i n ' s c o n s t r u c t i v i s m ,

cf.

[ShZ],

i n [G2] s e t t h e o r i e s were i n t e r p r e t e d i n s u i t a b l e 1 - s t o r d e r e x t e n s i o n s o f HA+ECTO+MPO (see [ G l ] ) . The p r e s e n t m o d i f i c a t i o n g e n e r a l i z e s [ G Z ]

i n o r d e r t o demonstrate t h a t

v a r i o u s CST's a r e i n t e r p r e t a b l e a c c o r d i n g t o t h e t r a d i t i o n a l c o n s t r u c t i v e p o i n t o f view, as e x p l a i n e d ( p a r t i c u l a r l y ) i n [ F 2 ] . Our Main Theorem (see P a r t 11.3. below) g i v e s t h e p r e c i s e i n f o r m a t i o n about a r i t h m e t i c s t a t e m e n t s p r o v a b l e i n v a r i o u s CST's extended by ECTO+MOO. The f o r m a l body of t h e p r e s e n t m o d i f i c a t i o n i s based on K i e e n e ' s 1945 n o t i o n o f r e a l i z a b i l i t y , c o n c e p t s from [ S h l ] Shanin's p r i n c i p l e , formulated i n P a r t 11.1.

(see

b e l o w ) , methods f r o m [FZ] and [BFPS],

and assumes some f a m i l i a r i t y w i t h [ T I .

PART I .

BACKGROUND

1.1. Let -

a structure

S

= (M,E)

be g i v e n f o r an a r b i t r a r y s e t M and an

a r b i t r a r y (not necessarily well-founded)

relation

ECMxM. Consider i n f o r m a l l y t h e

f o l l o w i n g problem. L e t us c a l l M t h e u n i v e r s e o f (codes o f ) s e t s . We s h a l l v i e w

I ) I am i n d e b t e d t o t h e o r g a n i z e r s o f t h i s c o n f e r e n c e f o r p r o p o s i n g t h e i n v i t e d paper and f o r t h e i r h e l p i n many ways. I want t o t h a n k P. A c z e l , M. Beeson, W. Buchholz, S. Feferman, G. J l g e r , W. P o h l e r s f o r s t i m u l a t i n g c o n v e r s a t i o n s . S p e c i a l thanks a r e due t o H. Schwichtenberg.

L.GORDEEV

124.

xEy as the following intensional membership relation on M: "x (codes a set which) is an element of (a set coded by) y " . So our sets are described by their codes

(M) and intensional membership relation (E) between them. We have to find some suitable "extensional cover" 5 of S. To be more precise, we are looking for suitable E-extensional equality =s on M, i.e. (E-ext.)(= S )

:

( V X , ~ () X~ = ~ Ycf (VuEx 3vEy(u= S v) and VvEy 3uE~(u=~v))).

Then we take our E-extensional membership relation ES to be E modulo =s. Then the model 5 = (M,=s, f) is the solution to our problem provided that =s is sufficiently nice. Consider the following definitions Definition 1 . For any a,b from M put a = S b :3WCMxM (aWb and EQR(W))

where

EQR(W) : (Vx/xy)~(xWy+ (VuEx 3vEy(uWv) and (VvEy 3uEx(uWv))). A s mentioned above, we put ~ Definition 2. a E S b : ( 3 x , ~ )(xEy

6

a =S x

&

b = S y), which is in fact equival-

equivalent to Definition 2'. Claim 1 .

sb

a € b : 3xEb(a=Sx). S

(axioms of equality together with the axiom of extensionality).

Particularly, =s is an E-extensional equality. (Cf. the extensional equality in [FrZ]. The related notion of equality as an Eisomorphism has been introduced in [Fi]; according to M. v. Rimscha, it was also in [Mi]).

PJLOO~. (0)

5

(1)

5

b (a=a) holds with W = {<x,x> : x belongs to MI.

b (a= b b=a). Suppose s C (a=b), i.e. a =S b , i.e. aWb and EQR(W) for some relation W C M x M . Put W' := {<x,y> : yWx) and observe that bW'a and

EQR(W'),

-f

so

b = S a, and hence 5 b (b=a).

(2) $ C ( a = b & b = c + a=c). Suppose C (a=b) and S C (b=c), i.e. aWlb, bW2c and EQR(Wl),EQR(W2) for some relations W1 , W 2 C M x M. Put W' :={<x,y>: 3z(xWlz&zW2y)} and observe that aW'c and EQR(W'), i.e. S b (a=c).

( 3 ) s k ( a = b & b E c + a E c ) . Indeed, O C * 3xEc. s C (a=x) * ' S + (a=c).

(a=b&bEc)=*3xEc.^S!=(a=bEb=x)

125

Constructive models for set theory

(4)

f C

(a=b & c E b + cEa).

3xEb.

f I=

(a=b & c=x)

s

3xEb 3yEa.

(5)

C (a= b

3 C (Vx(xEa

++

W ' := { < a , b }

U {<x,y>

s

xEb)

-+

C (a=b

& cEb)

*

(by D e f i n i t i o n 1 . )

& c=x & y=x)

* 3yEa. $

a = b ) . Suppose

C (c=y)

=*

(cEa).

S

f != V x ( x E a * x E b ) .

Put

:X = ~ Y } O . f c o u r s e aW'b. We have t o p r o v e EQR(W').

Suppose uEx and xW'y.

I f <x,y>

t h e n uEx -uEa

i s j u s t

=*

f C

(uEa)

*

*?

C ( u E b ) = * 3 ~ E b ( u = ~ v- )3 ~ E y ( u = ~ v ) . O t h e r w i s e xW'y means X = ~ Y then ,

UEX

*f

& x=y) f C ( u E y ) =* 3 v E y ( ~ = ~ v )S.i m i l a r l y , * 3uEx(uW'v). Thus EQR(W') h o l d s and hence f C

C (uEx

(vEy and xW'y)

Remarks 1.

Indeed,

=*

=$

i n b o t h cases (a = b ) .

.

T h i s p r o o f i s i n t u i t i o n i s t i c , b u t n o t e l e m e n t a r y because i n (5) we use C comprehension.

( ( X ;-

CA)

I t c o u l d s t r a i g h t f o r w a r d l y be f o r m a l i z e d i n t h e subsystem

I ' ) - o f i n t u i t i o n i s t i c a n a l y s i s (as u s u a l ,

"P

"

and

' I - ' '

are

r e s p e c t i v e l y t o denote t h e r e s t r i c t e d i n d u c t i o n and t h e i n t u i t i o n i s t i c l o g i c ) . I n the Part I I .

we w i l l show t h a t i t c o u l d be a l s o f o r m a l i z e d i n a s u i t a b l e

constructive version of ( ( A o

(S

f)

2.

The o p e r a t i o n

3.

In order t o i l l u s t r a t e

+-+

- CA) F ) - .

i s i n v a r i a n t w i t h r e s p e c t t o isomorphisms o f S and

(S I-+

5)

s3?Vb am

c o n s i d e r t h e f o l l o w i n g s i m p l e examples

(sometimes we w i l l use t h e a l t e r n a t i v e a b b r e v i a t i o n

s1 s1 ::

:

s2: s2:

b

0-

@

S5:

<

a

0.

'0

f o r xEy).

S 3 V b s? 4:

d

C

C

d\ :@ 8bs,

e

d

e

d

b One can e a s i l y v e r i f y t h a t : (x=y

++

si

= ( x = y ) ) and ( x E i y

s e t s coded by a,b,v,d,e

0 , l ,2,3,4

s6 f7

s8

* si C

(xEy))

f o r each i < 5 ;

i n t h e s e S i ' s become v. Neumann's n a t u r a l numbers

i n the s i n s ;

C ( b = c ) and a g a i n b o t h b and c a r e v . Neumann's 2 ; C (b#f#afb

observe t h a t

= c = d = e) ;

C ( a E c E b E a and a f b + c # a ) .

L. GORDEEV

126 I. 1 . 1 .

C l a i m 2.

Two nice pfiopemZu

06

(S ++ 3 ) .

Suppose S i s w e l l - f o u n d e d ( w i t h r e s p e c t t o

E). Then t r i v i a l ( r e -

s t r i c t e d ) E - i n d u c t i o n y i e l d s t h e uniqueness o f E - e x t e n s i o n a l e q u a l i t y on M.

s

i n t h i s case o u r d e f i n i t i o n o f

Hence

i s adequate.

C l a s s i c a l l y , g i v e n any w e l l - f o u n d e d s t r u c t u r e S one can v i e w t h e c o n d i t i o n (E-at)

t o be an i n d u c t i v e d e f i n i t i o n of ( s a y ) s t a n d a r d E - e x t e n s i o n a l e q u a l i t y on

M. Hence f r o m t h e c l a s s i c a l p o i n t o f v i e w o u r d e f i n i t i o n o f s t a n d a r d one

coincides w i t h the

- f o r any w e l l - f o u n d e d s t r u c t u r e S .

O f course, we cannot e x p e c t t h e uniqueness p r o p e r t y t o be p r e s e r v e d i n t h e g e n e r a l case. For, c o n s i d e r some two-element

'0,

s t r u c t u r e S which looks l i k e

and o b s e r v e t h a t 5 C ( a = b and a E b ) , i . e . o u r 5 l o o k s l i k e t h e a b u t o f c o u r s e S i t s e l f i s e x t e n s i o n a l i n f a c t o u r problem t r i v i a l cycle

6-

(formulated i n 1.1. t h a t o u r ( S ++

s)

above) has many d i f f e r e n t s o l u t i o n s .

However,

i t can be shown

i s t h e u n i v e r s a l one, w h i c h means t h a t t h e f o l l o w i n g i s p r o v a b l e

i n a s u f f i c i e n t l y r i c h i n t u i t i o n i s t i c metathory.

C l a i m 3.

( f o l l o w s f r o m Lemma 1 below).

e x t e n s i o n a l e q u a l i t y =;

Consider any S and any a l t e r n a t i v e E-

on M. D e f i n e €& by D e f i n i t i o n Z / D e f i n i t i o n 2 ' and s e t

.-

._(M, = s , , E' 1. Then t h e r e i s some s t r u c t u r e S , S p h i c a l l y embedded i n t o 5 , .

Sl

such t h a t S ' can be isomor-

Now c a l l a m e t a t h e o r y s u f f i c i e n t l y r i c h i f i t p r o v e s t h a t f o r e v e r y s e t M t h e r e i s some s t r u c t u r e S o = (M ,E ) and a f u n c t i o n f : M I Mo such t h a t O D into (Vx,ylM ( x = Y ++ C ( f ( x ) = f ( y ) ) ) . Note t h a t a m e t a t h e o r y w h i c h p r o v e s e v e r y

so

s e t M t o be w e l l - o r d e r a b l e ,

i s s u f f i c i e n t l y r i c h ( l o o k a t t h e f i r s t f i v e examples

f r o m Remark 3 above and o b s e r v e t h a t t h e same c o n s t r u c t i o n c o u l d be a p p l i e d t o an arbitrary ordinal). Particularly,

v a r i o u s subsystems o f a n a l y s i s a r e s u f f i c i e n t l y

r i c h since they deal w i t h sets o f n a t u r a l s ;

i t does n o t m a t t e r whether t h e y a r e

i n t u i t i o n i s t i c or classical.

Lemma 1. Suppose o u r m e t a t h e o r y i s s u f f i c i e n t l y r i c h .

S = (M,E) and any e q u i v a l e n c e r e l a t i o n

Consider any s t r u c t u r e

- ( w i t h r e s p e c t t o E) on M.

Then t h e r e i s

some s t r u c t u r e S 1 = (M ,E ) t o g e t h e r w i t h a 1-1 f u n c t i o n g : M + M1 such t h a t 1 1 C ( g ( x ) = g ( y ) ) ) and ( x E y t+ C ( g ( x ) E g ( y ) ) ) , f o r a l l x,y f r o m M. (x-y ++

z1

s1

Ph006. ( a s i m p l e e x e r c i s e i n K u r a t o w s k i ' s o r d e r e d p a i r t e c h n i q u e ) . F i x some So = (MO,EO) Step 1 .

and some f : M

*

M

0

with (VX,~) ( x = y M

D e f i n e S2 = (M ,E ) as f o l l o w s . 2 2

c+

5, C

P u t M2 : = & q ( { i } x M )

( f ( x ) = f(y))). U { 5 } x M and l e t

127

Constructive models for set theory E2 be a r e l a t i o n w h i c h o r d e r s M2 as f o l l o w s ,


,x>

A?> I

x E M 9 and f o r a l l uEx.

<1 ,x>

for

P .

, <5, t>> : s E o t }

...... Note t h a t a s e t coded by < 1 ,x> p a i r w i t h f ( x )

5,

in

i s nothing but Kuratowski's ordered

so.

regarded as an element o f

We show t h a t f o r a l l x , y E M: (i) ( x = y

cf

5,

+

(

Both imp1 i c a t i o n s ( i ) Suppose

s2

C (

= ))

and ( i i ) ( x E y

S2

( = ).

= < l , y > o r 4 ,x> = )

i s t i c ) proof of

=

(

= <5,f(y)>),

(<5,f(x)>

+

(<3,v>~<2,v>))

( i i ) Suppose

5,

5,

(E).

5, C

(E)

.

(E)

i .e.

(<1 ,x>E),

o t h e r hand,

s2

(<1 ,x> = <1 ,y>)

= <5,f(y)>)

5, C

* 0=1

<3,x>)

2

Hence

S2

*

NOW

( s i n c e <3,x>

i s empty, a l t h o u g h

c ( = <4,x>) * 0 . = 1 * x = y . 5, C ( = < l , y > o r = )

+

(

= )

*x=v.

Hence i n any

*xEy.

Step 2. P u t S1 := ( M ,E ) , where M1 := M2 and E := E U {<, > 1 1 2 1 (x(x-y)). Put g := Ax.. I t f o l l o w s from t h e above r e a s o n i n g that.,

51 C 51 b

(

= )

(E)

1.2. Consider -

c-f

++

3y'(y'-y 3v(v-x

E. x = y ' )

& vEy)

.tf

s m e universe U o f structures

x-y,

++

t h a t a belongs t o M and (M,E)

belongs t o U.

:

and

xEy.

S

= (M,E).

Define the corre-

sponding u n i v e r s e S e t ( U ) o f s e t s as t h e c o l l e c t i o n o f a l l t r i p l e s (a,M,E)

Set(U)

*

*

(<1 ,x> = )

= o r = <4,x>).

f o r some vEy. We have a l r e a d y seen t h a t case

+

C (<5,f(x)>

~ i m i l a r l y ,5

*x=y.

5, C

f o r some vEy. Now, K u r a t o w s k i ' s ( i n t u i t i o n -

a = b & c = d ) yields

while

* 5, * so C ( f ( x ) = f ( y ) ) * x = y . On t h e * S, c ( E < i ,x>) * S, C ( f 2 C ( = <2,x>) * s2 C (<2,v> =

5, c

S2

Hence

t+

"+".

a r e o b v i o u s . Consider

T h i s means t h a t each (a,M,E)

i s regarded as a s e t coded by a i n s i d e (M,E).

such from

Extend o u r t r e a t m e n t o f =

S

and ES i n o r d e r ' t o o b t a i n e x t e n s i o n a l e q u a l i t y and membership r e l a t i o n s between t r i p l e s from Set(U).

L. GORDEEV

128

Definition and (a,M,E)

S1 = (M1,El) E

0

3. W r i t e (a,M,E) = ( a ,M ,E ) f o r ( S F S ) != 1

E (al,Ml,E,)

1 1 1 + E ,

and ( S T S 1 ) = (M ,E ) w i t h Mo =

= {<,>

Claim

f o r (S1 ; S 2 )

u

: xEy}

0

0

{<<1 ,x>,
,y>>

{O}XM

= 2 1 where S = (M,E),

U {l}xM,

and

: xE1y}.

4. Set(U) C (axioms o f e q u a l i t y t o g e t h e r w i t h t h e axiom o f e x t e n -

sionality). The p r o o f i s q u i t e s i m i l a r t o t h a t o f C l a i m 1 above.

1

-

ma1 i z e d i n ((8, CA)

1 I-,see

I t c o u l d a l s o be f o r -

Remark 1 above. We w i l 1 show i n P a r t I I t h a t

((Z; - C A ) P ) - c o u l d be r e p l a c e d by t h e s u i t a b l e c o n s t r u c t i v e m o d i f i c a t i o n o f (

- CA) r

)-.

We s h a l l use t h e o p e r a t i o n ( U

I+

Set(U))

i n o r d e r t o produce v a r i o u s e x -

t e n s i o n a l models o f v a r i o u s ( i n t u i t i o n i s t i c o r c l a s s i c a l ) t h e o r i e s o f s e t s . There a r e a t l e a s t two r e a s o n a b l e d i r e c t i o n s .

T h a t i s , g i v e n some 2-nd o r d e r m e t a t h e o r y

(T) c o n s i d e r : ( a ) a u n i v e r s e U o f a l l s t r u c t u r e s d e s c r i b e d by T, ( b ) i t s w e l l WF founded p a r t UwFCU, i n o r d e r t o g e t e i t h e r a w e l l - f o u n d e d model S e t ( U ) o r a "complete"

model Set(U).

s t r u c t u r e (x,-,r)

o f Set(U) means t h a t S e t ( U ) C ( f o r e v e r y

"Completeness"

w i t h a r b i t r a r y r C x x x and a r b i t r a r y e q u i v a l e n c e r e l a t i o n - o n

x , w h i c h p r e s e r v e s r , t h e r e i s some y such t h a t (x,-,r) ( y , = Fy,EFy)).

O f c o u r s e "completeness"

is i s o m o r p h i c t o

and w e l l - f o u n d n e s s a r e i n c o m p a t i b l e .

These models w i l l be used i n o r d e r t o show t h a t v a r i o u s e x t e n s i o n a l t h e o r i e s o f s e t s a r e c o n s e r v a t i v e o v e r t h e i r s u i t a b l e i n t e n s i o n a l f i r s t o r second-order v e r sions, f o r a l l a r i t h m e t i c statements. We s t a r t w i t h Z e r m e l o - F r a e n k e l ' s

s e t theory.

L e t ZF be t h e c l a s s i c a l f i r s t - o r d e r p r e d i c a t e c a l c u l u s w i t h e q u a l i t y extended by t h e f o l l o w i n g sequence 1 )

- 8)

o f axioms f o r m u l a t e d i n t h e o r d i n a r y = ,€-language

o f s e t t h e o r y e n r i c h e d by t h e c o n s t a n t s :

0 (empty s e t ) and w (a s e t o f a l l n a t u r a l

numbers). (1)

Extensionality

(2)

VX(XF0)

(3)

Pairing

(4)

Union

(5) (6)

CA:Vx3yVu(uEy

+-f

u E x & cp(u)) f o r a l l cp

SC ( s t r o n g c o l l e c t i o n ) : Va(VxEa3ycp

-+

3 b ( V x E a 3 y E b c p A V y E b 3xEacp))

f o r a l l cp

(7)

Power: v x 3 y v u ( u E y

(8)

I n f : ( 0 E w F. V x E w ( x + 1 E w ) ) and V y ( 0 E y & V x E y ( x + 1 E y ) -+ w c y ) ,

++

ucx)

where x + 1 a b b r e v i a t e s X U{XI.

129

Constructive models for set theory

I n a d d i t i o n c o n s i d e r t h e f o l l o w i n g two axioms: I n d ( E ) : ( V x ( V u E x cp(u)

-t cp(x)) -t Vx cp(x)) f o r all cp, 1-1 3z,g(g : xz and V u , v E x ( < u , v > E y * g ( u ) E g ( v ) ) ) ) . onto Note t h a t Cpl proves o u r "completeness" whenever t h e axiom o f f a c t o r i z a t i o n i s

Cpl : V x , y ( y C x x x

provable

T

+

- o f c o u r s e t h i s i s t h e case f o r ZF- ( r e c a l l t h a t T - denotes t h e t h e o r y

w i t h i n t u i t i o n i s t i c l o g i c ) . Obviously, Z F + Ind(E) i s equivalent t o the c l a s s i c a l s e t t h e o r y w i t h t h e axiom o f f o u n d a t i o n , w h i l e ( Z F + I n d ( E ) ) -

Zermelo-Fraenkel

is

equivalent t o i t s i n t u i t i o n i s t i c version presented i n [ F r 2 ] . w i l l be t h e c l a s s i c a l second-order p r e d i c a t e c a l c u l u s w i t h e q u a l i t y ZF2 extended by t h e f o l l o w i n g sequence 1 ) 2 8 ) 2 o f axioms f o r m u l a t e d i n t h e second-

-

o r d e r language whose a t o m i c f o r m u l a s a r e (xEX),

(x=y),

(oc,y>=z),

( x + l =y),

which i n a d d i t i o n c o n t a i n s f i r s t - o r d e r c o n s t a n t s 0.1 and one second-order

constant

n a t u r a l numbers). P a i r i n g : V X , ~3 ! z ( u ~ , y > = z )

A

( u ~ , y > f O ) h V u , v ( o ~ , y > =

x = u E y=v)), P r o d u c t : VX,Y3ZVx(zEZ

t+

3xEX 3yEY(OC,y> = z ) ) ,

Oom, Rg : VX3YVy(yEY ++ 3X(EX))

A

VXilYVy(yEY

t-t

3X(a,y>EX)),

Union: V X , Y ~ Z V Z ( Z E c-t ~ (ZEX v Z E Y ) ) , CA: VX~YVY(YEY

++

YEX 8 cp(y)) f o r all cp,

C ( c o l l e c t i o n ) : VA(VXEA~Ycp(x,Y)

+

~ Z V X E AW ( X , ( Z ) ~ ) ) f o r a l l cp, where

( Z l x a b b r e v i a t e s { y : <x,y>E 21,

1, where ( Z C X ) a b b r e v i a t e s Y zEX) and ( Z E X ) a b b r e v i a t e s ( Z C X and X C Z ) .

Power: VX3YVZCX 3yeDom(Y) ( Z E (Y) Vz(zEZ

-t

( 0 , l ~ N ) and ( 0 + 1 = 1 ) and VXEN 3 ! y ~ N ( x + l = y ) and Vx N ( x + 1 # 0 )

Inf:

and V X ( O E X

& VXEN(XEX + X + lEX)

+

NCX).

The f o l l o w i n g theorem e x t e n d s theorem 1 f r o m [ F r 2

1,

Theorem 1 .

(a) ( Z F + I n d ( E ) / C p l ) (b)

ZF+ I n d ( E ) / C p l

i s c o n s e r v a t i v e o v e r ZF- f o r a r i t h m e t i c a l s t a t e m e n t s . 2 i s c o n s e r v a t i v e o v e r ZF2 f o r a r i t h m e t i c a l statemen'ts.

Paood. Step 1.

O b v i o u s l y , ZF;

We show t h a t ZF; Step 2. WF Set(U ) I= ( Z F + I n d ( E ) ) - , t o g e t h e r w i t h Se.t(UWF)

i s c o n t a i n e d i n ZF- and ZF2 i s c o n t a i n e d i n ZF.

proves S e t ( U )

!= (ZF+ C p l ) - t o g e t h e r w i t h

and t h e n c o n c l u d e t h a t ZF2 p r o v e s Set(U) (ZF+ I n d ( E ) ) .

++

For s i m p l i c i t y we s h a l l d e n o t e by x,y,..

...

.

.

I= ( Z F + C p l )

r e s p e c t i v e l y t r i p l e s ( x 0 ' X 1 ,X 2 ) ,

-+

( y ,Y , Y ) , Thus o u r Set(U) w i l l c o n s i s t o f a l l x w i t h (xOEX1 and X2cX1xX1), O 1 2 W F w h i l e Set(U ) w i l l c o n s i s t o f a l l f r o m S e t ( U ) such t h a t a l s o

L. GORDEEV

130

WF(Z) : VZ(VuEX1 (VvEX1 (EX2+

vEZ)

+

V E Z ) + X 1 C Z ) holds.

O ) , 1 ) Axioms o f t h e i n t u i t i o n i s t i c l o g i c w i t h e q u a l i t y t o g e t h e r w i t h t h e WF axiom o f e x t e n s i o n a l i t y a r e a l l t r u e i n b o t h S e t ( U ) and Set(U ) s i n c e Claim 4 above i s p r o v a b l e i n t h e second-order s e t t h e o r y ( ( 2 - CA) i n ZF;

0

For t h e empty s e t

2) y o :=

which i s contained

we t a k e (O,Xo,Yo)

w i t h Xo := {xEN : x = O l and

0=11.

{XEN:

3)

++

Pairing.

z1

Put z o := 0 ,

z2

P ) -,

( r e c a l l t h a t o n l y e x t e n s i o n a l i t y needs a s t r o n g v e r s i o n o f CA).

Given any x , y f r o m S e t ( U ) / S e t ( U

:= I 0 1

u

:= {<,O>}

u

101xxl

u

{<<1 ,y0>,O>}

WF

+ ++ ) d e f i n e z = {x,y}

as f o l l o w s

{ l } X Y 1 ,

u

{<,>

: EX2}

u I<<1

, u > , < l ,v>> :

+

t

< u , v > ~ Y 1 . Observe t h a t i s w e l l - f o u n d e d p r o v i d e d t h i s i s t h e case f o r b o t h x + + + + + + 2 and y. One can e a s i l y v e r i f y t h a t has t h e d e s i r e d p r o p e r t y V u ( u E z ++ u = x o r

t

+ +

u=y).

+

4)

Union. Given any x f r o m S e t ( U ) / S e t ( U := { O }

:= 0 , Y 1

YO {<,>: v e r i f y that

U {O}xX1,

Y

WF

:= {<,O>:

+

) define y =

t h e d e s i r e d p r o p e r t y VG(;€q

as f o l l o w s .

Put

3 v ( < u , v > ~ X ~and ) < v , x o > ~X 2 }

2 + < u , v > ~ X ~ } Observe . t h a t WF(x) i m p l i e s WF($).

7 has

+

ux

++

U

One can e a s i l y

3;:Ez(cE;)).

5 ) CA. Consider any f o r m u l a cp w i t h t h e d i s t i n g u i s h e d f r e e v a r i a b l e u, w h i c h 1 may a l s o c o n t a i n o t h e r parameters f r o m t h e l i s t z z". Given any sequence + + + WF + x,z1, zn o f t r i p l e s f r o m S e t ( U ) / S e t ( U ) d e f i n e y as f o l l o w s . Put y o := 0,

,...,

...,

Y 1 := { O }

u

{O} x WF

)

Set(U)/Set(U

xl,

Y2

:=

{<,>

C v,,,Z; ,..., z',[(v,X1

WF(q). One can e a s i l y v e r i f y t h a t

cpm

$

:
+

,X2),z1

E

x2} u

+ ,... ,zn] 1.

{<,O>

:

x2

E

&

Observe t h a t WF(T) i m p l i e s

++ +

has t h e d e s i r e d p r o p e r t y V u ( u 6 y

+

t+

;Ex

&

1. 6 ) SC.

Consider any f o r m u l a v, w i t h t h e d i s t i n g u i s h e d f r e e v a r i a b l e s x , y

w h i c h may a l s o c o n t a i n o t h e r parameters f r o m t h e l i s t zl,...,zn.

+ +

t h e case o f S e t ( U ) . Given any sequence a , z l ,

+

t h a t s e t ( U ) C (VxE;3ycpZ1,...,Zn holds f o r the formula Ji(u,Y)

Ji

:= 3 y o ( v v ( v E ( Y ) o

+

,...,z n ] ) .

o f t r i p l e s f r o m S e t ( U ) , assume

Then Vu(EA2

c-+

V = y0

~x,y,zl,

A

(yOE(Y)l)

A

c(y!lx ( y ) , ) A Y,..., ~ ~z n ,l ) .~ Y ~ a p~ p l~y

, A , ~ , ~ ~ ~ , ~

+

Using t h i s Z d e f i n e t h e r e q u i r e d t r i p l e b as f o l l o w s .

B2 :=

U {>>

{<
,v>>>,O>

3Y$(u,Y))

((y),

NOW

c o l l e c t i o n t o conclude t h a t t h e r e i s a 2 such t h a t Vu(EA2

B1 := { O }

+

d e f i n e d as f o l l o w s :

...,, , [ ( u , ~ ~

set(u)

[zl

F i r s t consider

+

...,zn

+

,ZF;-~

Ji(u,(Z),,)).

Put b0 := 0 ,

: ~A2 & V E ( ( Z ) ~ ) ~ } , : EA2 & V E ( ( Z ) ~ ) ~U}

& < < u , < l ,v>>,>>E ,w>>>> : € A O +2 I t i s n o t h a r d t o check t h a t ( V ~ € z 3 ~ : E ~ A vV,y E b b Z E z v , ) h o l d s f o r o u r +++j + zn. a,b,z

{<
,

,v>>>,
((Z)u)2}

,...,

I n t h e case o f Set(UWF) one s h o u l d r e p l a c e t h e above f o r m u l a

Ji by ( J i

and

131

Constructive models for set theory

W F ( Y ~ , ( Y ) ~( Y , )2)),

7) 1

2

Power. Given any

.' < V , X ~ > E

{VEX

2 becomes

and o b s e r v e t h a t t h e r e s u l t i n g from Set(U)/Set(U

From t h i s X ,

X2}.

u s i n g ZF;

WF

we1 I - f o u n d e d .

) d e f i n e t h e c l a s s X as

- p o w e r and comprehension axioms we

o b t a i n some Y such t h a t VZCX 3 y ~ D o m ( Y ) ( Z e ( ( Y ) ) and VyEDom(Y)((Y)

*

+

Y

Y

CX).

Using t h i s Y d e f i n e t h e r e q u i r e d y = P ( x ) as f o l l o w s : Put y o := 0,

u

{O} u {O}xDom(Y)

Y1 :=

{l}x(Dom(Y)XX1),

Y2 := {<,O>

: ysDom(Y)}U

: yEDom(Y) & VE(Y) } u {<>,>> : yEDom(Y) & y + Observe t h a t WF(2) imp1 i e s WF(y). I t i s n o t h a r d t o show t h a t

{<<1 ,>,> E X z } .

+ + +

Vu(uEy

tf

8) Y

+ +

UCX). ,Y ) , w i t h Xw := IN and o w O f course w i s well-founded.

I n f . D e f i n e t h e c o n s t a n t w as (0,X

:= {<m,n>

: O<m
U {<m,O>

t h a t N i s i s o m o r p h i c t o w;

: m>O}.

indeed, p u t i ( n ) := ( n + 1 ,Xw,Yu)

Observe

f o r each nEN and

look a t t h e f i r s t f i v e examples f r o m Remark 3 above.

9)

Set(U

WF

) C I n d ( E ) . The p r o o f i s s t a n d a r d : use t h e d e f i n i t i o n o f UwF and

2 - comprehension. -++ 1 0 ) Set(U) C C p l . We a d a p t t h e above p r o o f o f Lemma 1 . Given any x,y f r o m

ZF-

+

Set(U) w i t h ; C z x z

+

we have t o show t h a t t h e r e i s some z f r o m Set(U) and some g

from Set(U) ( r e g a r d e d as a s e t o f o r d e r e d p a i r s f r o m S e t ( U ) as u s u a l i n t h e s e t t h e o r y ) such t h a t

$: 2

2

++ ++

and Vu,v(Ey

F i r s t define a structure Put M :=

-

{VEX

1

:=

++

g(u) E g ( v ) )

.

S = (M,E) and e q u i v a l e n c e ( w i t h r e s p e c t t o E)

-

>

= ( V , X ~ , X ~ ) }o .f c o u r s e

U-v

+

u

{5}xx1,

z2

:= {<,o>

,v>>

: VEM} U {<,
{<<3,v>,<2,~>>

: VEM} U {<<3,~>,<4,v>>:

{<<5,s>,<5,t>>

: <s,t>&X

:

VEM}

u

(Xl,x2)-C

U {<<5,~>,<4,v>>

-

on M .

(u=v)

Put z o := 0,

{<,>

: VEM} U {<<4,v>,
VEM}

+

: <(u,X1,X2),(V,X,,X2)>EY},

W i t h t h e s e M,E , d e f i n e t h e d e s i r e d z as f o l l o w s .

KyJ({i}x~)

U {<,

tf

E := {EMxM

: (u,x1,x2)

:= { Q , V > E M X M

f o r a l l U,VEM.

Z,

: EX2},

++

+

,v>> :

: EE}

: VEM} U

VEM}

U

23 (compare w i t h t h e p i c t u r e f r o m t h e above p r o o f o f

Lemma 1 ) . A s has a l r e a d y been demonstrated (see t h e p r o o f o f Lemma 1 and r e p l a c e t h e i d e n t i t y ,=, (i),(ii)

by o u r -),

:((u-/Ev

*

t h e f o l l o w i n g h o l d s f o r a l l U,VEM (<0,u>,Z,,Z2)

(i),

( (u,Xl,X2)

(ii),

(<(u,Xl,X2),(v,X1,X2)>E;f

f o r a l l u,vsM.

= (v,X1,X2)

++

= /E(<0,v>,Z1,Z2)).

tf

Hence,

= (<0,v>,Zl,Z2))

(<0,u>,Z,,Z2)

(<0,u>,Z1,Z2)

and

E (<0,v>,Z1,Z2))

both hold

+

S i n c e , by t h e d e f i n i t i o n o f M and Z , t r i p l e s (,Z1 ,Z2)

+

j u s t code t h e s e t z, w h i l e t r i p l e s (v,Xl,X2)

+

f o r v&M code t h e s e t x,

f o r VEM

In o r d e r t o +

complete t h e p r o o f i t i s s u f f i c i e n t t o c o n s t r u c t t h e s e t t h e o r e t i c analogue g o f t h e f u n c t i o n g d e f i n e d by g(v,X

X ) := (<0,v>,Z1,Z2) 1' 2

f o r a l l VEM.

I n order t o

produce such g we i t e r a t e o u r p r e v i o u s c o n s t r u c t i o n o f an unordered p a i r . o b t a i n f o r e v e r y VEM t h e c o r r e s p o n d i n g o r d e r e d p a i r <(v,Xl,X2),(<0,v>,Z

First

1 ' Z 2 )>,

L. GORDEEV

132 i.e.

{{(v,Xl,X2)},

{(v,X1,X2),(<0,v>,Z

p a i r l o o k s l i k e (O,V,Wv)

(W)v f o r some f i x e d c l a s s W. d e f i n a b l e i n ZF;

1

,Z2)}};

o b s e r v e t h a t e v e r y such o r d e r e d

f o r some f i x e d c l a s s V , w h i l e W

c o u l d be regarded as

We w i s h t o emphasize t h a t b o t h V and W a r e e x p l i c i t l y

by u s i n g o n l y ( X U Y ) , ( X X Y )

, Oom(X) , Rg(X)

and Ao-comprehension.

U s i n g t h e s e V,W we now d e f i n e t h e d e s i r e d t r i p l e g as f o l l o w s : p u t go := 0, G1

{o} u

:=

( M X V ) , G~ := {<,o>

Step 3. As mentioned above,

: VEM} U {<,>

: <X,Y>E

t h e n a t u r a l numbers f r o m S e t ( U ) / S e t ( U

WF

( w ) ~ E VEM}.

) a r e isomor-

p h i c t o t h e ground ones. The p r o o f o f t h e theorem i s now f i n i s h e d . To complete t h i s s e c t i o n c o n s i d e r t h e f o l l o w i n g n i c e example. universe o f a l l f i n i t e structures.

L e t U be t h e

A c c o r d i n g t o o u r n o t a t i o n s f r o m Remark 3 , one

can r e g a r d U as t h e c o l l e c t i o n o f a l l f i n i t e o r i e n t e d graphs.

Then members o f

UwF a r e t h o s e w i t h o u t c y c l e s . The c o n s t r u c t i o n s i n t h e p r o o f o f Theorem 1 show WF t h a t S e t ( U ) and S e t ( U ) a r e r e s p e c t i v e l y models o f Z F - I n f + C p l and

-

I n f + I n d ( E ) . R e p l a c i n g t h e m e t a t h e o r i e s ZF; and ZF by r e s p e c t i v e l y c l a s s i c a l 2 (PA) and i n t u i t i o n i s t i c (HA) f i r s t - o r d e r a r i t h m e t i c (and u s i n g gadelnumbers w h i c h

ZF

code t h e s t r u c t u r e s ) conclude,

( Z F - Inf+Ind(E)/Cpl)-,(ZF-

t h a t every a r i t h m e t i c statement provable i n

Inf+lnd(E)/Cpl),

i s p r o v a b l e r e s p e c t i v e l y i n HA,PA-

w h i c h extends t h e f a m i l i a r r e s u l t o f W . Ackermann.

PART I I .

MODELS FOR CONSTRUCTIVE SET THEORY

According t o

[MI

(see a l s o [ F r 3 ] ) , t h e main d i f f e r e n c e between ZF- and a

c o n s t r u c t i v e s e t t h e o r y i s t h a t i n t h e l a t t e r axioms CA and Power a r e t o be r e p l a c e d r e s p e c t i v e l y by (Z -CA)

0

and E x p o n e n t a t i o n ( i . e .

g i v e n any two s e t s x and y

t h e r e i s a s e t z o f a l l f u n c t i o n s f r o m x i n t o y ) . T u r n i n g t o t h e above r e a s o n i n g f r o m P a r t l , n o t e t h a t i n t h i s case we can assume t h a t a l l o u r s t r u c t u r e s a r e c o u n t a b l e . To be more p r e c i s e , e v e r y

M from

S = (M,E)

i s supposed t o be a s e t o f

n a t u r a l numbers. W i t h t h i s r e s t r i c t i o n we can produce o u r e x t e n s i o n a l models w i t h i n ((Z; -CA)P)-,

as has been a l r e a d y mentioned i n P a r t I .

1

Note t h a t Z1 -comprehension

i s e s s e n t i a l from t h e c l a s s i c a l s t a n d p o i n t - o f

c o u r s e w i t h t h e e x c e p t i o n of t h e case o f f i n i t e S. To r e c o g n i z e t h i s , c o n s i d e r

S = (N,V)

w i t h N = s e t o f a l l n a t u r a l numbers and V = t h e u n i v e r s a l r e c u r s i v e l y

enumerable b i n a r y r e l a t i o n . Then n e i t h e r h x , y . ( x

=s y ) , nor A x , y . ( x E S y )

is a

hyperarithmetic relation. F o r t u n a t e l y , t h e s i t u a t i o n i s q u i t e d i f f e r e n t i n t h e case o f i n t u i t i o n i s t i c logic,

I t c o u l d be e x p l a i n e d as f o l l o w s .

11.1. Consider

t h e f a m i l i a r 2-nd o r d e r i n t u i t i o n i s t i c t h e o r y Elementary

133

Constructive models for set theory

- CA)r)-,

A n a l y s i s , w h i c h i s ((Z,

( = e l e m e n t a r y = bounded 0 and o n l y r e s t r i c t e d i n d u c t i o n on N . We

i t contains only A

i.e.

= p r e d i c a t i v e ) axiom o f comprehension,

s h a l l d e n o t e t h i s t h e o r y by EA. L e t EA* w i l l be EA extended by t h e f o l l o w i n g axioms.

ECTO : (Vn(,cp

+ 3mii))

+.

3eVn(-@

-t

3m(m= { e } ( n )

& @))) f o r a l l cp,@,

where { e l de-

notes (as, u s u a l ) t h e p a r t i a l r e c u r s i v e f u n c t i o n w i t h i n d e x e. SP(Shanin's p r i n c i p l e ) : VX3YVn(nEX

MPO : 1l(3mcp

3m T(E

c+

f i x e d p a i r i n g function, f o r instance

-

Y ) ), where hn,m.
i s some

Xr1,m.(2".3~).

f o r a l l p r i m i t i v e r e c u r s i v e cp.

-+ 3mcp)

4. EA* i s c o n s e r v a t i v e o v e r HA+ECTO+MPO.

Remark To see t h i s ,

introduce the notion of

Define formula

e(71

realizes)cp

( 1 1 r e a l i z a b i l i t y ) w h i c h i s as f o l l o w s .

f o r a l l f o r m u l a s cp o f t h e language o f EA i n such

a way t h a t :

(i)

( e ( 1 1 r e a l i z e s ) ( x E X ) ) := T l ( < x , e > E X ) ,

(ii)

(e(7-

realizes)3Xcp/VXcp)

:= %/vX

e ( l 1 rea]izes)cp),

( i i i ) i n t h e case o f a t o m i c a r i t h m e t i c a l f o r m u l a s , l o g i c a l c o n n e c t i v e s and f i r s t - o r d e r q u a n t i f i c a t i o n s i t i s d e f i n e d j u s t as K l e e n e ' s 1945 n o t i o n o f r e a l i z a b i l i t y . One can e a s i l y v e r i f y t h a t EA*

i s provably ( 7 1 r e a l i z a b l e ) w i t h i n

EA+MPO.

Hence f o r e v e r y a r i t h m e t i c a l s t a t e m e n t A , (EA* b A) i m p l i e s

(EA+MPO

k <(-,-

r e a l i z e s ) A ) f o r some numeral

e r e a l i z e s A)

(EA+MPO b

c l u s i o n (HA+ECT + M P

0

0

e, w h i c h

i s equivalent t o

w i t h K l e e n e ' s 1945 n o t i o n o f r e a l i z a b i l i t y .

The con-

k A) f o l l o w s f r o m f a m i l i a r r e s u l t s , i . e . c o n s e r v a t i v i t y o f

EA o v e r HA and t h e c h a r a c t e r i z a t i o n o f K l e e n e ' s 1945 r e a l i z a b i l i ' t y by p r o v a b i l i t y i n HA+ECTO+MPO. The c r u c i a l o b s e r v a t i o n i s

5. (A

Claim

0

-definability

l a t i o n s on S w i t h i n E A * - M P O ) . f r e e v a r i a b l e s (n,m,Z) (01

c-f

5 b

o f t h e e x t e n s i o n a l e q u a l i t y and membership r e There a r e two e l e m e n t a r y f o r m u l a s O o and 0 1 w i t h

such t h a t VS = (M,E)3ZVn,m€M((Q0

( n E m ) ) ) i s p r o v a b J e i n EA*

f-t

Pa006. We d e f i n e an e l e m e n t a r y f o r m u l a P ( r , d l ,d2,d3,el

then p u t

OO(n,m,Z)

: 3 r 3 d 1d 2d 33e 1e 2e 3Vx,y(Y+

3V(3q(<,q>F!z

& yx

3u(3p(<,p>IIZ

& Yx,y~u,vI

o1 (n,m,z) Our

Y

,Y

[U,Vl)))

5 k

(n=m)).

A

- MPO.

A

,e2,e3,x,y,n,m,Z)

(Vu(3p(<,p>IIZ)

vV(3q(<,q>Cz)

+.

+

I))),

: 3~3p(<,p>F!Z

QO(n,R,Z)

.

i s g i v e n by

Y : 3 k ( 3 f i v e f i n i t e sequences: r = zO,zl,

..., zk , n

= uO,u1,

...,uk

= x

,

and

L.GORDEEV

134

m = v,,vl

,...,vk

,

= y

p1 ,..., pk

E < , q ~ +Z ~and > t ( (vi+l

zi+l (ui+l pi+l

= {el}(
,zi ,vi+,

= Ie3}(
,qi+l>)

*

&

& qi+l

= {d3}(
,pi+l>))

c zi+l

= te23(
,qi+l>)

or &

,qi+l>)))).

Now, g i v e n any S = (M,E)

+

= {dl } ( a i ,vi ,zi , u ~ ,pi+l>) + ~

= I d 2 } ( < u i ,vi ,zi , u ~ + ,pi+l>) ~

Vx,y(uc,y>~E

?

q1 ,...,qk such t h a t ) V i < k ( < c u i + l , u i > , p i + l > $ Z

,

a p p l y SP i n o r d e r t o g e t a

3p(,p>$Z)).

? b

( n = m ) ) and (Ql(n,m,Z)

Z such t h a t

W i t h t h i s Z we show t h a t (@,(n,m,Z) (nEm)),

t+

f o r a l l n,meM.

Obviously,

t-t

t h e second

e q u i v a l e n c e f o l l o w s f r o m t h e f i r s t one because o f t h e d e f i n i t i o n o f O1. t h e f i r s t one. The i m p l i c a t i o n

"+"

Consider

Y is just

i s t r i v i a l s i n c e o u r r e l a t i o n hx,y.

t h e r e q u i r e d W from Dfn. 1 p r o v i d e d t h a t @,(n,m,Z)

h o l d s . Now suppose t h a t t h e r e

i s some W C M x M such t h a t eW and EQR(W). A p p l y SP i n o r d e r t o g e t a t h a t Vx,y(<x,y>eW <$Q.

3z(<<x,y>,z>$8)).

++

w) i n o r d e r

Vx,y,z,u,p((,z>tE

& z ' = {d2}(<x,y,z,u,p>)

2)

g

such

t h e r e i s an r such t h a t

Now a p p l y ECTO t o t h e c o n d i t i o n EQR(W) ( b y u s i n g t h e above c o r r e -

spondence between W and

1)

S i n c e EW,

t o get d

d

1 ' 2'

& <,p>$Z)

~x,y,z,v,q((,z>$B

Q

<,q>$z)

3v,z',q(v

such t h a t :

= {dl}(<x,y,z,u,p>)

+

3u,z',p(u

& <,q>$Z))

= {el}((x,y,z,v,q>)

E <,z'>tW

3

,d2,d3,el

and el,e2,e3

E <,z'>tn

p = {e }(oc,y,z,v,q>)

W i t h t h e s e parameters Z,n,m,r,dl

3 +

& q = {d3l(a,y,z,u,p>)

E z l = {e2)(oc,y,z,v,q>)

d

,e2,e3

and

w now a p p l y

E <,p>t~)).

r e s t r i c t e d in-

d u c t i o n i n o r d e r t o p r o v e t h a t @ (n,m,Z) h o l d s . 0 To i l l u s t r a t e t h e s i t u a t i o n c o n s i d e r o u r example of t h e u n i v e r s a l r e c u r s i v e l y enumerable s t r u c t u r e

S

= (N,V).

R e c a l l t h a t i n t h i s case b o t h r e l a t i o n s

Xx,y. ( x = y~) and h x , y . ( x E 5 y ) a r e e s s e n t i a l l y

1

El f r o m t h e c l a s s i c a l v i e w p o i n t .

A c c o r d i n g t o t h e above p r o o f , f r o m a s u i t a b l e c o n s t r u c t i v e p o i n t o f v i e w t h e s e 0 r e l a t i o n s are both B

3'

11.2. Consider -

H. F r i e d m a n ' s b a s i c c o n s t r u c t i v e s e t t h e o r y I B - see [ F r 3 ] ,

b u t remember t h a t we s t i l l use t h e o r d i n a r y = , E - language o f s e t t h e o r y e n r i c h e d by t h e c o n s t a n t s 0,w.

instead o f Let

R e c a l l t h a t IB c o n t a i n s o u r ZF- axioms 1 )

- 41, 8)

, (A, - C A I

5 ) , ( E x p o n e n t a t i o n ) i n s t e a d o f 71, ( A b s t r a c t i o n ) ( = F a c t o r i z a t i o n )

instead o f

61,

and (A,

- DC).

IB* a b b r e v i a t e t h e t h e o r y

IB extended by

Enm : Vx3y(Fne(y)

& Dorn(y) C w & Rg(y) = x )

E C T ~: (vnEo(-,cp

+

3rnEoq)

S P : V X C ~ L J ~ Y (=X R g ( w X w - y ) ) MP:(VnEw(cpv7cp)

,

-+ 3 e E w v n E o ( + p

+

3m(m = { e } ( n ) E

$))I

& 7-,3nEwcp+3nEwcp)

f o r a l l cp.

I n a d d i t i o n we c o n s i d e r t h e axiom o f f o u n d a t i o n f r o m [ F r 2 ] , Fnd : V x , z ( T r a n s ( x )

f o r a l l cp,$

,

& VyEx(yCz

+

yEz)

+

xcz).

i.e.

,

135

Constructive models for set theory

Theorem 2.

PI LOO^.

IB*

5

Fnd/Cpl

i s c o n s e r v a t i v e o v e r HA+ECT + M P 0 0

We adapt t h e p r e c e d i n g p r o o f o f Theorem 1 u s i n g r e s u t s from s e c t i o n

11.1.

IB*.

Step 1 . O b v i o u s l y , HA+ECTO+MPO i s c o n t a i n e d i n

WF as o u r m e t a t h e o r y and d e f i n e models S e t ( U ) and S e t ( U ) as

Step 2. We t a k e EA above

- see p r o o f o f Theorem 1. We show t h a t EA

proves Set(U) C ( IB*+Cpl)

to-

g e t h e r w i t h Set(UWF) b ( IB*+ Fnd).

0),

- 4),

1)

8 ) , 5 ) . To c o n c l u d e t h a t EA*

- MP,

proves Set(U)/Set(U

WF

) C

(axioms o f t h e i n t u i t i o n i s t i c l o g i c w i t h e q u a l i t y t o g e t h e r w i t h t h e axioms o f i n f i n i t y , and A -comprehension), r e 0 p l a c e i n t h e c o r r e s p o n d i n g s u b p r o o f o f Theorem 1 b o t h ( ( Z - C A ) P I - and ZF; by

e x t e n s i o n a l i t y , empty s e t , p a i r i n g , u n i o n ,

EA* - M P O .

Claim

5 shows t h a t

6 ) A b s t r a c t i o n : Vx3y f o r m u l a s cp.

1

Consider any

i t does n o t harm t h e p r o o f .

= { { u E x : cp(u,z

1

,...,z k ) }

:

z

1

,...,z k E x }

for a l l

A,

A, f o r m u l a cp w i t h t h e d i s t i n g u i s h e d f r e e v a r i a b l e s

1 w h i c h may a l s o c o n t a i n o t h e r parameters f r o m t h e l i s t v ,...,vn. + + -+ WF ) c o n s i d e r an a r Given any sequence x,vl, vn o f t r i p l e s f r o m S e t ( U ) / S e t ( U 1 k -+ b i t r a r y sequence z o , z o o f elements o f x , i . e . one such t h a t f o r e v e r y i i 1 k w i t h l Q i Q k , < z 0 , x O > ~ X 2 . For e v e r y such sequence z, zo define the correWF ) by p u t t i n g o f t r i p l e s from Set(U)/Set(U sponding sequence +. *. + -+ -+ 21 := ( z i X X ) ; o b v i o u s l y z ' E x . W i t h t h e s e z I ,..., v l ,...,vn, u s i n g (A,-CA) 0' 1 ' 2 -t WF from Set(U)/Set(U ) such t h a t we can o b t a i n t h e c o r r e s p o n d i n g y ,z; z;, u,z

,...,zk,

...,

...,

...,

3 ,...,3

+.

&,

...

;v

€ ;(

y-+l zo,.

. . ,zi

looks l i k e (O,V,W W z;,

... , z i

+-+

z;,

zez

,. . . ,zT) 1.

+

y = {{u€x : cp(u,a

o+

...,z k

,...,z k ) }

moreover, C l a i m 5 shows t h a t f o r some f i x e d W. W i t h t h e s e V,W we

. .. , z ;

z;,

,..., 2;

2;

together i n order t o obtain the required t r i p l e

0, : z',

-+

...,z k € z } -

p a r t o f t h e p r o o f o f Theorem 1.10).

+

+

Observe t h a t e v e r y such y

) f o r some f i x e d V ;

c o u l d be regarded as (W)

+ z',

+ - +

& cp(u,zl

... , z ;

can c o l l e c t a l l o u r y

+

+-+

c.f.

t h e s i t u a t i o n from t h e f i n a l

Observe t h a t WF(2) i m p l i e s WF(q).

+-+ WF 7 ) E x p o n e n t i a t i o n . Given any x , y f r o m S e t ( U ) / S e t ( U ) we have t o c o n s t r u c t a WF 3-i- + -+

z from Set(U)/Set(U

) such t h a t Vg'sgEz

F i r s t d e f i n e t h e c l a s s e s M1,M2,Rl,R2

t-f

F n c ( g ) & Dom(G) =

2&

Rg($)Cq).

as f o l l o w s .

:= { U : E X 3 , M2 := { V : ~ Y 2 } , 1 0 2 R1 := {~MlxMl : (u 1 ,X 1 ,X 2 ) = (u2,X1 , X 2 ) } ,

M

(vl ,Y1 ,Y2) = (v2,Y1 ,Y2)}

-

here we use C l a i m

e l e m e n t a r y f o r m u l a w h i c h says t h a t F

c

.

R2 := {al, v 2 > € M 2xM2 '

5. L e t Map (F ; M1 ,R, ; M2,R2) be t h e

M1xM2 i s a mapping f r o m M,

i n t o M2 b u t w i t h

L. GORDEEV

136

r e s p e c t t o e q u i v a l e n c e s R1 (on M1) and R (on M2). Observe t h a t t h e f o l l o w i n g 2 2-nd o r d e r v e r s i o n o f t h e axiom o f e x p o n e n t i a t i o n 3ZVF(Map(F;M1,R1

; M2,R2)

Indeed, a p p l y SP i n o r d e r t o g e t such

i s provable i n EA*-MPO. V U ( U E M ~f-t 3 p ( < u , p > t K l ) ) . Z := { < e , a , p > : (tKl

3eEOom(Z)(F?(Z),)

f-t

With t h i s

Vu,p(tM,

*

& < u , u ' > ~ R1

~ , u > ER1 & )>E

VF(Map (F ; M1 ,R1 ; M2,R2)

-

13. 2

Kl

that

put

= veM2 E V U ' , ~ '

3v({el()

< v , { e l ( < u ' , p ' > ) > ~ R2))) R

+

*

Kl

and 3 u , p ( < u , p > t K ,

8

Then ECTO shows t h a t

3 e E Dom(Z) (F:

+ +

Now observe t h a t t h e c o n d i t i o n ( g : x

; t h e imp1 i c a t i o n " + "

(Z),))

.z0 +

i s obvious.

y ) i s e q u i v a l e n t t o 3F(Map ( F ; M1 ,R,

+

;

M ,R ) and V U E M ) M / E M ~eF f-t <(u,X,.X;) , (v,Y1,Y2)>Eg)), w h i c h shows t h a t 2 2 we can proceed as f o l l o w s . F i x a Z as j u s t o b t a i n e d , t h e n f o r e v e r y e E O o m ( Z ) de-f

f i n e t h e s e t t h e o r e t i c analogue g

o f t h e mapping (Z)

f i n a l p a r t o f t h e p r o o f o f Theorem 1 , l O ) .

+

as has been shown i n t h e

F i n a l l y , u s i n g t h e same method, c o l l e c t

*

a l l these g ' s together i n order t o o b t a i n t h e required t r i p l e Z s i t u a t i o n i:

t h e above p r o o f o f A b s t r a c t i o n .

v i d e d t h i s i s t h e case f o r b o t h

-

2 is

-

c.f.

the

well-founded pro-

;and .;

+

9) Enm. Consider any x f r o m S e t ( U ) / S e t ( U t r i p l e s (n+l,N,Yu)

Observe t h a t

WF

f o r a l l n E N , and ( m + l , N , Y

1.

w

Recall t h a t our naturals a r e

) i f and o n l y i f

) = (m+l,N,Y

m = n . Now f o r e v e r y ( 1 - s t o r d e r o b j e c t ) n such t h a t € X 2 d e f i n e t h e c o r r e 0 sponding o r d e r e d p a i r < ( n + l , N , Y ) , (n,X ,X )>. Then c o l l e c t a l l t h e s e o r d e r e d 1 2 p a i r s t o g e t h e r as has been shown i n t h e f i n a l p a r t o f t h e p r o o f o f Theorem 1 , 1 0 1 .

+

O b v i o u s l y , t h e r e s u l t i n g t r i p l e y i s j u s t a f u n c t i o n w i t h maps some s u b s e t o f

+

o n t o x . Since w i s w e l l - f o u n d e d ,

WF();

) b (ECT +MPO) s i n c e o u r n a t u r a l numbers 0 MP i s o f c o u r s e d e r i v a b l e f r o m ECT + M P 0 0'

l o ) , 11) ECTo, MP. S e t ( U ) / S e t ( U a r e i s o m o r p h i c t o t h e ground ones. 12) Set(U

WF

i m p l i e s WF(q). WF

) b Fnd. The p r o o f i s s t a n d a r d . Use C l a i m 5 and A. -comprehension.

1 3 ) Set(U) I= C p l . Indeed, t h e o n l y p l a c e i n t h e c o r r e s p o n d i n g s u b p r o o f o f Theorem 1 i n w h i c h a non-elementary axiom has been used, was t h e d e f i n i t i o n o f E and

-.

But now we can a p p l y C l a i m 5 t o d e f i n e w i t h i n E A * - M P

F i n a l l y , observe t h a t SP

-

(Ao

- OC)

i s d e r i v a b l e f r o m (A,

0'

- CA) +

I n f + Enm+ECTO+

the proof i s t r i v i a l .

Note t h a t MPO was used o n l y t o p r o v e S e t ( U ) / S e t ( U

Step 3. Since n a t u r a l numbers f r o m S e t ( U ) / S e t ( U f r o m t h e ground model, statements.

IB*+Fnd/Cpl

WF

WF

) C- MP.

) are isomorphic t o the n a t u r a l s

i s c o n s e r v a t i v e o v e r EA* f o r a r i t h m e t i c a l

To conclude t h e p r o o f see Remark 4 above.

137

Constructive models for set theory

The main Theohem.

I I . 3.

IG(E)

1

: Vx3y i ( E ; x , y ) ,

(y =

v

n

Since i ( € ; x , y ) Cpl t h e s e IG(E)E

where i ( E ; x , y ) a b b r e v i a t e s t h e f o r m u l a

~

E

~

means “y

(

~+ uE ~ ~z + () v~~ zE) } ~) ,

~

i s t h e € - a c c e s s i b l e p a r t o f x”,

i n t h e presence o f

and IG(E) a r e j u s t t h e s e t t h e o r e t i c v e r s i o n s o f S .

i n d u c t i v e g e n e r a t i o n axioms IGE

and IG

Feferman’s

- see [ F l 1 , [ F Z ] . O t h e r w i s e , i f t h e

g i v e n s e t t h e o r y i s supposed t o c o n t a i n any v e r s i o n o f t h e axiom o f f o u n d a t i o n then b o t h IG(E) 1 and IG(E) do n o t c o n t a i n any s p e c i a l i n f o r m a t i o n , s i n c e obv i o u s l y Fnd i m p l i e s IG(E) 1 and Ind(€)

i m p l i e s IG(E).

Now c o n s i d e r t h e f o l l o w i n g l i s t o f c o n s t r u c t i v e s e t t h e o r i e s . IB*

5

SC

IB*

+

Ind(w)

2

I II.

IB*

+

Ind(w)

+ Sc 5

IV.

IB*

+

Ind(w)

+

V.

IB*

+

IG(E)

VI.

IB*

VII.

IB*

VIII.

IB*

IX.

IB*

Fnd/Cpl

F+ + Ind(w) + + Ind(w) + + Ind(w) + + Ind(w) +

R e c a l l t h a t H. respectively

5

.

I.

I I.

Fnd/Cpl

+

SC

Cpl

.

RDC

5

Ind(E)

5

Fnd/Cpl.

5

RDC.

SC.

IG(E) 1

+ Cpl.

+ Cpl. I G ( E ) I + Cpl + SC 5 RDC. IG(E) + Cpl + SC 2 RDC. IG(E)

Friedman’s c o n s t r u c t i v e s e t t h e o r i e s T1 ,T2,T3 f r o m [ F r

3 1 are

IB+SC+Fnd, I B + Ind(w) +SC+RDC+Fnd, I B + I n d ( w ) + S C + Ind(E)+ROC.

The Main Theorem says t h a t e v e r y t h e o r y f r o m t h e above l i s t I , s e r v a t i v e over the corresponding theory from t h e f o l l o w i n g l i s t .

I.

HA

II.

HA

+ +

ECTO + MP

0’

ECTO + MPO

1.

+ 0

I I I.

HA

+

ECTO + MPO

+

T I (
IV.

HA

+

ECTO + MPO

+

TI ( < ~ E ~ , + ~ O ) .

V.

HA

+

ECTO + MPO

+

T I (<m w D ) .

- I X . i s con-

L. GORDEEV

138

V VI VI I

. . .

HA HA

IX.

+ +

ECTO + MPO

+

ECTO + MPO

+ T I (<

HA

+

ECTO + MPO

+

HA

+

ECTO + MPO

+

Here T l ( < a )

TI(<

8(nU.EO)0). +

T I (< g

0).

w

0). €0TI(<~((BIEl+lO)O).

denotes t h e t r a n s f i n i t e i n d u c t i o n scheme o v e r t h e s p e c i a l p r i m i -

t i v e r e c u r s i v e system c1 of o r d i n a l n o t a t i o n s , based on t h e t h e o r y o f

-collapsing

Note t h a t ~ E o O < , . . . , < ~ ( ~ ~ E + l O ) Oa r e r e s p e c 1 t i v e l y t h e p r o o f t h e o r e t i c o r d i n a l s o f subsystems (El - A C ) , ID1, (2: - C A I I , 1 1 1 1 (C1 - L A ) , (El C A ) + B I , ( E 2 - A C ) , (E2 - A C ) + B I o f c l a s s i c a l a n a l y s i s the reader

functions,

see [ B ] ,

[ J ] , [JP].

-

-

w i l l f i n d enough i n f o r m a t i o n i n [ BFPS].

P r o o f o f t h e theorem. I I . The r e s u l t f o l l o w s f r o m t h e p r o o f o f Theorem 2 ( i t i s i n f a c t c o n t a i n e d

i n [ G 2 ] , b u t now t h e m e t a t h e o r y EA* I n d ( N ) . O f course,

i s t o be extended by t h e f u l l i n d u c t i o n scheme

Ind(N) i s p r o v a b l y (-,T

r e a l i z a b l e ) w i t h i n E A + Ind(N), w h i l e

) . Hence E A * + Ind(N) i s

t h e l a t t e r i s known t o be c o n s e r v a t i v e o v e r H A + T I ( < E € c o n s e r v a t i v e o v e r HA+ECT + M P + T I ( < € 0 0

)

-

see Remark

0

4

above.

€0

I . Compared t o t h e p r o o f o f Theorem 2, we d e a l w i t h t h e m e t a t h e o r y EA* enr i c h e d by t h e 2-nd o r d e r c o l l e c t i o n axiom C

-

c.f.

axiom

6 ) , o f ZF2 above. Observe

t h a t c o n s t r u c t i o n 6) from t h e p r o o f o f Theorem 1 shows t h a t E A * + C p r o v e s WF S e t ( U ) / S e t ( U ) C S C . Hence we o n l y have t o show t h a t E A * + C s t i l l i s c o n s e r v a t i v e o v e r HA+ECTo+MPO : To see t h i s , a p p l y o u r argument f r o m t h e p r o o f o f Remark 4. F i r s t note that C i s

(77

r e a l i z a b l e ) i n t h e presence o f C . Hence (EA*+C

i m p l i e s (EA+MP + C I-b r e a l i z e s A ) w i t h some numeral

0

s t a t e m e n t s A. Since (B r e a l i z e s A) i s n e g a t i v e ,

e for

I-

A)

arbitrary arithmetical

i t i s s u f f i c i e n t t o show t h a t

every negative a r i t h m e t i c a l statement provable i n E A + M P

0

+C,

i s p r o v a b l e i n PA.

T h i s r e s u l t i s i n f a c t c o n t a i n e d i n [ F r 3 ] , b u t we want t o be a l i t t l e more con1 s t r u c t i v e by u s i n g t h e n o t i o n o f (A r e a l i z a b i l i t y , w h i c h i n t u r n i s i m p l i c i t l y 1 1 1 c o n t a i n e d i n [F2]. T h a t i s , f i x s t a n d a r d u n i v e r s a l r e s p e c t i v e l y II and Z1 p r e d i 1 1 c a t e s P (m,x) and S (m,x) and p u t A, (,X) : Vx(P1 ( p , x ) tf S1 ( q , x ) t+ x E X I . 1 l 1 Then d e f i n e f o r m u l a s ( e ( A r e a l i z e s ) c p ) f o r a l l f o r m u l a s cp o f t h e language o f EA 1 i n such a way t h a t : (i)

(,(A;

(ii)

1 1 ( e ( A l rea1izes)Wcp) := %Vp,q(A, (,X)

reaIizes)(xEX))

Ie}()(Al

( ii i ) ( e ( 4

1

:= ( x E X ) , +

Iel(t

&

r e a l izeslcp),

r e a l izes)gXcp) := %Vp,q,n(e

1 n(A, r e a l i z e s l c p ) ,

= <,n>

1

& A1 (,X)

139

Constructive models for set theory

(iv)

i n t h e case o f a t o m i c a r i t h m e t i c a l f o r m u l a s ,

l o g i c a l c o n n e c t i v e s and

1-st o r d e r q u a n t i ' f i c a t i o n i t i s d e f i n e d e x a c t l y as K l e e n e ' s 1945 n o t i o n o f 1

r e a l i z a b i l i t y . One can e a s i l y v e r i f y t h a t E A + M P O + C i s p r o v a b l y (A, r e a l i z a b l e ) 1 1 w i t h i n t h e c l a s s i c a l t h e o r y (Z, -AC) 1 - c . f . t h e p r o o f o f ( E M O P + J G ( 8 , - A C ) ) 1 i n [ F21. F i n a l l y , (C - AC) F i s known t o be c o n s e r v a t i v e o v e r PA see t h e same

-

1

paper f o r p r e c i s e r e f e r e n c e s .

1945

To c o n c l u d e t h e p r o o f r e c a l l t h a t K l e e n e ' s

n o t i o n o f r e a l i z a b i l i t y does n o t change n e g a t i v e f o r m u l a s .

Ill. All v a b l e i n RA-

i n s t a n c e s o f t h e a r i t h m e t i c a l scheme Tl(
(rumified analysis i n levels

< 6 0 ),

are o f course pro-

which i s obviously contained

in

IB+ I n d ( w ) + S C . Hence HA+ECT + M P O + T l ( < g ~ o O ) i s c o n t a i n e d i n IB*+ Ind(w)+SC. 0 To o b t a i n t h e e s s e n t i a l p a r t o f t h e p r o o f we a p p l y t h e above r e a s o n i n g from I . But now, i n s t e a d o f E A * + C ,

RDC: (t/x(cp(X)

+

we c o n s i d e r t h e m e t a t h e o r y E A * + I n d ( N ) + C + R O C , where

3Y(cp(Y) E $ ( X , Y ) ) )

$((Z)n,(Z)n+l)))))

-+

W(cp(X)

-+

3'Z((Z)o

E X E V n E N (cp(Z),,)

6

f o r a l l cp,$

i s t h e 2-nd o r d e r v e r s i o n o f o u r s e t t h e o r e t i c ROC. T h i s m e t a t h e o r y proves WF Set(U)/Set(U ) ( I n d ( w ) + S C + R O C ) . To see t h i s i n t h e case o f ROC, f i r s t p r e s e n t WF 2-nd o r d e r f o r m u l a S e t ( U ) / S e t ( U ) C Vx(w(x) + 3y(cp(y) & $ ( x , y ) ) ) as VX(G(X) + 3'Y(G(Y) E G(X,Y)))

with suitable

t h e 2-nd o r d e r axiom RDC i n o r d e r t o g e t a Vn((P((Z),)

;((Z)",

&

Set(U)/Set(U

WF

(Z)n+l)),

Y

G,;,

then, g i v e n any X w i t h cp(X), a p p l y E X and

Z such t h a t ( 2 )

and t h e n c o n s t r u c t t h e s e t : h e o r e t i c

) of t h e f u n c t i o n hn.(Z),

p a r t o f t h e p r o o f o f Theorem 1 , 1 0 ) .

a

o

r e a l i z a b l e ) i n t h e presence o f Ind(N)+C+ROC.

+ Ind(N)+C+RDC k 0

from

i s provable i n

+ T I ( < ~ E ~ O ) To . see t h i s , f i r s t n o t e t h a t Ind(N)+C+RDC

i m p l i e s (EA+MP

2

To complete t h e p r o o f we have t o show t h a t

every a r i t h m e t i c a l statement A provable i n EA*+ Ind(N)+C+RDC, HA+ECT + M P

analogue

by u s i n g t h e above r e a s o n i n g from t h e f i n a l

is

Hence, ( E A * + I n d ( N ) + C + R O C

e realizes A)

w i t h some numeral

g. Then

(77

k A) apply

t h e above argument f r o m t h e p r o o f o f I i n o r d e r t o show t h a t E A + M P o + I n d ( N ) + C + 1 1 (A, r e a l i z a b l e ) w i t h i n c l a s s i c a l (2, - A C ) . As above, t h i s y i e l d s

RDC i s p r o v a b l e

t h e r e s u l t s i n c e gEoO =

I (El1 -AC) 1 .

V I , V I I . A l 1 i n s t a n c e s o f t h e a r i t h m e t i c a l schemes T I ( < ~ ( ~ w 2 W . ~ O ) O and ) TI

( < g ~ 0)

are provable respectively i n

%+I

observe t h a t t h e c l a s s i c a l systems

(II;

1 - CA)

(111

and (

-CAI-

(II;

, (II,1 - CA) + BI

t h e r e by t h e i r i n t u i t i o n i s t i c v e r s i o n s a s w e l l . Now, a r e d i r e c t l y c o n t a i n e d r e s p e c t i v e l y i n 1B+ I n d ( o )

+

(IT:

- C A I + BI)--see [ BFPSI,

c o u l d be r e p l a c e d

-CAI-

and

IG(E)F + C p l and

((II;

-CA)+BI)-

IB+ Ind(w)

+

I G ( E ) + C p l , a c c o r d i n g t o t h e t r e e theorem. For t h e second ( e s s e n t i a l ) p a r t o f t h e p r o o f , c o n s i d e r r e s p e c t i v e l y t h e metat h e o r i e s EA*+ Ind(N)

+

IGP : VX,R3Y i ( R ; X , Y ) ,

I G I ' and E A * + Ind(N) where i ( R ; X,Y)

+

I G , where

abbreviates the formula

( Y E ~ { ~ : V ~ E X ( V ~ ( < + ~ U , *E Z~ ) R-t ~ E z ) ) ) ,

L. GORDEEV

140

VXR3Y(i(R;X,Y)

IG:

and ( V v ~ X ( V u ( ( < u , v > ~ R + c p ( u ) ) + c p ( v ) ) + V V E Y cp(y))) f o r

a l l cp. I t s h o u l d be c l e a r , t h a t o u r I G P and I G have t h e same meaning a s t h o s e f r o m

,

[Fl]

[FZ]

( t h e o n l y d i f f e r e n c e i s t h a t i n o u r m e t a t h e o r i e s no 1 - s t o r d e r c o d i n g

of classes i s required).

(U,X,.X~)~(V,X~.X~)~.

a p p l y I G E i n o r d e r t o g e t Y such t h a t i ( R ; X , Y )

7 from Set(U)

define the required t r i p l e

Yz := {<,O>:

by y o := 0, Y

V E Y } U {<,:

k i(€;z,;);

Set(U)

Indeed, g i v e n any x

and R := { < u , v > € X :

from Set(U), p u t X := { v E X 1 : < v , x o > E X 2 } W i t h these X,R

-+

k IG(E).

Now, I G I proves Set(U) C I G ( E ) F and I G proves Set(U)

1

holds. F i n a l l y ,

{O} U { O } x X , ,

:=

< u , v > ~ X ~ } .I t i s c l e a r t h a t

moreover, t h e second a s s e r t i o n o f t h e d e f i n i t i o n o f I G ( E )

also

h o l d s i n t h e presence o f t h a t o f I G . We c o n c l u d e t h a t e v e r y a r i t h m e t i c a l s t a t e m e n t A provable i n

IB*+ Ind(w)

+ IG(E)

1 + Cpl,

IB*+ Ind(w)

+

I G ( E ) +Cpl

i s provable,

r e s p e c t i v e l y , i n EA*+ I n d ( N ) + I G P , E A * + I n d ( N ) + I G . i s (-,7 r e a l i z a b l e ) i n t h e presence o f I G / I G .

Now o b s e r v e t h a t I G F / I G

Indeed,

g i v e n X,R d e f i n e Y t o be t h e Y i n d u c t i v e l y g e n e r a t e d by

-X

-

:= {>

: - ~ ( < V , Z > E X ) & VU,W(T-I

R := {<,>>

-

(c.f.

--

is

the d e f i n i t i o n o f I G i s also

(77

(77

+

{{e}(u)}(w)

k)}.

Using t h e r e c u r s i o n theorem i t i s n o t

-

t h e second a s s e r t i o n of

r e a l i z a b l e ) ; moreover,

r e a l i z a b l e ) provided t h a t Y s a t i s f i e s the f u l l

Hence, f o r e v e r y a r i t h m e t i c a l s t a t e m e n t A,

I G scheme w i t h t h e s e X , R .

e realizes A)

I- A ) i m p l i e s ( E A + M P 0 + I n d ( N ) + I G P / I G I(and K l e e n e ' s 1945 n o t i o n o f r e a l i z a b i 1 i t y ) .

+

R)

<,W>E

: -,~(C,w>~ K ) ]

the s i m i l a r construction from [FZ]).

h a r d t o show t h a t i ( R ; X , Y )

(

IGI/IG

(EA*+ Ind(N)

e

f o r some numeral

+ I n d ( N ) + I G P and E A + M P + I n d ( N ) + I G a r e i n an o b v i o u s way 0 O 1 1 c o n t a i n e d r e s p e c t i v e l y i n t h e c l a s s i c a l t h e o r i e s (Itl -CA) = (El - C A ) and

Now observe t h a t E A + M P

- CA) + B I

(I

1

= (Z

1

- CA) + 81.

As above,

since

(e r e a l i z e s A )

i s negative, t h i s

and GE y i e l d s t h e r e s u l t because c(S2w.~O)0 1 1 Qw+1 d i n a l s of CA) and (II, CA) + B I r e s p e c t i v e l y .

0 are j u s t the proof theoretic o r -

(n, -

-

IX. For s i m p l i c i t y l e t a , p , y denote r e s p e c t i v e l y t h e o r d i n a l s ( s y s -

V, V I I I ,

mwO

, p (aE1+,0)0.

,gS2 0 €0 HA + E C T +~ M P +~ T I (
+

spectively contained i n

IB*

+

Ind(w)

+

IG(E)

IB*+ I G ( E )

+ Cpl + SC.

E +

We f i r s t have t o show t h a t

H A + E C T +~ M P +~ T I ( < y ) a r e r e -

(
IB*+ I n d ( w )

Cpl,

I n t h e case o f

a = I (I(

+

E+

IG(E)

- CA)

E

I

=

can a p p l y t h e argument f r o m t h e f i r s t p a r t o f t h e p r o o f o f V I .

6=

l(Zi-AC)l

= IIO-+o(W)l

d i r e c t l y contained i n

(see [BFPS],

[BPI

1,

IB+ I n d ( w ) + I G ( E ) +Cpl + S C .

Cpl

+ SC,

1

1 (El - CA) F I

one

Next, s i n c e

o b s e r v e t h a t t h e o r y 10-Go(W) Finally,

t h e r e s u l t f r o m [ J ] w h i c h says t h a t a l l i n s t a n c e s o f t h e a r i t h m e t i c a l scheme

Tl(
is

i n t h e case o f y a p p l y

a r e p r o v a b l e i n t h e i n t u i t i o n i s t i c t h e o r y To f r o m [ F Z I . Note t h a t t h e

Constructive models for set theory

p r o o f f r o m [ J ] c o u l d be f o r m a l i z e d i n

I B + Ind(w)

+

141

IG(E) + C p l + S C as w e l l , by r e -

placing i n [J] the 'recursion theorem-definition'of

the operation ' w '

by t h e o r -

d i n a r y s e t t h e o r e t i c d e f i n i t i o n by i n d u c t i o n ( h e r e SC p l a y s t h e r o l e o f t h e J o i n axiom f r o m To): I n t h e second p a r t o f t h e p r o o f we proceed as i n t h e above p r o o f o f V I , V I I , but now we c o n s i d e r r e s p e c t i v e l y m e t a t h e o r i e s EA*+ IG1

+

ROC, E A * + I n d ( N ) + I G + C + R D C .

Then we a p p l y

(77

show t h a t f o r e v e r y a r i t h m e t i c a l s t a t e m e n t A,

+

C,

E A * + Ind(N)

realizability)

(EA*+ I G I

+

+

+ C

IGF

i n order t o

C k A), ( E A * + Ind(N)+

I G ! + C + R O C k A ) , ( E A * + I n d ( N ) + IG+C+RDC k A ) i m p l y r e s p e c t i v e l y (EA+MPO+ IG

e r e a l i z e s A ) , ( E A + M P 0 + Ind(N)+ I G F + C + R D C k e r e a l i z e s A), (EA+MPO+ I n d ( N ) + I G + R D C t e r e a l i z e s A). Next we m o d i f y i n an o b v i o u s way t h e above

+Ck

1 d e f i n i t i o n o f (A, r e a l i z a b i l i t y ) i n o r d e r t o d e f i n e t h e c o r r e s p o n d i n g n o t i o n o f 1 (Az r e a l i z a b i l i t y ) , w h i c h i s a l s o i m p l i c i t l y c o n t a i n e d i n [ F Z ] . One can e a s i l y EA+MPO+ Ind(N)+ I G 1 + C + R D C , EA+MPO+ Ind(N)+ I G 1 1 a r e p r o v a b l y (A' r e a l i z a b l e ) w i t h i n t h e c l a s s i c a l systems (X2-AC)P,(Sz-K), 2 1 ( Z 2 - A C ) + B I r e s p e c t i v e l y . As above, t h i s y i e l d s t h e r e s u l t s i n c e B and y a r e 1 1 j u s t t h e p r o o f t h e o r e t i c o r d i n a l s o f r e s p e c t i v e l y (E2- A C ) and (Ez -AC) + B I , w h i l e 1 1 ( E i - A C ) E i s known t o be c o n s e r v a t i v e o v e r (Il - C A I ! = (2, - C A I E f o r a r i t h m e t i c a l 1 1 statements ( [ F r 1 1 and ct = (Z1 - C A I

v e r i f y t h a t E A + M P O + IGF + C , +C+ROC

I

IV. ID;(W)

I t i s known t h a t ~

E

I.

~

~= IID;(W)I + ~ O

i n [ F r 3 1 , where t h e o r y

lO-(O)

(see [BFPS], [ B P I ) . Now, t h e t h e o r y

I B + Ind(w) + S C + I n d ( E )

i s d i r e c t l y contained i n

-

a p p l y t h e same method as

was shown t o be c o n t a i n e d i n T3 (one can r e g a r d W

as an a c c e s s i b l e p a r t o f some s p e c i a l p r i m i t i v e r e c u r s i v e system o f o r d i n a l notations,

+

s i m i l a r t o t h e one w h i c h a c c e s s i b l e p a r t i s K l e e n e ' s

MPO+TI(<&nl+lO)

i s contained i n

0 ) . Hence, HA+ECTO

IB*+ Ind(w) + S C + I n d ( E ) .

To g e t t h e e s s e n t i a l p a r t o f t h e p r o o f one c o u l d proceed as above by u s i n g t h e metatheory EA*+ Ind(N) + C + R D C + B I ,

and then showing t h a t t h e l a t t e r i s i n f a c t

c o n s e r v a t i v e o v e r H A + E C T O + M P O + T I(
by u s i n g t h e method d e s c r i b e d i n

[ZI . I t s h o u l d be i n t u i t i v e l y c l e a r

s l i g h t l y more c o n s t r u c t i v e way

-

however,

t h a t we can g e t t h e r e s u l t i n a

namely, by a s u i t a b l e c o n s t r u c t i v i z a t i o n o f t h e

s t a n d a r d e x t e n s i o n a l v e r s i o n o f t h e whole h y p e r a r i t h m e t i c h i e r a r c h y o f subsets o f N. T h i s can be done by u s i n g an a p p r o p r i a t e m o d i f i c a t i o n

O*

o f K l e e n e ' s 0, i n

o r d e r t o i n t e r p r e t SC i n t h e c o r r e s p o n d i n g e x t e n s i o n a l model. From t h e c l a s s i c a l v i e w p o i n t t h e m o d i f i e d system (i.e.

O*

w i l l d e s c r i b e t h e same s e t o f o r d i n a l s as 0

t h e r e c u r s i v e o r d i n a l s ) . So o u r p r o o f r u n s as f o l l o w s . F i r s t c o n s i d e r ( i n o r d e r t o i l l u s t r a t e t h e d i f f e r e n c e between 0 and

f o l l o w i n g 1 - s t o r d e r e x t e n s i o n ID-(O,H) (1)

O ( 0 ) & Vk(O(k)

+

O(Zk))

&

o f HA,

Vm(Vi(O({m)(i))

+ 0(3.Sm)),

O*)

the

L. GORDEEV

142

(do)

(3)

& Vk(cp(k)

+

~ ( 2 ~ 6) V) m ( V i c p ( { m } ( i ) )

+

~ ( 3 . 5 ~ ) ) Vn(O(n)+cp(n))) +

for a l l cp, where 0 and H a r e new r e s p e c t i v e l y u n a r y and t e r n a r y f i r s t - o r d e r p r e d i c a t e s ( f o r t h e s i m p l i c i t y we w r i t e cp({m)(i)) proves Vn(O(n)

+

V P , X ( H(p,n,x) ~ ~

f o r {m>(i)+ & cp({m}(i))).

Obviously,

IO-(O,H)

+H(p,n,x)).

Now o u r second-order o b j e c t s ( c l a s s e s ) a r e t o be i n t e r p r e t e d i n t h e language o f ID-(O,H)

f o r a l l p,n w i t h O(n).

as Xx.3zH(p,n,<x,z>)'s

Since the universe U o f

a l l s t r u c t u r e s (M,E) i s d e t e r m i n e d by t h i s i n t e r p r e t a t i o n , b o t h t h e u n i v e r s e s WF Set(U) and S e t ( U ) a r e d e t e r m i n e d as w e l l . However, t h e l a t t e r c o u l d be d e f i n e d WF i n a more i n f o r m a t i v e way. T h a t i s , i n s t e a d o f S e t ( U ) we w i l l c o n s i d e r Set(UWF(0)) by p u t t i n g WF(O)(Z) t o be t h e i n t e r p r e t a t i o n o f t h e f o r m u l a : 3m,s(O(m)

& VU,VEX,(EX~

{n : n + O m).

+

t o t h e ( r e c u r s i v e l y enumerable) s e t

Having i n mind o u r i n t e r p r e t a t i o n o f t h e second-order

s i d e r t h e m e t a t h e o r y IO-(O,H) Vn(T-,O(n)

{ s ) ( u ) 4 O p m { s } ( v ) ) ) , where d o r mdenotes

+

t h e r e s t r i c t i o n o f standard r e l a t i o n

O(n))).

+

+ ECTO+MPO(0), where

Observe t h a t ID-(O,H) +ECTO+MPO(0) p r o v e s S e t ( U ) / S e t ( U

C ( IB* + I nd ( w ) + R O C ) .

Set (UWF(') )

language, con-

MPO(0) a b b r e v i a t e s ( M P O +

Indeed, (

I=

IB*

+ I nd ( w ) )

WF

)/

f o l lows f r o m t h e p r o o f

of Theorem 2 s i n c e o u r i n t e r p r e t a t i o n o f E A * + Ind(N) i s p r o v a b l e i n ID-((I,H)+ ECTo+MPO(0) (remember t h a t H(p,n,x)

i s provably negative f o r O(n)),

s t r u c t i o n s ( e x c e p t t h a t f o r C p l ) p r e s e r v e WF(O) as w e l l as WF.

t h i s i s s t i l l t r u e because t h e r e (see t h e p r o o f o f I I I ) we g e t a f u n c t i o n w i t h range C O , w h i c h codes t h e d e s i r e d f u n c t i o n

and a l l con-

I n t h e case o f ROC

total r e c u r s i v e

"t";t h e r e f o r e

the

supremum o f t h i s f u n c t i o n does e x i s t as an element o f 0, w h i c h shows t h a t t h e structure

ID-(O,H)

t

i s i n t e r p r e t a b l e and p r e s e r v e s a l s o WF(O).

F i n a l l y , axiom 3 ) o f

t o g e t h e r w i t h t h e r e c u r s i o n theorem p r o v e s Set(UWF(o))

every a r i t h m e t i c a l statement provable i n

IB*+ Ind(w)

+

b

Ind(E) + R D C ,

I n d ( O ) . Hence, i s provable i n

ID-(O,H) + E C T ~ + M P ~ ( ~ ) . However, i n o r d e r t o i n t e r p r e t SC by u s i n g t h e same r e a s o n i n g we need t h e boundedness theorem ( t h e r e c u r s i v e image i n 0 o f an a r b i t r a r y h y p e r a r i t h m e t i c s e t i s e f f e c t i v e l y bounded i n

O),

w h i c h does n o t seem t o be p r o v a b l e i n I O - ( O , H ) +

ECTO+MPO(0). T h i s i s t h e reason why i n s t e a d o f ID-(O,H) version (1)*

IO-(O*,H*),

O*(O) +

we c o n s i d e r t h e m o d i f i e d

w h i c h i s g i v e n by

& Vk(0*(k)

O*(39.5R.7m))

( 2 . 1 ) * Vp,x(H*(p,O,x)

+

-

O*(2k))

& Vk,q,m(O*(k)

,

{p}(x) = 01,

&

Vi(H*(q,R,i))+0*({m}(i))

143

Constructive models for set theory k

) +(H*(p,Zk,x)

(2.2)*

Vp,k,x(O*(Z

(2.3)*

V p , q , R , r n , ~ ( O * ( 3 ~ . 5 ~ . 7 ~ )-+ ( H * ( ~ , 3 ~ . 5.7 , x )

R

u = {m}(i)

(3)*

(cp(0)

& H*(q,L,i)

-+

* cp(2 1) vn(O*(n)

Observe t h a t ID-(O*,H*)

{m} : Ai.H*(q,R,i)

-+

O*

m

c-t

173i,u,v(x

=

&

E H*(p,u,v)))),

k

& Vk-(cp(k)

cp(3q.5R.7m))

O*,H*

++Vu~~3vH*(p,k,-%,u,v>))),

& Vq,R,m(cp(R)

+

& V i (H*(q,R,i)

+

v({m}(i)))

-+

9 ( n ) ) ) f o r a l l cp.

p r o v e s t h e boundedness theorem, s i n c e now e v e r y

i s bounded i n

O*

by 3q.5R.7m.

Hence, r e p l a c i n g O,H by

one can a p p l y t h e above argument i n o r d e r t o show t h a t e v e r y a r i t h m e t i c a l IB*+ Ind(w)

statement p r o v a b l e i n

+ SC+

Ind(E)

+ RDC,

i s p r o v a b l e i n ID-(O*,H*)

+

ECTO+ MPO (O*). The n e x t s t e p i s r o u t i n e .

(O*

Define the notion o f

r e a l i z a b i l i t y ) t o be t h e t r i v -

i a l e x t e n s i o n o f K l e e n e ' s 1945 r e a l i z a b i l i t y by ( e ( O * r e a l i z e s ) O * ( n ) ) := O*(n) := H*(p,n,x).

and (e(O* r e a l i z e s ) H * ( p , n , x ) ) i s p r o v a b l y (0* r e a l i z a b l e )

i n ID-(O*,H*)

a r i t h m e t i c a l s t a t e m e n t A, (ID-(O*,H*) MP

(O*) I-

e realizes A)

Observe t h a t ID-(O*,H*)

+ECTO+MPO(O*)

f o r some numeral

e and

I- A )

Lemma 2. L e t ID(O*,H*)

I- A ) ,

be ID-(O*,H*)

m e t i c a l s t a t e m e n t p r o v a b l e i n IO(O*,H*),

P/r00,4.

+ECTO+MPO(O*)

w i t h classical logic.

I- A )

Then e v e r y a r i t h -

i s p r o v a b l e i n PA+TI(
By a f a m i l i a r t r i c k ( s u g g e s t e d t o t h e a u t h o r by S . Feferman and ( i n d e i s contained i n ID,.

( (a) v (b) v (c)

1,

( b ) : 3 k ( r 1 = 2 ~& O*(k) & ( j = O

X

E. v i ( x ( l , q , R , i )

j = o E, l i , u , v ( x = < u , v > & u = { m } ( i ) j = 1 E. V i , u , v ( x = < u , v >

E u=Cm}(i)

X

( i i ) (vj,p,n,x(A(cp;

*

- CA)

-+

x(I,q,Il,i)

E,

x(o,p,u,v)).v.

v X(I,p,u,v)))),

v X ( l ,z,n,y)

).

- e x t e n d e d by

P(j,p,n,x)

j,p,n,x)

v Oz({m}(i)))

E, X(O,q,R,i)

where O*(n) a b b r e v i a t e s Vz,y(X(O,z,n,y) L e t I D1 (A) be (A,

.v.

I),

E. o,*(L)

= 3q.5L.7m

&4{pI(x)=o))),

& Vu3vX(O,p,k,<x,u,v>)

j = 1 6 3uVvX(1 ,p,k,*,u,v>)

A(P ; j , p , n , x )

E

where

( n = ~ & ( j = ~ & { p } ( =x o ) . v . j = l

( c ) : 3q,R,m(n

Consider t h e

:= ( j = O v j = 1 )

f o l l o w i n g p o s i t i v e second-order p r e d i c a t e A ( X ; j , p , n , x )

(i)

+

i t i s s u f f i c i e n t t o prove the following

p e n d e n t l y ) P . Aczel ), we show t h a t IO(O*,H*)

(a):

i m p l i e s (ID-(O*,H*)

K l e e n e ' s 1945 r e a l i z a b i l i t y . As

0 u s u a l , i n o r d e r t o g e t t h e r e q u i r e d c o n c l u s i o n : (ID-(O*,H*) implies ( H A + E C T O + M P O + T I ( < ~ ~ S i l + , O )

+ECTO+MPO(O*)

+MPO(O*) and c o n c l u d e t h a t f o r e v e r y

+

, cp(j,p,n,x))

f o r a l l cp a r i t h m e t i c a l i n P,

-+

vj,p,n,x(P(j,p,n,x)

* cp(j,p,n,x))

L. GORDEEV

144

where P i s a new f i r s t - o r d e r & - p l a c e p r e d i c a t e . Now p u t O*(n) proves ID(O*,H*),

:= O:(n)

and H*(p,n,x)

:= O*(n)

& P(O,p,n,x).

F i r s t use ( i ) i n o r d e r t o d e r i v e ( 1 ) * o f IO(O*,H*). O*(n) & P ( l , p , n , x )

and use ( i ) ,

*

Vp,x(H*(p,O,x)

(2.1)'

(2.2)'

k

Vp,k,x(H*(p,2

,x)

* Vu3vH*(p,k,o(,u,v>)

v i ,u,v(x= Then p u t O+*(n) := O*(n)

(2.2)',

tf

(2.2)'

o*(P,)

O*

vi(~(l,q,E,i) v +

,

*

O*

v

{m}(i)))

&

and

{m}(i)))

E,

v Fir(p,u,v))).

F(q,k,i)

& Vz,y~(P(O,z,n,y)

v O*({m}(i)))

,x)

& F(p,u,v))

& P(l ,z,n,y))

( 2 . 3 ) ' , a p p l y ( i i ) i n o r d e r t o d e r i v e Vn(O*(n)

i m p l y (2.1)*,

k

o*(k) & Vi(P(l,q,k,i)

tf

E u = Cm}(i)

i n o r d e r t o d e r i v e ( 3 ) * o f ID(O*,H*) Vi(P(l,q,k,i)

~ ( { p } ( x )= 0))

and F ( p , Z

& u = { m } ( i ) & H*(q,k,i)

~(p,3q.5'.7~,x)

*

,

Vp,q,k,rn,~(H*(p,3~.5'.7~,x) 3i,u,v(x=

:=

Then p u t F i r ( p , n , x )

( i i ) i n order t o derive

{ p } ( x ) = 0 and F ( p , O , x )

3 u V v F ( p , k ,o(,u ,v>)) (2.3)'

Then ID,(A)

w h i c h can be shown s t e p by s t e p as f o l l o w s .

+

and, u s i n g (2.1)', OT(n)).

Then a p p l y ( i i )

- use t h e above i m p l i c a t i o n t o t r a n s f o r m

i n t o Vi(P(O,q,R,i)

( 2 . 2 ) * o f ID(O*,H*),

+

O*({m}(i))).

Finally,

w h i l e ( 1 ) * + ( 3 ) * + (2.3)'

(2.1)O,

implies (2.3)*

o f ID(O*,H*). To complete t h e p r o o f o f t h e lemma r e c a l l t h a t e v e r y a r i t h m e t i c a l s t a t e m e n t

-

p r o v a b l e i n ID1, i s p r o v a b l e i n P A + T I ( < ~ E ~ ~ + ~ see O ) [ BFPS]. T h i s f i n i s h e s t h e p r o o f o f t h e theorem.

I I .4. Find -

RematLh6.

1. The s t r o n g e s t s e t t h e o r y i n [ F r 3 ] was T4 = T 3 + ( f u l l ) CA.

Combining ( i n

f a c t , s i m p l i f y i n g ) t h e p r o o f s o f t h e above theorems one can e a s i l y p r o v e t h a t

T$-= lB_*+ I n d ( w ) + S C + R D C + I n d ( E ) + C A , as w e l l as any o t h e r c o n s i s t e n t e x t e n s i o n of

IBf+CA by axioms f r o m t h e above l i s t I - I X .

m e t i c a l s t a t e m e n t s as ( f u l l ) second-order

,

a l l have t h e same p r o v a b l e a r i t h -

i n t u i t i o n i s t i c a r i t h m e t i c HA2 extended

by ECTO+MPO. (N.B. IBT = B * - S P ) .

2 . A l l c o n s e r v a t i o n r e s u l t s from t h i s p a r t ( i n c l u s i n g t h e above remark) a r e preserved i f TC :

IB* i s supposed t o i n c l u d e a l s o Vx3y(Trans(y) & x C y & Vz(Trans(z) & x C z

Exp, :

Vx,y3zVg(gEz

AC! :

Vx,y(VuEx3!

t-f

ycz))

,

Fnc(g) & D o m ( g ) = x & R g ( g ) C y ) ,

v E y cp

cpvP,[g(u)l)) , f o r

-t

-+

3 g ( F n c ( g ) & Dom(g) = x & Rg(g) C y & V u E x

a l l cp.

The p r o o f i s an o b v i o u s m o d i f i c a t i o n o f t h a t o f Theorem 2.

145

Constructive models for set theory

3. A l l

s e t t h e o r i e s f r o m t h i s p a r t ( i n c l u d i n g t h e ones i n remark 1 , 2 )

have

t h e d i s j u n c t i o n p r o p e r t y , e x i s t e n t i a l d e f i n a b i l i t y p r o p e r t y f o r n a t u r a l numbers, and a r e c l o s e d under C h u r c h ' s R u l e CRo.

T h i s s i m p l y f o l l o w s f r o m t h e above p r o o f s

o f t h e theorems and t h e well-known p r o p e r t i e s o f t h e n o t i o n o f r e a l i z a b i l i t y .

4. I n a few words, t h e c r u c i a l s t e p i n t h e above p r o o f s f r o m P a r t I I . 2 , 3 . was as f o l l o w s .

Given an e x t e n s i o n a l c o n s t r u c t i v e s e t t h e o r y T ,

i t s suitable

S was d e f i n e d i n such a way t h a t f o r e v e r y a r i t h m e t i c a l s t a t e m e n t A , (T+ECTO+MPO k A ) * (S+ECTO+SP+MPO I- A) * r e a l i z e s A ) , w h i l e ECT + SP was used t o p r o v e t h e A. - d e f i n a b i 1 i t y o f (S+MP 0 k 0 e x t e n s i o n a l e q u a l i t y and membership r e l a t i o n s ( C l a i m 5 ) o f S e t ( U ) , f o r s u i t a b l e U. i n t e n s i o n a l ( s a y ) second-order v e r s i o n

e

The n o t i o n o f r e a l i z a b i l i t y was based on t h e p a r t i a l r e c u r s i v e f u n c t i o n s .

k

observe t h a t t h e d e s i r e d c o n c l u s i o n ( S + M P o

e r e a l i z e s A)

Now

c o u l d be proved d i -

r e c t l y , w i t h o u t u s i n g ECTO+ SP, because t h e A - d e f i n a b i l i t y o f = , E i s p r o v a b l y 0 r e a l i z a b l e i n S -which f o l l o w s d i r e c t l y f r o m t h e p r o o f o f Claim 5. Moreover, t h e l a t t e r does n o t depend on any s p e c i a l model o f A - c a l c u l u s w h i c h i s assumed t o determine t h e c o r r e s p o n d i n g n o t i o n o f r e a l i z a b i l i t y . choose i t s Feferman - s t y l e e x t e n s i o n SApp

Hence,

i n s t e a d o f S one can

based on t h e a b s t r a c t n o t i o n o f a p p l i -

c a b i l i t y (compared t o [ F l ] , [ F2] no f i r s t - o r d e r c o d i n g o f c l a s s e s i s r e q u i r e d i n SApp). Then c o n c l u d e t h a t f o r e v e r y a r i t h m e t i c a l s t a t e m e n t A , (sAPp F

t

r e a l i z e s A ) f o r some a p p l i c a t i v e t e r m

According t o t h e " r e a l i z a b i l i t y t h e l a t t e r i m p l i e s (SApp

+

+

I- A) *

p r o v a b i l i t y " arguments f r o m [ B e 1

1,

I- A ) . Next t h e m a i n s t e p i s t o show t h a t e v e r y a r i t h -

m e t i c a l s t a t e m e n t p r o v a b l e i n SApp, w i t h t h e corresponding o r d i n a l M.

forcing

(T

t.

i s p r o v a b l e e i t h e r i n HA o r i n H A + T l ( < a )

a ( c f . t h e Main Theorem). ( T h i s method, due t o

Beeson, was used ( i n a l i t t l e more c o m p l i c a t e d f o r m ) i n [ B e 2 1 i n o r d e r t o

prove t h a t

IB i s c o n s e r v a t i v e o v e r HA).

The(proof-theoretic) proof o f t h i s re-

d u c t i o n does n o t seem t o be e s p e c i a l l y d i f f i c u l t e x c e p t f o r t h e case when S con1 t a i n s C o l l e c t i o n ; one c o u l d n o t succeed i n (A. - r e a l i z a b i l i t y ) because o f t h e

1

e s s e n t i a l r o l e o f c l a s s i c a l l o g i c i n t h e p r o o f o f (Xi -AC)

*

1

(Ai - C A I .

The p a r t i a l r e s u l t a l o n g t h e s e l i n e s i s as f o l l o w s I.

lB+SC+Fnd/Cpl

II

,

i s c o n s e r v a t i v e o v e r HA.

V I , V I I . Unstarred versions o f s e t t h e o r i e s from the f i r s t l i s t

I I , V I , V I I ( c f . t h e Main Theorem) a r e r e s p e c t i v e l y c o n s e r v a t i v e o v e r HA+Tl(
w i t h the corresponding o r d i n a l

c1

f r o m t h e second l i s t I I ,

VI, VII.

O f course a l 1 these conservation r e s u l t s a r e preserved f o r TC, Expl,

AC!.

Moreover,

this i s s t i l l true ( [ G 3 ] ) with

IB extended by Enm,

IB extended a l s o by t h e

axiom o f c h o i c e f o r a l l f i n i t e t y p e s , w h i c h i s known t o be i n c o m p a t i b l e w i t h CTO.

L.GORDEEV

146

B I BL I OGRAPHY. Beeson, M.

Goodman's Theorem and beyond. Pac. Journ. of Math.

(841,

1979, 1-16. Beeson, M. E x t e n s i o n a l i t y and choice i n c o n s t r u c t i v e mathematics Pac. Journ. o f Math., (88), 1980, 1-28. Buchholz, W. Normalfunktionen und K o n s t r u k t i v e Systeme von Ordinalzahlen. Proof Theory Symposium, K i e l 1974. Springer L e c t u r e Notes i n Math., i+ 500, 1975, Springer V e r l a g B e r l i n , 4 -25. Buchholz, W., Pohlers, W. Provable w e l l - o r d e r i n g s o f formal t h e o r i e s f o r t r a n s f i n i t e l y i t e r a t e d i n d u c t i v e d e f i n i t i o n s . J . o f Symb. L o g i c , (431,

1978, 118 - 125.

Buchholz, W., Feferman, S., Pohlers, W., Sieg, W. Iterated inductive d e f i n i t i o n s and subsystems o f a n a l y s i s ; recent p r o o f - t h e o r e t i c a l studies. Springer L e c t u r e Notes i n Math. # 897, Springer Verlag B e r l i n , 1981. Feferman, S. A language and axioms f o r e x p l i c i t mathematics. Algebra 450, 1975, 87 -139. and Logic. Springer Lecture Notes i n Math., i# Feferman, S. C o n s t r u c t i v e t h e o r i e s o f f u n c t i o n s and classes. In M. Boffa, 0. van Oalen, K. McAloon (eds.). L o g i c Colloquium '78. North-Holland Publ.

Co., Amsterdam, F i n s l e r , P. ( 2 5 ) , 1926,

1979, 159 - 224.

uber d i e Grundlagung der Mengenlehre. Math. Z e i t s c h r i f t ,

683-713, 1

Friedman, H. I t e r a t e d i n d u c t i v e d e f i n i t i o n s and Z2 - A C . I n A. Kino, J. M y h i l l , R.E. Vesley (eds.), I n t u i t i o n i s m and Proof Theory. NorthHolland Publ. Co., Amsterdam, 1970, 435 - 442. Friedman, H. The consistency of c l a s s i c a l s e t theory r e l a t i v e t o a s e t t h e o r y w i t h i n t u i t i o n i s t i c l o g i c . J. o f Symbolic Logic, (381, 1973,

315-319. Friedman, H. Set t h e o r e t i c foundations f o r c o n s t r u c t i v e a n a l y s i s , Ann. o f Math., (l05), 1977, 1-28. Gordeev, L. A m a j o r i z i n g semantics f o r h y p e r a r i t h m e t i c sentences. Nauchn. Semin. LOMl, (68). 1977 (Russian).

Zap.

Gordeev, L. An approach t o c o n s t r u c t i v i z a t i o n o f C a n t o r ' s Theory o f Sets. The same volume. Gordeev, L .

IB+ACFT

i s s t i l l c o n s e r v a t i v e over HA. (unpublished n o t e ) ,

1980. J l g e r , G. A w e l l - o r d e r i n g p r o o f f o r Feferman's theory T o . To appear i n A r c h i v f a r Math. L o g i k und Grundl. Jzger, G. & Pohlers, W.

(4-

CA)

+ 81

Eine beweistheoretische Untersuchung von

und verwandter Systeme. To appear i n S i t z u n g s b e r i c h t e der

Bayerischen Akad. der Wissenschaft, MUnchen. [Mi

1

M i r i m a n o f f , D. Les antinomies de Russell e t de B u r a l i - F o r t i e t l e probliime fundamental de l a t h e o r i e des ensembles. L'Ens. Math., (191,

1917, 37-52. M y h i l l , J.

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Shanin, N. On a hierarchy of methods of interpreting propositions in constructive mathematics. Trudy Math. Inst. Steklov, (129), 1973, 203 266.

[TI

Troelstra, A. Metamathematical investigation of intoitionistic arithmetic and analysis. Springer Lecture Notes in Math., 344, Springer Verlag Berlin, 1973.

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147

THE LXJ. BROUWER CENTENARY SYMPOSIm A S . Troelsira and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

149

A NOTE ON BAR INDUCTION RULE susumu

HAY AS HI^)

Institute of Mathematics, University of Tsukuba Sakura-mura, Niihari-gun, Ibaraki, Japan

The bar induction rule (BIR) is a derived rule corresponding to the principle of bar induction. BIR for HA(f) was noted by Kreisel [19651, 3.341. BIR for HAS and HAH was proved by Hayashi [1977, 19801 and BIR for HAw was proved by Howard [1980]. On the other hand, Schwichtenberg [1977] proved the bar recursion rule for HAw. The purpose of this paper is to prove BIR for the systems which are introduced

by Feferman and Friedman to formalize Bishop's constructive analysis (BCM) and also for some stronger systems of intuitionistic set theory. The proofs of BIR mentioned above are syntactical. The proof of Hayashi [1977] is very similar to Brouwer's 1 'proof' of the bar theorem, i.e. any normal derivations of 111 theorems of HAS or HAH are very similar to the so-called fully analyzed proofs. This will be an advan-

tage of the syntactical method. However, the proof theoretical method (Gentzen's method) is not flexible. It seems to be impossible to apply such a method directly to the systems considered in this paper. We will overcome this difficulty in the following two methods. For the systems without sufficent inductive definitions we will reduce BIR for them to BIR for subsystems of analysis by the methods in Feferman [1978], Friedman [1977], and prove BIR for such systems by Gentzen's method. For the systems with sufficent inductive definitions we will use semantical methods. In Beeson [1977], Beeson used a forcing method (uniform forcing) to prove the rule of uniform continuity, which is a generalization of the fan rule. The semantical proofs of BIR for various systems were found by some people independently. Their ideas were essentially same, but the ways of presenting the ideas were different. Beeson used K-forcing, Grayson used Heyting valued models over formal Baire space, Troelstra used the elimination of choice sequences and the author used a topos theoretic method, which is equivalent to Grayson's method. Fourman and Joyal used a beautiful topos theoretic method which is a refinement of the method of Grayson and the author. Unfortunately, none of them is suitable for all systems considered in this paper. For this reason, all versions of the proof will be used in this paper.

I would like to thank the people who helped me. Especially, I would like to 1) The author is partly supported by Grant in Aid for Scientic Research Proj.

No. 434007.

1so

S. HAYASHI

thank Prof. M.J. Beeson, Prof. M.P. Fourman, Dr. R.J. Grayson and Prof. A.S. Troelstra for information on their unpublished work on BIR and for their permission to make use of it. I am grateful to Dr. Grayson and Prof. Troelstra for critical comments and suggestions on this paper.

I . PRELIMI;\IARI€S The systems considered in this paper are divided into three groups as follows: 1. Feferman’s systems for BCM,

2. Friedman’s systems for BCM, 3. proof theoretically strong subsystems of IZF+RDC. Feferman’s systems are based on the notions of functions and classifications.

A ECA 5

The systems considered in this paper are EN;-’/ and EM:-)

+J+ IG(-)r?

t ECA

IG(-)

J , EML-)l+

variants of these with choice principles will be studied EM:-)

+ J + IG(-)

by Th-) and TA-’

J 4. IG(-)r+

ECA

in the notation of Feferman [1978]. Some

+ ECA by

too.

We will denote

To as in Feferman [19781.

Friedman’s systems for BCM are subsystems of IZF

+

RDC (intuitionistic ZF set

theory with the relativized dependent choice). They are distinguished from the systems in the third group by their considerable lack of proof theoretic strength. The systems in the second group considered in this paper are B, T1’ T2’ T3, T4 in the notation of Friedman [1977]. The systems in the third group are not adequate for BCM, but adequate for topos theoretic investigiations. We will study IZF, IZ, 1Z and their variants with 0

or without ACC or RDC. IZF is the intuitionistic ZF set theory with collection, IZ is intuitionistic Zermelo set: theory, IZ is the system obtained from I2 by 0 restricting separation to A -separation, ACC is the axiom of countable choice and 0

RDC is the axiom of relativized dependent choice. We will consider the following forms of BIR for these systems. Let S1 and

S

2

be classes of formulae of a theory T. (S1, S2)-BIR: if Vf 3n A(&) & A & U < ~ is a theorem of T and A is a unary formula in the set S1 then the following is also provable in T: (AcB where Ind(B) is wzu

Vu E

w

<w

&

A : monotone &

( Va

E

w(u*

L E X)

+

Ind(B)) u

E

-f

W < ~ ~ B ,

X) , B belongs to the set S2 and

is the set of finite sequences from w. If S1 and S2 are both the class of arbitrary formulae, then (S1,

be denoted by FBIR. If S1 is the class of arbitrary formulae and

S2

S2)-BIR will is the class

of formulae which are permitted for the comprehension or separation in the system T, then (S1,

S2)-BIR will be denoted by FBIRO. Hence the conclusion of FBIRO is

equivalent to the one which is obtained through replacing the formula B by a new class or set variable. For the systems B, T1, T2 in the group 2, we will consider

151

Bar induction rule

(D, S2)-BIR where D is the class of definable formulae in the sense of Beeson [1978] and S2 is the same in the case of FBIRO. We will denote this by DBIRO. Note that the full bar induction rule FBIR does not hold for all systems considered in this paper. We list here the systems for which FBIR does not hold. PROPOSITION 1. FBIR does not hold for the following systems: 1. all of the systems in the group 1 except Ti-), To, 2. B, T1, T2 in the group 2 , 3 . I2 in the group 3 . 0

PROOF. Let T be one of those systems in this proposition. Let< be a pnnitive recursive ordering on the natural numbers. Define Ax by x€w<' & x=<xl, ,xn> & l(xl Z

...

... > xn).

If WF(<) is provable in T and BIR holds for T, then TI(< ,F) is provable in T for any formula F. (WF and TI are defined as in Feferman [1977],

&

<

be the although we use the following version of WF: Vf 3n( lf(n+l)< fn).) Let standard ordering on the natural numbers. Of course 1WF(<). If FBIR holds for EMi-)b, then the schema of full induction holds for it. Then EM;-)\ to EM;-).

However, EM:-)

is stronger than EMi-'r.

is identical

Indeed we can prove the consist-

ency of the latter in the former. (Cf. Feferman 119781). Similarly for the systems without full induction, e.g. B, T1, IZo. The proof theoretic strength of each of these systems was estimated in terms of a subsystem of analysis with the same proof thoretic strength in Feferman [1978], Friedman [1977]. By these results we can see that full induction increases the proof theoretic strength of these systems. For I20 the proof thoretic strength is just equivalent to that of HAH. Indeed we can prove Consis(HAH)

Consis(IZo) in HAH by means of the semantics in Hayashi [1981].

Since a model of HAH is constructed in IZ0 system. Let T be the system EMo

+J+

+

full induction, IZ0 is weaker than this

IG(-)r.

The elementary comprehension ECA for

all elementary formulae can be replaced by a finite number of special cases without loss of effect. (The special cases of ECA are essentially equal to the axioms generating classes in BG.) So we may assume that T is finitely axiomatized except for full induction. So we can use the well-known argument of Kreisel and Levy [1968]. Namely, there is an instance of a formula F such that TI(< ,F) is not provable in T, where

<

is the cannonical ordering up to

FBIR does not hold for the systems EM(-) 0

+

J

cO.

Since WF(<) is provable in EM(-) 0 ' ECA. Finally we deal the

+ IG(-)b

system T2, which is not finitely axiomatized. Obviously WF(<)

is provable in it.

Hence, if FBIR holds for it, then the schema of full induction up to cb holds for it. Then we can construct than

E )

a

model of R A ( < E ~ )(ramified analysis with levels less

in T2. However, T2 and RA(
152

S. HAYASHI

other systems mentioned in Proposition 1, we will prove the full bar induction rule FBIR. For the above three systems DBIR holds. 0

Now we prove an important lemma, by which we can reduce our monotone BIR to a recursive BIR. LEMMA 1 (recursive continuity rule). Let T be one of our systems except B, T1' T2. If Vf 3n A(5n) is a theorem of T, then there is a natural number e such that the following formula is also provable in T: Vf cww(f={e1

&

Vgcww(!f(g) & A(f(g)))), where f(g) and !f(g) are defined as in Troelstra [1977]. For B, T1 and T2 this rule holds provided that the formula A is definable in the sense of Beeson [1978]. PROOF. This lemma is a direct consequence of the results on the formalized explicit definibility property in Beeson [1977, 19781. Some systems in the group 1 and 3 do not appear in Beeson 11977, 19781, but this is not a problem. Note that the method of constructing the model of IZF + RDC in the corresponding non-extens i o n a l theory in Beeson [1978] does work even for T3 and T4, although Beeson used

other models for these two systems. Thus we can prove this lema for these two systems without restriction on the formula A. By means of this l e m a we can reduce the monotone BIR to a recursive BIR. Assume that the assumption of the l e m a holds and e is the number obtained by the lemma. If we set R = {uE w<(": 3f(uL{e)(f))>, then the following formulae are provable in T: <w R is a monotone subset of w , Vf iIn(Fn E R,) , if A is monotone then RegA. Hence it is sufficent to prove the following versions of BIR. RBIR (RBIRO): Let R be a formula representing a primitive recursive predicate. We denote the formula Vf 3n(Fnc R) & R:monotone by WM(R). If WM(R) is a theorem of T then the following formula is also provable in T: (RGF & Ind(F)) -+ U < ~ S F , where F is an arbitrary unary formula (F is a formula permitted in the rule BIRO). We will prove these versions of BIR in the following sections. 2 . WEAK SYSTEMS: SYiWACTlCAL PROOFS

In this section we will prove RBIRO for EM:-)I'

kJ

ECA, EM:-)

+ J 5 ECA,

B, T1 and T2. We will consider also some variants with choice principles. First we prove a lemma, by which we can reduce BIR for systems with choice to BIR for systems without choice.

Bar induction rule

153

2 IG(-) 5 IG(-) 1 ACFT is conservative over T for Ill formulae.

2

LEMMA 2. Let T be one of the systems EMi-)F+ J tion). Then T

2

5

ECA

(full induc-

1 PROOF. By Ill formulae we mean the formulae which are logically equivalent to one of the following form: Vf 3n t(’in)=O, where t is an application term of EM:-) representing a primitive recursive function. Since formulae of the form f

Yl F

are essentially (V,3)-free, the soundness theorem of 71-realizability for To* in Feferman [1978], IV, 11.1-12.2 can be refined so that T,,*

+

(AC~~)*CF +

Furthermore, we can find a term

3f T:-) s

t

f tl F

for

L*.

FE

such that

(-+- Vf E 8 (f,sf)E#. TO 1 So the i- -realizability does not change the essential meaning of I$ formulae. Hence 1 Lemma 2 holds for T:-’, and similarly for the other systems. It should be noted that this result is implicit in Beeson [1977], Theorem 1.6.5.

PROPOSITION 2. RBIRO holds for EMi-)p

5

J

5

ECA

+ J 2 ECA 5

2 ACFT, EM;-)

ACFT, B, T1 and T2. PROOF. By Lemma 2 it is sufficent to prove RBIRO for EMi-)r

5 J,

EM:-)

+ J,

B,

T1 and T2. Firstly we deal with Feferman’s systems. Assume that WM(R) is provable in EM(-) + J, where R is a formula representing a primitive recursive predicate. 0 1 Feferman [1978] constructed a model of EM$-)r + J in (I1-AC)rby interpreting the 1 classes as Al-indices. We may relativize this construction to any number theoretic 1 1 functions. Hence WM(R) is provable in (X1-AC)r. It is known that (X1-AC)l is conservative over PA. Furthermore we can show that (E:-AC)P

is conservative over

PA(f),

where PA(f) is the system obtained from PA by adding a function variable f. 1 Note that we consider f as a variable of (Z1-AC)t. (We will use the notation T(f) for other systems with the same meaning, and we will also use the notation T(X), where X is a predicate variable.) This conservation result will be proved by various methods. We adopt Friedman’s argument. In Friedman [1977] he showed Kj which 0 1 is an extension of (Z1-AC)r is a conservative extension of PA for II2 sentences. It is fairly easy to relativize this result. Namely K3 (f) is a conservative extension

0

of PA(f) for 112 formulae. Let M be the set I~Ew<~:uERVlth(u)<2}. Note that M is infinite and recursive. Let i be a bi-recursive isomorphism from w to M such that iy, i< >=O. We define a recursive ordering R on w such that x XR y iff ix where * is the Brouwer-Kleene ordering on Then the ordering 4R on w has

<*

<

the maximum element 0. Let TI(
be the sequent Vx( Vy(ydRx

It is not so difficult to see that TI(IR,X) is provable in PA(X).

+

Xy)

+

Xx)=?XO.

Let Zm(X) be the

infinitary system of first order arithmetic Zm with X. (Cf. Schwichtenberg [19771.) Let Il be a cut free derivation of TI(XR,X) in the system Z_(X).

As in Hayashi

[1977a], we analyze the structure of this derivation. First we define a subtree P of Il as follows:

S. HAYASHI

154

1. The end sequent of II belongs to P. 2. If the lower sequent of the following inference belongs to P:

r * x, s
r then an upper sequent

r

=)

=)

X,

x,

s<

+

for all s e

Xs

vy(y
+

w

XY)

t + Xs belongs to P iff s X R t is true.

R

3 . If a lower sequent of an occurrence of another kind of inference belongs

to P, then all upper sequents of it belong to P. Second we replace each occurrence of formula in P according to the following translation F b F*: Xt* s<

R

=

Xt, ( Vy(y<,t

-f

Xy))*

=

Xt, ( s d R t

-f

Xs)*

=

Xs, where

t is true, and F* is empty, i.e. F is eliminated, for the other formulae. Let

11* be the resulting tree of sequents. Then it is a recursive infinitary derivation

with the following inferences: 1. If s and t represent the same number, then Xs 2.

* Xt is an initial sequent.

The usual structural rules, the rule of cut, and the rules of repetition,

i.e., S/S and

r

-e A

0

-:/re

A n , are permitted.

3 . The progression rule, i.e.

r *TI,

Xs

for all s such that s i R t

r * n,

xt

I

is permitted. If t represents the minimum element, then this rule turns into an initial sequent

r * II,

Xt. (Note that we can find the minimum element

effectively.) Weeliminateall cuts in TI* by the standard method (cf. Schwichtenberg [ 1 9 7 7 ] , 3.5) and eliminate all rules of repetition by the trivial method. Let l’l 0 be the resultXsn* ing derivation of *XO. Then all sequents in IIo have the form * Xsl, Set min( *Xsl,...,Xs ) = minIXsl, Xs }. Assume that {x:FxI is a class and min( * A ) + Fz) for all Vx( Vy‘yiRxFy Px) holds. Then we can prove V z ( z

...,

...,

-f

sequents * A

in IIo, since llo has an ordinal assinment less than e O . Hence we can

conclude VxFx. These arguments can be formalized in EMk-’p.

Hence we have proved

V X(TI(
+

(C

U < ~ E X is ) prov1 -AC) instead of 1

1 (hl-AC)p. Next we use the following result of Friedman 119701 : (Zt-AC) is a conser1 vative extension of RA(<E 0) for 112 formulae. Then we see that TI(dR,X) is provable

in R A ( < E ~ ) if Vf 3n R(%)

is provable in EM-:’

+ J.

By SchUtte 119771, Theorem

22.7 and the inequality $aB < $ E ~ O for any a, B < e O , we can prove RBIRO for EM;-’

+J

if TI(a,X) is provable in EMo

+J

for each a < 0 ~ ~ This 0 . condition is

easily verified, since SchUtte [ 1 9 6 5 ] , L e m a 5 is easily formalized in EM:-) + J. Note that Schiitte formalized his proof of TI(0e.O) in an intuitionistic version of RA(
Using ECA(-)

+

Ga <

E ~ .

and the join axiom, we can construct a model of the intuitionistic

Bar inductionrule

155

RA(
contain existential quantifiers and there are no essential uses of disjunctions in his proof of the transfinite induction. So ECA(-) is sufficient for the model 1 construction. The II2 theorems in B, T1, T 2 are also provable in PA(f), PA(f), RA(<E~), respectively. This is proved by relativizing the proof of Friedman [1977],

Theorem (p. 2 3 ) . Hence by the same arguments as for Feferman's systems, we can prove RBIR,,

for these systems. Note that we can perform the above formalization of

Schiitte's proof of the transfinite induction in T2, using A 0 -separation and strong collection. As a byproduct of the above proof, we can estimate the suprema of provably recursive well-orderings of those systems.

+

COROLLARY 1. The suprema of provably recursive well-ordeings of EMi-)b+ J + J ECA f. ACFT, B, T1 and T 2 are, respectively, E ~ , E

ECA f. ACFT, EM:-) E~

and

~ ,

$goo.

3 . STRONG SYSTEMS: SEMANTICAL PROOFS

In this section we will prove RBIRO for EMh-)f & J

+

IG(-)E+

+ ACFT

ECA

(full inducrion) and RBIR for the remaining systems by semanticel methods. We will prove RBIR for IZF precisely and will give brief sketches of proofs for the other systems. To prove RBIR we will use the cHa of formal opens of Baire space. We will denote this cHa by 0. The theory of formal spaces is developed by Fourman and Grayson [198?]. We do not consider the general theory. We will define Cl directly. In the following u, v, w... are used to denote finite sequences of natural numbers, uzv means 'u is an extension of v' and [u] denotes {v:v
Set

Ju = {R: R is a well-founded monotone bar of u, i.e.

Vv
&

Then J is a Grothendieck topology on the poset

w<',

VX((R5X

&

Ind(X))

+

vu~R([u]cR)

&

RS[u]I.

i.e. it satisfies the following

four conditions:

1. Ju is a family of monotone subsets of [u], 2.

[u] E J U ,

3. if R

E

Ju and v
4 . if R E Ju, R' is monotone subset of [u] and Vv ER(R' n [v] E Jv), then R' E Ju. Note that Ju is a filter by these properties, i.e. R n S E J U for R, S s J u and if S Z R E J U and

S

is a monotone subset of [u], then S E J U . Set 0 = {S:Scw

<w

Let R be a monotone subset of

& w

S:monotone

. Set

<w

&

V R E J U (Rc_S-tu~S)l.

S.HAYASN

156

jR = {u:Rfl[u] E J u I .

By the above properties of J. we see that R G j R and jR belongs to Q. Furthermore, jR is the smallest set in R that contains R as a subset. Hence j is identical on R. LEMMA 3. R is a cHa, i.e. Q, a,

A

,v exist and vx (RA

A

S)=VRA

x

S.

A

PROOF. If R, S E Q , then R n S E n . Hence RAS=RES. If RA E R , then j( U R ) E R . Hence \ / R a = j (

u

x

Rx).

Let Rh,

If u belongs to

S EQ.

VRa and S ,

A E

X

R n S n [u]

V(RAh S).

x

LEMMA 4 . Let RA, R E G and XSw<'.

If V R A = R , U R C X and Ind(X),

x

A UE

U

a x

A

belongs to Ju, i.e. u

PROOF. If

A

then

then RSX.

A-

R, then u RAn [u] belongs to Ju, i.e. it is a well-founded bar

of u. Hence u belongs to X.

x

Note that these facts are provable in IZ0. Hence we can construct)'(V

in IZF.

(Cf. Grayson [1979] for V(n).) Assume that a unary formula R satisfies the assumption of RBIR. By the formalized soundness theorem, IZF k'V(')k WM(R)'. From now on i.e. we will work in IZF. Let f be the generic element of uw in V"), dom(f)=~x~, f(<&ii>)={u:u= & u =n}. m

,...

Then it is easy to check (f:w + d = 1. Assume that Ind(X) and R E X . Let n be a natural number and let v be an finite sequence of natural numbers. Set Rn,v = f ? R~ & >?n'n. Then

v

Rn,v = 1, E K where K is the set w x .'
v

g./

V

v

Thus u is an extension ,l[fO=v & fl=vl &...& f(n-l)=vn-lJ, where V=. 0 of v. Since R is monotone, u belongs to X. We have proved that R CX. E K n,v By Lemma 4 we see that w<'EX. This completes the proof of RBIR for IZF. We will show IZF [Vx

E

+ ACC k

'V(')k

ACC'. Assume that u belongs to the set

w 3y E a A(x,y)l.

Then u belongs X ={u:

to

Fix a natural number m and set R = aynBA(g,y)] for y dom(a). Y Y R y , where y runs through dom(a). Set

31viIi wIyili ,(i#j

+

[vil A [v, 1 4 .

v Ivil=Iul and vi

E

1A(g,yi)l)l.

i

Then obviously

h,Ii

w,

hiIi

5 X and Ind(X) by ACC. By Lemma 4 u belongs to X. Take Y Y such that they satisfies the condition in the definition of X

U R

for U. We denote these by {vm,ilm,i E u , on m. (ACC is used!) Set

1m,iE

to indicate the dependence

Bar induction rule

157

v

v

dom(f)=wxdom(a),

f(<m,y>)= V[vm,ilh UY=Y~,~]. i Then u belongs to the set [f:w-+a & Vx cwA(x,fx)D. Similarly RDC is valid in)'(V by RLIC itself. So RBIR holds for IZF t ACC RDC. Similarly for the other systems. For the system T4, there are some problems. Since T does not have the axiom 4 of power set, is not a set but a class. So we must modify the definition of V"). For technical reasons, we will use the sheaf theoretic formulation of V(n) in Grayson [1975], Takeuti and Titani [19811. (See Grayson [1981] for a general theory of forcing in intuitionistic systems without power-set.) In T4 the supremum of any subclass { R X I A E Aof Sa is also given as in Lemma 3 , since there exists a set I s h such that U R = U R Note that the recursion ?,€A AcI theorem in Grayson [1975] holds in T4. So by transfinite induction on E , we can define V F ) E TC({a}) for any set a as follows: x=cVF) iff Fnc(lx1) & dom(lxl)cVF) & range(Ixl)cQ &

.

&

Ex€

xyLExA Ey.

&

Define V(n) by iff a E V y ) . aE We can define [at:bl , b=bj as in Takeuti and Titani [1981]. The axioms of T 4 except the axiom of exponentiation are satisfied as usual in the Heyting valued model, and this is easily verified in T 4 . We will show the validity of the axiom of exponentiation in T4. and {RiIiE IgSa such that [u]= V R i , there i €1 I such that [u]= Rfn. And this fact is provable in T 4' n EW

LEMMA 5. For any element u of w<' exists a function f:w

-+

v

PROpF. Standard and left for the reader. Note that we need ACC to prove this. Let a, bc V").

Without loss of generality, we may assume that a is a presheaf,

i.e. a is definite in the sense of Takeuti and Titani 119811 and for any p e wcw and x E dom(a) the element x b of V(n) also belongs to dom(a). Furthermore, we assume that b is definite. By the exponentiation axiom the collection of all denumerable subsets of dom(b) which are compatible in the sense of Takeuti and Titani [1981] is a set. We denote it by X. Set

n =n,

B = {VA:A< X I / [ = obtained glueing the elements of A (cf. Takeuti where V A is the element of)'(V and Titani [1981]). Of course B is a set, and E and 1 = 1 on the elements of B are induced from E and 1 = 1 on V(*).

For any a E V(n) we define a* E V")

dom (a*)=Ix cdom(a*):Ex=[u]}, dom(a*)= Then ua*=a]= 2 . Let Y be the set

"w

UE

<w

{
For any cY,

[u]7:

u <wdomu(a*), a*(x)=ax, UE

w

Ey=Efy, Ef A [y=y'I(

we define Ffc V(n),

nfy=fy']l 1 .

say g, by

by

Ea*=Ea.

158

S. HAYASHI Eg=Ef=[u], dorn(g)={<x,y>':

Define

C E)'(V

[c=ba] =

by dom(c)=F"Y,

1. This

x cdomu(a*) & y cfx), u(<x,Y>)=ExA Ey. cx=Ex, and Ec= %. Then it is straightforward to show

completes the proof of FBIR for T4. Note that the fundamental

ideas of this verification is due to Beeson [1978]. REMARK 1. Since we needed ACC in Lemma 5. The above proof is not applicable to T -RDC. However, T4 is reducible to HAS and BIR for HAS is known. So we can 4 prove BIR for T -RDC by a syntactical method.

4

REMARK 2. The proof of BIR using the cHa

is due to Grayson. The author's

original proof was topos theoretic. He used models in the Grothendieck topos over the site (w<',J).

By the aid of the comparison lemma in SGA 4, it is easily <w ,J) is equivalent to the topos over 62. Hence

checked that the topos over (w

Grayson's method and the author's method are practically equivalent. However, Grayson's approach is more understandable for logicians. REMARK 3. Fourman and Joyal [198?] have obtained a refinement of the above proof. Their proof does not use Lemma 1. They use the glueing method which was used by Freyd to prove the explicit definability property for intuitionistic simple type theory. Roughly speaking, Fourman and Joyal used the rule of continuous local choice in Hayashi 119801 instead of Lemma 1 and performed the two steps in our proof at once by using more sophiscated cHa. However, it seems impossible to

apply their method to T4 and weaker systems. It is impossible to apply the above proof to T3, since full separation is

necessary to show the validity of A -separation in 0

So we cannot prove BIR of

it directly as for the above systems. Friedman [1977] reduced T3 to ID(o), which 1 does n o t have separation axioms. As in section 2 we can prove that 111 theorems of T are also provable in the system BI in Friedman [1977] by using a relativized 3 version of the [ I-realizability in Friedman [1977], Lemma 3.4. By the reduction 1 steps in Zucker [1973], p.443, we see that 111 theorems of T3 are also provable in EL

+ BIO. Assume

that a formula R satisfies the assumption of RBIR for Tg. Then we

see that CS

t

Va 3n(ineR)

by the above argument. Using the elimination theorem (cf. Troelstra [1977], 6.16), we see that IDBl I- 3e E K Vn(en#O

-f

Va

E

n(a(en-1)

E R)).

We can construct K in T just as in constructed in T3 in Friedman [1977]. So we 3 can interpret IDBl in T3. Hence we can prove the conclusion of RBIR f o r R by using the induction principle for K in T 3' REMARK

4. By the above,argumentusing the elimination theorem, we can prove

RBIR for IDB1. This is due to Troelstra. Note that ACC in IDBl is necessary for the

Bar inductionrule

elimination theorem,

so

159

also in this case we needed ACC. In general ACC is used in

the weak systems,in which the topology J is not small or does not exist,to replace J by more convenient covering families. It should be noted that the elimination method is not equivalent to the semantics over the cHa of formal opens of Baire space. G. van der Hoeven and I. Moerdijk have shown that it is practically equivalent to a monoid forcing. Finally we will prove BIR for Feferman's systems with inductive generation axioms with the aid of Beeson's K-forcing. K is the class of the neighbourhood functions defined by the inductive definition IG(-)r

or IG(-).

K-forcing is essen-

tially equivalent to sheaf semantics (Kripke-Joyal semantics) over the site (w<',J),

if we assume ACC. In K-forcing, only the covering families induced by the

members in K are used, so we need ACC to prove the soundness theorem for it. However, Feferman's systems considered here have no choice principles. So we need the following lemma.

J + IG(-)t'? IG(-) 5 (full LEMMA 6. Let T be one of the systems EMi-)I + ECA + ACFT is conservative over T for formulae of the form

induction). Then T Af.{e}(f)

E K , where e is a numeral.

PROOF. Assume that Af.{e)(f)

E

K is provable in T

+ ECA + ACFT.

As in Lemma 2,

u ,

we see that 3z((z,hf.{e}(f))

EK) is provable in T, where

is defined as in

Feferman [1978], 11.3. Note that K is inductively defined. Take the classes A and R such that K=i(A,R). XE

A

iff 3z((x,z)

It is easily checked that A and R are self realizable, i.e.

e l # ) , since they are defined by essentially .(V,3)-free

It is straighforward t o prove the following in T: (x,z) self realizable A and R. So hf.{e}(f)

E

formulae.

~ r f i , x )+ X E i(A,R)

for any

K is provable in T.

K-forcing is defined like the uniform forcing in Beeson [19771, 1.8 except for the following modifications: P IkA B i s Vqsp(qikA + 3aE K(q,alkB)), p Ik VxA i s Vx3a E K(p,a IF A ) ) , +

where p,abA means Vq(aq#O

-+

W I C h ) . Then the following soundness theorem holds.

PROPOSITION 3 (Beeson). Let S be one of the systems in Lemma 6, and let T be

S

+ ECA + ACN.

If TaCA, then T k 3 a ~ K ( < >,alkA), where Ta is defined as in

Beeson [1977], 0.3. We may add ACFT to T in this soundness theorem. PROOF. The soundness theorem is proved as in the case of uniform forcing. We need ACN to prove the K-forcing version of Lemma 1.8.2 of Beeson [1977]. With the above lemma and proposition, we can easily prove RBIR

0

or RBIR for

the systems considered here. The details ere left for the reader. It should be

S. HAYASHI

160

noted that BIR for the systems with ACN are due to Beeson and only Lemma 6 is due to the auther. Thus the following proposition has been proved in this section.

2J

PROPOSITION 4. RBIRO holds for FM:-r'

+

IG(-)r 2 ECA k ACN A ACFT

2 J +IG(-) % ECA induction) and RBIR holds for EM:-) T k ACC 2 RDC, where T is IZo, 12 or IZF.

t ACN 2 ACFT,

2 (full

Tj, T4 and

4 . SUMMARY

We have proved the following theorem. THEOREM. FBIR or FBIR holds for the systems considered in this paper except 0

B, T1, T2. For these three systems DBIRO holds. Furthermore, there are counterexamples of FBIR for the systems for which FBIR is not proved in this paper. FBIR for the above three systems, EM:-) 0

2 ECA and FBIR for intuitionistic set

theories with only replacement are open problems.

APPENDIX

Troelstra showed in an unpublished note that a generalized bar induction rule (GBIR) for IDB* is an easy consequence of Troelstra [1980]. This appendix was written to present his proof and sketch how to extend it to some of the systems in this paper. Let IDB* be the system in Troelstra [19801 and let A be an almost negative arithmetical formula. Then Troelstra's GBIR is as follows: if k if e AW 3nP(h), then

k(

h/n'E'A

Vxx'e.A(Pn+. P(n*p))

where P c A Q is VxcACW(Px

+

&

P c A Q & IndA(Q))

Qx) and IndA(Q) is VncA<'(

-+

A<Wc_P,

Vyc A(P(n*9))

-+Pn).

THEOREM (Troelstra). IDB* is closed under GBIR. PROOF. Assume that Vf E A" 3nP(?n)

is provable in IDB*. In Troelstra [19801

it was shown how to handle realizability by numbers for IDB*; exactly the same methods can be used for functional q-realizability to obtain a generalized recursive continuity rule for IDB*. So the following is provable in IDB*: 3f ew"(f-{e) & Vge AW(!f(g) & A(f(g))). Hence we may assume that P is a recursive predicate. By Troelstra [1980], 5.8 we see that CS* t vu E A~ jnP(in). By the generalized elimination result Troelstra [19801, 4 . 3

161

Bar induction rule

IDB*k 3e E K Vn E A<"(en#O -+ Va E nP(a(enL1))). [A] Then by the principle of induction over unsecured sequences, we can prove the conclusion of GBIR. The semantical proof of BIR which does not use ACC can be used to prove GBIR without any modifications except that it uses the generalized recusive continuity rule instead of Lemma 1. Indeed we can prove recursive GBIR (i.e. P is recursive) for any arithmetical formula A for IZo, IZ and IZF. The semantical proofs which use ACC can be used to prove GBIR, if ACC is replaced by the axiom of partial choice APC. For example, let APCar be the schema b={xE w:Bx)

&

V x E b 3yc cA(x,y)

-+

3f:b+cA(x,fa),

where B is an almost negative arithmetical formula. Indeed T4 + APCar is closed under GBIR. Furthermore we can prove GBIR for Feferman's systems in Proposition 4 without adding APC. Let ACm be the choice schema in Feferman's language corresponding to APCar. Let T be one of Feferman's systems in Proposition 4. Then we can prove GBIR for T + ACm by KLA1-forcing.Similarly to Lemma 6, we can prove the conservation result on T with respect to the sentences Af.{e}(f) eKLA1,where A is ess. ( v

, 3)-free. Hence we can prove GBIR for T as in Proposition 4 .

On the other hand, we cannot extend the proofs of BIR which use the conserva1 tion results. We used the conservation results with respect to ll -formulae for 1 1 such systems. Now we need the conservation of extended IIl-formulae of the form

vf E A" gnP(?n),

where P is recursive and A is almost negative and arithmetical.

The author does not know how to prove this except for EMOF and B. Beeson [1980] has proved that these two systems are conservative over HA. His proof can be relativized, so these systems are conservative over HA(f). EMh-)r+

Hence.GBIR holds for

ECA and B. The rules GBIR for the other systems are still open problems.

BIBLIOGRAPHY Beeson, M.J. [1977] Principles of continuous choice and continuity of functions in fqrmal systems for constructive mathematics, Annals of Math. Logic 12, 249-395. [1978] Continuity in intuitionistic set theories, in: D. van Dalen, M.Boffa and K. MacAloon, Eds., Logic Colloq. 78 (North-Holland, Amsterdam), 1-52. [1980] Extensionality and choice in constructive mathematics, Pacific J. Math. 88, 1-28. Feferman, S . [19771 Theories of finite type related to mathematical practice, in:

S. HAYASHI

162

J. Barwise, Ed., Handbook of mathematical logic (North-Holland, Amsterdam), 913-971. [1978] Constructive theories of functions and classes, in: D. van Dalen, M. Boffa and K. MacAloon, Eds., Logic Colloq. 78 (North-Holland), 159-224. Fourman, M.P. and Grayson, R.J. [198?] Formal spaces, in this volume. Fourman, M.P. and Joyal, A. [198?] Metamathematical applications of sheaf theory, in preparation. Friedman, H 1

[19701 Iterated inductive definitions and C1-AC, in: A. Kino, J. Myhill and R.E. Vesley, Eds., Intuitionism and Proof theory (North-Holland, Amsterdam) 435-442. [1977] Set-theoretic foundations for constructive analysis, Annals of Math. 105, 1-28. Grayson, R.J. [1975] Sheaf appraoch to models of set theory, M. SC. thesis, Oxford. [1979] Heyting-valued models for intuitionistic set theory, Springer Lecture Notes 753, 402-404. [1981] Forcing in intuitionistic systems without power-set, Report August 1981, Institut fiir Math. Logik, Mlinster. Hayashi, S. [19771 On derived rules of intuitiinistic second order arithmetic, Comment. Math. Univ. St. Pauli 26, 77-103. [1977a] A note on provable well-orderings in first order systems with infinitary inference rules, Tsukuba J. Math. 1, 125-135. [19801 Derived rule related to a constructive theory of metric spaces in intuitionistic higher order arithmetic without countable choice, Annals of Math. Logic 19, 33-65. [19811 On set theories in toposes, Springer Lecture Notes 891, 23-29. Howard, W.A. [19801 Ordinal analysis of terms of finite type, J. of Symbolic Logic 45, 493-504. Kreisel, G. [19651 Mathematical logic, in: T.L. Saaty, Ed., Lectures on modern mathematic vol I11 (John Wiely and Sons, New York), 95-195. Kreisel, G. and LBvy, A.

[19681 Reflection principles and their use for establishing the complexity of axiomatic systems, 2. Math. Logik Grundlagen Math. 14, 97-142.

Bar induction rule

163

Schiitte, K. [19651 Predicative well-orderings, in: J.N. Crossely and M.A.E. Dummet, Eds., Formal systems and Recursive Functions (North-Holland, Amsterdam) 280-303. [1977] Proof theory (Springer, Berlin). Schwichtenberg, H. [1977] Proof theory: Some applications of cut elimination, in: J. Barwise, Eds., Handbook of mathematical logic (Norith-Holland, Amsterdam), 867-895. [1979] On bar recursion of type 0 and 1, J. of Symbolic Logic 44, 325-329. Takeuti,

G.

and Titani, S .

[1981] Heyting valued universes of intuitiionistic set theory, Springer Lecture Notes 891, 189-306. Troelstra, A . S . [1977] Aspects of constructive mathematics in: J. Barwise, Ed., Handbook of mathematical logic (North-Holland), 973-1052. [1980] Extended bar induction of type zero, in: J . Barwise, H.J. Keisler and

K. Kunen, Eds., The Kleene Symposium (North-Holland, Amsterdam), 277-316. Zucker, J. [19731 Iterated inductive definitions, trees and ordinals, Springer Lecture Notes 344, 435-450.

THE L J J . B R O W E R CENTENARY SYMPOSIOM A.S. TroehRa and D. van Dalen (editors) 0North-HolhndPubiiShing C O m p l ~ ~1982 y,

165

THE EFFECTIVE TOPOS

J. M . E .

Hyland

Department of Pure Mathematics, Cambridge, England.

10

Introduction. The s u b j e c t of t h i s paper i s t h e most a c c e s s i b l e of a series

of toposes which can be c o n s t r u c t e d from n o t i o n s of r e a l i z a b i l i t y : it i s t h a t based on t h e o r i g i n a l n o t i o n of r e c u r s i v e r e a l i z a b i l i t y

i n Kleene C19451.

O f c o u r s e t h e r e a r e many o t h e r k i n d s of r e a l i z a -

b i l i t y (see Kleene-Vesley C19651, K r e i s e l C19591, T a i t C19751).

All

t h e s e (and even t h e D i a l e c t i c a I n t e r p r e t a t i o n ) f i t i n t o a very abs t r a c t framework d e s c r i b e d i n Hyland-Johnstone-Pitts C19801.

(Since

w e w i l l r e f e r t o t h i s paper f r e q u e n t l y , w e s h o r t e n t h e r e f e r e n c e t o H J P C19801.)

I n t h i s a b s t r a c t framework one p a s s e s e a s i l y ( a s i s

becoming customary, see Fourman C19771, Makkai-Reyes C19771, BoileauJ o y a l C198ll) between l o g i c a l and c a t e g o r y t h e o r e t i c f o r m u l a t i o n s , using whichever i s most a p p r o p r i a t e .

One good example i s worth a

h o s t of g e n e r a l i t i e s , so i t i s t h e aim of t h i s paper t o p r e s e n t t h i s a b s t r a c t approach t o r e c u r s i v e r e a l i z a b i l i t y i n some d e t a i l .

The

b a s i c s t r a t e g y r e a d i l y extends t o o t h e r cases. Many p e o p l e , most n o t a b l y Beeson (see f o r example Beeson C19771, have c o n s i d e r e d r e a l i z a b i l i t y extended t o g i v e i n t e r p r e t a t i o n s of complicated formal systems.

The f l a v o u r of t h e more c a t e g o r y theo-

r e t i c t r e a t m e n t i s t o have one t h i n k i n t e r m s of models.

Thus t h e

approach l o o k s l i k e sheaf models f o r i n t u i t i o n i s t i c l o g i c (see Fourman-Scott C 1 9 79 1 , where one only h a s n a t u r a l a c c e s s t o the models. (This p a r a l l e l between r e a l i z a b i l i t y and s h e a f models was f i r s t made e x p l i c i t , f o r set t h e o r y , i n an u n t i t l e d manuscript, by Powell.)

J.M.E. HYLAND

166

A s i n t h e c a s e of s h e a v e s , w e w i l l f i n d o u r s e l v e s l o o k i n g a t genuine

mathematical s t r u c t u r e s ( w i t h t h e i r non-standard l o g i c ) when w e investigate t r u t h i n the e f f e c t i v e topos.

W e w i l l be p r e s e n t i n g

" t h e world of e f f e c t i v e mathematics" a s it a p p e a r s t o t h e c l a s s i c a l (Of c o u r s e , it i s p o s s i b l e t o p r e s e n t t h e i d e a s i n

mathematician.

t h e c o n t e x t of more o r less any m a t h e m a t i c a l i d e o l o g y . ) While t h e l o g i c a l approach t o c a t e g o r i e s e n a b l e s us t o work w i t h c o n c r e t e s t r u c t u r e s and a p p l y o u r e x p e r i e n c e of e l e m e n t a r y l o g i c , t h e c a t e g o r y t h e o r e t i c approach t o l o g i c e n a b l e s us t o do away w i t h much l o g i c a l c a l c u l a t i o n and t o use i n s t e a d s i m p l e f a c t s about c a t e g o r i e s ( i n p a r t i c u l a r f a c t s a b o u t t o p o s e s and g e o m e t r i c morphisms).

I t h a s become c l e a r i n r e c e n t y e a r s t h a t much of con-

s t r u c t i v e l o g i c can be t r e a t e d v e r y e l e g a n t l y i n t h e c o n t e x t of topos theory.

This i s i n harmony w i t h work i n t h e i n t u i t i o n i s t

t r a d i t i o n on Beth a n d Kripke models (see van Dalen C19781, and t h e r e w e r e many c o n t r i b u t i o n s t o t h e Brouwer Centenary Conference i n t h i s

area.

T h i s p a p e r s i m p l y does t h e same k i n d of t h i n g f o r r e a l i z a -

bility.

Of c o u r s e t h e r e

&

a s u r p r i s e h e r e : t h e t o p o s of t h i s p a p e r

i s most u n l i k e a Grothendieck t o p o s , and it i s n o t i n i t i a l l y

p l a u s i b l e t h a t t h e o r y a b s t r a c t e d from n o t i o n s of c o n t i n u i t y s h o u l d have any a p p l i c a t i o n i n t h i s most n o n - t o p o l o g i c a l

setting.

The f i r s t t h r e e s e c t i o n s of t h e p a p e r s e r v e t o i n t r o d u c e t h e e f f e c t i v e t o p o s a s a world b u i l t o u t o f t h e l o g i c of r e c u r s i v e realizability. f o r the subject.

Much d e t a i l i s o m i t t e d i n t h e hope of g i v i n g a f e e l The main c a t e g o r y - t h e o r e t i c i d e a s axe e x p l a i n e d

and i n t e r p r e t e d i n 5 8 4 - 6 .

I n p a r t i c u l a r we show why t h e n o t i o n o f

a n e g a t i v e formula a r i s e s n a t u r a l l y i n t h e t h e o r y o f s h e a v e s .

In

887-13, w e apply t h i s work t o a s t u d y of a n a l y s i s i n t h e e f f e c t i v e topos.

W e show t h a t i n e s s e n c e it i s c o n s t r u c t i v e r e a l a n a l y s i s

( i n t h e s e n s e of Markov).

I am g r a t e f u l t o P r o f e s s o r T r o e l s t r a f o r

some a d v i c e on t h i s t o p i c ( I f i n d t h e p u b l i s h e d m a t e r i a l unreadable:

The effective topos

167

and i n p a r t i c u l a r f o r d e t e c t i n g a n e r r o r i n an e a r l y d r a f t o f t h i s paper.

5514-17 a r e c o n c e r n e d w i t h f e a t u r e s of t h e e f f e c t i v e t o p o s

where t h e power s e t m a t t e r s : u n i f o r m i t y p r i n c i p l e s and p r o p e r t i e s of j-operators.

The p a p e r c l o s e s w i t h some g e n e r a l remarks on the

mathematical s i g n i f i c a n c e o f t h e e f f e c t i v e t o p o s . F i n a l l y I would l i k e t o thank t h e o r g a n i z e r s o f t h e Brouwer Centenary Conference f o r t h e o p p o r t u n i t y t o p r e s e n t t h i s p a p e r ( i n such p l e a s a n t s u r r o u n d i n g s : ) and t o a p o l o g i z e t o everyone f o r b e i n g so long i n w r i t i n g it.

51 R e c u r s i v e r e a l i z a b i l i t y . R e c u r s i v e r e a l i z a b i l i t y i s b a s e d on t h e p a r t i a l a p p l i c a t i v e where a s i n H J P 119803 w e w r i t e n.m = n(m) f o r t h e

s t r u c t u r e (IN ,.)

r e s u l t o f a p p l y i n g t h e n ' t h p a r t i a l r e c u r s i v e f u n c t i o n t o m. s a v e s on b r a c k e t s compared w i t h t h e n o t a t i o n { n l n . ) a n o t i o n of A-abstraction

i n ( I N ,.)

(This

One c a n d e f i n e

i n t h e u s u a l way from t h e

com-

b i n a t o r s , and w e w i l l u s e it f r e e l y i n what f o l l o w s , s o t h a t ( f o r example) Ax.x w i l l d e n o t e a n i n d e x f o r t h e i d e n t i t y f u n c t i o n .

We

a l s o t a k e f o r convenience a r e c u r s i v e p a i r i n g f u n c t i o n <

,

>:

INxIN->

IN; (n,m)

-->

,

and l e t n1,n2 be ( r e c u r s i v e i n d i c e s f o r ) t h e c o r r e s p o n d i n g u n p a i r i n g €unctions. R e c u r s i v e r e a l i z a b i l i t y i s u s u a l l y f o r m u l a t e d i n terms o f t h e notion e realizes

0

where e i s a n a t u r a l number and @ i s a s e n t e n c e o f ( H e y t i n g ' s ) arithmetic.

The c r i t i c a l c l a u s e s i n t h e i n d u c t i v e d e f i n i t i o n a r e

implication

e r e a l i z e s 4 ->

$ i f f f o r a l l n, i f n realizes @ then e ( n ) i s

d e f i n e d and r e a l i z e s $,

JM.E. HYLAND

168

universal quantification e r e a l i z e s Vn.$(n) i f f f o r a l l n , e ( n ) i s d e f i n e d and r e a l i z e s

@ ( g )Cg t h e numeral f o r n l . The o t h e r i n d u c t i v e c l a u s e s a r e and -

e realizes

$A$

i f f n ( e ) r e a l i z e s $ and n 2 ( e ) r e a l i z e s $,

1

or e r e a l i z e s @vJ, i f f e i t h e r a , ( e )

or

IT^ ( e )

= 0

and n 2 ( e ) r e a l i z e s $

= 1 and n 2 ( e ) r e a l i z e s $,

falsity no numbers r e a l i z e

1,

existential quantification e r e a l i z e s h . $ ( n ) i f f n 2 ( e ) r e a l i z e s @ ( r l ( e )) [ n l ( e ) t h e

- -

numerical f o r

I T ~ ( I ~. )

F i n a l l y w e g i v e t h e i n i t i a l c l a u s e f o r e q u a l i t i e s between c l o s e d terms

e r e a l i z e s s = t i f f b o t h s and t d e n o t e e . For a c a r e f u l t r e a t m e n t o f t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n of a r i t h m e t i c t h e r e a d e r may c o n s u l t T r o e l s t r a C19731.

W e w i l l see i n

53, t h a t t h i s i s t h e i n t e r p r e t a t i o n of a r i t h m e t i c w i t h i n t h e

e f f e c t i v e topos.

For an a c c o u n t of t h e o r i g i n a l m o t i v a t i o n see

Kleene 119731; it i s i n t e r e s t i n g t o t r y t o u n d e r s t a n d i t i n t e r m s o f t h e p r e s e n t paper. Apparently Dana S c o t t f i r s t n o t i c e d t h a t r e a l i z a b i l i t y c o u l d be understood " m o d e l - t h e o r e t i c a l l y ' '

{ e ( erealizes @ I . truth-values,

i n terms of t h e t r u t h - v a l u e s

T h i s g i v e s u s a s e t C = P ( l N ) of non-standard

and so f o r e a c h s e t X , a s e t Zx of non-standard p r e -

d i c a t e s on X.

W e w r i t e $ = ($,lx

e l e m e n t s o f C"

and can r e f o r m u l a t e o u r e a r l i e r d e f i n i t i o n f o r t h e

E

X) and J,

=

( Q X \ x E X) f o r

p r o p o r t i o n a l c o n n e c t i v e s by d e f i n i n g o p e r a t i o n s p o i n t w i s e on Zx as follows:

169

The effective topos

$ x ~ $ x= { < n , m > l n E $x and m

( $ A $ ) ~ =

( $ v $ l X = $xv$x

$x+$x

($+I))~ =

e(n)

$,I,

E

=

t < ~ , n > l nE $ x I u t < l , n > j n

=

{el i f n

t h e n e ( n ) i s d e f i n e d and

,$,

E

E

E

= t h e empty s e t .

IX

The r e a d e r may a l s o l i k e t o have TX = IN.

lk

There i s a r e l a t i o n

of e n t a i l m e n t ( a p r e - o r d e r ) d e f i n e d on each

Cx by

I-,$

i f f n { ( $->

$),lx

E

XI i s non-empty.

The soundness of t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n of i n t u i

-

t i o n i s t i c propositional logic i s the following proposition. X

P r o p o s i t i o n 1.1.

(Z

i s a Heyting p r e - a l g e b r a :

as a c a t e g o r y

t h e p r e o r d e r h a s f i n i t e l i m i t s ( m e e t s ) , f i n i t e c o l i m i t s ( j o i n s ) and

i s C a r t e s i a n c l o s e d (Heyting i m p l i c a t i o n ) . Proof:

The s t r u c t u r e i s g i v e n e x p l i c i t l y i n t h e d e f i n i t i o n s above.

W e now i n t r o d u c e t h e a b s t r a c t n o t i o n of q u a n t i f i c a t i o n from

categorical logic.

Y of sets w e d e f i n e

For any map f : X ->

s u b s t i t u t i o n a l o n g f , f * : C y ->

CX a s composition w i t h f :

( f * $ ) , = $ f (x) f * i s a f u n c t o r ( i n f a c t a map of Heyting p r e - a l g e b r a s ) from Y X ( Z , I - y ) t o ( C , I-x ) and q u a n t i f i c a t i o n a l o n g f i s g i v e n by t h e adjoints t o f*.

As shown i n HJP C19801 t h e s e are d e f i n e d by

right adjoint (vf.+)y = n t f ( x ) = y

->

$X

IX

E

xl,

E

XI,

l e f t adjoint (3f.$)y = U { f(X) = y

A

$xlX

where T,

f f ( x ) = Y]l

= '-JITlf(X)

= yj =

tL,

i f f ( x ) = y, otherwise,

i s t h e n a t u r a l i n t e r p r e t a t i o n a s a non-standard p r e d i c a t e of

J.M.E. HYLAND

170

Note t h a t w h i l e U { ~ $ ~ l f ( = x )y} i s a s a t i s f a c t o r y a l t e r -

f ( x ) = y.

n a t i v e d e f i n i t i o n of t h e l e f t a d j o i n t , r l { $ x l f ( x ) = y} i s d e f i n i t i o n of t h e r i g h t a d j o i n t u n l e s s f : X -->

=

a

Y is surjective.

However u s u a l q u a n t i f i c a t i o n i s q u a n t i f i c a t i o n a l o n g t h e obvious p r o j e c t i o n , and almost a l l p r o j e c t i o n s are s u r j e c t i v e , so t h i s nuance w i l l c a u s e t h e r e a d e r (and a u t h o r ) no f u r t h e r t r o u b l e . The r e a d e r w i l l see t h a t what w e have j u s t d e s c r i b e d i s an i n t e r p r e t a t i o n of i n t u i t i o n i s t i c p r e d i c a t e l o g i c : w e have s t a n d a r d f u n c t i o n s and sets, a (non-standard r e p r e s e n t a t i o n o f ) s t a n d a r d e q u a l i t y and a c o l l e c t i o n of non-standard p r e d i c a t e s . a " g e n e r i c p r e d i c a t e " namely t h e i d e n t i t y i n E L .

W e a l s o have

W e can e n c a p s u l a t e

a l l t h i s structure i n the following proposition. Proposition 1 . 2 .

The

t o g e t h e r w i t h t h e f * and t h e i r a d j o i n t s

3f and Vf and t h e " g e n e r i c p r e d i c a t e " , form a t r i p o s on t h e c a t e g o r y of

Sets

Proof:

( i n t h e s e n s e of H J P C19801. See H J P C19801.

I n what w e have s a i d , w e have n o t needed t o d i s t i n g u i s h f o r mulae from t h e i r i n t e r p r e t a t i o n s , d i s t i n c t i o n as f a r as p o s s i b l e . i n d i c a t e an

and w e w i l l c o n t i n u e t o b l u r t h i s ( W e w i l l u s e open f a c e b r a c k e t s t o

i n t e r p r e t a t i o n when n e c e s s a r y t o p r e v e n t c o n f u s i o n . )

W e say t h a t $

I-x$.

E

.Ex i s v a l i d i f f

E

Zx i s v a l i d i f f T l-lVX.$,

T

By a d j o i n t n e s s w e have

9 where X:

X -->

1 i s a unique map from X t o a one element s e t .

That

i s , I$ i s v a l i d i f f Vx.I$(x) , t h e u n i v e r s a l g e n e r a l i z a t i o n of $ is

valid or realizable.

W e w i l l use t h i s n o t i o n b o t h t o d e s c r i b e and

s t u d y t h e t o p o s which w e can c o n s t r u c t on t h e b a s i s of ( 1 . 2 ) . 12

D e s c r i p t i o n of t h e e f f e c t i v e t o p o s . When c o n s t r u c t i n g a t o p o s from a t r i p o s a s i n H J P C19801, one

The effective topos

171

must ( i )add new s u b o b j e c t s of t h e s e t s one h a s s t a r t e d w i t h t o

r e p r e s e n t t h e non-standard p r e d i c a t e s , and (ii) t a k e q u o t i e n t s of t h e s e by t h e non-standard

relations.

equivalence

T h i s l e a d s t o t h e d e s c r i p t i o n of t h e o b j e c t s of e f f e c An o b j e c t of t h e e f f e c t i v e t o p o s i s a s e t X with a

t i v e topos.

non-standard p r e d i c a t e = on X x X such t h a t symmetry

x = y -->

transitivity

x = Y A Y = z ->

are v a l i d .

y = x x = z

Note t h a t w e do n o t have r e f l e x i v i t y :

( a s i s t h e case

f o r Heyting a r i t h m e t i c ) t h e r e need be no uniform r e a l i z a t i o n of W e r e g a r d and w i l l w r i t e t h e p r e d i c a t e x = x

(reason why) x = x.

a s an e x i s t e n c e p r e d i c a t e , Ex, and a s a membership p r e d i c a t e , x

E

X.

There i s a u s e f u l d i s c u s s i o n of t h e l o g i c of e x i s t e n c e p r e d i c a t e s i n S c o t t C19791.

Of c o u r s e w e need t o c o n s i d e r a l l non-standard maps t o o b t a i n t h e e f f e c t i v e t o p o s , and t o do t h a t w e a r e reduced t o c o n s i d e r i n g functional r e l a t i o n s .

The - m

from (X, =) t o ( Y , = ) i n t h e

e f f e c t i v e topos a r e e q u i v a l e n c e c l a s s e s of f u n c t i o n a l r e l a t i o n s where (a) G

6

cxXy

i s a functional relation i f f = y ' ->

relational

G(x,y)AX =

X'AY

strict

G ( x , y ) ->

E XAE Y

single-valued

G ( x , y ) ~ G ( x , y ' )-->

total

E X -->

G(X',y')

y = y'

+y.G(x,y)

are a l l valid, (b) 6 i s e q u i v a l e n t t o H i f f G(x,y) <-->

is valid.

H(xrY)

W e w i l l s a y t h a t G r e p r e s e n t s t h e map [GI: ( X , = )

-->(Y,=).

I t i s u s e f u l t o n o t e t h a t i f G and H a r e both f u n c t i o n a l r e l a t i o n s

J.M.E. HYLAND

112

(from ( X I = )

to (Y,=),

t h e n t o show G and H e q u i v a l e n t , i t s u f f i c e s

t o show t h a t an i m p l i c a t i o n i n one d i r e c t i o n i s v a l i d . F u n c t i o n a l r e l a t i o n s can be composed: i f G

E

Xxxy

are f u n c t i o n a l r e l a t i o n s , t h e n s o i s 3 y . G ( x l y ) A H ( y , z ) =

i s a f u n c t i o n a l r e l a t i o n from ( X , = )

to itself.

and H E

Cyxz

E

Cxxz.

Also

These g i v e t h e

composition and i d e n t i t i e s , and s o w e have a c a t e g o r y .

I n view of

( 2 . 1 ) , w e c a l l t h i s c a t e g o r y t h e e f f e c t i v e t o p o s and d e n o t e it by

Eff h e r e a f t e r . Theorem 2 . 1 . Proof:

Eff

is a topos.

See H J P C19801 f o r d e t a i l s .

W e can e x t e n d t h e non-standard

i n t e r p r e t a t i o n of 91 t o g i v e an

a c c o u n t of t h e i n t e r n a l l o g i c o f t h e c a t e g o r y

Eff.

T h i s goes a s f o r

t h e l o g i c of s h e a v e s e x c e p t f o r obvious m o d i f i c a t i o n s t o d e a l w i t h t h e f a c t t h a t f u n c t i o n s are ( o n l y ) r e p r e s e n t e d by f u n c t i o n a l r e l a tions.

A g e n e r a l account of t h e i n t e r n a l f i r s t - o r d e r

l o g i c of

c a t e g o r i e s i s g i v e n i n Makkai and Reyes C19771, and a c c o u n t s o f t h e h i g h e r o r d e r l o g i c of t o p o s e s can be found i n Fourman C19771 and B o i l e a u - J o y a l C19811.

A s t h e s e a c c o u n t s make c l e a r , c a t e g o r i c a l

c o n s t r u c t i o n s can be d e f i n e d by means of t h e i n t e r n a l l o g i c .

Thus,

n o t o n l y can ( a n e x t e n s i o n o f ) v a l i d i t y i n t h e s e n s e o f 9 1 , be used t o d e t e r m i n e what i s t r u e i n categorical constructs.

Eff,

b u t i t can a l s o be used t o d e f i n e

(Now c o n t i n u i n g t h e i n t e r p l a y , t h e s e c a t e g o r i -

c a l c o n s t r u c t s can t h e n b e used t o e s t a b l i s h f u r t h e r f a c t s a b o u t what is true i n

Eff.)

W e now g i v e some s i m p l e examples o f t h e l o g i c a l

description of the structure of

Eff.

(On a few o c c a s i o n s w e w i l l

need t o q u o t e some more c o m p l i c a t e d f a c t s o f t h e same k i n d . ) 1) A map [GI:

(X,=) -->

(Y,=) i s monic i f f

G ( x , y ) ~ G ( x ' , y )-->

x =

X'

is valid. A s u b o b j e c t of

(X,=) can always b e r e p r e s e n t e d (though n o t

u n i q u e l y ) by a c a n o n i c a l monic of t h e form

The effective topos

[='I:

(X,=') -->

173

(X,=)

where I[ x = ' x q

f o r some A

E

=

ux

A(X)A

=

x l n

Zx s t r i c t and r e l a t i o n a l f o r (X,=).

Thus s u b o b j e c t s

always a r i s e by r e s t r i c t i n g t h e membership p r e d i c a t e w h i l e (as f a r as p o s s i b l e ) l e a v i n g t h e e q u a l i t y a l o n e . 2 ) Given two maps CG1,CHI:

(Y,=)

(X,=) -->

,

t h e i r e q u a l i z e r is

r e p r e s e n t e d by t h e c a n o n i c a l monic o b t a i n e d from t h e s t r i c t and relational 3y.G(x,y)hH(xfy)

E

Ex.

The c o n s t r u c t i o n of o t h e r f i n i t e l i m i t s i s a n a l o g o u s . The diagram

W,=)

1

[H'I

[G">

V

(Xr=)

->

(Z,=)

1

[HI

(Yr=)

[GI

i s a p u l l b a c k i f f CHIo[G'I = CGIo[H'I,

(CG'1,CH'I):W ->

ZXX is

a

monic and G ( x , y ) ~ H ( z , y )-->

is v a l i d .

3W.G'

(W,Z)AH'

(w,x)

The c o n d i t i o n t h a t o t h e r diagrams g i v e f i n i t e l i m i t s can

be e x p r e s s e d s i m i l a r l y i n t h e l o g i c . 3 ) A map [GI:

(Y,=) i s s u r j e c t i v e i f f

(X,=) --> Ey ->

3x.G(xry)

is valid. A q u o t i e n t c a n always be r e p r e s e n t e d as [-I: where

(Xr=)

-->

(Xv-)

- i s s t r i c t r e l a t i o n a l f o r (X,=) and s u c h t h a t "- i s an e q u i v a l e n c e r e l a t i o n on (X,=)"

is valid.

Thus q u o t i e n t s are a m a t t e r of e x t e n d i n g t h e e q u a l i t y

r e l a t i o n and l e a v i n g t h e membership p r e d i c a t e a l o n e .

J.M.E. HYLAND

174

W e can now show t h a t any o b j e c t ( X I = )

Eff i s a q u o t i e n t of a

of

s u b o b j e c t of an " o r d i n a r y s e t " , j u s t i f y i n g t h e e x p l a n a t i o n a t t h e s t a r t of t h i s s e c t i o n . o b j e c t of

Eff

For a s e t X w e l e t AX ( a s i n 5 4 ) be t h e

with underlying s e t X and (non-standard r e p r e s e n t a t i o n

o f ) standard e q u a l i t y . Proposition 2 . 2 .

Any o b j e c t ( X , = )

of

Eff

i s a q u o t i e n t by

s u b o b j e c t EX of AX o b t a i n e d from t h e e x i s t e n c e p r e d i c a t e of Proof:

Note.

=

of t h e (X,=)

.

Obvious i n view of 1) and 3 ) above. W e have s t a r t e d using open f a c e b r a c k e t s t o e n s u r e r e a d a b i l i t y

( e s p e c i a l l y i n connection w i t h e q u a l i t y ) , a s promised i n §I. W e a l s o abuse n o t a t i o n and w r i t e X f o r ( X , = )

where c o n t e x t makes t h e

meaning obvious. 53.

-Some

Eff.

o b j e c t s and maps i n

W e can e a s i l y d e s c r i b e a t e r m i n a l o b j e c t 1 i n

52, 1 is

( I * } , = ) where { * I [I*

=*I

= T

1*

p would do a s t h e v a l u e

T

=

in Z{*'

.

*I

{*I is

Since

W e now c a l c u l a t e t h e g l o b a l

Ef f ,

t h a t i s t h e maps from

a s i n g l e t o n , such maps a r e r e p r e s e n t e d by

degenerate f u n c t i o n a l r e l a t i o n s G G ( Y ) A Y = y ' -->

G(y)

t h a t i s , any non-empty

= Z,

s e c t i o n s of an a r b i t r a r y o b j e c t (Y,=) of

.

I n view of

i s a s i n g l e t o n , and

Of c o u r s e any p e q u i v a l e n t t o

1 t o (Y,=)

Eff.

E

Zy,

such t h a t

G(y') Ey

-->

G(y)AG(y') -->

y = Y'

3y.G(y) are a l l valid.

The t o t a l c o n d i t i o n t e l l s us t h a t f o r some y , yo s a y ,

G(yo) i s non-empty.

The r e l a t i o n a l and s i n g l e - v a l u e d c o n d i t i o n s

imply t h a t G(Yo)

-->

(G(y) <-->

and hence ( s i n c e G(yo) i s non-empty)

yo

=

y)

175

The effective topos

G(y) <--> are valid.

i s non-empty,

[I yo = y l l

Clearly i f

Yo = Y <->

is valid.

yo = y then

Y1 = Y

W e deduce a t once t h e f o l l o w i n g c h a r a c t e r i z a t i o n .

Proposition 3.1.

Each map [ G I :

(Y,=) d e t e r m i n e s and i s com-

1 ->

p l e t e l y determined by [y I G(y) non-empty}, which i s an e q u i v a l e n c e class f o r the ( p a r t i a l ) equivalence r e l a t i o n

Uy

"

i s non-empty".

= y'l

Conversely any such e q u i v a l e n c e c l a s s d e t e r m i n e s a map from 1 t o (Y,=).

Finite colimits i n

Eff

a r e h a r d t o g e t used t o because f o r a

s t a r t c o p r o d u c t s a r e odd: t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n of d i s j u n c t i o n is v e r y r e s t r i c t i v e .

I n p a r t i c u l a r , t h e c o p r o d u c t 2 of 1

with i t s e l f i s n o t t h e obvious o b j e c t A2 w i t h s t a n d a r d e q u a l i t y (see 54). of

Eff.

L e t us look a t maps from A2 t o an a r b i t r a r y o b j e c t IY,=)

Suppose G ( i , y ) r e p r e s e n t s such a map (where 2 = { O , l } ) .

Then s i n c e EO yo,yl

= E l = T,

t h e t o t a l c o n d i t i o n t e l l s us t h a t t h e r e a r e

such t h a t G(O,yo)nG(l,yl) i s non-empty.

Arguing a s f o r t h e

terminal object we find t h a t G ( i , y ) <--> is valid.

yi = y

However [GI does n o t c o r r e s p o n d simply t o a p a i r o f

e q u i v a l e n c e c l a s s e s i n IylEy non-empty?:the union of t h e e x i s t e n c e of t h e two e q u i v a l e n c e c l a s s e s must i n t e r s e c t n o n - t r i v i a l l y ,

and t h i s i s

a real restriction. In f a c t the object 2 i n EO =

101, E l

(Of c o u r s e any p,,pl An

=

{l},

Eff

can be r e p r e s e n t e d as ( 2 , = ) where

[li = j]l

= EinEj.

w i t h p0 npl empty would do as t h e v a l u e s EO, E l . )

argument a s above shows t h a t maps from 2 a r e p a i r s of maps from 1.

Note a l s o t h a t t h e o n l y maps from A2 t o 2 are c o n s t a n t ( t h a t i s , f a c t o r through 1 ) . There i s an obvious monic from 2 t o A2.

I n 516

J.M.E. HYLAND

176

w e w i l l show t h a t t h e whole s t r u c t u r e of

Eff

depends on 2 -->

A2

n o t b e i n g i s o , i n t h e s e n s e t h a t t h e topology i n v e r t i n g 2 -->

A2

collapses

Eff

Sets.

back t o

S i n c e 2 i s n o t A2, w e would h a r d l y e x p e c t t h e n a t u r a l number

Eff

object N i n

Am.

t o be

I n f a c t it i s t h e o b j e c t ( I N , =) where

En = i n ] , There are maps 0: 1 -->

[n

=

= EnnEm.

m]

and s : IN->

IN

IN

in

Eff

r e p r e s e n t e d res-

p e c t i v e l y by Go and Gs where G o ( * , n ) = I O j n I n ) and G,(n,m) Proposition 3.2.

IN

g:

(X,=)

G

E

4

Eff.

Suppose t h a t w e are g i v e n maps a : 1 -->

1'.

-->

(X,=)

IN i s a

IN and s : IN->

t o g e t h e r w i t h 0: 1 -->

n a t u r a l number o b j e c t i n Proof:

= In+l)n{m).

(X,=)

r e p r e s e n t e d r e s p e c t i v e l y by Ga

E

and

Cx and

W e can d e f i n e r e p r e s e n t a t i v e s Gn f o r gn i n d u c t i v e l y by

GO(X,XI)

=

U X

=

x8n

,

G n + l( x , x l )

g 9 Now w e can d e f i n e a f u n c t i o n f : IN->

9

= ~ X ~ ~ . G ~ ( X , X ~( X ~ ~) *A, X G').

(X,=)

9

9

r e p r e s e n t e d by

G f ( n , x ) = EnA3x'.Ga(x')AGn(x',x). 9

W e claim t h a t

fi

a

commutes.

(X,=)->(X,=) 9

fl

T h i s amounts t o showing t h a t G ( x ) <-->

3n.Go(*,n)AGf(n,x)

and

3 x ' . G f ( n , x ' ) A G ( x ' , x ) <--> 9

are both valid.

3m.Gs (n,m) A G f (m,x)

These can b o t h be e s t a b l i s h e d by use of e l e m e n t a r y

logic.

It remains t o show t h a t f i s unique s u c h t h a t ( * ) commutes. suppose t h a t f

'

r e p r e s e n t e d by Gf

l o g i c w e see r e a d i l y t h a t

I

i s a n o t h e r such map.

SO

By use of

177

The effective topos G f ( 0 , x ) <->

Gf

I

(0,x)

i s v a l i d and t h a t

G f ( n + l , x ) <-> G f , (n+l,x)<->

a r e both v a l i d .

3x'.Gf(n,x')AG ( x ' , x ) g 3 x ' . G f , ( n , x ' ) AG ( x ' , x ) g

But i n t e r m s o f t h i s d a t a w e can d e f i n e , by p r i m i -

t i v e r e c u r s i o n , a p a r t i a l r e c u r s i v e f u n c t i o n u n i f o r m l y mapping Gf ( n , x ) t o G f , ( n , x )

Remark.

,

and t h i s i s enough t o show t h a t f

=

f',

S i n c e q u a n t i f i c a t i o n i n o u r l o g i c (see Fourman-Scott

C 19791

and S c o t t C19791) i n v o l v e s t h e e x i s t e n c e p r e d i c a t e , w e see a t once on t h e b a s i s o f ( 3 . 2 ) t h a t t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n c o r r e s ponds t o t h e l o g i c of t h e n a t u r a l number o b j e c t i n Corollary 3.3.

Eff.

A s e n t e n c e of Heyting a r i t h m e t i c i s r e c u r s i v e l y

r e a l i z e d i f f i t i s t r u e of t h e n a t u r a l number o b j e c t i n

Eff.

A l l t h e s p e c i f i c o b j e c t s w e have looked a t s o f a r have been

o b j e c t s (X,=) where

[x = x'J

non-empty i m p l i e s x = x '

( i n X).

Indeed a l l t h e o b j e c t s w e c o n s i d e r u n t i l 1 1 4 w i l l be ( i s o m o r p h i c t o ) ones o f t h i s s o r t (see 5 6 f o r a d i s c u s s i o n of what t h e c o n d i t i o n means). form.

I t i s a s w e l l t o have an example of an o b j e c t n o t o f t h i s

The most o b v i o u s example i s t h e s u b o b j e c t c l a s s i f i e r i n

( t h a t i s , t h e o b j e c t of t r u t h v a l u e s ) . t h i s i s t h e o b j e c t (X ,<->),

A s i n d i c a t e d i n H J P C19801

t h a t is, the set X

= P(JN )

e q u a l i t y g i v e n by t h e non-standard b i - i m p l i c a t i o n . an e a s y e x e r c i s e t o show t h a t (I,<-->) o b j e c t (X,=) where

I[x = x ' l

Eff

W e leave it a s

non-empty i m p l i e s x = x ' .

examples i t i s n a t u r a l t o c o n s i d e r (EA,<->)

of which w e c o n s t r u c t e d

with

i s n o t i s o m o r p h i c t o any

f i r s t t h a t X would have t o have j u s t two e l e m e n t s . )

the pointwise bi-implication.

Eff

(Show

As f u r t h e r

where now e q u a l i t y i s

These are e s s e n t i a l l y t h e o b j e c t s o u t

i n the f i r s t place; they a r e i n f a c t the

power s e t s of t h e o b j e c t s A X (see 9 4 ) .

J.M.E. HYLAND

178

54.

The i n c l u s i o n of t h e c a t e g o r y o f s e t s i n t h e e f f e c t i v e t o p o s . I n t h e l a s t s e c t i o n w e s a w glimpses of a f u n c t o r A from t h e

Sets o f

category

Definition.

sets t o

Eff,

t h e e f f e c t i v e topos.

For a s e t X, d e f i n e AX t o be ( X , = A x ) l

i s t h e n a t u r a l i n t e r p r e t a t i o n of t h e e q u a l i t y i n

f : X ->

Y in

Sets, d e f i n e

A f : AX ->

where

Sets.

For a map

AY t o be t h e map r e p r e s e n t e d

by t h e f u n c t i o n a l r e l a t i o n ,

This d e f i n i t i o n can be made whenever w e c o n s t r u c t a t o p o s from a t r i p o s (see P i t t s [ 1 9 8 l l ) , and w e always have o w f i r s t r e s u l t . Proposition 4 . 1 .

A:

Sets ->

Eff

functor preserving f i n i t e l i m i t s ) Proof:

is a Cartesian functor ( t h a t i s ,

.

F u n c t o r i a l i t y i s obvious.

That A i s C a r t e s i a n f o l l o w s

e a s i l y from t h e way f i n i t e l i m i t s a r e d e f i n e d l o g i c a l l y i n 53.

For

d e t a i l s see H J P C19801 o r P i t t s C19811. The n e x t r e s u l t i s a g e n e r a l f e a t u r e of r e a l i z a b i l i t y t o p o s e s .

Propcsition Proof:

4.2.

A:

Sets ->

i s va1i.d. =

Then

y i f f g(x)

yn

AX t o A Y .

=

T

y: t h u s f = g.

To show t h a t A i s f u l l , l e t G

E

Sets, and

ug(x)

<-->

I[f(x) = y l =

i s f u l l and f a i t h f u l .

Y in

Suppose t h a t f , g : X --> =

f(x)

Eff

iff

=

that

y~

f g ( x ) = yl] =

T

whence

This shows t h a t A i s f a i t h f u l .

Zxxy

be a f u n c t i o n a l r e l a t i o n from

The r e l a t i o n a l and s t r i c t c o n d i t i o n s axe a u t o m a t i c a l l y

s a t i s f i e d , t h e s i n g l e - v a l u e d c o n d i t i o n i m p l i e s t h a t f o r given x t h e r e i s a t most one y w i t h G ( x , y ) non-empty

( c o n s t r u c t i v e l y , in-

h a b i t e d ) , and t h e t o t a l c o n d i t i o n imples t h a t t h e r e is a t l e a s t one such y f o r g i v e n x. non-empty

Thus w e have g: X ->

i f f g(x) = y .

Then c l e a r l y

~ ( x , y )-->

u g w = yn

Y such t h a t G ( x , y )

is

179

The effective topos

i s v a l i d : an i n d e x f o r t h e i d e n t i t y r e a l i z e s i t .

But t h e t o t a l

c o n d i t i o n becomes E x ->

G(x,g(x))

i s v a l i d , whence

Ig ( x ) is valid.

Remark.

=

yn

->

G

Thus G r e p r e s e n t s t h e map Ag. ndr6 Joyal has pointed out t h a

XIY)

T h i s shows t h a t A is f u l l . A i s analogous t o t h e

Yoneda embedding: it i s C a r t e s i a n , f u l l and f a i t h f u l , and ( s o ) p r e serves exponentiation.

But I do n o t u n d e r s t a n d t h e f o r c e of t h i s

analogy. The main r e s u l t o f t h i s s e c t i o n i s a n o t h e r g e n e r a l f e a t u r e o f r e a l i z a b i l i t y toposes.

R e c a l l t h a t i n ( 3 . 1 ) w e showed i n e f f e c t

t h a t t h e g l o b a l s e c t i o n f u n c t o r on

r : Eff

-->

Sets d e f i n e d

(i)r ( X , = )

= {XIEX

Eff

i s n a t u r a l l y isomorphic t o

by

i s non-empty]/_

where x

- x'

if

[Ix = x i ]

'

i s non-empty; ( i i ) i f G i s a f u n c t i o n a l r e l a t i o n from ( X , = )

t o (Y,=)

representing g, then T ( g ) ([XI)

= {ylG(x,y)

i s non-empty} where [ X I d e n o t e s

t h e equivalence c l a s s of x.

r

i s a c o n c r e t e v e r s i o n of t h e g l o b a l s e c t i o n f u n c t o r , w i t h which

w e can work, even c o n s t r u c t i v e l y : s t a r t i n g from an a r b i t r a r y b a s e

topos E , T ( X , = )

s t i l l makes s e n s e a s t h e i n t e r p r e t a t i o n i n E of

" t h e set o f maps from 1 t o ( X I = ) " . r e p l a c e d by " i n h a b i t e d " ) Theorem 4 . 3 .

.

A is t h e d i r e c t image f u n c t o r of a g e o m e t r i c morphism,

whose i n v e r s e image f u n c t o r i s Proof:

r

(Of c o u r s e , "non-empty" must be

r.

The g l o b a l s e c t i o n f u n c t o r i s always C a r t e s i a n : a l t e r n a t i v e l y ,

a s d e f i n e d i s C a r t e s i a n by t h e l o g i c a l c o n s t r u c t i o n of f i n i t e

l i m i t s d e s c r i b e d i n 53.

So w e c o n c e n t r a t e on the a d j o i n t n e s s .

J.M.E. HYLAND

180

W e define t h e u n i t of t h e adjunction

ny:

ATY f o r (Y,=) i n

Y -->

Eff

by t h e f u n c t i o n a l r e l a t i o n (y,[y'l)-->

U{Eyly

Cy'l}={yy'

E

,

-

if y y' otherwise.

Now l e t G b e a f u n c t i o n a l r e l a t i o n from (Y,=) t o AX.

The t o t a l and

s i n g l e - v a l u e d c o n d i t i o n s imply t h a t i f Ey i s non-empty, i s a unique x

X w i t h G ( y , x ) non-empty.

E

implies t h a t i f

y

=

y']

then t h e r e

The r e l a t i o n a l c o n d i t i o n t h e n w e g e t t h e same x

is non-empty,

Thus w e have a w e l l - d e f i n e d map g: rY -->

for y' as for y.

X.

By

l o g i c , t h e composite A ( 9 ) o r l y i s r e p r e s e n t e d by ~ ( y , x )= U I E ~ Aug(cy11)

=

xn

~y

E

iy11

E

~YI.

By t h e s t r i c t c o n d i t i o n G(y,X)

-->

EY

i s v a l i d ; s o s i n c e G ( y , x ) i s non-empty i f f x

=

q(Cy1) and s i n c e

clearly Ey

H(y,g(cyl))

-->

is v a l i d , w e deduce t h a t

G ( y , x ) --> is valid.

H(y,x)

S i n c e b o t h G and H a r e f u n c t i o n a l r e l a t i o n s , t h i s shows

( a s remarked i n 5 2 ) t h a t t h e y r e p r e s e n t t h e same f u n c t i o n , and w e have o u r f a c t o r i z a t i o n [GI

=

A(q) o n y .

I t remains t o show t h a t q i s unique w i t h t h i s p r o p e r t y .

9 ' : TY -->

But i f

X is such t h a t G(Y,X)

<-->

U{EYA u g i ( c y i i ) =

xn

i s v a l i d , t h e n G ( y , g ' ( C y l ) ) i s non-empty,

Iy

E

cy'i

E

TY}

so t h a t g ' = q .

This

completes t h e p r o o f . Remark.

I t is an e a s y c o r o l l a r y of t h e proof of

Eff

(4.3)

t h a t maps

have a s i m p l e c a n o n i c a l r e p r e s e n t a t i v e .

( Y , = ) -->

AX i n

g: TY -->

X c o r r e s p o n d under t h e a d j u n c t i o n t o a map IY,=) ->

Let

Then t h i s l a t t e r map i s r e p r e s e n t e d by t h e f u n c t i o n a l r e l a t i o n

AX.

T h e effective topos ( y , x ) -->

XI

UiEylg([yl) =

181

{y:

=

W e can now i n d i c a t e how c a t e g o r y t h e o r y may be a p p l i e d t o s t u d y

realizability.

( 4 - . 2 ) and ( 4 . 3 ) t o g e t h e r s a y t h a t A :

Sets -->

i s an i n c l u s i o n of t o p o s e s (see Johnstone C19771) so t h a t j-sheaves

on

Eff

f o r a s u i t a b l e topology j .

c a t i o n of j , which depends on t h e f a c t t h a t Proposition 4 . 4 .

The topology j such t h a t

Eff

Sets

is

W e g i v e an i d e n t i f i -

Sets h a s c l a s s i c a l l o g i c , Eff. Sets i s t h e double -J

negation topology Proof:

In the f i r s t place,

Sets i s

dense i n

Eff s i n c e

A

p r e s e r v e s t h e i n i t i a l o b j e c t (see ( 8 . 1 ) ) ; s o j is a t most g r e a t e s t dense t o p o l o g y ) . But

it f o l l o w s t h a t j must be

T-,

Sets i s

71

(the

b o o l e a n , and from t h i s

.

W e can now d e s c r i b e what t h e use o f " c l a s s i c a l o b j e c t s " i n

i n t u i t i o n i s m amounts t o i n our c o n t e x t : s i n c e t h e y are d e f i n e d by l i b e r a l use o f 7-,, t h e y are when i n t e r p r e t e d i n the image o f A .

Thus A ( = )

c l a s s i c a l mathematics w i t h i n 15.

Eff,

the objects i n

s h o u l d be r e g a r d e d a s t h e world of

Eff.

B a s i c f a c t s from t h e l o g i c o f s h e a v e s . While the material p r e s e n t e d i n this s e c t i o n i s i m p l i c i t i n t h e

topos t h e o r e t i c l i t e r a t u r e , i t can n o t be found i n t h e form w e r e quire.

With Grothendieck t o p o s e s one h a s t y p i c a l l y a subtopos E

j of a t o p o s E which one u n d e r s t a n d s ( E i s u s u a l l y a f u n c t o r c a t e g o r y ) and one r e q u i r e s r e s u l t s which e n a b l e one t o d i s c u s s E E and t h e topology j . It is t h e topos E

j

j

i n terms of

For us however t h e s i t u a t i o n i s d i f f e r e n t .

( t h a t is

Sets) which

obtain information about E ( t h a t i s

w e understand and w e wish t o

Eff)

i n terms o f E

and j . j W e p r e s e n t t h e material i n t h e following general context.

E is

a t o p o s w i t h a topology j , E . i s t h e f u l l s u b c a t e g o r y of E c o n s i s t i n g J of j-sheaves and L: E -> E j i s t h e sheafification functor l e f t

182

J.M.E. HYLAND

adjoint t o the inclusion E

E.

->

W e give t h e b a s i c d e f i n i t i o n s

i n a number o f u s e f u l e q u i v a l e n t forms which a r e i m p l i c i t e i t h e r i n Johnstone C19771 o r i n Fourman-Scott Definition.

119791.

o b j e c t F of E i s j - s e p a r a t e d

An

i f f any of t h e f o l l o w i n g

equivalent conditions is s a t i s f i e d : ( i )f o r any j-dense monic m:

with fm = gm, w e have f (ii)t h e u n i t (iii) E

I=

nF:

Vf,f'

A subobject

Y' >-->

F

Y -->

g;

=

L ( F ) of t h e a d j u n c t i o n i s monic;

F -->

~ . j ( f= f ' ) -->

E

Y and maps f , g :

(monic) A

(f = f ' ) .

E o f an o b j e c t E of

>--Z

E i s j-closed

i f f any one o f t h e f o l l o w i n g e q u i v a l e n t c o n d i t i o n s i s s a t i s f i e d : ( i )i f a: E -->

R c l a s s i f i e s A >-->

( i i ) t h e commutative s q u a r e

(iii)E Of

I=

Ve.j(e

E

A) -->

e

E

these d i f f e r e n t formulations,

t h e o r e t i c one,

E , then j a = a;

A

-->

L(A)

E

-->

L(Ej

i s a pullback;

A.

(i) i s the t r a d i t i o n a l category

(ii)i s p a r t i c u l a r l y u s e f u l f o r understanding

Eff

and (iii) i s t h e l o g i c a l f o r m u l a t i o n ( t r e a t i n g j a s a p r o p o s i t i o n a l operator)

.

I t i s obvious from t h e d e f i n i t i o n s t h a t F i s j - s e p a r a t e d

iff

t h e e q u a l i t y on F i s j - c l o s e d ,

and t h a t a s u b o b j e c t of a j - s e p a r a t e d

o b j e c t i s i t s e l f j-separated.

W e c o l l e c t some f u r t h e r f o l k l o r i c

f a c t s about t h e s e n o t i o n s i n t h e n e x t theorem. Theorem 5 . l . ( a ) I f E and F are j - s e p a r a t e d ,

nExF:

ExF -->

L(ExF) = L ( E ) x L ( F ) i s (b) I f F i s j-separated,

t h e composite of nFE: L ( F ~ )%>

->

FE

--5

t h e n SO i s EXF.

Also

nEXnF.

t h e n s o i s FE f o r any E .

Also

L(FE) w i t h t h e n a t u r a l map

L ( F ) L ( E ) followed by t h e isomorphism L ( F ) n E : L ( F ) L ( E )

E L ( F ) E i s t h e monic q F : FE -->

L ( F ) E , and t h e e v a l u a t i o n map

183

The effective topos E F X E -->

F i s o b t a i n e d by f a c t o r i n g evo(aon

Xn

)

through q F .

FE ( c ) If

c

i s j-closed.

F i s j - c l o s e d and a: E -->

>-->

E.

t h e n SO is

E.

E i s j - c l o s e d t h e n s o i s ( A -->

2->

Also L ( A -->

B) = L ( A ) -->

E i s j-closed

( f ) I f A >--> j-closed.

E are j - c l o s e d

E and B >-->

A l s o L(AAB) = L ( A ) A L ( B ) .

(e) I f B any A >-->

E

Also L(a*(C)) = L(a)*(L(C)).

( d ) I f A >--> AAB >-->

F t h e n Cr*(C) >-->

>-->

E for

L(B).

and a: E -->

F t h e n V a . A >-->

F is

= tlL(a) .L(A).

A l s o L(Va.A)

( 9 ) I f R >--z

B)

ExE i s a j - c l o s e d e q u i v a l e n c e r e l a t i o n on E l

Also the image ( o r s u r j e c t i v e

t h e n t h e q u o t i e n t E/R i s j - s e p a r a t e d . monic) f a c t o r i z a t i o n of E-

> L(E)

->

L(E) /L ( R )

> E/R

->

L(E/R)

nE is E-

n E/R

Proof: A l l t r i v i a l by t h e l o g i c of j - o p e r a t o r s of Fourman-Scott

C19791).

= L ( E ) / ~ ( ~ ) .

( s k e t c h e d a t t h e end

Category t h e o r e t i c p r o o f s are ( i m p l i c i t )

i n Johnstone C19771. L e t us now e x p l a i n why w e a r e i n t e r e s t e d i n c l o s e d s u b o b j e c t s .

Our u n d e r s t a n d i n g of Grothendieck t o p o s e s rests on t h e f a c t t h a t i n v e r s e image f u n c t o r s p r e s e r v e c o h e r e n t l o g i c ( t h a t i s t h e i n c l u s i o n of see some of t h e

A , v , ~ ) .

But

Sets i n Eff i s i n t h e wrong d i r e c t i o n i f W e wish t o l o g i c o f Sets p r e s e r v e d i n Eff. I n g e n e r a l a d i r e c t

image f u n c t o r p r e s e r v e s l i t t l e , b u t w e can g e t r a t h e r s t r o n g r e s u l t s , --> E l by r e s t r i c t i n g a t t e n t i o n t o j T h i s is s i g n i f i c a n t because a j - c l o s e d s u b o b j e c t

when d e a l i n g w i t h i n c l u s i o n s E j-closed subobjects. A >-->

E " a g r e e s w i t h i t s meaning i n

nE*(LA)

(This i s v e r s i o n

E" j

= A.

(ii) of t h e d e f i n i t i o n . )

i n t h e sense t h a t

J.M.E. HYLAND

184

Given an i n t e r p r e t a t i o n of t h e atomic formulae of a f i r s t o r d e r language i n E w e g e t

U $ 1 of an a r b i t r a r y formula i n E

( i )an i n t e r p r e t a t i o n

, and

by applying L an i n t e r p r e t a t i o n of t h e atomic formula i n E hence an i n t e r p r e t a t i o n

.

3'

W e a r e i n t e r e s t e d i n when

tation

II $ ]I

in E

j

"

j

and

j

of an a r b i t r a r y formula i n E .

II $ 1 i s a s u b o b j e c t of E , t h e n

Clearly i f L(E)

[$I

(ii)

I[ $ ]I

B$

i s a s u b o b j e c t of

j "agrees with t h e i n t e r p r e -

i n t h e sense t h a t

nE*( 1 4

n j)

=

u$n .

The r e l e v a n t d e f i n i t i o n i s of a form f a m i l i a r from T r o e l s t r a C19731. Definition.

I n a f i r s t o r d e r language, t h e n e g a t i v e formulae ( o r

formulae i n t h e n e g a t i v e fragment) a r e t h o s e b u i l t up from atomic formulae u s i n g Theorem 5 . 2 .

->,V.

A,

I f an i n t e r p r e t a t i o n of a f i r s t o r d e r language i n E

i n t e r p r e t s t h e atomic formulae a s j - c l o s e d s u b o b j e c t s and $ i s a

I$ 1

negativeformula with

>->

E, then

r l E * ( v ~ n j =) ~ $ 1 . Proof:

I n d u c t i o n on t h e complexity of $ u s i n g (5.1) (c) ( d ) ( e ) and

(f).

Remarks

1) W e can only have e q u a l i t y f o r j - s e p a r a t e d o b j e c t s . 2) As

to

n+nj

$

n

i s j-closed,

nE*(

I$

1j)

=

f

$ ]I i s e q u i v a l e n t

= L ( u $ D ) .

3 ) The r e s u l t i s j u s t a consequence of t h e " j - i n t e r p r e t a t i o d ' of t h e l o g i c of E.

For n e g a t i v e formulae w e a r e r e a d i n g it n o t a s a

p r e s c r i p t i o n f o r d e r i v i n g t h e l o g i c of E

j

from t h a t of E , b u t a s t h e

s t a t e m e n t t h a t t h e l o g i c of E a g r e e s w i t h t h a t of E

16

S e p a r a t e d o b j e c t s and c l o s e d s u b o b j e c t s i n

j'

Eff.

I n t h i s s e c t i o n , w e d e s c r i b e what (5.1) means f o r t h e p a r t i c u l a r

topology.

Eff

i s Sets so t h a t j i s t h e double n e g a t i o n j ( I n f a c t w e do n o t use t h i s l a s t f a c t , so t h a t t h e

c a s e when E i s

and E

The effective topos

185

m a t e r i a l r e l a t i v i z e s t o an a r b i t r a r y base t o p o s E . ) . W e w i l l say J t h a t an o b j e c t of Eff is s e p a r a t e d when it i s j - s e p a r a t e d and t h a t a subobject of an o b j e c t is c l o s e d when i t i s j - c l o s e d .

t o one of t h e form ( X , = ) Proof:

i s s e p a r a t e d i f f it i s isomorph

An o b j e c t of Eff

Proposition 6.1.

where

fi

x = x' 1

non-empty

C

implies x = x '

By v e r s i o n (ii) of t h e d e f i n i t i o n of j - s e p a r a t e d ,

w e see

t h a t i f an o b j e c t i s s e p a r a t e d , it i s a s u b o b j e c t of some A X . any c a n o n i c a l monic i n t o A X i s of t h e r e q u i r e d form.

But

Conversely any

o b j e c t of t h e r e q u i r e d form is a s u b o b j e c t of a AX Cthe obvious map

is monic), and s u b o b j e c t s of s e p a r a t e d o b j e c t s a r e s e p a r a t e d . Definition.

An

object (X,=)

of

Eff,

where

1[ x = x ' i

i m p l i e s x = x', i s a c a n o n i c a l l y s e p a r a t e d o b j e c t of

non-empty

Eff.

(Such an o b j e c t i s completely determined by t h e v a l u e s

[Ix E X i

for

each x i n X, and i s e s s e n t i a l l y ( t h a t i s , modulo t r i v i a l coding) given a s a c a n o n i c a l monic i n t o A X . ) Proposition 6 . 2 .

If

(X,=)

and (Y,=) a r e c a n o n i c a l l y s e p a r a t e d , t h e n

so i s t h e u s u a l product ( X x Y , = )

u Proof:

(x,y)

=

where

( x l , y ~ ) n

=

IIX

=

x l n

A

uy =

y l n

.

Immediate from ( 5 . 1 ) ( a ) and t h e d e f i n i t i o n of t h e product of

maps i n t h e l o g i c . The c a s e of f u n c t i o n s p a c e s i s more complex t h a n t h a t o f p r o d u c t s . Since t h e g e n e r a l d e s c r i p t i o n of a f u n c t i o n space (see HJP C19801 i s too clumsy, w e must use ( 5 . 1 ) ( b ) t o c o n s t r u c t a s u i t a b l e r e p r e s e n tation. Proposition 6.3.

L e t (Y,=)

canonically separated.

and (Z,=)

be o b j e c t s o f

Then t h e f u n c t i o n s p a c e (Z,=)

Eff ("=)

w i t h (Z,=) may be

taken t o be t h e c a n o n i c a l l y s e p a r a t e d o b j e c t ( r Z r Y , = ) where ( t a k i n g

r ( z , = ) 5 2) I n = II V

(cYn n

~ ~ Y . ~ ( C ~ I ) = ~ I =

n~ II

Ey-->a(Cyl)=a' ( c y i ) i l y c ~ i

and where t h e e v a l u a t i o n map i s r e p r e s e n t e d by t h e f u n c t i o n a l r e l a t i o n

J.M.E. HYLAND

186

I [ E ~A

EY

A

n .

a(cy1) = z

( 5 . 1 ) ( b ) g i v e s us a monic from (Z,=)

Proof:

("=)

t o rZrY d e f i n e d

i n t h e l o g i c by

uvY

Y . ~ C C ~ I )z

n

which i s e q u i v a l e n t t o t h e formulae g i v e n a s CZ,=) separated.

is canonically

The r e p r e s e n t a t i o n of t h e e v a l u a t i o n map f o l l o w s from

t h e d e f i n i t i o n i n t h e l o g i c o f t h e map d e s c r i b e d i n ( 5 . 1 ) ( b ) by elementary l o g i c . Remark.

I f f o r e v e r y y i n Y , Ey i s non-empty

(and w e may a s w e l l

disregard the o t h e r s ) , then the following a l t e r n a t i v e representation of t h e f u n c t i o n s p a c e i s c a n o n i c a l l y s e p a r a t e d : I Z y , = ) a = a l n

=

n{

where

n

I I ~ = ~ ~ - - > ~ ( ~ ) l =y , ~y l~ E (~ ~~ ~ )

and where t h e e v a l u a t i o n map i s r e p r e s e n t e d a s above.

(We g e t t h i s

a l t e r n a t i v e r e p r e s e n t a t i o n by c o n s i d e r i n g t h e obvious map from rZ(yl=)

(Y,=)

t o Z E Y l where EY i s t h e c a n o n i c a l s u b o b j e c t o f AY o f which

is a quotient,)

E a is non-empty,

Then i f w e d i s r e g a r d t h o s e a i n Z y such t h a t

w e can c o n t i n u e t h i s p r o c e s s and o b t a i n a s i m p l e

d e s c r i p t i o n of i t e r a t e d f u n c t i o n s p a c e s of s e p a r a t e d o b j e c t s .

We

c o n s i d e r t h i s f u r t h e r i n 5 5 7 and 11. W e next consider closed subobjects i n

Proposition 6.4.

A s u b o b j e c t o f an o b j e c t

Eff. (X,=) of

Eff i s

i f f i t i s r e p r e s e n t e d by a c a n o n i c a l monic determined by A

closed E

Cn o f

f o r s o m e A 5 I'(X,=). ( I t does no harm t o l e t A d e n o t e t h e s u b s e t o f r ( X , = )

,

the canonical

monic as d e f i n e d and t h e c l o s e d s u b o b j e c t which it r e p r e s e n t s ) . Proof: By v e r s i o n (ii) of t h e d e f i n i t i o n of j - c l o s e d , o b j e c t of (X,=) must b e of t h e form rlx

-1

a c l o s e d sub-

(AA) f o r some A

TIXI=).

The effective topos

187

But what w e have d e s c r i b e d i s e a s i l y s e e n t o b e e q u i v a l e n t t o t h e d e f i n i t i o n of

nx -1 (AA)

Definition.

A monic o f form (X,=')

Ix='x']

i n the logic.

= U{ I[x=x'JJ

f o r some A 5 T ( X , = )

[=">

IlxleA} = {

(X,=) where

[x=xD, i f [ X I E A , I , otherwise,

i s a c a n o n i c a l c l o s e d monic.

( ( 6 . 4 ) shows

e s s e n t i a l l y t h a t t h e closed subobjects a r e j u s t those represented by c a n o n i c a l c l o s e d m o n i c s ) . On many o c c a s i o n s i t i s more n a t u r a l t o d i s r e g a r d i n (X,=')

Remark.

t h e x which a r e n o t i n A.

W e s h a l l s u i t t e r m i n o l o g y t o need and

r e f e r t o t h i s m o d i f i c a t i o n a l s o a s a c a n o n i c a l c l o s e d monic.

Note

t h a t t h e n o t i o n becomes p a r t i c u l a r l y s i m p l e i n c a s e (X,=) i s c a n o n i c a l l y s e p a r a t e d , a s t h e n w e may t a k e A 5 X ( t a k i n g r(X,=)

5 X again).

We now s a y what ( 5 . 1 ) ( c ) , ( d ) , ( f ) mean f o r t h e e f f e c t i v e t o p o s . Proposition 6.5. of

(x,=), C

a map i n

>-->

L e t A >->

(X,=) and B >->

(Y,=) a s u b o b j e c t o f

(X,=) be s u b o b j e c t s

(Y,=) and [GI:

Eff.

I f C i s a c a n o n i c a l c l o s e d monic ( d e f i n e d from C 5 r ( Y , = ) ) , [Gl-l(C)

(Y,=)

(X,=) -->

then

i s t h e c a n o n i c a l c l o s e d monic d e f i n e d from (r(G)-l(C)

=

{ C x l l { y l G ( x , y ) non-empty)

E

C).

I f A , B are c a n o n i c a l c l o s e d monics ( d e f i n e d from A , B 5 r ( X , = ) ) ,

A A B i s t h e c a n o n i c a l c l o s e d monic d e f i n e d from AnB. c a n o n i c a l c l o s e d monic ( d e f i n e d from B 5 r ( X , = ) ) ,

If B

then

is a

t h e n A ->

B is

t h e c a n o n i c a l c l o s e d monic d e f i n e d from

TA -> and VCG1.B

B = {Cxll i f [ X I

rA t h e n [ X I

E

B},

i s t h e c a n o n i c a l c l o s e d monic d e f i n e d from V r ( C G 1 ) .B =

Proof:

E

{Cyll i f G(x,y) non-empty t h e n [-XI

E

B}.

( 5 . 1 ) t e l l s us t h a t t h e r e l e v a n t s u b o b j e c t s are c l o s e d and

t h a t w e g e t a r e p r e s e n t a t i o n by a p p l y i n g T, d o i n g t h e r e q u i r e d construction i n monic.

Sets, and

taking t h e corresponding canonical closed

J.M.E. HYLAND

188

Remark.

The c o n s t r u c t i o n s d e s c r i b e d i n 6 . 5 are p a r t i c u l a r l y s i m p l e and (Y,=) are c a n o n i c a l l y s e p a r a t e d .

i n case the objects (X,=)

F i n a l l y w e c o n s i d e r t h e meaning o f ( 5 . 1 ) (9) f o r t h e e f f e c t i v e topos.

I t g i v e s a c o n v e r s e t o t h e o b v i o u s remark t h a t i f

is

(XI=)

canonically separated] then the equality ( o r diagonal) i n (Xl=)x(Xl=)

i s t h e c a n o n i c a l c l o s e d monic d e f i n e d by t h e d i a g o n a l i n

xxx. Proposition 6.6.

-

Suppose t h a t

l e n c e r e l a t i o n on ( X , = )

Eff.

E

in

Cxxx

r e p r e s e n t s a c l o s e d equiva-

Then t h e q u o t i e n t ( X , - )

is iso-

morphic ( i n t h e o b v i o u s way) t o t h e c a n o n i c a l l y s e p a r a t e d o b j e c t

( r (XI-)

, z ) where

1 ~ x l ~ C x Jl l = U { Proof:

I[ x ' - x l ' J

Ix'

->

The composite ( X , = )

E

[ X I and xl' > rl ( X I - )

(X,-)

[-I

t e d by H(x,Cxll)

x-xl')

=U{

Ixl'

E

E

[xll}.

Ar(Xl-)

i s represen-

Ixll).

By ( 5 . 1 ) (9) w e r e q u i r e t h e image f a c t o r i z a t i o n of [HI, and what w e have i s a s t a n d a r d d e f i n i t i o n of t h i s f a c t o r i z a t i o n i n t h e l o g i c .

57.

The e f f e c t i v e o b j e c t s . Since

Sets i s

i n c l u d e d i n Eff,

c o n t a i n s c l a s s i c a l mathema-

Ef

t i c s so much of it i s n o t p a r t i c u l a r l y " e f f e c t i v e " .

In t h i s section

w e c o n s i d e r o b j e c t s whose c l o s e r e l a t i o n t o t h e a p p l i c a t i v e s t r u c t u r e (IN ,. )

e n s u r e s t h a t o p e r a t i o n s on them a r e g e n u i n e l y " e f f e c t i v e " .

I n l a t e r s e c t i o n s w e w i l l show t h a t t h e o b j e c t s of a n a l y s i s i n

Eff

a r e ( q u i t e f a m i l i a r ) o b j e c t s of t h i s k i n d . Definition.

An o b j e c t

1x

E

XJ

(ii) U x

E

XJ n fx'

(i)

(X,=)

is (strictly) effective i f f

i s non-empty each x

and (iii)U x = x ' l = II x

E

X,

X J non-empty i m p l i e s x = x ' ,

E

E

XI

n Ux'

E

XI

.

( O c c a s i o n a l l y w e may d e s c r i b e an o b j e c t as e f f e c t i v e when it i s isomorphic t o one of t h e above form. l o o s e s e n s e i s meant.)

I t w i l l be obvious when t h i s

189

The effective topos

C l e a r l y e f f e c t i v e o b j e c t s a r e ( c a n o n i c a l l y ) s e p a r a t e d , and w e can e a s i l y show t h a t t h e y s h a r e t h e c l o s u r e p r o p e r t i e s of s e p a r a t e d

.

objects

Proposition 7 . 1 .

(a) I f (X,=)

and (Y,=) a r e e f f e c t i v e , then so i s

(b) I f

i s effect'ive, then so i s t h e function

t h e i r product. (Z,=)

f o r any (Y,=) i n ~ f f .

space (z,=) (',=)

( c ) A s u b o b j e c t of an e f f e c t i v e o b j e c t i s effective. ( d ) A q u o t i e n t of an e f f e c t i v e o b j e c t by a c l o s e d equivalence r e l a t i o n i s e f f e c t i v e . ( a ) i s t r i v i a l : look a t ( 6 . 2 ) .

Proof:

( b ) f o l l o w s by i n s p e c t i o n of (6.3). a

E

r Z r Y with

I f w e restrict t o those

non-empty, t h e n w e g e t an o b j e c t s a t i s f y i n g

Ea

( i ) (ii) , and (iii)above.

( c ) r e q u i r e s more work. l e t ( X I = ' ) --> [ x

='XI

Write x

Since

L e t ( X I = ) be s t r i c t l y e f f e c t i v e and

be a c a n o n i c a l monic w i t h

(X,=)

Il x=x'n

II

= R(x)A

E'

X f o r x ='x and p u t X ' = Ix

for some s t r i c t r e l a t i o n a l R

[ x E ' x n n [x' € ' X I

E

E

cX

.

XI Ux E ' X 1 i s non-empty}.

non-empty i m p l i e s

Ilx

E

xDn [ x u E xn

non-empty which i m p l i e s x = x', w e g e t a s t r i c t l y e f f e c t i v e o b j e c t (XI,=) with

[x

E

= I[ x E ' X 1

X'l

.

I t i s isomorphic t o ( X I = ' )

because

n ( (ux is non-empty

E

x'I

n Ux'

.

( d ) f o l l o w s from ( 6 . 6 ) .

If

E

xi] 1

<-->

I[x

I t follows t h a t

x

E

X

I

( r ( X I = ),%)

n I[ X'E X

)

- i s a closed equivalence

r e l a t i o n on (X,=) which i s s t r i c t l y e f f e c t i v e , t h e n non-empty imples

='x'II

[x-xi n [xl-x'D

1 non-empty which i m p l i e s x

=

is strictly effective.

The f u l l s u b c a t e g o r y of

Eff

whose o b j e c t s a r e t h e e f f e c t i v e

ones has a c o n c r e t e r e p r e s e n t a t i o n f a m i l i a r t o l o g i c i a n s i n

x'.

J.M.E. HYLAND

190

connection w i t h t h e e f f e c t i v e o p e r a t i o n s . r e l a t i o n s on IN R,S,

( t h a t i s e q u i v a l e n c e r e l a t i o n s on t h e i r f i e l d s )

... and w r i t e

m/R

= { C n l R ( n E F i e l d ( R ) } f o r t h e set of

e q u i v a l e n c e c l a s s e s of R . such t h a t t h e r e i s f

E

L e t a m x F: R ->

Field(R).

E

S be a map F:

N/R->lN/S

IN w i t h

F(CnlR) for a l l n

Take p a r t i a l e q u i v a l e n c e

=

[f(n)lS

C l e a r l y w e have a c a t e g o r y .

Each p a r t i a l e q u i v a l e n c e r e l a t i o n R g i v e s r i s e t o a s t r i c t l y effective object ( B / R , = ) F: R -->

of

Eff

where E ( l n l R )

S g i v e s r i s e t o a map (IN/R ,=)

->

=

Cnl,.

( B / S ,=)

A map

represented

by F(CnlR,[mlS) = U{CnlRh[mlSIF([nlR) = [ m l s l , and so w e have a f u n c t o r i n t o

Eff

which i s c l e a r l y f a i t h f u l and i s

f u l l by a p p l y i n g g l o b a l s e c t i o n s t o ( 7 . 1 ) ( b ) .

C l e a r l y any s t r i c t l y

e f f e c t i v e o b j e c t i s isomorphic t o one o b t a i n e d from a p a r t i a l equivalence r e l a t i o n .

L e t u s d e s c r i b e t h e f u n c t i o n s p a c e SR i n t h e

c a t e g o r y of p a r t i a l e q u i v a l e n c e r e l a t i o n . R

I t i s g i v e n by

eS f i f f nRm i m p l i e s e ( n ) S f ( m )

.

A moment's thought shows t h a t t h i s c o r r e s p o n d s t o t h e p r e s c r i p t i o n

f o r f i n d i n g t h e s p a c e of f u n c t i o n s from ( B / R ,=)

.

t o ( N / S ,=)

given

by ( 7 . 1 ) (b)

T h i s i s a u s e f u l way t o t h i n k o f t h e m a t e r i a l i n

§ § l o and

( I n f a c t t h e embedding of t h e p a r t i a l e q u i v a l e n c e

11.

relations in

Eff

p r e s e r v e s t h e l o c a l C a r t e s i a n c l o s e d s t r u c t u r e of

t h e former c a t e g o r y . ) One p a r t i c u l a r e f f e c t i v e o b j e c t i s c r y i n g o u t f o r a t t e n t i o n : t h a t c o r r e s p o n d i n g t o t h e e q u a l i t y r e l a t i o n on N . object. I N =

(IN ,=)

This i s t h e

where [In = mll

=

inlnIm}.

As w e n o t e d i n 13, t h i s i s t h e n a t u r a l number o b j e c t ; w e c o n s i d e r some of i t s p r o p e r t i e s i n l a t e r s e c t i o n s .

F i r s t however, w e w i l l use it

t o g i v e a c h a r a c t e r i z a t i o n of e f f e c t i v e o b j e c t s .

R e c a l l t h a t any

191

The effective topos

object (X,=)

i s a q u o t i e n t o f a s u b o b j e c t o f AX.

o b j e c t s w e c a n r e p l a c e AX by ( I N ,=) Proposition 7.2.

.

Every e f f e c t i v e o b j e c t i s a q u o t i e n t by a c l o s e d

e q u i v a l e n c e r e l a t i o n of a c l o s e d s u b o b j e c t of Proof:

If

(X,=)

For e f f e c t i v e

(IN ,=)

.

c o r r e s p o n d s a s above t o t h e p a r t i a l e q u i v a l e n c e

r e l a t i o n R on I N , t h e n t h e c l o s e d s u b o b j e c t o f

is that

LIN ,=)

determined by F l d ( R ) 5 I N and t h e c l o s e d e q u i v a l e n c e r e l a t i o n

- is

given by

That t h e r e s u l t i n g q u o t i e n t of a s u b o b j e c t of t h e same R i s immediate i n view of

( I N ,=)

g i v e s rise t o

( 6 . 4 ) and ( 6 . 6 ) .

Now w e can s t a t e o u r c h a r a c t e r i z a t i o n theorem. Theorem 7 . 3 .

Eff

The f o l l o w i n g c o n d i t i o n s on a o b j e c t X of

are

equivalent: ( i )X i s isomorphic t o a s t r i c t l y e f f e c t i v e o b j e c t ; (ii)X i s a c l o s e d q u o t i e n t of a c l o s e d s u b o b j e c t of

(iii) X i s a c l o s e d q u o t i e n t o f a s u b o b j e c t of

Proof:

( i )i m p l i e s

(ii) i s ( 7 . 2 ) ,

(ii) i m p l i e s

(IN

,=)

:

(IN , = I . (iii)i s t r i v i a l

and (iii) i m p l i e s ( i ) f o l l o w s from ( 7 . 1 ) ( c ) and ( a ) . Remark.

Since

(m,=)i s

t h e n a t u r a l number o b j e c t , w e have shown

t h a t t h e e f f e c t i v e o b j e c t s are t h o s e subnumerable i n a c e r t a i n way. However t h e e q u a l i t y on an e f f e c t i v e o b j e c t must be c l o s e d (as i t

i s a s e p a r a t e d o b j e c t ) and t h e r e a r e q u o t i e n t s o f

(m,=)

e q u i v a l e n c e r e l a t i o n s which are e m p h a t i c a l l y n o t c l o s e d .

by (The

r e a d e r w i l l know where t o look a f t e r r e a d i n g t h e n e x t s e c t i o n ! ) So t h e e f f e c t i v e o b j e c t s a r e a p r o p e r s u b c l a s s of t h e q u o t i e n t s of

d e c i d a b l e o b j e c t s r e c e n t l y s t u d i e d by P e t e r Johnstone i n a g e n e r a l context.

J.M.E. HYLAND

192 18.

Markov's p r i n c i p l e and almost n e g a t i v e formulae. I n t h i s s e c t i o n w e see how t h e g e n e r a l r e s u l t of (5.2) can be

extended i n t h e c a s e of t h e t o p o s

Effj

j with

Lemma 8.1.

Sets. A: Sets ->

Eff

and (double n e g a t i o n ) topology

=

Eff

preserves t h e i n i t i a l object.

Thus

Il i s always a c l o s e d s u b o b j e c t , and hence d e c i d a b l e s u b o b j e c t s a r e closed. Proof:

T r i v i a l category theory.

Lemma 8.2.

Markov's p r i n c i p l e

VR L P ( N ) . ( V n . R ( n ) v ~ R ( n ) h ~ l 3 n ( R ( n -> )

Eff.

holds i n Proof:

3nRCn))

A s t h e a r i t h m e t i c a l s t a t e m e n t s h o l d i n g i n Ef

a r e those

r e a l i z e d i n t h e o r i g i n a l s e n s e of Kleene (see 13) t h i s i s t h e Note t h a t w e do n o t need t o

s t a n d a r d argument ( T r o e l s t r a C19731). know about P ( N ) 1 Lemma 8 . 3 . 3n.R(n,x) >-> Proof:

I f R >-> X

N x X is a d e c i d a b l e s u b o b j e c t i n E X , t h e n

i s c l o s e d and r ( 3 n . R ( n f x ) ) = 3 n . r ( R ( n f x ) ) .

This amounts t o --3n.R(n,x)

5

3n.R(n,x) which f o l l o w s by

(8.2).

Remark.

Though ( 8 . 2 )

depends on Markov's p r i n c i p l e i n

does n o t r e l a t i v i z e t o an a r b i t r a r y t o p o s ,

w e w i l l always have j ( 3 n . R ( n , x ) )

5

w, and

so

( 8 . 3 ) does r e l a t i v i z e :

3n.R(n,x).

( 8 . 1 ) and ( 8 . 3 ) s u g g e s t t h a t w e e x t e n d t h e c l a s s of n e g a t i v e formulae. Definition.

A formula is c a l l e d almost n e g a t i v e i f f it i s b u i l t up

from atomic formulae u s i n g

A,

->,V,i,

and sequences of 3n a p p l i e d

t o d e c i d a b l e formulae ( t y p i c a l l y e q u a t i o n s between numerical-valued

terms)

.

W e now g i v e o u r e x t e n s i o n of ( 5 . 2 ) . Theorem 8 . 4 .

I f t h e atomic formulae of a f i r s t o r d e r language a r e

193

The effective topos

i n t e r p r e t e d as c l o s e d s u b o b j e c t s i n with

[@

1 >->

Eff

and @ i s almost n e g a t i v e

then

E,

nE*(u+nj) =

t@n.

As f o r ( 5 . 2 ) u s i n g ( 8 . 1 ) and ( 8 . 3 ) a s w e l l .

Proof:

The f o r c e o f (8.4) i s t h a t , f o r @ a l m o s t n e g a t i v e @ i s t r u e in

Eff

i f f t h e c o r r e s p o n d i n g i n t e r p r e t a t i o n of @ i n Sets i s t r u e :

t h a t i s , t h e meaning of @ i n Ef

" a g r e e s w i t h " i t s c l a s s i c a l meaning.

( 8 . 4 ) i s a v e r s i o n of 3 . 2 . 1 1 ( i ) and (ii)of T r o e l s t r a 119731;

w e could o b t a i n a more p r o o f - t h e o r e t i c v e r s i o n by s e l a t i v i z i n g t o t h e f r e e t o p o s ( w i t h n a t u r a l number o b j e c t ) .

For a language which

can " e x p r e s s i t s own r e a l i z a b i l i t y " we c o u l d o b v i o u s l y o b t a i n v e r s i o n s of 3.2.12

and 3.2.13 of T s o e l s t r a 119731.

For t h e sake of

completeness w e g i v e a v e r s i o n of 3.6.5 of T r o e l s t r a C19731. Definition

i s t h e l e a s t class C of

( c f . Hyland C19771) PR(a.n.1

formulae such t h a t ( i )C c o n t a i n s a l l atomic formulae; ( i i )C i s c l o s e d under ~ , v , V , 3 ; (iii)i f @ is almost n e g a t i v e

(more g e n e r a l l y almost n e g a t i v e

preceded by e x i s t e n t i a l q u a n t i f i e r s ) and $ i s i n C, t h e n ( @ ->

$1

i s i n C. P r o p o s i t i o n 8.5.

I@ 1 >->

with

I n t h e s i t u a t i o n of

i f @ i s i n PR(a.n)

then

E,

n+n Proof:

(8.41,

5

nE*(

I@ n

j).

By i n d u c t i o n on t h e complexity of

@.(Note

t h a t v and + ' a r e

from t h e way t h e y are i n E). j Remark. For a g e n e r a l s h e a f subtopos E of E w e have j [ @ I 5 nE*( fi @ 1 . ) f o r a l l @ i n P R ( j - c l o s e d ) . So i f atomics are calculated differently i n E

3

i n t e r p r e t e d as j - c l o s e d , PR(negative)

then w e get the r e s u l t f o r a l l @ i n

.

The f o r c e of

(8.5) i s t h a t , f o r @ i n PR(a.n), i f @ i s t r u e i n

J.M.E. HYLAND

194

Eff,

Sets i s

then t h e c o r r e s p o n d i n g i n t e r p r e t a t i o n of @ i n

true.

T h i s g i v e s r i s e t o a c o n s e r v a t i v e e x t e n s i o n r e s u l t (when r e l a t i v i z e d ) a s i n T r o e l s t r a C19731 53.6. §9

Choice p r i n c i p l e s and t h e r e a l numbers. I n t h i s s e c t i o n w e make a s t a r t towards showing t h a t a n a l y s i s

in

Eff

is j u s t constructive recursive analysis.

Markov's p r i n c i p l e ( 8 . 2 ) . )

(We a l r e a d y have

W e do t h i s i n two s t e p s .

F i r s t w e show

t h a t w e have t h e c h o i c e p r i n c i p l e s t o e n s u r e t h a t t h e Dedekind r e a l s ( t h e r i g h t r e a l s i n a t o p o s ) a r e Cauchy (see Fourman-Hyland Then w e u s e t h e re-

C19791 and a l s o Fourman-Grayson t h i s volume).

Eff

s u l t s of 5 7 t o show t h a t t h e Cauchy r e a l s i n

c a n be i d e n t i f i e d

w i t h a f a m i l i a r s t r i c t l y e f f e c t i v e o b j e c t used i n c o n s t r u c t i v e recursive analysis. F i r s t w e need t o know what t h e s p a c e of f u n c t i o n s from IN

looks l i k e i n

an a r b i t r a r y ( X , = )

Eff.

to

A s s t a t e d i n H J P C19801,

by

l o g i c a l c o n s i d e r a t i o n s it i s

(ZIN x x , = ) where [G =

HI

= EGAn{G(n,x) <->

H ( n , x ) In

TN , x

E

d

XI,

w i t h EG t h e n o n - s t a n d a r d v a l u e of "G i s a f u n c t i o n a l r e l a t i o n " . Suppose now t h a t e r e a l i z e s Vn f o r every n, e ( n ) such t h a t e ( n )

E

E

E

U I E X A[ @ ( n , x ) ] Ix

E X ~ A[ @ ( n , x n ) l

.

IN , 3 x E

E

XI.

S e t G(n,x)

(XI=)

. @ ( n , x ) . Then

F o r e a c h n p i c k xn =

EnA [ x = x n 1

.

Now ( u n i f o r m l y i n e ) w e can f i n d numbers r e a l i z i n g EG and V n . 3 x . G ( n I x ) h $ ( n , x ) : G i s r e l a t i o n a l , s t r i c t and s i n g l e v a l u e d i n a s t a n d a r d way from i t s d e f i n i t i o n : A n . < n , n l ( e ( n ) ) > r e a l i z e s G i s t o t a l ; Xn. , a 2 ( e , n ) > > realizes V n . 3 x . G ( x , n ) h @ ( x l n ) . Thus ( u n i f o r m l y i n e ) w e have a number realizing

195

The effective topos 3g: Ri->

(X,=) .Vn.$(n,g(n))

and so w e have proved t h e f o l l o w i n g r e s u l t . Proposition 9 . 1 .

t h e axiom o f c h o i c e from t h e n a t u r a l

AC(IN,X),

numbers t o an a r b i t r a r y t y p e X , h o l d s i n Remark.

W e used AC ( I N , X )

in

sets i n

Eff.

t h e above p r o o f .

But i f

(X,=)

i s e f f e c t i v e , t h e n no use o f a c h o i c e p r i n c i p l e i n t h e base t o p o s i s needed (compare ( 7 . 1 ) ( b ) : i n t h i s c a s e t h e argument i s c o n t a i n e d

w i t h i n T r o e l s t r a 119731 3.2 1 5 . ( l e f t t o t h e r e a d e r ) w e a l s o have t h e s t r o n g e r

By a s i m i l a r proof result. Proposition 9 . 2 .

DC(X),

t h e axiom of dependent c h o i c e s on a n

a r b i t r a r y type X, holds i n

Eff.

Again D C ( X ) i s used i n t h e p r o o f , b u t i s n o t needed f o r

Remark.

e f f e c t i v e o b j e c t s X. AC(W,IN)

are the same.

is enough t o show t h a t t h e Cauchy and Dedekind r e a l s To g e t an e x p l i c i t r e p r e s e n t a t i o n of

R as a

s t r i c t l y e f f e c t i v e o b j e c t , w e use t h e Cauchy sequence d e f i n i t i o n . Lemma 9.3.

The i n t e g e r s Z and r a t i o n a l s Q i n

s t r i c t l y effective objects

Ex = Proof:

a,=)and

Eff

( Q , = ) where f o r x i n Z o r Q ,

# x i s an e l e m e n t a r y code f o r

# x) where

can be t a k e n a s

X.

They are o b t a i n e d s u c c e s s i v e l y from (IN ,=)

by t a k i n g c l o s e d

( d e c i d a b l e ) q u o t i e n t s of c l o s e d ( d e c i d a b l e ) s u b o b j e c t s o f p r o d u c t s :

so t h e r e s u l t f o l l o w s from t h e p r e s c r i p t i o n s i n v o l v e d i n (7.1) ( a ) , ( c ) [ e a s y c a s e of c l o s e d s u b o b j e c t s l and Lemma 9 . 4 .

The s p a c e of maps from IN t o Q i n

e f f e c t i v e o b j e c t (Qm ,=) Qm

and

[[ c1

E

(a).

QIN!

Eff

is the s t r i c t l y

where

= t h e r e c u r s i v e f u n c t i o n s from

= {eje(n) = # a ( n )1

IN t o Q

, t h e s e t of i n d i c e s f o r a .

J.M.E. HYLAND

196

This i s t h e p r e s c r i p t i o n i m p l i c i t i n ( 7 . 1 ) ( b )

Proof:

.

Since we

have enough c h o i c e t o show t h a t any r e a s o n a b l e n o t i o n s of Cauchy

Eff w e d e f i n e

sequence g i v e t h e same r e a l s i n

C S , t h e c o l l e c t i o n of

( r e s t r i c t e d ) Cauchy sequences by

Thus d e f i n i t i o n i s i n t h e n e g a t i v e fragment and so s i n c e < i s d e c i d a b l e on t h e r a t i o n a l s and hence by (8.1) c l o s e d , d e f i n e s a c l o s e d s u b o b j e c t of Qm

in

Eff.

I n view of (8.4) w e can i d e n t i f y

it.

Lemma 9.5.

The space of Cauchy sequences i n

Eff

is the s t r i c t l y

e f f e c t i v e o b j e c t ( C S , = ) where CS i s t h e set of r e c u r s i v e Cauchy sequences and Proof:

Ur

E

i s t h e s e t of i n d i c e s f o r r .

CS]I

By t h e d i s c u s s i o n above.

To o b t a i n t h e r e a l s IR, w e t a k e t h e q u o t i e n t of CS by t h e equivalence r e l a t i o n r

-

s i f f vn.lrn-snl

<

1 p-3 '

(This c h o i c e of d e f i n i t i o n g i v e s one p l e n t y of "elbow room".) Proposition 9.6.

The space IR of r e a l s i n

e f f e c t i v e o b j e c t (IR ,=)

Eff

is the s t r i c t l y

where

IR = t h e r e c u r s i v e r e a l s ( t h a t i s r e a l s w i t h r e c u r s i v e Cauchy sequences converging t o them) and

Ux

E

IR ]I = t h e s e t of i n d i c e s f o r Cauchy sequences converging t o x.

Proof:

As b e f o r e

- defines a

i s by t h e p r e s c r i p t i o n of

c l o s e d e q u i v a l e n c e r e l a t i o n s o this

(7.1)

(a).

W e have shown t h a t t h e r e a l s i n

Eff

a r e represented j u s t a s

t h e y are i n ( c o n s t r u c t i v e ) r e c u r s i v e a n a l y s i s .

Of c o u r s e , as t h e y

t o o a r e d e f i n e d i n t h e n e g a t i v e fragment, t h e o p e r a t i o n s of a d d i t i o n , m u l t i p l i c a t i o n and so f o r t h a r e what t h e y should b e .

To

do s e r i o u s a n a l y s i s however w e need t o c o n s i d e r f u n c t i o n s which w e

do i n t h e next few s e c t i o n s .

The effective topos

197

I t i s seldom e f f i c i e n t t o g r i n d t h i n g s o u t i n models f o r

Remark.

c o n s t r u c t i v e a n a l y s i s : where p o s s i b l e one should use t h e axiomatic p o i n t of view.

Consider f o r example t h e q u e s t i o n of t h e fundamental

theorem of a l g e b r a i n

Eff.

This theorem i s proved i n Bishop C19671.

One way of r e a d i n g B i s h o p ' s c o n s t r u c t i v e mathematics (though n o t t h e intended one!) i s t o r e g a r d it a s f o r m a l i z e d i n an i n t u i t i o n i s t i c type t h e o r y with e x t e n s i o n a l e q u a l i t y and u s i n g ( D C ) .

Hence i n view

of ( 9 . 2 ) t h e fundamental theorem of a l g e b r a i s t r u e i n

Eff

in other realizability toposes). t a t i o n of C i n

"al,

Eff d e r i v e d from

..., a n

( a s it i s

I n view of t h e obvious r e p r e s e n (9.6)

, and

the fact that

a r e t h e r o o t s of zn+an-lzn-l+.

..+ao = 0"

d e f i n e s a c l o s e d s u b o b j e c t ( i n C 2n) w e can i n t e r p r e t t h i s f a c t a s There i s an e f f e c t i v e p r o c e s s t a k i n g i n d i c e s f o r t h e

follows.

r e c u r s i v e complex c o e f f i c i e n t s of a monic polynomial of d e g r e e n over t h e r e c u r s i v e complex numbers t o i n d i c e s f o r t h e r e c u r s i v e roots.

I t is n o t t r i v i a l t h a t a r e c u r s i v e polynomial has r e c u r s i v e

r o o t s and any n a t u r a l proof would s e e m t o e s t a b l i s h t h e s t r o n g e r r e s u l t and a s such would have t h e (DC)

form

of an a b s t r a c t proof using

E f f e c t i v i t y and Church's T h e s i s .

510.

I t i s t i m e t o g i v e s u b s t a n c e t o t h e c l a i m made i n 17 t h a t W e f i r s t consider

o p e r a t i o n s on e f f e c t i v e o b j e c t s a r e " e f f e c t i v e " . t h e s p e c i a l c a s e of Church's T h e s i s . Lemma 10.1.

The space of maps from IN t o IN i n

Eff

is the s t r i c t l y

e f f e c t i v e o b j e c t ( I N r n ,=) where ININ

and Proof:

6a

E

= t h e r e c u r s i v e f u n c t i o n s from

ININ]

IN t o IN

= { e l e ( n ) = a ( n ) 1 , t h e s e t of i n d i c e s f o r a

This i s t h e p r e s c r i p t i o n i m p l i c i t i n (7.1) ( b )

Proposition 10.2.

,

.

"Church's T h e s i s " t h a t a l l f u n c t i o n s a r e r e c u r s i v e ,

J.M.E. HYLAND

198

mIN . 3 e . V n . 3 y ( T ( e , n , y ) ~ U ( y ) =

~ c Ei

Eff.

holds i n

CT i s X l e e n e ' s T - p r e d i c a t e

Proof: in

ci(n))

and U h i s o u t p u t f u n c t i o n . ]

I n view o f ( 3 . 3 ) e l e m e n t a r y r e c u r s i o n t h e o r y c a n be developed

Eff

as i n T r o e l s t r a E19731.

Eff

in

a(n))

So by

Vn. 3 y . ( T ( e , n , y ) A U ( y ) =

(8.4)

a g r e e s w i t h i t s meaning i n

Sets.

Then Xe.<e,<e,e>>

r e a l i z e s "Church's T h e s i s " . Remark.

Church's T h e s i s a s t r a d i t i o n a l l y f o r m u l a t e d i n H e y t i n g ' s

A r i t h m e t i c (see T r o e l s t r a C19731) i s a n amalgam o f o u r "Church's T h e s i s " and AC(IN ,IN). W e can hope t o g e n e r a l i z e ( 1 0 . 2 ) t o a l l e f f e c t i v e o b j e c t s i n

( 7 . 3 ) which s t a t e s t h a t t h e y can i n a c e r t a i n way be sub-

view of

numerated (by t h e codes f o r t h e i r e l e m e n t s ) . Lemma 10.3. in

Eff,

If

i s s t r i c t l y e f f e c t i v e and ( Y , = )

(Z,=)

t h e n t h e s p a c e of maps from (Y,=) t o ( Z , = )

s t r i c t l y e f f e c t i v e object (ZrY,=)

Eff

i s the

where

= t h e " r e c u r s i v e " maps from T Y t o Z

Zry

in

is a r b i t r a r y

( t h a t i s , t h e maps

with i n d i c e s ) , and

ci E

= {ele(n)

ZTy

E

Eci(y) f o r a l l n

E

Ey}, t h e set o f

i n d i c e s f o r a. Proof:

This i s t h e prescription i m p l i c i t i n (7.1) ( b ) .

Here t h e n i s a g e n e r a l i z a t i o n of Proposition 10.4.

L e t (Y,=) <-

(10.2). B >->

IN r e p r e s e n t t h e e f f e c t i v e

sY

o b j e c t (Y,=) as a q u o t i e n t of a c l o s e d s u b o b j e c t of I N , and l e t (X,=)

A >-->

<s

r e p r e s e n t (X,=) as a q u o t i e n t o f a c l o s e d

IN

X

s u b o b j e c t of I N .

Yci holds i n

E

Then a " g e n e r a l i z e d Church's T h e s i s "

YX.3e.Va

E

A . 3 z ( T ( e , a , a ) h a ( S X ( a ) ) = Sy(U(z)))

Eff.

(One can u s e f u l l y compare t h i s r e s u l t w i t h t h e t r e a t m e n t of t h e extended Church's T h e s i s i n T r o e l s t r a C19731.)

The effective topos Proof:

199

The c o n d i t i o n s g i v e n e n s u r e t h a t a ( S x ( a ) ) = Sy(U(z)) i n t e r -

p r e t s as a c l o s e d s u b o b j e c t . need B c l o s e d . )

(Note t h a t U(y)

E

B is implicit, so we

Since

3 z ( T ( e , a , z ) A a ( S X ( a ) )= Sy(U(z))) is equivalent t o

a ( S , ( a ) ) = Sy(U(z)),

3~.T(e,a,z)AVz.(T(e,a,z) -->

it a l s o i n t e r p r e t s a s a c l o s e d s u b o b j e c t . e from a n i n d e x f o r a . t a k i n g any a

E

I t remains t o d e t e r m i n e

The t o t a l c o n d i t i o n f o r Sx g i v e s a map

A t o an e l e m e n t o f E S X ( a ) :

a n i n d e x f o r a maps t h i s

t o E a S X ( a ) ; t h e c o n d i t i o n t h a t S y i s o n t o p r o v i d e s a map from t h i s t o some b

E

B with Sy(b) = a S x ( a ) .

e i s a n i n d e x f o r t h i s composite

which can c l e a r l y b e chosen e f f e c t i v e l y i n t h e i n d e x f o r a . I n p a r t i c u l a r , w e can see t h a t when e f f e c t i v e o b j e c t s a r e presented ( v i a p a r t i a l equivalence r e l a t i o n s ) a s closed quotients of c l o s e d s u b o b j e c t s o f I N , t h e n maps between them a r e e f f e c t i v e i n t h e i n d i c e s (and t h i s h o l d s i n

Eff).

This i s t y p i c a l l y t h e s i t u a t i o n

i n constructive recursive analysis. 111.

The e f f e c t i v e o p e r a t i o n s . I n t h i s s e c t i o n w e use ( 1 0 . 4 )

a s t h e i n d u c t i o n s t e p t o show

t h a t t h e s t a t e m e n t t h a t t h e f i n i t e t y p e s o v e r IN a r e t h e h e r e d i t a r i l y (extensional) e f f e c t i v e operations holds i n

Eff.

A s s u m e f o r n o t a t i o n a l p u r p o s e s a c o l l e c t i o n of type symbols g e n e r a t e d from 0 by x ( f o r p r o d u c t s ) and --> The f i n i t e t y p e s o v e r t h e n a t u r a l numbers d e f i n e d i n d u c t i v e l y by INo

( f o r function spaces).

( m u la

a t y p e symbol) a r e

= IN,

IN U X T

=

INu+= =

m'xm u 7'

(rnT )

El 0 .

The h e r e d i t a r i l y e f f e c t i v e o p e r a t i o n s ( H E O u l u a t y p e symbol)

(see Kreisel 119591 and T s o e l s t r a C19731) may be d e f i n e d by f i r s t d e f i n i n g a c o l l e c t i o n ( R u I a a t y p e symbol) of p a r t i a l e q u i v a l e n c e

JM.E. HYLAND

200 r e l a t i o n s i n d u c t i v e l y by nROm

iff

n = m,

nRaxTm

iff

r1 ( n )Raml ( m ) and r 2 ( n ) R T r 2(m)

eRa+.rf

i f f i f nRam t h e n e ( n )

, f (m) are

,

d e f i n e d and

e(n)RTf(m). W e can t h e n r e g a r d HEOa a s t h e e q u i v a l e n c e classes I N / R a .

I n view

o f t h e d i s c u s s i o n i n 9 7 , w e can e q u a l l y r e g a r d HEOa a s b u i l t up ( t o g e t h e r w i t h i n d i c e s f o r i t s e l e m e n t s ) from t h e n a t u r a l number

Eff,

object i n i n (10.3).

by t a k i n g t h e u s u a l p r o d u c t s and f u n c t i o n s p a c e s as

Thus i n

Eff

No

= (HEOaI=)where

Ex = t h e i n d i c e s f o r x,

and where nRam i f f n,m a r e i n d i c e s f o r t h e same x

E

HEOg.

Then w e

can r e g a r d HEOa as t h e g l o b a l s e c t i o n s o f t h e f i n i t e t y p e s o v e r JN in

Eff. These d e f i n i t i o n s a l l r e l a t i v i z e and o u r n e x t r e s u l t s t a t e s

that

Eff

"knows t h a t i t s f i n i t e t y p e s a r e t h e e f f e c t i v e o p e r a t i o n s " .

Theorem -11.1. For each a , INa

= HEOa h o l d s i n

Eff,

i n such a way

t h a t t h e p r o d u c t s and f u n c t i o n s p a c e s c o r r e s p o n d . Proof:

The R a t s are d e f i n e d by n e g a t i v e formulae, and so by 18.4)

interpret in

Eff

as c l o s e d p a r t i a l e q u i v a l e n c e r e l a t i o n s a g r e e i n g

w i t h t h e i r meaning i n

Sets.

I f we c a l c u l a t e the equivalence

c l a s s e s i n t h e obvious way u s i n g (7.1) ( d ) w e j u s t g e t (HEOg,=) t h a t i s INa

as i t ought.

in

Eff.

C l e a r l y t h e rest o f t h e s t r u c t u r e c o r r e s p o n d s

( I f t h i s i s t o o a b s t r a c t , t h e r e a d e r can use a

laborious induction, with (10.4)

d e a l i n g w i t h t h e main i n d u c t i o n

step.) Remark.

Something q u i t e deep i s going on behind (11.1) which i s

connected w i t h i t e r a t i o n s of t h e e f f e c t i v e t o p o s c o n s t r u c t i o n a: s t u d i e d i n B i t t s C19811.

I t i s i n connection w i t h t h e e f f e c t i v e

o b j e c t s t h a t w e can g e t a g e n e r a l e x p r e s s i o n of t h e idempotency o f

20 1

The effective topos of r e a l i z a b i l i t y (see T r o e l s t r a (1973) 3 . 2 . 1 6 ) . Sequential continuity.

512.

From ( 8 . 2 ) , ( 9 . 2 ) , ( 1 0 . 4 ) and t h e d i s c u s s i o n i n 19, it s h o u l d be clear t h a t analysis i n

Eff i s

j u s t constructive recursive analysis.

So w e have t h e u s u a l c o n t i n u i t y r e s u l t s which are v e r s i o n s of t h e

Kreisel-Lacombe-Shoenfield theorem. Theorem 1 2 . 1 .

"Brouwer's Theorem" t h a t e v e r y map from IR t o IR i s

Eff.

continuous h o l d s i n

The r e a d e r w i l l have t o do t h i s h i m s e l f

Proof:

(along t h e l i n e s of

below) o r e l s e f i n d ( a s I have f a i l e d t o do) a r e a d a b l e

(12.4)

account from t h e Russian s c h o o l . ( 1 2 . 1 ) i s o n l y moderately s p e c t a c u l a r .

Recursive maps on t h e

r e c u r s i v e r e a l s , w h i l e n o t t h e r e s t r i c t i o n of c o n t i n u o u s f u n c t i o n s on t h e ( c l a s s i c a l ) reals (see ( 1 3 . 4 ) ) , domain.

are c o n t i n u o u s

on t h e i r

( T h i s i s s t a t e d as E x e r c i s e 15.35 i n Rogers 119671.)

w e j u s t need e f f e c t i v i t y t o g e t ( 1 2 . 1 ) .

So

By p a s s i n g t o h i g h e r t y p e s

w e g e t a more i n t e r e s t i n g phenomenon: we g e t e f f e c t i v e maps, which are n o t c o n t i n u o u s on t h e i r e f f e c t i v e domain, b u t which a r e s t i l l continuous from t h e p o i n t of view o f

Eff.

W e consider t h e h e r e d i t a r i l y e f f e c t i v e operations.

in

Eff

t h e s e are j u s t t h e f i n i t e t y p e s .

By (11.1),

(The r e a d e r w i l l see t h a t

much of t h e material can be developed f o r an a r b i t r a r y " t y p e s t r u c t u r e " over IN .)

W e d e f i n e a n o t i o n of sequence c w v e r g e n c e on

each HEOu i n d u c t i v e l y a s f o l l o w s : on HEOO

= IN,

xn ->

x i f f 3k,Vn 2 k.xn = x;

on HEOuxT = HEOuxHEOT, ), on H E O ~ + ~( H= E O ~HE'

W e say t h a t a function f

sequence convergence.

(xn,yn) ->

,f n->f E

( x , y ) i f f xn - > x a n d

i f f xn->x

i m p l i e s f n (xn)->f

yn-->y; (x)

HEOO+T i s c o n t i n u o u s i f f f p r e s e r v e s

.

JM.E. HYLAND

202 Remark.

The meaning o f t h e s e d e f i n i t i o n s i n

t h e meaning i n

Eff

does

not

agree with

Sets.

L e t us i n i t i a l l y r e s t r i c t a t t e n t i o n t o t h e h e r e d i t a r i l y e f f e c -

t i v e o p e r a t i o n s o f p u r e t y p e (HEOklk a p u r e t y p e symbol) where e a c h k + l d e n o t e s (k ->

For f n , f i n HEOk+l,

0).

i s a modulus f o r f n --> Vx

E

f

HEOk.Vn

p(x).fn(x) = f (x).

2

Eff.) x.

L e t xn -->

for a l l n

t

t h i s is f a l s e i n

A s s u m e f u n c t i o n s i n HEOk+l

I f 1.1 i s a modulus f o r f n -> Proof:

f i n HEOk+l:

Eff.)

S e t s , though t r u e i n (In

HEOk+l

E

iff

( W e do n o t assume h e r e t h a t f n -->

Lemma 1 2 . 2 .

we say t h a t p

f i n HEOk+l,

axe continuous.

t h e n f n ->

f.

S i n c e 1.1 i s c o n t i n u o u s , t h e r e i s a k s u c h t h a t

k , p ( x n ) = ~ ( x =) k '

k" s u c h t h a t f o r a l l n

k',

L

say.

f (x,)

A s f i s c o n t i n u o u s t h e r e is

= f (x)

.

Then f o r a l l

n 2 max(k,k',k"), f n (xn) = f ( x n ) = f ( x ) . Remark.

T h i s argument i s e n t i r e l y e l e m e n t a r y and h a s u s e f u l

application t o a v a r i e t y of type s t r u c t u r e s i n a v a r i e t y of toposes. Lemma 12.3.

I f f n -> Proof:

(In

Eff.)

A s s u m e a l l f u n c t i o n s i n HEOk a r e c o n t i n u o u s . t h e n t h e r e i s a modulus 1.1 f o x f n -->

f i n HEOk+l,

The s e q u e n c e w i t h c o n s t a n t v a l u e x c o n v e r g e s t o x i n HEOk,

so w e c a n deduce f n ( x ) ->

f (x), t h a t i s

Vx3kVm 2 k f m ( x ) = f ( x )

.

By b a s i c a r i t h m e t i c c h o o s e k minimal f o r e a c h x . f u n c t i o n p : HEOk -> Remark.

f.

This gives us a

IN which by (11.1) o r ( 1 0 . 4 ) i s i n HEOk+l.

T h i s argument depends on e f f e c t i v i t y i n

Eff.

Again t h e r e

a r e many u s e f u l v e r s i o n s o f i t . Lemma 1 2 . 4 .

F

E

HEOk+2,

(In

Eff.)

I f p i s a modulus f o r f n -->

t h e n t h e r e i s an r such t h a t Vn t r F ( f n ) = F ( f ) .

f i n HEOk+l

and

203

The effective top-

Proof:

f o r f n l p r f , F r e s p e c t i v e l y w e wish

From i n d i c e s bn,m,b,c

t o f i n d an r s u c h t h a t Vn

t

r. F ( f n ) = F ( f ) .

F o l l o w i n g Gandy, use

t h e second r e c u r s i o n theorem t o d e f i n e an i n d e x b ' by, b ( a ) i f m ( a ) < l e a s t y.y shows c ( b ) = c ( b ' ) C =yo s a y ] , b'(a) =

{

bn(a) for n least

t

yo w i t h c ( b n ) # c ( b ) , o t h e r w i s e .

l y shows c ( b ) = c ( b ' ) i f f T ( c , b , ~ l ( y ) ) A T ( c , b ' , ~ ~ ( y ) ) ~ U ( ~ l=( y ) ) U(T2

(Y) )

.I

W e see e a s i l y t h a t yo e x i s t s and t h a t Vn t y o . c ( b n ) = c ( b ) ( u s i n g

Markov's p r i n c i p l e ) . Remark.

Thus yo i s c l e a r l y what w e want.

T h i s e s s e n t i a l l y i s Gandy's p r o o f of t h e Kreisel-lacombe-

S h o e n f i e l d theorem. modulus f o r f n ->

S i n c e b o t h Vn

k F ( f n ) = F ( f ) and p i s a

t

f a r e interpreted a s closed subobjects i n

it makes n o d i f f e r e n c e w h e t h e r w e do ( 1 2 . 4 )

internally in

Eff.

Theorem 1 2 . 5 .

( i )I n

f n ->

f i n HEOr+l

Eff

i f f t h e r e i s a modulus p f o r f n ->

( i i )I n

Proof:

Eff

Sets o r

it h o l d s f o r a l l p u r e t y p e s r + l t h a t

and t h a t a l l m e m b e r s of HEOr+l

members of HEOa+T

externally i n

Eff,

f i n HEOr+l,

are c o n t i n u o u s .

i t h o l d s f o r any t y p e s a , ~ , t h a t a l l

a r e continuous.

( i )f o l l o w s by i n d u c t i o n u s i n g ( 1 2 . 2 ) , ( 1 2 . 3 ) and ( 1 2 . 4 ) .

( i i ) f o l l o w s by e x t e n s i o n u s i n g C a r t e s i a n c l o s e d n e s s o f t h e h e r e -

d i t a r i l y e f f e c t i v e o p e r a t i o n s and o f t h e c o n t i n u o u s f u n c t i o n a l s ( i n t h e s e q u e n t i a l v e r s i o n , see

Hyland C1979 1 ) i

The r e a d e r s h o u l d compare ( 1 2 . 5 ) w i t h t h e example of Gandy (see Gandy-Hyland C19771 o f a t y p e 3 e f f e c t i v e o p e r a t i o n n o t con-

t i n u o u s on t h e t y p e 2 e f f e c t i v e o p e r a t i o n s ) . C o n t i n u i t y h a s a q u i t e d i f f e r e n t meaning i n t e r n a l l y i n Remark.

Eff.

The f i n i t e t y p e s o v e r R i n

hereditarily effective operations i n t i a l l y continuous f u n c t i o n a l s i n

Eff

coincide not only with t h e

E f f , b u t a l s o w i t h t h e sequen-

Eff.

The u s e o f t h e modulus was

i n t r o d u c e d o r i g i n a l l y i n t h e c o n t e x t of r e c u r s i o n t h e o r y on t h e

J.M.E. HYLAND

204

( s e q u e n t i a l l y ) c o n t i n u o u s f u n c t i o n a l s by S t a n Wainer. 513.

F a i l u r e of compactness. A s i s w e l l known, t h e r e a r e d e c i d a b l e s u b s e t s R o f 2'B,

the

s e t o f f i n i t e b i n a r y sequences such t h a t ( i ) any r e c u r s i v e a (ii)t h e r e are a S

E

E

2B

2 R w i l l s a t i s f y (i)).

e x t e n d s some u

2m

which e x t e n d no u

R,

E E

R ( s o t h a t no f i n i t e

T h i s h a s a n immediate consequence f o r

Eff. P r o p o s i t i o n 13.1.

In

Eff

t h e r e i s a d e c i d a b l e s u b o b j e c t R of 2

B

such t h a t ( i ) any a e x t e n d s some u

R,

E

( i i ) f o r any k , t h e r e i s an a which e x t e n d s no u

length

6

Thus i n

E

R of

k.

Eff

t h e r e i s a d e c i d a b l e cover of 2 N , Cantor s p a c e , by

b a s i c clopen s e t s , w i t h no f i n i t e subcover. EITHER ( i ) and ( i i ) a r e almost n e g a t i v e and h o l d i n

Proof:

sets of

the recursive r e a l s , OR immediate from Church's T h e s i s .

Corollary 13.2. F: 2m-> Proof:

I n Eff t h e r e i s a c o n t i n u o u s b u t unbounded f u n c t i o n

IN.

Set F(a)

=

l e a s t l e n g t h of u

E

R with a extending u.

(13.1) shows t h a t t h e Fan Theorem f a i l s a s b a d l y as p o s s i b l e i n

Eff.

T h i s i s why w e g e t (13.2)

.

There a r e Grothendieck t o p o s e s i n

which (13.1) h o l d s w i t h o u t t h e s t i p u l a t i o n t h a t R i s d e c i d a b l e . t h e known e x a n p l e s , a l l c o n t i n u o u s f u n c t i o n s from 2 m uniformly c o n t i n u o u s and so bounded.

In

t o JN are

I t i s n o t known whether t h e r e

are Grothendieck t o p o s e s i n which (13.1) h o l d s . The t r a d i t i o n a l way t o o b t a i n r e s u l t s analogous t o (13.1) and (13.2) f o r t h e reals i s t o use " s i n g u l a r c o v e r i n g s " a s s t u d i e d i n Zaslavskii-Ceitin

C19621.

(Of c o u r s e one c a n s e t up (13.1) and

(13.2) i n a n analogous f a s h i o n . )

The effective topos

P r o p o s i t i o n 13.3.

In

Eff,

205

t h e r e i s a sequence of r a t i o n a l i n t e r -

v a l s c o v e r i n g JR, b u t of a r b i t r a r i l y s m a l l measure. E s s e n t i a l l y a d i a g o n a l enumeration, see Z a s l a v s k i i - C e i t i n

Proof: C19621.

The proof i s a l s o s k e t c h e d i n Rogers C19671 E x e r c i s e s

15.36, w i t h o u t c o n s i d e r a t i o n s of e f f e c t i v i t y .

By t h e c o n d i t i o n s , t o

be s a t i s f i e d by t h e sequence o f r a t i o n a l i n t e r v a l s , can be e x p r e s s e d a s almost n e g a t i v e formulae, so by ( 8 . 4 ) t h i s does n o t matter. C o r o l l a r y 13.4.

Eff

In

t h e r e i s a c o n t i n u o u s f u n c t i o n from IR t o IR

which i s unbounded on some c l o s e d bounded i n t e r v a l , and so i n p a r t i c u l a r i s n o t uniformly c o n t i n u o u s on some c l o s e d bounded i n t e r v a l Proof:

Same r e f e r e n c e s a s f o r ( 1 3 . 3 ) .

Remark.

The r e s u l t s o f t h i s s e c t i o n can a l l be regarded a s proved

internally i n

Eff,

t h a t i s , t h e y f o l l o w from t h e e f f e c t i v i t y w e

e s t a b l i s h e d i n 510. Though w e know Grothendieck t o p o s e s i n which Il? f a i l s t o be l o c a l l y compact (Fourman-Hyland C19791), i n a l l known examples, t h e t y p i c a l consequences of l o c a l compactness f o r a n a l y s i s s t i l l h o l d . C e r t a i n l y c o n t i n u o u s f u n c t i o n s on bounded c l o s e d i n t e r v a l s are uniformly c o n t i n u o u s .

So t h e e f f e c t i v e t o p o s opens up p o s s i b i l i t i e s

unknown amongst Grothendieck t o p o s e s . found i n Z a s l a v s k i y - C e l t i n 814.

F u r t h e r examples can be

C19621.

Q u o t i e n t s o f c l a s s i c a l o b j e c t s , and power o b j e c t s . I t i s a f a m i l i a r f e a t u r e o f i n t u i t i o n i s t i c mathematics t h a t

c o l l e c t i o n s o f s e t s ( s p e c i e s ) can appear f a r more amorphous t h a n c o l l e c t i o n s of f u n c t i o n s .

W e have s e e n i n

Eff

t h a t t h e o b j e c t of

f u n c t i o n s between " w e l l - b e h a v e d " . o b j e c t s i s i t s e l f "well-behaved'' (16.3) and ( 7 . 1 ) ( b ) )

.

W e have s e e n t h i s good behaviour i n o t h e r

c o n t e x t s (Moschovakis C19731, S c o t t C19701, are t h e e a r l y r e f e r e n c e s ) , and it c a n be made t h e b a s i s f o r n i c e p r o o f - t h e o r e t i c r e s u l t s . However, when t h e s u b o b j e c t c l a s s i f i e r i s i t s e l f c o m p l i c a t e d , t h e

J.M.E. HYLAND

206

power s e t of however simple a ( t o some e x t e n t i n h a b i t e d ) o b j e c t w i l l I t i s t i m e t o look a t such o b j e c t s i n

be complex.

Eff. R i n Eff can be

A s w e mentioned i n 13, t h e s u b o b j e c t c l a s s i f i e r

t a k e n a s (Z,<-->) where <--> C = P(W).

i s t h e r e a l i z a b i l i t y bi-implication

on

W e may and so do t h i n k of t h e members o f C a s e x i s t i n g

"globally".

This w i l l

C l e a r l y t h e n (Z,<-->) i s a q u o t i e n t of A C .

Eff

mean t h a t w e can o b t a i n maps t o R i n

from s u i t a b l e maps t o C i n

Sets. Lemma 1 4 . 1 .

Suppose AY -->

-

f : X -->

Y i n d u c e s a map f :

( Y , = )i s a s u r j e c t i o n . (Y,=) i n

-->

(X,=)

such t h a t

J

4

-

(X,=)

Eff

Then a map

>

(Y,=)

commutes

f

f ( x ) = f ( x ' ) is valid.

i f f x = x ' -->

-

Under t h e s e c i r c u m s t a n c e s f i s r e p r e s e n t e d by

equality relation.)

t h e f u n c t i o n a l r e l a t i o n EXA [ I f ( x ) Proof:

(That i s , i f f f p r e s e r v e s t h e

=

y

1

.

By a r o u t i n e use of l o g i c .

I n t h e c a s e of t h e s u r j e c t i o n A C -->

R , e v e r y map a r i s e s a s i n

(14.1).

Proposition 1 4 . 2 .

Any map from (X,=) t o R

a s defined i n ( 1 4 . 1 ) ( i )x = x ' ->

and

(ii)f ( x )

-->

f o r an f : X --> ( f ( x ) <-->

=

(Z,<-->)

in Eff i s

Z such t h a t both

f(x'))

Ex

are valid. Remark.

Given f : X ->

g: X -->

Z with both

g ( x ) = ExAf ( x ) Proof:

C with

( i )v a l i d , one can e a s i l y d e f i n e

( i )and (ii)v a l i d , and such t h a t

z

=

s.

Set

.

S i n c e maps from (X,=) t o R a r e i n b i j e c t i v e correspondence

with maps 1 -->

P(X,=)), ( 1 4 . 2 ) i s immediate from t h e d e s c r i p t i o n

of t h e power s e t i n ( 2 . 1 2 ) o f H J P C19801.

A r e a d e r who f i n d s t h a t

The effective topos

207

proof u n p a l a t a b l e , can t a k e a r e p r e s e n t a t i v e G ( x , p ) f o r a map ( X , = ) t o f2, s e t f ( x ) = 1 v q . ( v p ( G ( x , p )hp --> G ( x , p ) <-->

q ) ->

p) is valid:

Exh ( f ( x ) <-->

q ]I

,

since (i)i s v a l i d f o r f , the

remark above a p p l i e s t o g i v e (ii) f o r g ( x ) = Exhf ( x ) r e l a t i v i z e d , t h i s i s a proof of From ( 1 4 . 2 ) w e see t h a t i f

within

.

(Suitably

( 2 . 1 2 ) of HJP L19801.)

i s s e p a r a t e d t h e n any map

(X,=)

t o R f a c t o r s through q : A X -->

(X,=)

and check t h a t

Our n e x t r e s u l t g i v e s t h i s

R.

Eff.

P r o p o s i t i o n 14.3.

The map q ( ' ' = ) :

f o r any s e p a r a t e d (X,=) i n Proof:

Eff.

From ( 6 . 3 ) w e see t h a t AZ"")

A ( Z rx)

Then q ( x ' ' ) : H(f,R)

P((X,=))

-->

ERhn{R(x) <->

=

is a surjection

R (',=)

-->

AC(xr=)

i s (isomorphic t o ) A ( Z r X ) .

i s r e p r e s e n t e d by

E x A f ( [ x l ) Ix

But w e can t a k e I'X 5 X (assuming ( X , = )

E

XI.

c a n o n i c a l l y s e p a r a t e d ) and so

by s e t t i n g f t o b e t h e r e s t r i c t i o n of R t o I ' X , we see a t once t h a t 3f .H(f , R )

ER -->

i s v a l i d , s o t h a t [HI i s s u r j e c t i v e .

§15.

The Uniformity P r i n c i p l e . F i r s t a general uniformity principle f o r

P r o p o s i t i o n 15.1.

L e t AX ->

an e f f e c t i v e o b j e c t . VtJlVX

holds i n Proof: (X,=)

be a s u r j e c t i o n and l e t ( Y , = )

be

Then

( X , = ) .3y

E

(X,=)

Eff.

E

( Y , = ) $ J ( x , y )-->

3y

E

(Y,=) .vx

E

(X,=)$J(x,y)l

Eff. Take (Y,=) s t r i c t e f f e c t i v e and c o n s i d e r f i r s t t h e c a s e when Let e

i s AX.

b = vl(e(0)) t i v e ; and c

=

E

E

[Vx.i'y.$J(x,y)

Ey f o r some y i n Y ,

v2(e(0)) is in

1

.

Then 0

E

Ex each x i n X ,

unique as ( Y , = )

1$(x,y) 1

.

is s t r i c t effec-

B u t t h e n i f d = Xn.c, w e

f i n d t h a t h e . < b , d > r e a l i z e s t h e formula i n s q u a r e b r a c k e t s .

i s no dependence on E + . )

(There

The r e s u l t f o r a q u o t i e n t of AX is an

immediate consequence of t h e s p e c i a l case.

so

J.M.E. HYLAND

208

W e have an immediate c o r o l l a r y .

C o r o l l a r y 15.2.

The "Uniformity P r i n c i p l e "

\d$"dX E P ( I N ) .3n

IN $ (X,n) -->

3n

E

IN .VX

E

P(IN ) $ ( x , n I l

Eff.

holds i n Proof:

E

i s an e f f e c t i v e o b j e c t and by (14.3) P ( R ) i s a q u o t i e n t

IN

of A ( Z m ) . The u n i f o r m i t y p r i n c i p l e i s an extreme form o f c h o i c e p r i n c i p l e : t h e c h o i c e f u n c t i o n i s c o n s t a n t because t h e domain i s amorphous w h i l e t h e range i s well-behaved. domain a r e n e c e s s a r y . from IN t o N .

C o n d i t i o n s on both t h e range and t h e

Obviously t h e r e are non-constant f u n c t i o n s

As r e g a r d s c o n d i t i o n s on t h e r a n g e , t h e r e a d e r may

l i k e t o show t h a t t h e q u o t i e n t map from A E t o 62 does n o t s p l i t . 916.

j-operators:

I n a topos, j-operators P

5

A2 t o be i s o .

f o r c i n g 2 -->

j(p)

a r e maps j : 62 -->

o r equivalently

p -->

j(phq) = j ( p ) h j ( q )

q i j ( p ) -->

j(j(p))

j(p).

5

c o u r s e t h e r e i s a l s o an i n t e r n a l o b j e c t o f j - o p e r a t o r s , n

of 62"

Eff

which w e can d e s c r i b e i n

P r o p o s i t i o n 1 6 . 1 . ( i )The o b j e c t '26

j(q)

5 j ( T )

T

j ( j( p ) ) = j ( p ) Of

62 s a t i s f y i n g

a subobject

as f o l l o w s . in

Eff

can be t a k e n as ( E x , = )

where Bf =

n

=

t v p . f ( p ) <-->

gip) 1

(ii)The o b j e c t o f j - o p e r a t o r s

of

in

Eff i s

t h e subobject

r e p r e s e n t e d by t h e c a n o n i c a l monic d e € i n e d by e i t h e r o f t h e

(Ex,=)

above ways of g i v i n g t h e n o t i o n of j - o p e r a t o r .

Alternatively it is

(J,=) where J i s t h e set of j - o p e r a t o r s and where

t j = k l with Ej = Proof:

= E j h [ V p . j ( p ) <-->

I[j i s a j - o p e r a t o r

]I

( i ) f o l l o w s from ( 1 4 . 2 )

t h e n immediate.

k(p)J

. i n t h e manner o f (14.3) and (.ii)i s

The effective topos

209

Remark. A s e x p l a i n e d i n J o h n s t o n e [ 1 9 7 7 ] j - o p e r a t o r s correspond t o t o p o l o g i e s and s o t o s u b t o p o s e s . j-operators

under p o i n t w i s e

(internally).

I t i s known t h a t t h e l a t t i c e of

i s a complete Heyting a l g e b r a

2

The r e a d e r s h o u l d r e f e r t o Fourman-Scott [ 1 9 7 9 ]

I t i s perhaps worth commenting

an e x p l i c i t c o n s t r u c t i v e t r e a t m e n t . f u r t h e r on t h e o r d e r r e l a t i o n . B j

6

kj

W e have

= EjAEkA

n Vp.j(p)

defining t h e appropriate subobject i n external j-operators,

then j

k(p)J

-->

Eff.

I f w e are l o o k i n g a t

k iff

6

k(p)

Vp.j ( p ) --> is valid.

for

F i n a l l y n o t e t h a t i f j ( i ) i s non-empty

d e g e n e r a t e topology which c o l l a p s e s t h e t o p o s

then j is t h e

.

L e t us look a g a i n a t t h e double n e g a t i o n topology.

( W e do n o t

bother with a c o n s t r u c t i v e v e r s i o n . ) ( 7 7 ) p = U I T i~ s ~ non-empty)

=

{I:

i f p non-empty,

C l e a r l y then w e have t h e f o l l o w i n g lemma. Lemma 1 6 . 2 . empty

For any j ,

(-,-,)

5

j i f f n { j ( p ) / p non-empty}

.

Proof:

Trivial.

W e now c o n s i d e r how t o f o r c e monics t o be i s o .

of

i s non-

L e t a subobject

(X,=) be given by a c a n o n i c a l monic A and d e f i n e a map R t o @

A

(p) =

C l e a r l y i f j f o r c e s A >-->

-->

p

n .

(X,=) t o be i s o , t h e n

@ A ((jp ) ) --> is valid.

(x,=). A ( X )

112x

by

j (p)

This g i v e s us a way t o d e s c r i b e t h e l e a s t j - o p e r a t o r

f o r c i n g A >-->

( X I = ) t o be i s o .

Proposition 16.3. f o r c i n g A >->

I n t h e above s i t u a t i o n , j A , t h e l e a s t j - o p e r a t o r

(X,=) t o be i s o , i s jA(p) =

Uvq. CC@,q ->

q ) h ( p -->

q ) -->

q) l

.

J.M.E. HYLAND

210

Proof:

Obvious, a s i n t h e l o g i c t h i s s a y s l A ( P ) = A {Sl@,q

a.

where A i s t a k e n i n t e r n a l l y i n (

(9A ( p ) ->

9) s

->

qAp

5

5

ql

I t i s e a s y t o check t h a t ( a s

@ , ( q ) ) i s v a l i d ) j,

W e now show t h a t f o r c i n g 2 >->

is a j-operator.

A2 t o be is0 c o l l a p s e s

Eff

to

Sets. The l e a s t j - o p e r a t o r f o r c i n g 2 >->

Proposition 16.4. is (

A2 t o b e is0

17).

Proof:

L e t j be t h e l e a s t j - o p e r a t o r

i n ( 1 6 . 3 ) from @ :

->

@ ( p )=

u

{01+p

a.

nI j I n 1 ) In

1

E

VP,q. ( p ->

Il1-t~

= Iele(0)

( 7 7 )

i s non-empty:

E

p o r e(1)

E

PI.

s j , t h a t i s by ( 1 6 . 2 )

i s non-empty.

I n f a c t it i s enough t o show t h a t

for i f a is i n

( j ( p ) -->

q ) ->

A2, o b t a i n e d a s

Here

C l e a r l y i t s u f f i c e s t o show n { j ( p ) Ip non-empty]

f o r c i n g 2 >->

j (9)

and x i s i n n { j { n I ) In

N

E

1,

t h e n ( a ( X n . n ) ) x i s i n n { j ( p ) lp non-empty].

,

c in

Now t a k e b i n

1 V p . p ->

t a k e as 2 >->

A2 i s j - d e n s e d i n j ( { O } ) n j ( { l l ) . Note t h a t

e

=

Xx.c( ( a x ) d ) i s i n

j(p)]

l[Vp.@( j ( p ) ) ->

" d p . j ( j ( p ) ) -->

.

j ( p )1

j(p)J

,

and

Define u s i n g t h e

s e c o n d r e c u r s i o n theorem an i n d e x f by (fk) (0)

=

b(k)

1 ( f k ) (1) = U ( l e a s t y . T ( e , S l ( f , k + l ) , y ) ) . Now by a s t a n d a r d k i n d o f argument, w e c a n show t h a t 1 Sl(f,k) (0) = (fk) (0) = b(k)

and

E

j(Ik1)

S1 l ( f , k ) (1) = ( f k ) (1) = e ( S1 l(f,k+l))

1 a r e a l l d e f i n e d , and t h e n w e see t h a t S l ( f , O ) i s i n @ ( j ( { n I ) )f o r 1 a l l n and so e ( S l ( f , O ) ) i s i n n { j ( { n } ) In E m l as r e q u i r e d .

517.

j - o p e r a t o r s and d e c i d a b i l i t y . (16.4)

appears t o restrict t h e j-operators i n

w e c a n show t h a t t h e y have a r i c h s t r u c t u r e .

Eff,

but i n fact

A p p a r e n t l y it w a s

21 1

The effective topos

Powell who f i r s t r e a l i z e d t h a t t h e r e i s a c o n n e c t i o n between n o t i o n s of d e g r e e and t h e f o r c i n g of d e c i d a b i l i t y i n r e c u r s i v e r e a l i z a b i l i t y .

W e c o n t e n t o u r s e l v e s w i t h a p r e c i s e s t a t e m e n t and a s k e t c h of a proof. F i r s t w e g i v e a lemma o f Andy P i t t s which s i m p l i f i e s t h e p r e -

s e n t a t i o n of t h e p r o o f . I n t h e s i t u a t i o n of (16.3) j, can a l t e r n a t i v e l y be de-

Lemma 1 7 . 1 .

f i n e d by @,*(p)

*

where

=

I p ~ { * l5 q and 0,Cq)

n{q 5

5 qlr

i s an i n d e x f o r t h e empty p a r t i a l f u n c t i o n , so l o n g as Ex

non-empty i m p l i e s A(x) non-empty. Proof: sion.

L e t us d r o p t h e s u b s c r i p t

A.

Note t h a t $ p r e s e r v e s i n c l u -

Hence because P -->

(PA{*))

5

$ ( P ) -->

@(PA{*))

i s v a l i d , w e can deduce t h a t @(P)5 @*(P) is v a l i d .

Also w e have

@ ( @ * ( P )5) n { @ ( q )IpAI*l 5 q and @ ( q ) 5 ql

-c n{qjpAt*} 5 q

and @ ( q )5 q l = @ * ( P I ,

so t h a t

@ ( $ * ( P ) )5 @ * ( P )

is v a l i d , r a t h e r t r i v i a l l y .

w e have j s

@ *i n Eff,

and i t remains t o show t h a t $ * s j .

W e can t a k e a

E

[ V p , p ->

w e can t a k e b

E

UVx

E

Thus by t h e d e f i n i t i o n of j i n ( 1 6 . 3 )

j (p)J

and s i n c e @ Cj Cp))

(X,=). ( A ( x ) ->

j C p ) ) -->

S

j (p) i n

jCp) 1

.

Eff,

NOW

d e f i n e an i n d e x e by t h e second r e c u r s i o n theorem a s f o l l o w s . a ( n ) , i f x = -a,*> e ( x ) = ' b ( m ) ) (Xy.e(z(y))), i f x = <m,z>, z Cons i de r f o r any p , t h e s e t SCp) = I x l e ( x ) t h a t ( i )PAC*) 2 S ( P ) , and (ii)$ ( S ( P ) ) 5 S ( P ) ,

E

j(p)}.

# *. W e see e a s i l y

J.M.E. HYLAND

212

s o w e can deduce t h a t $ * ( p ) 5 S ( p )

.

Thus c l e a r l y e r e a l i z e s

Vp.@*(p) 5 j ( p ) , and t h i s completes t h e p r o o f . Now f o r A 5 I N , l e t DA ( n ) = { < O , n > l n E A } u { < l , n > l n

4

A} s o t h a t

DA r e p r e s e n t s c a n o n i c a l l y t h e s u b o b j e c t AV

We write

t h e " d e c i d a b i l i t y of A " . $,(PI

m,

A >->

i

U3n

=

E

IN .DA(n) -->

n ,

p

t h a t is, the l e a s t

and kA f o r t h e l e a s t j - o p e r a t o r g e n e r a t e d by $,, j - o p e r a t o r f o r c i n g A t o be d e c i d a b l e . $A* i n

By ( 1 7 . 1 ) k A i s e q u a l t o

Eff.

Theorem 1 7 . 2 .

(External v e r s i o n ) .

kA 5 kB i n

Eff

i f f A i s Turing

r e d u c i b l e t o B. Proof:

Note f i r s t t h a t kA

(holds i n

Eff).

Suppose t h a t e ( n )

5

kg i n

Eff i f f Vn.$,*(DA(n)) is v a l i d

*

E

JIB ( D A ( n ) ) f o r each n.

W e wish t o show how t o

compute A from B , t h a t i s how t o determine DA(n) from a knowledge of DB ( m ) f o r f i n i t e l y many m.

Let x = e(n)

*

E

Q B (DA(n); t h e n

e i t h e r x i s of form < y , * > and w e e a s i l y see t h a t y must be DA(n) so

w e a r e home,

or x

i s of form < m , e l >

s a y i n which c a s e e l

so w e t a k e x1 t o be e l ( < O , m > ) x1

E

E

D B ( m ) -->

$B* (DA(n))

o r e l ( < l , m > ) a s appropriate,

JIB* ( D A ( n ) , ) and r e p e a t t h i s p r o c e s s .

From t h e d e f i n i t i o n of

qB*, t h i s t e r m i n a t e s i n a f i n i t e number of s t e p s g i v i n g DA(n) a s required. Suppose c o n v e r s e l y t h a t A i s Turing r e d u c i b l e t o B v i a an i n d e x f . Define using t h e second r e c u r s i o n theorem e Cn,y) where y i s ( a code f o r ) a f i n i t e set of numbers of form <m,O> o r < m , l > as f o l l o w s . < k , * > , i f t h e r e i s a computation { f } Y ( n ) ( u s i n g only i n f o r m a t i o n i n y ) , e(nfy) =

1 i

<m,q>,

=

k

i f t h e computation { f l Y ( n ) a s k s f o r a v a l u e n o t i n y f and g i s an i n d e x for < m , i > -> e(n,yuI<m,i>l).

213

The effective topos I t i s e a s y t o see t h a t f o r a l l n , and f o r y i n f o r m a t i o n t r u e of B,

I n p a r t i c u l a r f o r y t h e empty

e ( n , y ) i s d e f i n e d and i n @,*(D,(n)).

set, w e have f o r a l l n , e (n,y) i n

@*,

(DA(n))

.

Thus Vn.QB* ( D A ( n ) ) i s v a l i d . This completes t h e p r o o f . Remark. I n f a c t t h e r e i s a proof of t h e i m p l i c a t i o n from r i g h t t o l e f t along t h e following l i n e s : i f j f o r c e s B decidable, then the s t a t e m e n t t h a t A i s r e d u c i b l e t o B and t h a t t h e computation i s always d e f i n e d , a r e almost n e g a t i v e f o r E f f . and so h o l d i n E f f . . hence i n -1 I -1 ' E f f . A i s decidable, t h a t is j f o r c e s A decidable. -1 W e cannot f i n d a crude i n t e r n a l v e r s i o n of ( 1 7 . 2 ) i n view of Goodman C19781.

However t h e proof of

g e t something o u t of i t .

( 1 7 . 2 ) is e f f e c t i v e , so w e can

C l e a r l y t h e f u n c t i o n which a s s o c i a t e s kA

with A i s i n t e r n a l l y d e f i n e d i n

Eff.

W e must s a y what w e mean by

"A Turing r e d u c i b l e t o B": w e mean t h e n a t u r a l n o t i o n of c o m p u t a b i l i t y

r e l a t i v e t o ( p a r t i a l ) c h a r a c t e r i s t i c functions.

W e obtain a result

by r e s t r i c t i n g a t t e n t i o n t o c l o s e d subsets of I N , t h a t i s t o P ( I N ) = {A 5 IN IVn. 9 - n

P r o p o s i t i o n 17.3. VA,B

holds i n Proof: 118.

E

B

n

A -->

E

A].

The s t a t e m e n t

F(IN).

(A Turing r e d u c i b l e t o B ) <-->

kA

' kg

Eff. By t h e e f f e c t i v i t y of t h e proof o f ( 7 . 2 ) .

General remarks on t h e e f f e c t i v e t o p o s . The p l e a s i n g f e a t u r e of t h e e f f e c t i v e t o p o s i s t h a t i n i t ,

i d e a s about e f f e c t i v i t y i n mathematics,seem t o have t h e i r n a t u r a l home.

W e mention t h e two main examples.

1) C o n s t r u c t i v e r e a l a n a l y s i s . i s what a n a l y s i s i n

Eff

W e have t r i e d t o i n d i c a t e t h a t t h i s

i s i n e s s e n c e i n 118-13.

I t i s worth n o t i n g

how t h e r e a l i z a b i l i t y l o g i c makes d i s t i n c t i o n s f o r us.

Consider t h e

J.M.E. HYLAND

214

examples (Kreisel 119591 t h a t t h e i n t e r m e d i a t e v a l u e theorem h o l d s c l a s s i c a l l y but not e f f e c t i v e l y f o r recursive (continuous) functions on t h e r e c u r s i v e r e a l s . vf

E

In

Eff,

lRIR .f (O) 0 ->

3X

E

(0,l)f

(X)

= 0

is f a l s e while

vf

E

~ l R . f ( 0 ) < O A f ( l ) >-->O

i s t r u e ( a s it i s t r u e i n

2 ) Effective algebra.

- 1 - 1 3 xE

Sets and

(O,l)f(X) = 0

equivalent t o a negative formula).

W e have n o t d i s c u s s e d t h i s a t a l l , b u t i t

seems worth p o i n t i n g o u t t h a t t h e d e f i n i t i o n s have a n a t u r a l meaning in

Eff.

A r e c u r s i v e l y p r e s e n t e d f i e l d (see Metakides-Nerode

i s an enumerable ( d e c i d a b l e ) f i e l d i n

Eff.

C19791

I t has a s p l i t t i n g

a l g o r i t h m i f f i r r e d u c i b i l i t y of polynomials i s d e c i d a b l e i n

Eff.

Thus t h e e f f e c t i v e c o n t e n t o f a r e c u r s i v e l y p r e s e n t e d s t r u c t u x e c o r r e s p o n d s t o p r o p e r t i e s of it which h o l d i n

Eff.

This suggests

t h a t p o s i t i v e r e s u l t s i n e f f e c t i v e a l g e b r a s h o u l d be e s t a b l i s h e d by proving r e s u l t s i n c o n s t r u c t i v e l o g i c from axioms which h o l d i n and i n t e r p r e t i n g t h e r e s u l t s i n a x i o m a t i c method.

Eff.

Eff,

T h a t i s , one s h o u l d u s e t h e

Of c o u r s e , n e g a t i v e r e s u l t s o b t a i n e d i n e f f e c t i v e

a l g e b r a can be i n t e r p r e t e d i n

Eff t o g i v e independence r e s u l t s .

What w e l a c k , above a l l , i n o u r t r e a t m e n t o f t h e e f f e c t i v e topos,

i s any r e a l i n f o r m a t i o n a b o u t a x i o m a t i z a t i o n analogous t o t h e r e s u l t s o b t a i n e d i n T r o e l s t r a 119731 a x i o m a t i z i n g r e a l i z a b i l i t y o v e r b o t h Heyting and Peano a r i t h m e t i c .

Of c o u r s e , one would e x p e c t t o look a t

t h e e f f e c t i v e t o p o s d e f i n e d o v e r a t o p o s o t h e r t h a n Sets ( s a y o v e r t h e f r e e t o p o s w i t h n a t u r a l number o b j e c t ) t o g e t a r e s u l t c o r r e s ponding p r e c i s e l y t o a n a x i o m a t i z a t i o n .

But a l l I wish t o p o i n t o u t

i s t h a t ( d e s p i t e t h e s u g g e s t i v e work of P i t t s C19811 on i t e r a t i o n )

w e have no good i n f o r m a t i o n i n t h i s area.

W e can n o t p r o p e r l y be

s a i d t o u n d e r s t a n d r e a l i z a b i l i t y u n t i l w e do.

215

The effective topos REFERENCES

Beeson C19771, P r i n c i p l e s o f c o n t i n u o u s c h o i c e , Ann. Math. Log. 1 2 , 249-322.

M.J.

E. Bishop C19671, A . B o i l e a u & A.

D.

Foundations o f C o n s t r u c t i v e A n a l y s i s (McGraw-Hill).

J o y a l 119811, La l o g i q u e d e s t o p o s , J.S.L.

El

6-16.

van Dalen C19781, An i n t e r p r e t a t i o n of i n t u i t i o n i s t i c a n a l y s i s , Ann. Math. Log. 13, 1-43.

M.P.

Fourman C19771, The l o g i c o f t o p o i , i n Handbook of Mathematical Logic ( e d . Barwise) (North-Holland)

M.P.

Fourman & R . J .

M.P.

Fourman & J.M.E. Hyland 119791, Sheaf models f o r a n a l y s i s , i n A p p l i c a t i o n s of Sheaves ( S p r i n g e r L e c t u r e Notes 7 5 3 ) .

.

Grayson, t h i s volume, Formal s p a c e s .

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R.O.

.

N.D.

Goodman C1978 1, The n o n - c o n s t r u c t i v e c o n t e n t of s e n t e n c e s of a r i t h m e t i c , J.S.L. 43, 497-501.

J.M.E.

Hyland C19771, Aspects of c o n s t r u c t i v i t y i n mathematics, i n Logic Colloquium '76 led. Gandy, Hyland) (North-Holland)

J.M.E.

Hyland 119791, F i l t e r s p a c e s and c o n t i n u o u s f u n c t i o n a l s , Ann. Math. Log. 16,1 0 1 - 1 4 3 .

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Hyland, P . T . J o h n s t o n e & A.M. P i t t s C19801 CHJP C198011, T r i p o s t h e o r y , Math. P r o c . Carnb. P h i l . SOC. E, 205-232.

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Kleene C19731, R e a l i z a b i l i t y : A R e t r o s p e c t i v e Survey, i n Cambridge Summer School i n Mathematical Logic ( S p r i n g e r L e c t u r e Notes 3 3 7 ) .

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S c o t t C19701, E x t e n d i n g t h e t o p o l o g i c a l i n t e r p r e t a t i o n t o i n t u i t i o n i s t i c a n a l y s i s 11, i n I n t u i t i o n i s m and Proof Theory ( e d . Xino, M y h i l l , Vesley) North-Holland)

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'THEL.E.J. BROUWER CENTENARY SYMPOSIUM A S . TroeoeLrmand D. van Dalen (editors) 0 North-HollandPublishing Company, 1982

217

CONSTRUCTIVE LOGIC VERSUS ALGEBRAIZATION I

G. Kreisel

*

and A. MacIntyre

S t a n f o r d U n i v e r s i t y and Yale U n i v e r s i t y USA

The I n t r o d u c t i o n e x p l a i n s t h e i s s u e s r a i s e d by c o n s t r u c t i v e l o g i c f o r d i f f e r e n t t r a d i t i o n s ; p r i n c i p a l l y , f o r mainstream c o n s t r u c t i v e mathematics t r e a t e d h e r e , and f o r t h e f o u n d a t i o n a l t r a d i t i o n , t h e s u b j e c t of P a r t 11. The m a t e r i a l i s arranged under t h e following headings. 1. Flashy metatheorems and some n e g l e c t e d i m p l i c a t i o n s . For example, c l a s s i c a l proofs of I$theorems a r e e s s e n t i a l l y no h a r d e r t o unwind t h a n corresponding c o n s t r u c t i v e , more p r e c i s e l y , i n t u i t i o n i s t i c p r o o f s . In s h o r t , i n t u i t i o n i s t i c r e s t r i c t i o n s t e n d t o be t o o weak i n p r a c t i c e , i n c i d e n t a l l y d e s p i t e t h e ( l i t t l e known) 2. Weakelegance of t h e i r metatheory documented i n a d i g r e s s i o n . ness of t h e c o n s t r u c t i v e meaning of IIg + I$ This i s i l l u s t r a t e d by r e f e r e n c e t o v a r i o u s elementary i m p l i c a t i o n s of t h i s form between f a m i l i a r f i n i t e n e s s theorems i n a r i t h m e t i c . A p r o v i s o i s formulated which c o r r e c t s t h a t weakness. and i s o f t e n s a t i s f i e d i n practice. 3. Metamathematics of c l a s s i c a l systems: modern v a r i a n t s of H i l b e r t ' s program without detour v i a c o n s t r u c t i v e . l o g i c . For example, some r e c a l c i t r a n t proofs o f $I theorems have i n f a c t been unwound by s u i t a b l e f u n c t i o n a l i n t e r p r e t a t i o n s such a s t h e n . c . i . ; c u t e l i m i n a t i o n i s g e n e r a l l y l e s s e f f i c i e n t . 4. Applying t h e n.c.i.: r e a s s u r a n c e s (about a d d i t i o n a l s k i l l s needed for s u c c e s s f u l a p p l i c a t i o n s , beyond knowing t h e metatheory of t h e n . c . i . ) . A t t e n t i o n i s drawn t o proof s t r a t e g i e s used i n o r d i n a r y mathematics, o f comparable g e n e r a l i t y t o l o g i c a l metatheorems, where similar a d d i t i o n a l s k i l l s a r e needed, and have proved t o be a v a i l a b l e . Much t h e same a p p l i e s t o s u c c e s s f u l a p p l i c a t i o n s of model t h e o r y which, c o n t r a r y t o s u p e r f i c i a l i m p r e s s i o n s , a r e shown t o i n v o l v e o p e r a t i o n s on p r o o f s , and not 5. A l g e b r a i z a t i o n : reminders and only on t h e theorem proved. s u r p r i s e s ( p r i n c i p a l s e c t i o n ) . By r e f e r e n c e mainly t o r e c e n t l i t e r a t u r e , t h e p o t e n t i a l o f a l g e b r a i z a t i o n and a couple o f o t h e r g e n e r a l mathematical s t r a t e g i e s i n t h e s e n s e o f 4. above i s documented, and c o n t r a s t e d w i t h l o g i c a l c o n t r i b u t i o n s t o t h e c o n s t r u c t i v e t r a d i t i o n i n mathematics. Remarkably many s i g n i f i c a n t and g e n e r a l p o i n t s a r e i l l u s t r a t e d by work on sums o f squares and by t h e Appendix. The l a t t e r supplements s e c t i o n 2. by r e f e r e n c e t o t h e most s o p h i s t i c a t e d i m p l i c a t i o n between f i n i t e ness theorems i n t h e number-theoretic l i t e r a t u r e . P a r t i c d a r a t t e n t i o n i s given t o r e a l i z i n g t h e ' p r o v i s o ' mentioned i n 2 . , or, equiv a l e n t l y , t o making a proper choice between v a r i o u s meanings o f t h e popular phrase: from a counterexample t o t h e conclusion we g e t one t o t h e premise. - The Appendix can b e r e a d independently o f t h e main t e x t .

.

*

This work w a s p a r t i a l l y supported by a National Science Foundation g r a n t , MCS81-04869.

218

G. KREISEL, A. MACINTYRE

INTRODUCTION Many years ago Bourbaki s a i d somewhere t h a t i n t u i t i o n i s m would come t o be seen

as a mere h i s t o r i c a l c u r i o s i t y .

Though a b i t offhand t h e remark makes a good

suggestion: t o look at t h e s u b j e c t i n d i f f e r e n t h i s t o r i c a l contexts.

W e shall

consider t h r e e obvious candidates. The present paper ( P a r t I) concerns t h e 2000 year o l d t r a d i t i o n of s o c a l l e d main stream c o n s t r u c t i v e mathematics.

This picks out p a r t i c u l a r kinds of

operations o r constructions t o be used f o r solving problems; f o r example. t h e humble r i n g operations i n t h e days o f Pythagoras o r t h e r e c u r s i v e operations which, by now, c o n s t i t u t e an i n t e g r a l p a r t o f t h e modern main stream.

Inci-

d e n t a l l y , t h e s e 'extremes' a r e n e a t l y connected by i d e n t i f y i n g t h e diophantine and t h e r e c u r s i v e l y enumerable r e l a t i o n s .

T r a d i t i o n a l advertisements f o r those

two c l a s s e s were, indeed, exaggerated, but can be r e f i n e d . t o have claimed t h a t number i s t h e measure o f a l l things. s u f f i c i e n t l y many t h i n g s .

Not q u i t e ; b u t of

Recursive operations were s a i d t o correspond t o t h e

p o s s i b i l i t i e s of t h e i d e a l i z e d ( f i n i t e ) mind. i n g t o non-recursive laws.

Pythagoras is s a i d

Perhaps not, if it grows accord-

But r e c u r s i v e operations are c e r t a i n l y w e l l s u i t e d

f o r s t a t i n g f a c t s about f i n i t e l y presented groups, chosen t o 'grow' t o recursive laws.

according

I n c o n t r a s t , i n t u i t i o n i s m p r e s e n t s a c o n f l i c t f o r t h e tra-

d i t i o n of c o n s t r u c t i v e mathematics by emphasizing c o n s t r u c t i v e proofs (not only: o p e r a t i o n s ) , involved i n c o n s t r u c t i v e

m.

Quite simply, it i s t h i s .

Any

r e s t r i c t i o n on proofs may exclude t h e p o s s i b i l i t y o f e s t a b l i s h i n g optimal sol u t i o n s ; and i f t h e r e s t r i c t i o n s have l o g i c a l c h a r a c t e r , not geared t o any s p e c i f i c subject matter, they may b e t o o weak i n p r a c t i c e .

I n s h o r t , one i s

l i a b l e t o fall between two s t o o l s ; i n accordance with t h e o l d warning t h a t , what i s t r u e i n general i s l i a b l e t o be t r i v i a l i n most p a r t i c u l a r cases.

Put

a l i t t l e less brashly: e l a b o r a t i o n s of g e n e r a l i t i e s tend t o reach t h e p o i n t of diminishing r e t u r n s very soon. The two f i r s t s e c t i o n s below document t h e c o n f l i c t i n question i n two ways.

Quite generally, i f r e s t r i c t i o n s on proofs a r e r e l e v a n t f o r t h e c o n s t r u c t i v e t r a d i t i o n i n mathematics a t a l l , t h o s e made by ( i n t u i t i o n i s t i c ) c o n s t r u c t i v e l o g i c tend t o be t o o weak.

But so i s ( t h e extension of c o n s t r u c t i v i t y t o ) t h e

meaning o f l o g i c a l l y complicated expressions which have no p l a c e i n t h e main stream t r a d i t i o n at all.

So, f o r t h e h i s t o r i c a l context considered here,

Bourbaki's remark has not been r e f u t e d up t o now.

( A t t h e end o f t h e paper

reasons are given why it seems premature t o a s s e s s t h e p o t e n t i a l o f c o n s t r u c t i v e l o g i c ; more p r e c i s e l y , even f o r t h i s context.)

But unlike Bowbaki w e do not

rely on o b i t e r d i c t a , l e t alone, ignore c o n s t r u c t i v e l o g i c (as Bourbaki's remark

219

Constructive logic versus algebraization might s u g g e s t ) , b u t u s e metatheorems o r i g i n a l l y proved f o r q u i t e d i f f e r e n t purposes: give them rope

...

Here it i s t o b e emphasized t h a t t h e formal meta-

theory o f c o n s t r u c t i v e l o g i c i s e x c e p t i o n a l l y r i c h and e l e g a n t , i l l u s t r a t i n g t h e This p o i n t i s f u r t h e r under-

truism t h a t elegance does not guarantee r e l e v a n c e . lined i n t h e t h i r d section.

I t reminds t h e r e a d e r o f t h e much l e s s e l e g a n t

metamathematics o f c l a s s i c a l systems which unwinds non-constructive p r o o f s , f o r example, t o e x t r a c t bounds (without d e t o u r v i a c o n s t r u c t i v e l o g i c ) .

This kind

of metamathematics avoids t h e l o g i c a l symbolism o f c o n s t r u c t i v e l o g i c ; b u t u n l i k e

main stream c o n s t r u c t i v e mathematics it b u i l d s up i t s f u n c t i o n a l equations from a crude s o t o speak l o g i c a l s t o c k o f o p e r a t i o n s .

Nevertheless, used w i t h d i s -

c r e t i o n t h i s has helped s o l v e problems t o which s p e c i a l i s t s wanted an answer. The a d d i t i o n a l s k i l l s which c o n s t i t u t e t h e d i s c r e t i o n r e q u i r e d , i n c i d e n t a l l y , not t o o d i f f e r e n t from t h o s e needed t o apply ( f o r m a l ) t h e o r y elsewhere, a r e discussed i n t h e f o u r t h s e c t i o n .

Granted,

-

This l e a d s t o t h e p r i n c i p a l p o i n t o f t h e paper.

a s always: f o r t h e c o n s t r u c t i v e t r a d i t i o n o f mathematics

i v e l o g i c has f a i l e d on almost every s p e c i f i c s c o r e .

- construct-

To l e a r n from t h e s e f a i l -

ures o f what had appeared t o be an a t t r a c t i v e p r o j e c t , one a s k s : What l i m i t a t i o n s of any g e n e r a l a i m behind it can b e rewarding?

Must one switch t o completely

d i f f e r e n t methods? O r i s some combination w i t h knowledge o f c o n s t r u c t i v e l o g i c fruitful?

More c o n c r e t e l y , t h o s e o f us who know died-in-the-wool

o r born-again

c o n s t r u c t i v i s t s , can ask: What do t h e y imagine t h e y have c o n t r i b u t e d t o t h e broad c o n s t r u c t i v e t r a d i t i o n i n mathematics?

Without exaggeration: t h e y a r e

under t h e impression t h a t l o g i c has n o t only formulated s i g n i f i c a n t g e n e r a l p r i n c i p l e s o f mathematical reasoning b u t t h a t it i s t h e only p a r t o f mathematics which has them: perhaps t h e o n l y p a r t which has t h e means t o formulate such principles.

Viewed d i s p a s s i o n a t e l y , p r o o f s i n non-constructive mathematics

would be a p e r f e c t l y a p p r o p r i a t e s u b j e c t h e r e even f o r c o n s t r u c t i v i s t s s i n c e such p r o o f s a r e c o n s t r u c t i o n s t o o .

But i n o r d e r n o t t o o f f e n d s e n s i b i l i t i e s

(by mentioning t a i n t e d o b j e c t s ) , we c o n t i n u e t o r e l y on m a t e r i a l from t h e constructive tradition.

I n t h e l a s t and p r i n c i p a l s e c t i o n o f t h i s paper, con-

s o l i d a t e d by t h e Appendix, g e n e r a l proof s t r a t e g i e s i n main stream c o n s t r u c t i v e mathematics a r e shown t o b e , r e a l i s t i c a l l y speaking, of q u i t e comparable 5 e n e r a L i t y t o t h e l o g i c a l kind, b e s i d e s b e i n g more e f f i c i e n t .

The p r i n c i p a l

example i s a l g e b r a i z a t i o n , mentioned i n t h e t i t l e , w i t h a couple o f q t h e r s thrown i n .

This choice i s p e r f e c t l y orthodox; c f . t h e f i r s t sentence o f

Bourbaki's Algbbre: F a i r e de l ' A l g S b r e , c ' e s t e s s e n t i e l l e m e n t c a l c u l e r . - Our p r e s e n t a t i o n c o n f l i c t s w i t h t h e canons o f s o c a l l e d exact methodology which req u i r e such l i t e r a r y forms as metatheorems.

f o r formulating

I f t h e s e forms are not w e l l s u i t e d

t h o s e g e n e r a l proof s t r a t e g i e s , e x a c t methodology d r a w s

220

G. KREISEL, A. MACINTYRE

attention away from the latter, thereby introducing a systematic error; a hopelessly lopsided view, a false philosophy of mathematics in the sense of Schopenhauer. The talk above of a possible systematic error certainly has some empirical support. Not only constructivist logicians have the impression that logic is uniquely suited for formulating general insights. Many model theorists think of logical transfer principles as so to speak qualitatively more general than, say, Hasse's local/global principle though, by realistic standards, the latter is more than competitive. Also, there is a plausible theoretical reason. The impression is plainly fostered by the foundational tradition, that is, the project of logical foundations which goes back roughly a century. This is the second obvious candidate for an historical context suggested by Bourbaki's remark at the beginning of this introduction. of Part I1 of this paper.

It will be the principal subject

In this context Brouwer's constructive logic is by

no means a curiosity but an historic necessity; nothing less than an essential 'antithesis' to the then-dominant 'theses' of Frege or Russell.

(Viewed from

outside they are variants, not rivals since they are all much more like each other than like ordinary conceptions of mathematics, then or now; cf. the story of French politics in 1968 when each of the 6 Trotskyite parties reputedly considered itself more different from any of the others than from Pompidou's position.) Furthermore, Brouwer's constructive logic has a direct bearing on questions raised by Aristotle, whose specifically logical scientific

-

-

as opposed to broad

interests are taken up in the modern foundational tradition.

In

fact, constructive logic applies to propositions about such incompletely defined objects as choice sequences, the principal topic of Brouwer's mature work; giving at least one memorable answer to the issues, on incompletely defined terms, raised by Aristotle in Met I'8, 1012-1023 (which has survived 2000 years). This obvious interest of choice sequences for the foundational tradition does not ensure any broader significance. They do not belong to the mathematical tradition at all for the simple reason that propositions about them have been at least, so far: successfully ditions. NB.

-

-

paraphrased by appropriate continuity con-

This qualification, on the broader significance of principal

objects in the foundational tradition, is not a mere innocent aside.

It con-

flicts with the claim that the foundational version is a successful or, perhaps, even the only legitimate heir of the heroic philosophical tradition; in particular, of the broad philosophy of mathematics concerned with the nature of mathematical structures and our knowledge about them; or - as Bourbaki's manifesto put it

-

with understanding the architecture of mathematics and our

intuitive resonances to it, as heroic an aim (or claim) as any.

This heroic

tradition is the third candidate for an historical context adumbrated at the beginning of this paper.

We see no evidence that the foundational tradition,

22 1

Constructive logic versus algebraization not . t o speak o f i t s c o n s t r u c t i v e p a r t , i s more t h a n a c u r i o s i t y h e r e , i f one t a k e s modern knowledge o f ( h e r o i c q u e s t i o n s a b o u t ) t h e n a t u r e of m a t t e r o r I n o t h e r words, l o g i c a l o b j e c t s do not have a p l a c e i n

h e r e d i t y as a s t a n d a r d .

For t h e r e c o r d it should be

t h e h e r o i c t r a d i t i o n comparable t o atoms or DNA.

added t h a t , by t h e s e s t a n d a r d s , t h e e f f o r t s advocated i n Bourbaki's manifesto may come t o be seen as an h i s t o r i c a l c u r i o s i t y t o o .

There i s a p o s i t i v e s i d e t o a l l t h i s n e g a t i v e s t u f f

(which, i n c i d e n t a l l y , i s

not out o f p l a c e i n a s u b j e c t known f o r independence, incompleteness o r unde-

It e x p l a i n s t h e e s s e n t i a l r o l e o f t h e s e a r c h f o r

c i d a b i l i t y theorems!). broad a r e a s s u i t e d t o

-

-

a p p l i c a t i o n s o f l o g i c a l g e n e r a l i t i e s , and p r e p a r e s us

f o r t h e imagination demanded by t h i s s e a r c h .

When o b j e c t s a r e genuinely fund-

amental f o r t h e h e r o i c t r a d i t i o n , and t h u s g e n e r a l i n t h e s e n s e of a few s e r v i n g as b u i l d i n g blocks f o r many d i f f e r e n t t h i n g s , a p p l i c a t i o n s w i l l r e a l l y look a f t e r themselves.

One can h a r d l y know t o o much about such o b j e c t s , and so t h e l a w These o b j e c t s are f a r from our

of diminishing r e t u r n s can u s u a l l y be ignored. ordinary view, and so l i t e r a l l y n o t s u p e r f i c i a l . discovered, u s u a l l y by extending experience.

T h e i r p r o p e r t i e s have t o be

I f t h e y ' c o n f l i c t ' w i t h common

sense, it has t o be remembered t h a t t h e y a r e not p a r t o f o r d i n a r y , common experience e i t h e r .

Logical n o t i o n s a r e almost a t t h e o p p o s i t e extreme, and a

good p a r t o f t h e i r a t t r a c t i o n may f a i r l y be a t t r i b u t e d t o t h i s f a c t .

They a r e

i n t e l l i g i b l e from elementary experience o r , e q u i v a l e n t l y , from viewing b r o a d e r experience s u p e r f i c i a l l y .

To i n t e g r a t e such n o t i o n s s u c c e s s f u l l y with t h e r e s t

of knowledge, one must expect t o pay a p r i c e , f o r example, t h e imagination mentioned e a r l i e r i n t h i s paragraph.

Conversely, s e g r e g a t i n g t h e n o t i o n s , and

studying them ( r e l e n t l e s s l y ) as i f t h e y were fundamental, l e a d s u l t i m a t e l y t o a parody o f fundamental t h e o r y , however b e g u i l i n g l y e l e g a n t it may be.

Readers

w i l l f i n d many examples i n t h e l i t e r a t u r e on a l l branches o f l o g i c ; we recommend

t h e two gems, i n S e c t i o n

4 below,

concerning ' u n i v e r s a l c a l i b r a t i o n s ' i n proof

theory and ' t h e o r i e s o f t h e o r i e s ' i n model t h e o r y .

We do not s e e them as mere

a b e r r a t i o n s ; b u t as i n s t a n c e s of t h a t 'profound i n t e l l i g i b i l i t y ' of mathematics

so prominent i n Bourbaki's manifesto, p a r t i c u l a r l y because o f t h e many s t r i k i n g p a r a l l e l s i n d i f f e r e n t branches o f foundations.

Here a p r e - e s t a b l i s h e d harmony

of e r r o r s , o v e r s i g h t s and misjudgmnknts i s no l e s s r e l e v a n t t h a n t h e more popular variety.

FLASHY METATHEOREMS AND SOME NEGLECTED IMPLICATIONS The f i r s t p a r t of t h i s s e c t i o n concerns t h e r i s k , mentioned i n t h e i n t r o d u c t i o n , of f a l l i n g between two s t o o l s when ( v a l i d ) methods o f proof a r e r e s t r i c t e d .

i s obvious t h a t o p t i m a l . s o l u t i o n s m a y be excluded.

-

Reminder.

Trivially, i f

It

222

G . KREISEL, A. MACINTYRE A

' o p t i m a l ' means: minimal,

0

is

111

and undecided by t h e r e s t r i c t e d methods, 3.x [ x = 1 v ( x = 0

t h e optimal s o l u t i o n ( r e a l i z a t i o n ) o f but only

x 5 1 can be proved.

A

A)]

is:

x = 0

,

Less t r i v i a l l y , s e v e r a l algorithms a r e known t o 0

be c o r r e c t o r e f f i c i e n t only modulo some so f a r undecided

conjecture, for

111

example, algorithms f o r p r i m a l i t y t e s t i n g modulo t h e g e n e r a l i z e d Riemann hypoWhat i s t r u e f o r r e s t r i c t i o n s t o a c t u a l l y known p r o o f s a p p l i e s q u i t e

thesis.

well' t o p r o v a b i l i t y by r e s t r i c t e d methods o f t h e kind considered here.

The second r i s k , of choices which a r e t o o weak (when any r e s t r i c t i o n s a r e app r o p r i a t e a t a l l ) , i s e s t a b l i s h e d i n ( a ) and ( b ) f o r t h e c a s e o f H e y t i n g ' s r u l e s This i s seen by two

f o r c o n s t r u c t i v e o r , more p r e c i s e l y , i n t u i t i o n i s t i c l o g i c .

metatheorems which a r e w e l l known, b u t u s u a l l y p r e s e n t e d f o r q u i t e d i f f e r e n t reasons. (a)

The n e g a t i v e t r a n s l a t i o n goes back t o Kolmogorov, Gadel, Gentzen.

It i s

u s u a l l y p r e s e n t e d as e s t a b l i s h i n g t h e adequacy of t h e i n t u i t i o n i s t i c ideology:

a l l c l a s s i c a l mathematics (covered by t h e t r a n s l a t i o n ) can a l s o b e done by use of Heyting's r u l e s ( w i t h t h e corresponding axioms); i n c i d e n t a l l y , without i n o r -

But, by s t a n d a r d s a p p r o p r i a t e t o t h e

d i n a t e i n c r e a s e i n t h e l e n g t h of p r o o f s .

c o n s t r u c t i v e t r a d i t i o n , t h e t r a n s l a t i o n has simply uncovered a loophole: an 0 easy paraphrasing o f obviously non-constructive p r o o f s , f o r example, o f

111

theorems.

( i ) and ( i i )below come from two extremes o f mathematics.

( i ) E u c l i d ' s proof o f t h e i r r a t i o n a l i t y of

/?

by r e d u c t i o ad a b s u r d m .

As a

m a t t e r of h i s t o r i c a l f a c t t h i s has f i g u r e d as a paradigm o f a non-constructive proof (probably e v e r s i n c e E u c l i d ) . pq

# 0

+

o f mVqA

p2

# 2q2

, say

from 1 3 p q - 1 A

A

.

, is

Since, f o r i n t e g r a l

p

and

q

,

n e g a t i v e , Heyting's r u l e s permit t h e i n f e r e n c e

No changes i n t h e t r a d i t i o n a l proof need b e made.

-

On t h e o t h e r hand, it i s obvious what i s r e q u i r e d by t h e c o n s t r u c t i v e t r a d i t i o n : a measure o f i r r a t i o n a l i t y , t h a t i s , a lower bound f o r The missing s t e p i s :

if

, according

p2 to

# 2q2

, then

2

Ip -2q

2

1

lfi-p/q\

1. 1 .

i n terms o f q ,

The l o g i c a l c l a s s i -

q u a n t i f i e r complexity, d r a w s a t t e n t i o n away from t h e

e s s e n t i a l requirement. ( i i ) Deligne's proof o f (A. W e i l ' s v e r s i o n o f ) t h e Riemann h y p o t h e s i s f o r vari e t i e s over f i n i t e f i e l d s .

This can indeed b e formalized i n c l a s s i c a l f i r s t

o r d e r a r i t h m e t i c , c o n t r a r y t o s t r a n g e s p e c u l a t i o n s , f o r example, by Manin i n I ( a ) on p. 36 o f Browder (1976).

It!

Since Deligne's theorem, l i k e t h e o r d i n a r y RH, i s i n

form, t h e n e g a t i v e t r a n s l a t i o n a p p l i e s .

Whatever doubts t h e r e may b e about

t h e p r e c i s e e x t e n t o f t h e c o n s t r u c t i v e t r a d i t i o n , t h e r e i s no doubt t h a t t h e n e g a t i v e t r a n s l a t i o n o f Deligne's proof l i e s o u t s i d e i t ; c f . P a r t I1 f o r a l e s s s t e r i l e way o f a n a l y s i n g p r o o f s of

IIy

theorems.

223

Constructive logic versus algebraization

(b)

Closure under Markov's r u l e .

Over t h e l a s t 25 y e a r s o r s o t h i s has been

v e r i f i e d f o r l a r g e c l a s s e s of i n t u i t i o n i s t i c systems, t h e r e b y extending ( a ) t o

0

( t h e non-negative) 112

formulas.

The r e s u l t w a s u s u a l l y p r e s e n t e d as e s t a b l i s h -

ing t h e s t a b i l i t x of t h e n o t i o n o f a demonstrably t e r m i n a t i n g Turing program, and hence of: provably r e c u r s i v e f u n c t i o n .

Another i n t e r p r e t a t i o n , which i s much

more r e l e v a n t t o t h e c o n s t r u c t i v e t r a d i t i o n i n mathematics, uses t h e f a c t t h a t , for quantifier-free

(*I

A

, the

passage 'Jni'Jmi A

from a proof (by H e y t i n g ' s r u l e s ) of t h e n e g a t i v e formula t o a proof of

Vn 3m A

is quite short.

Corollary on e x t r a c t i n g an a l g o r i t h m

rx

:n

+

m

satisfying

tln A [ n , a ( n ) ]

.

This

job i s e s s e n t i a l l y e q u a l l y d i f f i c u l t whether one s t a r t s w i t h an i n t u i t i o n i s t i c

Vn 3 m A

proof o f

o r one i n t h e corresponding c l a s s i c a l system.

Now, e x t r a c t i n g such a l g o r i t h m s , a l s o c a l l e d : unwinding non-constructive e x i s t ence p r o o f s , i s a p r i n c i p a l concern of t h e c o n s t r u c t i v e t r a d i t i o n .

-

of r e l a t i v e l y simple ( c l a s s i c a l ) proofs can be t e d i o u s .

The unwinding

Thus t h e i n t u i t i o n i s t i c

r e s t r i c t i o n s a r e t o o weak; so weak t h a t a r e d u c t i o n t o i n t u i t i o n i s t i c r u l e s i s not even a u s e f u l i n t e r m e d i a t e s t e p towards unwinding c l a s s i c a l proofs (of theorems). structive

NB.

$

No more could b e expected i n any case s i n c e t h e use o f con-

i s c e r t a i n l y not p a r t of mainstream c o n s t r u c t i v e mathematics

where t h e o p e r a t i o n s 'hidden' i n t h e l o g i c a l symbolism a r e made e x p l i c i t . Remark.

I n t h e s p e c i a l case o f formal a r i t h m e t i c t h e c o r o l l a r y follows from

Gentzen's o b s e r v a t i o n , almost 40 y e a r s ago, t h a t , f o r each

c1

<

EO

, winduction

can b e proved q u i t e simply i n Heyting's a r i t h m e t i c ; so we have r e l a t i v e l y s h o r t proofs o f

IIE

theorems with bounding f u n c t i o n s which a r e not & r e c u r s i v e (and

hard t o c a l c u l a t e ) .

But nowadays t h e r e i s a much n e a t e r way o f e s t a b l i s h i n g t h e

c o r o l l a r y , by t h e very d i r e c t , g e n e r a l proof o f ( * ) , both by Dragalin and H. Friedman.

noticed i n t h e seventies

A s so o f t e n w i t h p r e t t y p r o o f s , people d i d

not look f o r such i m p l i c a t i o n s as i n t h e c o r o l l a r y above, although t h e t o p i c , e x t r a c t i n g a l g o r i t h m s , s t a r e s one i n t h e f a c e . Digression i l l u s t r a t i n g f u r t h e r t h e p o i n t above on p r e t t y p r o o f s .

It seems t o

be a w e l l k e p t s e c r e t t h a t t h e metamathematical p r o p e r t i e s o f Heyting's r u l e s a r e very e l e g a n t (and can be b e g u i l i n g ; c f . t h e g e n e r a l r e p o r t on mathematical l o g i c i n K r e i s e l (1965) where proof t h e o r y i s t r e a t e d p r i n c i p a l l y f o r Heyting's r u l e s ; f o r p u r e l y formal r e a s o n s , i n 3.242 on p. 156, without much concern f o r wider i s s u e s ) .

The following two e x c e p t i o n a l l y p r e t t y metatheorems, discovered

224

G. KREISEL, A. MACINTYRE

s i n c e t h e n , p i n p o i n t s t r i k i n g d i f f e r e n c e s between t h e s e r u l e s and t h e c l a s s i c a l rules. c f . Mints (1979).

( a ) S t a b i l i t y o f E-theorems;

This answers a q u e s t i o n , r a i s e d

a t t h e Buffalo conference i n 1968, which a r i s e s as follows. different

-

f u n c t i o n a l and r e a l i z a b i l i t y

i t i o n i s t i c systems.

other?

interpretations

I

They e x t r a c t r e a l i z a t i o n s

an e x i s t e n t i a l formula occurring i n

-

A ).

3xA

, possibly

How a r e t h o s e

I

xd

xd

There a r e s t r i k i n g l y I

o f formal i n t u -

from any d e r i v a t i o n

of

d

w i t h a d d i t i o n a l parameters ( b e s i d e t h o s e

, for

different

I

, related

t o each

Work by s e v e r a l a u t h o r s s e t t l e d t h e m a t t e r very s a t i s f a c t o r i l y i n terms

o f a p p r o p r i a t e equivalence r e l a t i o n s , b u t without r e g a r d f o r t h e f o l l o w i n g b a s i c limitations. For one t h i n g , mathematically t r i v i a l changes i n d can a f f e c t xd , t h e common I v a l u e o f xd (up t o e q u i v a l e n c e ) considerably; c f . p. 149 of Suppes (1981). I

More i n t e r e s t i n g l y , all t h e maps: d xd have t h e s y s t e m a t i c p r a c t i c a l d e f e c t o f r e t a i n i n g redundancies; c f . Goad ( 1 9 8 0 ) . S i g n i f i c a n t l y , t h e p a r t i c u l a r correction lor. cit.,

c a l l e d ' p r u n i n g , ' v i o l a t e s s e v e r a l familiar l o g i c a l i d e a l s i n

t h e s u b j e c t , f o r example, s o - c a l l e d s t r o n g normalization (which r e f u s e s t o d i s t i n g u i s h between s e n s i b l e and s i l l y r e d u c t i o n s ) .

(B)

Demonstrable a r i t h m e t i c t r a n s f i n i t e i n d u c t i o n on

well-foundedness o f

<

<

.

0 ( c1 -0rderings)

< and

F o r almost 30 y e a r s , it has been known t h a t , f o r some

which a r e not well-founded,

each non-empty a r i t h m e t i c s e t

P

can be proved

in f i r s t o r d e r c l a s s i c a l a r i t h m e t i c t o have a f i r s t element; i n o t h e r words, t h e contrapositive o f arithmetic t r a n s f i n i t e induction, can be proved f o r all

P

T I ( lP, < )

for short,

(demonstrably i n p r i m i t i v e r e c u r s i v e metamathematics);

1 1

c f . p. 128 o f K r e i s e l (1953) u s i n g aC - t r u t h d e f i n i t i o n f o r a r i t h m e t i c . t h i s i s c e r t a i n l y not s u r p r i s i n g . s p e c i f i c ( e v i d e n t l y : non-negative) systems

S

, if

All

In c o n t r a s t , by Friedman (19751, t h e r e i s a

Po

s.t.,

TI(PO,<) i s d e r i v a b l e i n

f o r a wide c l a s s of i n t u i t i o n i s t i c

S

, then <

i s well-founded;

in

f a c t , demonstrably embeddable i n t h e a s s o c i a t e d c a n o n i c a l o r d e r i n g ( f o r example,

of o r d i n a l

E~

i n t h e c a s e of a r i t h m e t i c ) .

- To

add t o t h e a e s t h e t i c a p p e a l ,

So

Po f u n c t i o n s l i k e a p r e d i c a t e

i s obtained by a d a p t i n g Sackst Po proof o f t h e d e n s i t y theorem f o r Turing degrees of r . e . s e t s ( a s u b j e c t n o t variable.

famous f o r i t s use i n o t h e r p a r t s o f l o g i c o r mathematics).

-

We have h e r e a

p r e t t y analogue t o t h e f a m i l i a r use i n a l g e b r a o f a l g e b r a i c a l l y independent s e t s o f p a r t i c u l a r numbers t h a t f u n c t i o n l i k e v a r i a b l e s .

But, a t l e a s t so f a r , t h e

most p e r t i n e n t q u e s t i o n has been n e g l e c t e d . Are t h e r e o r d e r i n g s

< f o r which a proof i n

S of

TI(PO,<) i s e a s i e r t o find

o r t o w r i t e dotm t h a n , s a y , a proof of an embedding of < i n t h e canonical ordering associated with

S ?

Constructive logic versus algebraization

225

In s h o r t , i s t h e p r e t t y r e s u l t a l s o a s c i e n t i f i c t o o l ? Reminder. ComFor example, F r i e d b e r g (1957) on P o s t ' s

parisons a r e not n e c e s s a r i l y odious.

problem r a i s e s t h e same q u e s t i o n a s Friedman ( 1 9 7 5 ) , s o t o speak on t h e s c i e n t i s t ' s r e s p o n s i b i l i t y o f u s i n g t o o l s f o r s e n s i b l e aims; a t l e a s t i n h i s own

-

domain of competence.

Conjecture.

The k i n d o f exact methodology, mentioned

i n t h e i n t r o d u c t i o n , draws a t t e n t i o n away from t h e a c t u a l p o s s i b i l i t i e s o f e s t a b l i s h i n g s i g n i f i c a n c e ; by t a l k about g e n e r a l ' p r i n c i p l e s ' For example,

one p o s s i b i l i t y

would be t o d i s c o v e r an equations i n

(1957)

s . t . t h e d e c i s i o n problem f o r q u a r t i c diophantine

N

v a r i a b l e s i s o f i n t e r m e d i a t e degree.

N

f o r doing t h i s .

o f enhancing t h e s i g n i f i c a n c e of F r i e d b e r g

( T h i s would not only be

f r u i t f u l f o r number t h e o r y , b u t a l s o f o r t h e t h e o r y o f Turing degrees.)

WEAKNESS OF THE CONSTRUCTIVE MEANING OF

ni

+

fl;

The weakness documented in t h e l a s t s e c t i o n concerned i n t u i t i o n i s t i c r e s t r i c t i o n s Here it w i l l b e shown t h a t , i n p r a c t i c e , t h e g e n e r a l re-

on r u l e s of proof.

flo 3

quirements on c o n s t r u c t i v e v a l i d i t y a r e i n a d e q u a t e , a l r e a d y f o r

$

(though not f o r

+

no3

) implications.

+

no

3

This i s done by r e f e r e n c e t o G8del A t t h e end o f t h e s e c t i o n t h e a n a l y s i s

(1958) which r e a d e r s a r e l i k e l y t o know.

w i l l be used t o e x p l a i n (and c o r r e c t ) t h e views on t h i s m a t t e r i m p l i c i t i n t h e informal l i t e r a t u r e on c o n s t r u c t i v e mathematics. - Reminder.

vx p m ( x , y , z )

from

+

Vu 3v VwD(u,v,w)

Vu Vy 3v Vw 3x 32 { A [ x , Y ( x ) , z ) ]

Y

where

: x

, and

y

G8del (1958) passes

t o t h e formally s t r o n g e r formula D(u,v,w)I

+

-

,

r e q u i r e s a r e a l i z a t i o n by c o n s t r u c t i v e f u n c t i o n a l s

v

:

such t h a t , f o r a l l u,y,w,

, x,z

(u,Y) I+ v (with

7,

: (u,Y,w)

x,z

x s h o r t f o r V(u,y) and X ( u , Y , w ) ,

etc.)

(t) NB.

holds.

The l e t t e r s

A

and

s t a n d f o r ( p r o p e r t i e s o f ) a l g e b r a i c numbers

D

and diophantine e q u a t i o n s considered below. F i n i t e n e s s theorems.

To judge t h e s i g n i f i c a n c e o f

( t ) ,it

i s b e s t t o apply it

to

f a m i l i a r examples where n e i t h e r a b s t r a c t nor formal nonsense h a s much chance t o t u r n our heads.

w

2v

-t

resp.

w

Let

7DO(u,w) indexed

A

and

D

be, resp.:

, where A. and by x , r e s p . u

Suppose f i r s t a bounding f u n c t i o n

for z

1. y ( x )

terms o f

y

.

.

Then

( t ) provides

All i s w e l l .

Do

z 'y

+-I

AO(x,z)

,

a r e thought o f a s decidable s e t s o f

. y

f o r t h e premise i s known:

1 AO(x,z)

a bounding f u n c t i o n f o r t h e conclusion i n

z

,

226

G. KREISEL, A. MACINTYRE

Suppose next we have no e x p e c t a t i o n o f f i n d i n g such a bounding f u n c t i o n . we t a k e t h e c o n t r a p o s i t i v e o f

I n words:

i f , f o r any

then we have

X

u

( t ) ,t h a t

, some

'sufficiently'

t . some 'Large'

S .

z

large

w

-

satisfies

Do(u,w)

y , no l o n g e r assumed t o be a

This i s u s e f u l i f , f o r example, s u f f i c i e n t l y l a r g e elements

{w : D 0 ( u , w ) } a r e a t hand ( f o r some u ), and l a r g e elements of x ) , w i t h an obvious p r o v i s o .

{ z : A O ( x , z ) } a r e i n demand ( f o r some

-

V ).

' l a r g e ' agrees w i t h t h e t e c h n i c a l meaning above (determined by more, t o pass from w

, one

x

I t must

y so t h a t t h e a p p r o p r i a t e meaning o f

b e p o s s i b l e t o choose t h e parameter

t i v e l y on

,

A O ( X , z ) ; b o t h terms i n in-

satisfies

v e r t e d c o m a s depend on t h e f u n c t i o n parameter bounding function. of

Then

is,

t o 'corresponding'

o f t h e arguments o f

X

u

.

,

X

Further-

must not depend t o o s e n s i -

The l i t e r a t u r e on i m p l i c a t i o n s

between f i n i t e n e s s theorems i l l u s t r a t e s convincingly how t h e proviso can be met; c f . b ( B ) of t h e l a s t s e c t i o n f o r an example which does & s a t i s f y

-

t h e proviso t o

(1958). Though i n comment (iii)o f t h e next s e c t i o n some s e v e r e l i m i t a t i o n s o f G8del (1958)f o r i t s own aims w i l l come up,

t h e same e x t e n t .

Remark on G8del

it should be remembered t h a t , for t h e p r e s e n t purpose, it has a l r e a d y p a i d o f f by b r i n g i n g out t h e need f o r t h e p r o v i s o above. Famous examples: bad approximation theorems w i t h f i n i t e l y many exceptions ( o f a l g e b r a i c by r a t i o n a l numbers) and f i n i t e n e s s theorems for c e r t a i n c l a s s e s , s a y ,

8

o f b i n a r y diophantine equations.

Here a l l r e f e r e n c e s w i l l b e t o t h e ex-

c e p t i o n a l l y elementary e x p o s i t i o n o f s t a n d a r d r e s u l t s i n Mordell

(1969), [MI f o r

s h o r t , while t h e Appendix goes i n t o t h e most r e c e n t and incomparably more so-

263 of [MI. We now

p h i s t i c a t e d discovery i n t h i s a r e a ; r e f i n i n g Theorem 2 on p . change n o t a t i o n :

a

x

above means now a p a i r

(a,m)

where t h e a l g e b r a i c number

i s given by t h e c o e f f i c i e n t s o f i t s d e f i n i n g equation,

o f any r a t i o n a l approximation simply, when

where

T

p/q

p/q

, and

y

z

t h e denominator

a bound on t h e denominator (most

i s p u t i n i t s lowest t e r m s ) .

Then

( f o r 'Thue') depends on t h e p a r t i c u l a r bad approximation theorem under

d i s c u s s i o n ( S i e g e l , Roth, e t c . ) .

Note t h a t

A.

i s decidable.

u codes ( t h e c o e f f i c i e n t s o f ) t h e diophantine equation, s a y , class

fiT

covered by t h e bad approximation theorem. DO(u,w) : 3w'[D1(w,wI)

D1

i n the

Then

= 01

i s a g a i n ( p o i m i t i y e ) r e c u r s i v e l y decidable f o r each p a i r u and

W

, provided

it

221

Constructive logic versus algebraization i s so decidable whether

,n D1

consists of binary equations

+

P

, and

> 2

1 'm

where

Q

Q )

(over

c f . [MI, Theorem of degree

4 on

-

188 ( a l s o , for s m a l l n

p.

< n/2-1 ) .

< n

i s of degree

Q

where

186 o f [MI; t h e

D1

Given

and i r r e d u c i b l e

, Theorem

5 on p . 198, f o r

Q

was s o l v e d e a r l i e r by Thue;

( t ) above

i s s a t i s f i e d beyond

.

P(l,a) = 0

To a p a i r

(w,w')

corresponds, i n t h e s i m p l e s t s e n s e p o s s i b l e , an ex-

D1

c e p t i o n a l l y good approximation

p/q

with ' l a r g e '

t r a p o s i t i v e o f ) ( ? ) , X and Z a r e independent o f

u and Y w1

n

, coded by u , t h e corresponding

a l g e b r a i c number i s simply one o f t h e zeros o f

depends on

is the

a c t u a l l y needed i s i n ( 5 ) on p. 187.

T

The p r o v i s o mentioned i n connection w i t h

o f l a r g e solutions of

n

The corresponding

2 6 ;

Q

The c a s e o f c o n s t a n t s

t h e most o p t i m i s t i c e x p e c t a t i o n s .

V

n

of t h e form

i s homogeneous of degree

P

and

cf. Theorem 1 on p.

and

5

a ; cf. [MI, Theorem 2 , p. 187 (due t o S i e g e l ) .

degree of

aT

.

codes a n element of &I

u

m + n/(m+l)

T(a,rn) =

Example 1.

&&

.

9 : q =

Y ,X

W

.

So, i n ( t h e con-

a l s o independent o f w

+

I n s ~ b o l s :for n-tuples x,n depending on u

, ,

.

ZIv WY k ( 3 x ~ z ) [ A ( x , y , z )+ D ( u , v , w ) ]

The example i l l u s t r a t e s most convincingly t h e s i g n i f i c a n c e o f reducing t h e s e t of p a r a m e t e r s , e s p e c i a l l y i n combination w i t h r e s t r i c t i o n s on v and z as f u n c t i o n s of y , r e s p . o f w (as above, where t h e y a r e e q u a l ) . - I n p l i f i c a t i o n p a r a l l e l s t h e case of

- r ( a , m ) = 2 + m-'

Example 2.

II;

l o g i c a l terms, t h e s i m -

implications.

, independent

o f t h e degree o f

c o n s i s t s of e q u a t i o n s

corresponding c l a s s P + Q Q

-f

a

.

By [MI, p. 189, 1. 5 (and r e p e a t e d i n 1. -10 on p.

due t o Roth.

Exercise.

1;

where

D1 o f t h e form

i s as i n Example 1 and

P

-2 , find

< n

i s of degree

For given

This form i s

198) t h e obvious

D1

.

t h e corresponding a ( e a s y ) , and t h e a p p r o p r i a t e

m. Warning concerning Theorem 22 on p. 278 o f [MI (due t o S c h i n z e l ) .

&IT o f Example 2 i s extended t o a l l ducible, t h a t i s ,

Q

i s o f degree

C

D1 n

Here t h e . c l a s s

whose p a r t o f h i g h e s t degree i s i r r e -

.

But t h e proof i s much more complicated,

and a s e p a r a t e i n v e s t i g a t i o n i s needed t o check whether t h e b a s i c p r o v i s o i s s a t i s f i e d ; perhaps along t h e l i n e s o f t h e Appendix where t h e a l g e b r a i c numbers corresponding t o

P + Q

a r e d e f i n e d by d e l i c a t e a l g e b r a (independently o f t h e

s i z e o f h y p o t h e t i c a l s o l u t i o n s of Reminders.

p

+

Q

1.

( i ) The d i s c u s s i o n o f t h e p r o v i s o e a r l i e r on a p p l i e s d i r e c t l y t o

Example 2 s i n c e t h e r e i s no h i n t i n s i g h t of an e f f e c t i v e bounding f u n c t i o n

for arbitrary

a , i n Roth's theorem.

Beyond t h i s t h e r e i s t h e q u e s t i o n o f

Y

,

G. KREISEL, A. MACINTYRE

228

s p o t t i n g diophantine e q u a t i o n s , s p e c i f i c a l l y , s u i t a b l e

Q

, which

have l a r g e

s o l u t i o n s , and t h u s e x c e p t i o n a l l y good approximations w i t h l a r g e denominators

up

t o a l g e b r a i c numbers

for which t h i s i s needed.

( i i ) The p a r t i c u l a r c a s e of Thue's equations: P = m ( i n Example 1) i s more

By ( B a k e r ' s ) Theorem 7 on p. 199 o f [MI, t h e o l d i m p l i c a t i o n i s

delicate.

superseded i n two w w s .

F i r s t , t h e conclusion i t s e l f i s now e f f e c t i v e l y es-

t a b l i s h e d , w i t h bounds on t h e s i z e o f t h e s o l u t i o n s of

P= m

.

Secondly,

Baker's argument d e r i v e s t h i s from an e f f e c t i v e v e r s i o n of a 'weakened' form o f Thue's approximation theorem, t h a t i s , for a l a r g e r

T

t h a n Thue's

So t h e c o n t r a p o s i t i v e provides even b e t t e r approximations t o

solutions of

.

P = m

%

n/2 + 1.

from l a r g e

So t h e i n t e r e s t o f t h e c o n s t r u c t i v e i m p l i c a t i o n i n

Example 1 depends on i t s s i m p l i c i t y , and i s , of course, confined t o c a s e s where a l s o l e s s p r e c i s e bounds:

la-p/ql

5

q-(n'2+1)

a r e good enough; c f . t h e

Appendix f o r a more c o n c r e t e d i s c u s s i o n , f o r diophantine e q u a t i o n s of genus

1

,

f o r which Baker and Coates provide ( o u t r i g h t ) e f f e c t i v e bounds.

(iii) Another way of p u t t i n g t h e i s s u e i n ( i i )i s t o r e c a l l t h a t a proof of an implication i s

not

always superseded by a proof o f t h e conclusion:

t o t h e c a s e where ' a p r o o f ' i s r e p l a c e d by ' t h e v a l i d i t y . '

i n contrast

E v i d e n t l y , some

obviously e s s e n t i a l information i n such p r o o f s i s l o s t by s t a t i n g t h e r e s u l t as

I I + ~

a mere

no3

implication.

Broad conclusions about t h e mathematical and l o g i c a l l i t e r a t u r e on c o n s t r u c t ivizations.

Mathematicians t e n d t o b e e x c e s s i v e l y s u s p i c i o u s of any b u t t h e

most s t r a i g h t f o r w a r d c o n s t r u c t i v e meaning, f o r

0

112

o r o t h e r prenex formulae.

The passage t o t h e c o n t r a p o s i t i v e , d i s c u s s e d above, i s c e r t a i n l y u n f a m i l i a r . Even t h e switch from bounds for t h e s i z e t o bounds for t h e number of s o l u t i o n s i s dismissed out o f hand simply because t h e l a t t e r ( o b v i o u s l y ) do not ensure t h e

former; c f . 1. 4-5 on p. 199 o f [MI.

But t h e r e can be imaginative u s e s f o r

-

bounds on t h e number t o o ; for example, i n Baker (1964) f o r provement i n Davenport and Roth (1955) o f s t r a i g h t f o r w a r d Even i f pure ideology l i k e Bishop

theorem.

-

t h e p a r t i c u l a r im-

bounds i n Roth's

(1967) i s d i s r e g a r d e d , t h e l o g i c a l

l i t e r a t u r e t e n d s t o e r r i n t h e o p p o s i t e d i r e c t i o n ; going a f t e r wholesale cons t r u c t i v i z a t i o n , and i g n o r i n g t h e need for d i s c r e t i o n , shown by t h e p r o v i s o above.

This a p p l i e s g e n e r a l l y t o v a r i o u s schemes o f s o c a l l e d f u n c t i o n a l i n t e r -

p r e t a t i o n s , b e f o r e and a f t e r G8del (1958). mathematics a r e i n t e r p r e t e d by also: ively).

A

follows from A '

A'

to

A'

There theorems A o f some branch of A'

can b e proved c o n s t r u c t i v e l y , b u t

q u i t e t r i v i a l l y (though g e n e r a l l y not c o n s t r u c t -

The l a t t e r i s b e g u i l i n g :

t h e passage from A

s . t.

.

viewed i n o r d i n a r y t e r m s , nothing i s l o s t by

However, except i n very s p e c i a l c a s e s ,

not

enough i s gained t o make t h e passage worthwhile f o r mainstream c o n s t r u c t i v e mathematics.

The next s e c t i o n c o n t a i n s examples o f such e x c e p t i o n s , b u t without

Constructive logic versus algebraization

229

a t a l l (which, by t h e l a s t s e c t i o n , i s unrewarding even

using c o n s t r u c t i v e a s an i n t e r m e d i a t e s t e p ) .

METAMATHEKATICS OF CLASSICAL SYSTEMS The s u c c e s s f u l exceptions J u s t mentioned belong t o a, by now w e l l known, v a r i a n t of H i l b e r t ' s program p r o v i d i n g c o n s t r u c t i v e ( f u n c t i o n a l ) i n t e r p r e t a t i o n s o f

formally non-constructive proofs.

Readers a r e assumed t o b e f a m i l i a r w i t h t h e

e a s i e s t example o f t h i s v a r i a n t , t h e s o c a l l e d no-counter example-interpretation, n.c.i.

The main purpose o f t h i s s e c t i o n i s t o give away some t r i c k s

for short.

of t h e t r a d e which a r e needed f o r an e f f i c i e n t u s e of t h e method i n p r a c t i c e .

'

( i ) The v a r i a n t uses techniques o r i g i n a l l y developed Reminders. f o r H i l b e r t ' s program, b u t w i t h a t o t a l change o f d i r e c t i o n . S p e c i f i c a l l y , f o r t h e o r i g i n a l program p r o o f s o f theorems were r e l e v a n t , w h i l e h e r e such theorems (and proofs o f IIO-lemmas) a r e ignored completely; f o r more d e t a i l , c f . r e c e n t p u h c a t i o n s by one o f us, f o r example, on pp. 131-163 o f Suppes (1981) o r i n viewed i n t h e forthcoming Herbrand volume. ( i i ) The n.c.i. = b e terms o f c o n s t r u c t i v e l o g i c by use o f G8del (1958) which came up a l r e a d y i n t h e l a s t s e c t i o n . G e n e r a l l y , i n a r i t h m e t i c o r pure l o g i c , t h e same negations o f prenex formulas can b e proved c l a s s i c a l l y or i n t u i t i o n i s t i c a l l y . I f A* i s prenex and c l a s s i c a l l y e q u i v a l e n t t o 1 A , t h e n t h e n.c.i. o f A reduces t o t h e i n t e r p r e t a t i o n o f -A* i n G8del (1958). But, as f i r s t shown i n Shoenfield (1967), t h i s view i s i n e f f i c i e n t s i n c e G8del (1958) i s v e r y simply adapted t o c l a s s i c a l systems. A s it happens, t h e pedagogic v a l u e o f t h i s a d a p t a t i o n can be documented by an ( i m p r e s s i v e ) example. G i r a r d had seen v a r i o u s e x p o s i t i o n s which followed a d e l (1958) c l o s e l y , b e f o r e r e a d i n g S h o e n f i e l d (1967), b u t l e a r n t t o use t h e i d e a s e f ( i i i ) Another d e c e p t i v e a t t r a c t i o n f e c t i v e l y only from t h e l a t t e r . i s i t s g e n e r a l i t y : by o f G8del ( 1 9 5 8 ) , compared t o t h e n.c.i., using f u n c t i o n s o f all f i n i t e t y p e s , t h e formal scheme provides a (To b e r e c u r s i v e i n t e r p r e t a t i o n o f second o r d e r a r i t h m e t i c t o o . p r e c i s e , t h i s does not apply t o t h e very s p e c i a l c l a s s o f c o n s t r u c t i v e rules d e s c r i b e d i n G8del (1958) b u t t o t h e more n a t u r a l c l a s s o f r e c u r s i v e l y continuous o r countable f u n c t i o n s . ) However, i n p r a c t i c e , h i g h e r t y p e s a r e not convenient f o r unwinding a t a l l ; reminiscent o f i t e r a t e d i m p l i c a t i o n s , which, according t o f o o t n o t e 2 on p. 109 o f K r e i s e l (1953), a r e l i a b l e t o make one's head s p i n . Worse s t i l l : when comprehension i s used i n a n e s s e n t i a l way ( t o prove a I@ theorem, n o t merely some I@ lemmas) t h e c o n s t r u c t i v e c o n t e n t i s t o o marginal t o be worth e x t r a c t i n g a t all, i n accordance w i t h t h e warning a t t h e end o f t h e l a s t s e c t i o n . (a)

The l i t e r a t u r e c o n t a i n s b o t h

Learning t o unwind by use o f t h e n . c . i .

formal metamathematical arguments, a p p l i e d t o f u l l y formalized p r o o f s , and f a i r l y e l a b o r a t e unwindings o f i n f o r m a l arguments which experienced mathematici a n s had found p a r t i c u l a r l y h a r d t o handle. be pedagogically u s e f u l . Example.

(*)

For

0

5

c < 1

Neither o f t h e s e extremes seems t o

We have had some s u c c e s s w i t h t h e following

, prove

the

Il02-formula

Vk 3n ( c n < k ' )

,

G. KREISEL, A. MACINTYRE

230 where

k

and

n

range o v e r p o s i t i v e i n t e g e r s , b y means o f t h e s t e p s ( i ) ,( i i ) ,

NB.

( i i i ' ) ,( i v ) below.

(m

n) (0

5

a

bounded, say = cn

a

(iii) i f

en

since

0

-t

cm 5 c " ) .

( i ) sequences

( i i )f o r

( * ) i s t r i v i a l l y equivalent t o :

converge, if t h e y a r e monotone, s a y , d e c r e a s i n g , and

5 an -<

0

1 ;

, ( i )a p p l i e s ;

i s t h e l i m i t of

F

,F =

{cn}

CF, hence

(l-c)T= 0

,

and s o

F = 0;

a l t e r n a t i v e l y , working w i t h subsequences i n s t e a d o f limits: ( i i i ' ) {en+'}

i s a subsequence o f

{c"} ; s o , by ( i i ) , c n ( l - c )

Vh 3n [ c n ( l - c ) < h - l ] (iv)

If, i n (iii), h

Now, t h e n . c . i .

1. k (l-c)-'

which i s s a t i s f i e d by t h e f u n c t i o n a l the least

a

operat?on: en of

II

u(i)

0 < a -a

P

m

, and

,

e n < k-'

, and

so ( * ) .

< k-l) i s

II

: (k,p)P p

, where

p

-+

0 < au(i+l)

< k-l

, and

is

H

i

cn+l

i n ( i i i ' )i s t h e successor, say

i s , by ( i v ) , t h e s m a l l e s t i n t e g e r

2 k(l-c)-'

U

.

.

The o t h e r argument

This y i e l d s a bound

for ( * ) : (3n

5

II[k(l-c)-',o])

(en < k-l)

.

u(j) = ~'(1)

, actually

< a k ( n o t only < k ) . When t h i s i s a p p l i e d 1 = cn , by ( i i ) ,t.hen t h e f u n c t i o n u which corresponds t o t h e s h i f t

There i s such an to:

i < k : u(i+l)

-t

0

.

cn(l-c) < hml

and

of ( i ) ,Vk Bp 3 m ( m > p

-+

.

Comments. ( a ) By an easy c a l c u l a t i o n t h i s bound i s not o p t i m a l ; Put d i f f e r e n t l y : f o r example, it i s o f o r d e r k i n s t e a d o f l o g k ( B ) t h e bound i s n o t a p r o p e r t y of t h e theorem ( * ) , b u t o f t h e p a r t i c u l a r proof above: ( i ) ,( i i ) ,( i i i l ) , ( i v ) . ( y ) Optimal bounds cannot b e expected s i n c e II a p p l i e s t o a r b i t r a r y monotone sequences, ( 6 ) Conversely, (y) i s not d e c i s i v e and n o t o d y t o : an = cn s i n c e bounds a r e c e r t a i n l y n o t t h e sum t o t a l o f a l l t h a t one wants t o know about (*I! R e a l i s t i c a l l y , optimal bounds from ( * ) a r e e a s i e r t o come by t h a n ( i ) ,l e t a l o n e any r e l a t i o n between ( * ) and ( i ) . More g e n e r a l l y , ( E ) proofs provide much more knowledge t h a n t h e v a l i d i t y , c o n s t r u c t i v e o r otherwise, o f t h e theorem proved; b u t v a l i d i t y i s t h e p a r t of t h i s knowledge which i s e a s i e s t t o s t a t e ; c f . t h e n e x t s e c t i o n f o r more d e t a i l s on such a d d i t i o n a l knowledge NB. Unwind( o v e r and above t h e v a l i d i t y of t h e theorem proved). i n g makes only a q u i t e s p e c i a l kind of t h a t knowledge e x p l i c i t . For t h i s purpose t h e r e i s a f a m i l i a r a l t e r n a t i v e t o t h e n . c . i . which i s t h e s u b j e c t o f ( b ) below.

.

.

Warning. Though t h e n.c.i. i s e s t a b l i s h e d - and t h u s t h e n o t i o n o f 'unwinding' d e f i n e d - f o r p r o o f s o f a r b i t r a r y formulas, i t s p r i n c i p a l uses come from $ formulas proved by means o f l o g i c a l l y more comp l i c a t e d lemmas, l i k e ( i )i n t h e proof above; f o r an exception, c f . c e r t a i n Z g formulas d i s c u s s e d by one o f us i n t h e Herbrand volume i n

23 1

Constructive logic versus algebraization t h e s p e c i a l c a s e where t h e f u n c t i o n s used i n t h e n.c.i. appropriate arithmetic conditions.

satis0

B i b l i o g r a p h i c a l remark. The n.c.i. of IIB formulas i s s u f f i c i e n t t o r e p l a c e t h e t r a d i t i o n a l h e a r t s e a r c h i n g concerning a s e n s i b l e cons t r u c t i v e meaning o f i m p l i c a t i o n s w i t h u n i v e r s a l premisses (and IIO2 c o n c l u s i o n s ) , f o r example, on pp. 358-361, e s p e c i a l l y f o o t n o t e 1 on p. 361, o f Hilbert-Bernays, v o l . 2 (second e d i t i o n ) . Since c l a s s i c a l l o g i c i s considered, Vx A + 3y B , w i t h parameters p , reduces t o a IIQ formula. Unwinding a proof provides bounds x yp s . t . P '

(Vx 5 xp)A

+

(3y

5 YP )B

.

I n s h o r t , one simply has a s t r o n g e r p r o p o s i t i o n ( b y bounding i n t h e premise). (b)

x

Functional i n t e r p r e t a t i o n s versus c u t e l i m i n a t i o n (and i t s v a r i a n t s , such

as n o r m a l i z a t i o n ) : needs of unwinding.

a c o n f l i c t between ' t h e o r e t i c a l ' e x p o s i t i o n s and p r a c t i c a l Generally speaking, it i s easy t o pass from c u t - f r e e

proofs t o i n t e r p r e t a t i o n s s i n c e t h e l a t t e r , l i k e Kleene's r e c u r s i v e r e a l i z a t i o n s , suppress many d e t a i l s of p r o o f s .

Thus if b o t h c u t e l i m i n a t i o n and i n t e r -

p r e t a t i o n s a r e t o be p r e s e n t e d , it i s most economical t o s t a r t w i t h t h e former; c f . Schwichtenberg

(1977). But i n p r a c t i c e , e s p e c i a l l y when o p e r a t i n g on

sketches o f p r o o f s , c u t e l i m i n a t i o n i s q u i t e inconvenient s i n c e it i s hard t o check whether o r not some proposed c u t - f r e e proof i s anything l i k e t h e normal form o f t h e given s k e t c h .

The reason i s t h a t c u t e l i m i n a t i o n r e b u i l d s a

given p r o o f , and key theorems, which a r e l o g i c a l l y more complicated t h a n t h e end r e s u l t , g e t l o s t i n t h e process.

In contrast, i n functional interpretations

t h e ( f u n c t i o n a l ) equations corresponding t o key theorems remain prominent, w i t h r e l a t i v e l y s h o r t l o g i c a l i n f e r e n c e s b e i n g r e p l a c e d by a s e r i e s o f d e f i n i t i o n s of f u n c t i o n a l s and v e r i f i c a t i o n s o f t h e r e l e v a n t i d e n t i t i e s .

The d i f f e r e n c e i s

s t r i k i n g l y i l l u s t r a t e d i n G i r a r d ' s unwinding o f t h e proof by t o p o l o g i c a l dynamics o f van d e r Waerden's theorem, s i n c e c u t e l i m i n a t i o n i s used t o unwind some o f t h e s t e p s , t h e n . c . i .

for others.

I n c i d e n t a l l y , h i s work a l s o e x h i b i t s

t h e phenomenon, a l r e a d y mentioned i n ( a ) o f t h e f i r s t s e c t i o n , how mathematica l l y t r i v i a l changes can a f f e c t d r a m a t i c a l l y t h e unwinding process and even t h e r e s u l t i n g bounds; c f . p. 1 5 3 o f Suppes (1981).

This i s i n accordance w i t h com-

ment ( y ) i n ( a ) above i f , by o r d i n a r y mathematical s t a n d a r d s , t h e b u s i n e s s of unwinding c o n s t i t u t e s a t r i v i a l a s p e c t o f t h e proof i n q u e s t i o n . p o s i t i v e s i d e of t h e phenomenon i s t h i s :

NB.

The

once one has l e a r n t t o s p o t s u i t a b l e

t r i v i a l changes ( i n informal p r o o f s ) t h e b u s i n e s s becomes more l u c r a t i v e : a b e t t e r r a t i o of r e s u l t t o e f f o r t .

G. KREISEL, A. MACINTYRE

232 APPLYING THE N . C . I . :

REASSURANCES

Experience shows t h a t t h e r e a r e at l e a s t two k i n d s of o b s t a c l e s t o applying t h e n.c.i.

o r related interpretations i n practice.

One i s t h e i d e a t h a t p r o o f s ,

i n c o n t r a s t t o theorems,are h o p e l e s s l y e l u s i v e o b j e c t s , n o t amenable t o any t h e o r e t i c a l processing a t a l l .

I n terms of

a ( E ) o f t h e l a s t s e c t i o n , t h e most

t h a t could be hoped f o r i s t o paraphrase a d d i t i o n a l knowledge c o n t a i n e d i n a proof i n terms o f g e n e r a l i z a t i o n s o f t h e theorem proved, a s i n ( a ) o r d i n a r y axiomatic mathematics, and ( b ) model t h e o r y , f o r example, o f f i r s t o r d e r axiom systems.

The i d e a , s u r e l y v a l i d f o r t h e b u l k o f q u e s t i o n s about proofs t h a t

occur t o us n a t u r a l l y , overlooks t h e p o s s i b i l i t y t h a t t h e very s p e c i a l p r o j e c t o f unwinding may be an exception (and obvious p a r a l l e l s i n o t h e r domains; f o r example, ' h e r o i c ' p r o g r e s s w a s needed t o answer such n a t u r a l q u e s t i o n s as: Why i s g l a s s t r a n s p a r e n t ? ) needed t o apply t h e n . c . i . metatheorems about it.

The second o b s t a c l e i s t h a t a d d i t i o n a l s k i l l s a r e e f f e c t i v e l y , over and above knowledge o f g e n e r a l

L i t t l e i s s a i d about t h o s e s k i l l s : t h e thought of them

c r e a t e s a c e r t a i n malaise. The p r i n c i p a l purpose o f t h i s s e c t i o n i s t o d i s p e l t h e malaise,emphasizing

( a ) t h o s e a s p e c t s of axiomatic a n a l y s i s which make it q u i t e a k i n t o a t h e o r y o f proofs i n t h e n a t u r a l s e n s e o f t h e word, and ( b ) t h o s e s t e p s i n a p p l i c a t i o n s of model t h e o r y which i n v o l v e digging i n t o p r o o f s , and not o n l y l o o k i n g a t t h e form o f t h e theorem proved.

It w i l l be apparent t h a t h e r e t o o a d d i t i o n a l s k i l l s

a r e needed ( o v e r and above t h e g e n e r a l scheme o f axiomatic a n a l y s i s o r g e n e r a l metatheorems about, say, u l t r a p r o d u c t s ) w i t h obvious p a r a l l e l s i n a l l t h e o r e t i c a l schemes, i n n a t u r a l s c i e n c e and technology; f o r example, t h e t h e o r y i t s e l f does not t e l l you whether it i s r e l e v a n t t o a g i v e n c o n c r e t e s i t u a t i o n . These f a c t s a r e r e a s s u r i n g because ( a ) and ( b ) have proved s u c c e s s f u l d e s p i t e t h e need f o r t h o s e a d d i t i o n a l s k i l l s s i n c e t h e y a r e i n f a c t a v a i l a b l e ; and d e s p i t e manuals on exact methodology which l e a v e us unprepared f o r t h a t need (and f o r t h e e x i s t e n c e and r e l i a b i l i t y o f t h e a d d i t i o n a l s k i l l s at is s ue).

( a ) Axiomatic a n a l y s i s :

neglected aspects.

Instead o f seeing t h e significance

o f an axiomatic p r e s e n t a t i o n p r i m a r i l y i n t h e g r e a t e r g e n e r a l i t y o f t h e theorem proved, l e t a l o n e i n some dubious d o c t r i n e o f g r e a t e r ' p r e c i s i o n , ' it as a s t r a t e g y b o t h f o r f i n d i n g and remembering proofs. Cartan

we recognize

Elementary example:

(1967),a f i r s t y e a r course i n c a l c u l u s , has t h e s u b t i t l e : d i f f e r e n t i a l

c a l c u l u s i n Banach spaces.

This means t h a t , f o r t h e p a r t of c a l c u l u s covered,

only few

a r e needed, v i z . t h o s e o f Banach s p a c e s , an e f -

p r o p e r t i e s o f lfi

f i c i e n t summary o f t h e key s t e p s i n t h e p r o o f s of t h e t e x t .

A t t h e same time

t h i s c o n s t i t u t e s a proof s t r a t e g y s i n c e one can c o n f i n e o n e s e l f t o u s i n g only those properties.

More g e n e r a l l y , r e l a t i v e l y few p r o p e r t i e s , Bourbaki's few,

s o c a l l e d b a s i c s t r u c t u r e s , have been found adequate f o r similar s t r a t e g i e s i n a very broad domain o f mathematics.

The need f o r ' a d d i t i o n a l s k i l l s ' i s e v i d e n t

Constructive logic Venus algebraization

233

s i n c e t h e proper choice of terms i n which some given proof o r branch o f mathematics i s t o be g e n e r a l i z e d , i s n o t o r i o u s l y d e l i c a t e .

For an understanding o f

t h e a c t u a l p o s s i b i l i t i e s o f mathematical reasoning t h e f a c t t h a t t h e s e s k i l l s a r e a v a i l a b l e i s more d i r e c t l y r e l e v a n t t h a n d i s c o v e r i e s , f o r example, i n a r t i f i c i a l i n t e l l i g e n c e which t r i e s t o t r e a t such s k i l l s a s ' b l a c k boxes' t o be replaced by e a s i l y mechanized processes with s i m i l a r end r e s u l t s . Comments. ( i ) The use of axiomatic a n a l y s i s a s a proof s t r a t e g y does not seem t o be w e l l known t o people w r i t i n g on h e u r i s t i c s , l i k e Polya, nor t o t h o s e i n t h e education b u s i n e s s i f pp. 170-171 of Suppes (1981)a r e r e p r e s e n t a t i v e . The l a t t e r i l l u s t r a t e , very convincingly, a t l e a s t two important consequences of t h a t ignorance. F i r s t , t h e s t a n d a r d s o f t h e 'new' p r o p o s a l s for proof s t r a t egies a r e d e s p e r a t e l y low, a s so o f t e n when one i s under t h e i l l u s i o n t h a t nobody e l s e has e v e r thought about o n e ' s t o p i c . More s p e c i f i c a l l y , on p. 170 axiomatic s e t t h e o r y i n t h e s t y l e o f t h e f i f t i e s i s p r e s e n t e d as i f it were even remotely t y p i c a l o f axiomatic mathematics although i n f a c t it i s a parody of t h e l a t t e r . What makes t h e s e ( m i s t a k e n ) views and much more l o c . c i t . o f genuine i n t e r e s t f o r t h e time b e i n g i s t h a t t h e same i d e a s a r e widely h e l d among l o g i c i a n s , b u t r a r e l y a r t i c u l a t e d s o i n n o c e n t l y . ( i i )N a t u r a l l y , t h e r e i s p l e n t y of r e l e v a n t l i t e r a t u r e on t h e m a t t e r , p a r t i c u l a r l y i n r e c e n t y e a r s ; f o r example, or t h o s e h i s t o r i e s where (some) survey l e c t u r e s and reviews i n t h e B u l l . A.M.S. t h e author l i k e s t o i d e n t i f y himself w i t h t h e hero (and so has a l o p s i d e d h i s t o r i c a l view), b u t happens t o have something s i g n i f i c a n t t o say about t h e (iii)P. 170 of Suppes (1981) mathematical c o n t e n t o f t h e h e r o ' s thoughts. explains why ( i i )has not become widely known, at l e a s t t o t h o s e accustomed t o t h e e x p o s i t o r y i d e a l s o f t h e s o c i a l s c i e n c e s o r o f academic philosophy d e s c r i b e d l o c . c i t . as ' s o p h i s t i c a t e d and s u s t a i n e d d i s c u s s i o n s , ' and by o t h e r s as pret e n t i o u s and longwinded d i s q u i s i t i o n s . Be t h a t as it may,such r e a d e r s w i l l simply not n o t i c e t h e r e l e v a n t p o i n t s i n ( i i ) ,formulated w i t h extreme d i s c r e t i o n a s i f t h e a u t h o r s were a f r a i d o f a l l t h e wrong people clapping at t h e wrong p l a c e s (as S a l i n g e r p u t it i n The Catcher i n t h e Rye). I n c i d e n t a l l y , p. 170 l o c . c i t . a l s o shows t h a t t h i s danger e x i s t s . (b)

Model t h e o r y : a c l o s e r look at i t s a p p l i c a t i o n s .

No doubt t h e r e a r e some

simple examples which i n v o l v e no more t h a n a p e r f e c t l y obvious s u b s t i t u t i o n i n some g e n e r a l metatheorem.

But i n most s u c c e s s f u l a p p l i c a t i o n s , g e n e r a l know-

ledge o f t h e a r e a , and s k i l l f u l d i s s e c t i o n o f informal p r o o f s were needed i n addition. Reminders.

( i ) Key p r o p e r t i e s o f a given informal proof a r e o f t e n formulated

i n higher o r d e r form, and r e c o n d i t e mathematical arguments =mentioned

in the

g i v e n proof a r e used t o f i n d an e q u i v a l e n t s e t of f i r s t o r d e r p r o p e r t i e s ; c f . group t h e o r e t i c a l arguments i n t h e a n a l y s i s o f Gromov's theorem by van den Dries and Wilkie ( A ) .

( i i ) Often it i s n e c e s s a r y t o modify a given proof ( i f model

t h e o r y i s a p p l i e d t o c r u c i a l lemmas) o r even t h e theorem i t s e l f i f p r o p e r t i e s which a r e demonstrably not of f i r s t o r d e r are involved.

An example occurs i n

t h e next s e c t i o n i n connection w i t h sums of s q u a r e s where it was necessary t o guess t h a t A r t i n ' s theorem on archimedean o r d e r i n g s , n o t a f i r s t o r d e r n o t i o n , could b e g e n e r a l i z e d (by modifying t h e old d e f i n i t i o n of: p o s i t i v e semi-definite). (iii) Depending on p e r s o n a l experience and i n t e r e s t r e a d e r s may wish t o r e c a l l now or l a t e r

-

analogues o u t s i d e mathematics, t h a t i s , t h e a d d i t i o n a l s k i l l s

needed f o r an e f f e c t i v e use o f t h e o r y beyond knowledge o f i t s formal p r o p e r t i e s .

-

234

G. KREISEL, A. MACINTYRE

Additional reassurances f o r l o g i c a l l y fastidious readers repelledby current proof theory because of i t s (continued, if t a c i t ) pretensions ; above a l l , the 'universal' scheme of calibrating formal theories by socalled canonical orderings, used originally by Gentzen i n t h e p a r t i c u l a r case of formal arithmetic. The whole subject seems suspect if it 'permits' t h i s s o r t of thing. F o r reassurance: there w a s a comparable episode i n t h e model theory of t h e f i f t i e s which, l i k e many f a i l u r e s of the past, i s (barely mentioned i n h i s t o r i e s , and) by now halfforgotten: T a r s k i ' s theory of theories, r e l a t i n g t h e syntactic form of axioms t o algebraic properties of t h e corresponding class of models. One o f us even toyed with the obvious analogue: relations between 'logical' r e f l e c t i o n principles and 'mathematical' t r a n s f i n i t e induction. - Ironically, there were equally pedantic discussions about the exact meaning of 'algebraic' and of 'canonical' (definitions Of Orderings Of the order type involved) ; i r o n i c since the r e s u l t s obtained were patently pointless i n most p a r t i c u l a r cases where there was no doubt about the meaning a t a l l . - The episode i s reassuring i n our v i e w . Quite soon sensible people dropped t h i s s o r t of caricature of a genuinely general theory, and retained a few special cases of special i n t e r e s t ; c f . t h e recent use i n van den Dries ( A ) of Lyndon's interpolation theorem, o r i g i n a l l y intended f o r Tarski's theory of theories. (Perhaps some o f t h o s e people got good practice with ultraproducts when using them t o establish algebraic equival e n t s t o t h e notion of f i r s t order formula.)

-

Broad conclusions on applying t h e n.c.i.

The principal additional s k i l l s needed

a r e , of course, t o spot informal proofs f o r which t h e n.c.i.

i s rewarding at

all, and, more s p e c i f i c a l l y , t o spot t h e , often small, parts of such proofs

which need attention.

Also one w i l l look f o r t h e kind o f trivial change i n t h e

proof, already mentioned ( f o r example, i n ( b ) i n t h e l a s t section) which can greatly simplify t h e use of t h e n.c.i. Another snag ( t o avoid): Though the n.c.i. i s not as 'universal' as t h e business of calibration above, i t i s not p a r t i c u l a r l y geared t o subject matter e i t h e r . Yet it aims t o extract bounds which a r e useful only i f they depend on approp r i a t e parameters. The proper choice o f parameters i s known t o be almost as delicate as t h a t of generalizations i n ( a ) above. Moreover, even f o r a given s u i t a b l e choice the r e l a t i v e weight of parameters i s usually of greater i n t e r e s t than t h e location o f t h e bounds i n some 'logical' hierarchy as a function of a l l t h e parameters i n sight; c f . the parameters c and k i n ( a ) of t h e l a s t section, or t h e number of variables and degree o f polynomials i n ( a ) of t h e next. Consequently, one cannot expect general metatheorems ( i n terms of l o g i c a l classific&ions) which a r e comparably useful t o t h e kind of mathematics i n t h e next section or t h e Appendix. However, once the unwinding by means of the n.c.i. has actually been carried out, t h e s i g n i f i c a n t parameters may s t a r e one i n t h e face; cf. footnote 2 on p. 139 of Suppes (1981) where lower bounds on L ( 1 , x ) are r e l a t e d t o t h e location of zeros o f t h e c - h c t i o n , though the l a t t e r a r e Reminder. Often t h e use of not mentioned i n t h e theorem s t a t e d origindlly. general theory i s even less d i r e c t , requiring a completely fresh start t o work out concretely a r e s u l t o f t h e form assured by t h e theory, f o r example, i n some cases of quantifier elimination.

-

D i a e s s i o n on deceptive terminology: proof theory and model theory. A s these words are used i n l o g i c , they do l i t t l e towards t h e aims ordinarily associated with them: finding proofs of given theorems, and models of some given object, say, t h e r e a l s . (NB. Ordinarily, one speaks of models of objects, retaining the e s s e n t i a l features of t h e l a t t e r , not: models of axioms expressing those features.) Logical proof theory t e l l s us what we gain i f some given proof happens t o have a r e s t r i c t e d syntactic form; f o r example, f o r unwinding i f a l l t h e heavy machinery i s used f o r l a m a s . Logical model theory s p e l l s out t h e gains, f o r example, f o r ultraproduct manipulations, i f t h e choice of modelling, t h e axioms, can be defined by f i r s t order formulas ( o r refinements such as

23 5

Constructive logic versus algebraization

p o s i t i v e o n e s ) . Without exaggeration: axiomatic a n a l y s i s p r a c t i s e d i n mathematics i s much c l o s e r t o t h e o r d i n a r y meanings o f 'proof t h e o r y ' and 'model t h e o r y ' as a l r e a d y explained above i n connection w i t h proof s t r a t e g i e s i m p l i c i t i n axiomatic p r e s e n t a t i o n s , f o r example, i n Cartan (1967). S i m i l a r l y , t h e l a t t e r would be s a i d t o propose t h e use o f v a r i o u s Banach spaces a s models o f t h e r e a l s f o r t h e purposes considered t h e r e . Other d e c e p t i v e terminology i n t h e l o g i c a l l i t e r a t u r e w i l l come up i n b ( a ) i n t h e next s e c t i o n on 'computation theory. ALGEBEUIZATION: REMINDERS AND SOME SURPRISES Algebraization

-

o f theorems, not n e c e s s a r i l y o f proofs!

ometry o r topology, logic

-

is

one o f

emphasized i n t h e l a s t s e c t i o n .

structive part

-

as i n a l g e b r a i c ge-

t h o s e g e n e r a l mathematical s t r a t e g i e s

o f mathematics.

-

outside

It i s p a r t i c u l a r l y u s e f u l i n t h e con-

Other s t r a t e g i e s which w i l l come up i n p a s s i n g

below a r e known from t h e ( d i s c r e e t ) mottoes: e n r i c h , r e s p . e n l a r g e your category. Reminders (from t h e i n t r o d u c t i o n ) . Once g e n e r a l mathematical s t r a t e g i e s a r e recognized a s b e i n g i n t h e running a t a l l , as comparable t o formal metatheorems, they w i l l e a s i l y come t o t h e r e a d e r ' s mind; n a t u r a l l y , s o m e background i n math(Otherwise one cannot hope - nor, probably, ematical reasoning i s assumed. even want - t o have a balanced view of mathematics, a philosophy, a s always, i n t h e sense o f Schopenhauer.) S i m i l a r l y , t h e s t y l e of e x p o s i t i o n below w i l l not be d i c t a t e d by any p a r t i c u l a r methodology, b u t simply by our purpose: t o convey, e f f i c i e n t l y and r e l i a b l y , where and how a l g e b r a i z a t i o n i s r e l e v a n t , f o r example, by memorable examples. This i s i n keeping w i t h t h e s t r e s s on ' a d d i t i o n a l s k i l l s ' ( a l s o i n t h e l a s t s e c t i o n ) : t h o s e needed f o r a p p l y i n g exact g e n e r a l p r i n c i p l e s s u c c e s s f u l l y seem q u i t e s i m i l a r t o t h o s e involved i n s e e i n g such p r i n c i p l e s i n a particular instance. Our two p r i n c i p a l examples a r e : ( a ) A r t i n ' s work on sums o f s q u a r e s , almost at t h e dawn o f modern a l g e b r a i z a t i o n , b u t s t i l l of i n t e r e s t , and ( b ) t h e e v o l u t i o n of t o o l s from group t h e o r y and a l g e b r a i c geometry o v e r t h e l a s t h a l f century t o give c o n s t r u c t i v e v e r s i o n s of v a r i o u s i m p l i c a t i o n s between f i n i t e n e s s theorems i n number t h e o r y needed f o r S i e g e l ' s theorem on d i o p h a n t i n e equations. amples i l l u s t r a t e a remarkable v a r i e t y o f g e n e r a l p o i n t s . Reminders.

Both ex-

( i ) Algebra provides computable s u b s t i t u t e s f o r b r u t a l completions

and c o m p a c t i f i c a t i o n s i n t r o d u c e d

-

by another g e n e r a l s t r a t e g y !

-

t o apply

t o p o l o g i c a l t o o l s ( l i m i t arguments, d u a l i t y theorems); u s u a l l y , by f i n d i n g minimal c l o s u r e c o n d i t i o n s , as i n t h e passage from Dedekind c u t s i n lR

t o real

c l o s u r e ( t h e a l g e b r a i c r e a l numbers, i n semi-algebraic topology) o r i n henseli z a t i o n o f complete n o e t h e r i a n r i n g s ; c f . a l s o b ( a ) below on t h e choice o f suitable presentations.

( i i ) More g e n e r a l l y , a s i m i l a r t r a n s f e r o f n o t i o n s t o

t h a t i n ( i ) p e r m i t s t h e a l g e b r a i c ( o r t o p o l o g i c a l ) i n t e r p r e t a t i o n o f computat i o n s ; f o r example, n o t i o n s from t h e t h e o r y o f ( f i n i t e ) f i e l d s i n t h e c a s e o f computations

mod p

c f . Chapter 2 o f

( f o r primes

[MI).

p

,a

stand-by i n t r a d i t i o n a l number t h e o r y ;

S t r i c t l y speaking, t h e a l g e b r a o f polynomials could b e

c i t e d h e r e , and i t s use f o r s t u d y i n g continuous and e s p e c i a l l y a n a l y t i c functions.

But a l l t h i s i s so p e r v a s i v e t h a t it goes a g a i n s t t h e g r a i n t o t h i n k of

it a s a p a r t i c u l a r s t r a t e g y .

(iii) A s f o r l i t e r a t u r e c o n t a i n i n g advice on t h e

G . KREISEL, A. MACINTYRE

236

good use o f ( i )and ( i i ) ,it i s b e s t t o look a t e s s a y reviews or l e c t u r e s t o

(1966), r a t h e r t h a n a t t e x t s ; c f . a l s o a ( i i i ) of

l e a r n e d s o c i e t i e s , l i k e Atiyah the last section.

(iv)

A s f o r s u r p r i s e s , probably t h e most famous r e c e n t

example i s t h e success o f a l g e b r a i c geometry f o r t r e a t i n g t h e e q u a t i o n s of Yang-

M i l l s though t h e p o s s i b i l i t y of something of t h i s s o r t i s q u i t e i n l i n e w i t h , say, pp. 66-68 of Atiyah (1966).

Again, l o g i c i a n s a r e sometimes s u r p r i s e d how

much high-brow a l g e b r a i s o c c a s i o n a l l y needed t o s o l v e an innocent looking dec i s i o n problem; c f . Grunewald and Segal (1979) who e x p l o i t a g e n e r a l s t r u c t u r e t h e o r y o f a r i t h m e t i c groups, presumably not developed p r i m a r i l y for a l g o r i t h m i c But t h e n , a f t e r a l l , knowing t h e a l g e b r a i c s t r u c t u r e of t h e ' s o l u t i o n

purposes.

-

Less t r i v i a l l y , for c o n s t r u c t i v e

refinements it pays t o e l i m i n a t e non-algebraic

c o n d i t i o n s l i k e well-foundedness

s e t ' should h e l p one decide i f it i s empty.

o r archimedean o r d e r i n g s from p r o o f s (about o b j e c t s which s a t i s f y t h e s e conFor t h e n one i s not even tempted t o d e f i n e s o l u t i o n s by use of t h o s e

ditions).

c o n d i t i o n s ; f o r example, d e f i n i t i o n s by r e c u r s i o n o r a r b i t r a r y f i n i t e decompos i t i o n s l e a d , almost i n e v i t a b l y , t o complicated s o l u t i o n s ; s p e c i f i c a l l y , funct i o n s d e f i n e d by r e c u r s i o n w i l l be h a r d t o compute a t arguments of h i g h rank.

-

F i n a l l y , f o r a s u r p r i s e on a q u i t e s u p e r f i c i a l l e v e l , r e c a l l t h e s u b t i t l e o f Cartan (1967), a l r e a d y d i s c u s s e d i n ( a ) of t h e l a s t s e c t i o n .

Incidentally,

even v i s u a l l y t h e t e x t has an a l g e b r a i c look s i n c e it c o n s i s t s o f e q u a t i o n s w i t h b u t few o p e r a t i o n s i n p l a c e of t h e usual i n e q u a l i t i e s and t h e l o g i c a l

€,6

Warning a g a i n s t r e g a r d i n g t h i s a s p e c t as a s p e c i a l c a s e of t h e t r i v -

symbolism.

i a l i t y t h a t t h e l o g i c a l symbolism can always be e l i m i n a t e d by i n t r o d u c i n g s u f f i c i e n t l y many Skolem o p e r a t i o n s .

This overlooks two genuine d i s c o v e r i e s .

Few

such o p e r a t i o n s t u r n e d o u t t o be enough f o r t h e whole c o u r s e , without any p a r t i c u l a r l y a r t i f i c i a l r e s t r i c t i o n on t h e m a t e r i a l , and t h e o p e r a t i o n s chosen have a d d i t i o n a l (mathematical, f o r example, c o n t i n u i t y ) p r o p e r t i e s r e l e v a n t t o t h e subject.

NB.

The e l i m i n a b i l i t y r e f e r r e d t o i s t r i v i a l not because it i s easy

t o establish but

-

t o r e p e a t a w e l l worn p h r a s e

away from t h o s e d i s c o v e r i e s .

-

because it draws a t t e n t i o n

A r e l a t e d m a t t e r comes up i n t h e next paragraph.

A l o E i c a l i s s u e concerning a l g e b r a i z a t i o n , when r e l a t i o n s

R

a r e defined on some

'complicated' o b j e c t , proved t o s a t i s f y ' s i m p l e ' a l g e b r a i c l a w s , and used t o answer q u e s t i o n s about t h a t o b j e c t from experience w i t h o t h e r s t r u c t u r e s obeying those l a w s .

Logic

could come

i n i f those proofs required

strong logical, i n particular, set-theoretical principles.

-

or simply used!

-

This i s not so i n

t h e examples ( a ) and ( b ) below. Reminders. ( i ) The most famous successes o f a l g e b r a i z a t i o n - i n v o l v i n g embeddings o f ZZ i n ordered, a l g e b r a i c a l l y c l o s e d o r p-adic f i e l d s , and t h e l i k e have nothing t o do w i t h t h e p o s s i b i l i t y a t i s s u e . The choice between t h o s e f i e l d s i s c r i t i c a l ; b u t t h e i r formal t h e o r i e s a r e s o elementary t h a t t h e y a r e a l l lumped t o g e t h e r a t t h e lowest l e v e l o f any p r o o f - t h e o r e t i c c l a s s i f i c a t i o n (remotely l i k e t h o s e considered i n t h e c u r r e n t l i t e r a t u r e ) . One reason for b r i n g i n g up t h e i s s u e a t a l l i s i t s obvious r e l a t i o n t o a c u r r e n t l y popular

-

231

Constructive logic versus algebraization

t o p i c : o f examining t h e formal need f o r s e t - t h e o r e t i c d i s t i n c t i o n s , between c o n s t r u c t i b l e and o t h e r s e t s , t h o s e o f low or high ( i n f i n i t e ) c a r d i n a l i t y , t o s e t t l e problems about groups o r Bore1 s e t s . ( i i ) Contrary t o an almost univers a l o v e r s i g h t , t h e need mentioned does not determine t h e p l a c e o f s e t - t h e o r e t i c d i s t i n c t i o n s . One has t o compare t h e i r r o l e w i t h t h e o t h e r i n g r e d i e n t s used i n t h e proofs involved, and, more d e l i c a t e l y , one has t o s e e what would b e l o s t ( f o r some given branches o r c o r n e r s o f mathematics) i f t h e problems mentioned were t o be ignored a l t o g e t h e r . The same a p p l i e s t o t h e l o g i c a l i s s u e above. I n s h o r t , t h e c a s e f o r a l o g i c a l a n a l y s i s of s u c c e s s f u l a l g e b r a i z a t i o n may be more convincingly r e f u t e d by l o o k i n g a t examples where s t r o n g l o g i c a l p r i n c i p l e s needed ( b u t seen t o b e , once a g a i n , t h e l e a s t i n t e r e s t i n g s i d e o f t h e m a t t e r ) .

are

(a)

Sums o f squares (SOS, f o r s h o r t ) ; f o r f u r t h e r background, b i b l i o g r a p h y and

o t h e r d e t a i l s , c f . t h e papers o f C. N. D e l z e l l i n t h i s volume and i n t h e f o r t h coming Herbrand volume.

The t o p i c i s f a m i l i a r from H i l b e r t ' s 1 7 t h problem con-

cerned w i t h a n a t u r a l normal form f o r 'showing' t h a t a polynomial semi-definite,

psd

is positive

p

f o r s h o r t ; c f . a l s o H i l b e r t ' s N u l l s t e l l e n s a t z which provides

a normal form f o r 'showing' t h a t a given polynomial vanishes a t a l l t h e common z e r o e s o f a g i v e n bunch o f polynomials.

More r e c e n t l y , D e l z e l l , van den D r i e s ( A )

and o t h e r s e s t a b l i s h e d a normal form proposed by Brumfiel f o r showing t h a t a semi-algebraic

s e t i s c l o s e d ( o r open).

Without having been o f much d i r e c t use,

work on some o f t h e s e p r e t t y normal forms i l l u s t r a t e s e x c e p t i o n a l l y w e l l u s e f u l general s t r a t e g i e s .

One o f them, i n t h e c a s e o f SOS, i s t h e s e l e c t i o n o f

i n a n t parameters among: t h e c h a r a c t e r o f t h e c o e f f i c i e n t f i e l d n

o f v a r i a b l e s and t h e degree o f

-choice

( p E K[xl,.

.., x n ] ) .

for

p c c.p.

,if

1 1

c. > 0 1 -

p

EK[xl

,...,x

1

or

K ( 5

,...,xn)

, and

I+

pi

if

similarly of:

i sums o f weighted s q u a r e s a r e considered, t h a t i s ,

.

e-

number

Another i s t h e proper

o f category f o r t h e s o l u t i o n , t h a t i s , o f t h e t y p e o f maps: p

p = cpi2

where

p

, the

K

p = Ccipi

2

A t h i r d s t r a t e g y concerns t h e p o s s i b l e need f o r e n r i c h i n g

d a t a , f o r example, by coding t h e f a c t t h a t

p

is

psd.

, the

For good measure, s i n c e

both model-theoretic and p r o o f - t h e o r e t i c metamathematical methods have been a p p l i e d t o t h e problem one a l s o g e t s an i d e a o f t h e i r c o n t r i b u t i o n s ; r e l a t i v e t o each o t h e r , and t o more a l g e b r a i c o r number-theoretic a n a l y s e s . ( i ) Back i n 1888 H i l b e r t Some d e t a i l s , documented i n f u l l i n D e l z e l l ' s papers. showed t h a t a some psd q u a r t i c ERCx, .x2.x31 w a s not a sum of s q u a r e s (SOS) of polynomials. R. M. Robinson w a s a b l e t o use t h e p u r e l y a l g e b r a i c c h a r a c t e r of ( a n a s i d e i n ) H i l b e r t ' s argument f o r a g e n e r a l method t o d e c i d e , f o r any x 1 , and an a r b i t r a r y r e a l c l o s e d f i e l d K , whether p i s (psd, p E K[xl, ( i i ) The a l g e b r a i c c h a r a c t e r o f A r t i n ' s and a l s o ) a S f ; S of polynomials. 1 f o r a r b i t r a r y ordered K (and r e s u l t , modified t o apply t o p E K[xl , . . . r ~ p b e i n g r e q u i r e d t o b e psd i n t h e real c l g s u r e o f K ) , ensures a r e p r e s e n t a t i o n as a SOS independent o f t h e c h a r a c t e r of K This i s a c o r o l l a r y o f (iii) More mathematical work o f Landau and model t h e o r y o r r e l a t e d 'nonsense.' and o f P f i s t e r f o r r e a l c l o s e d K , g i v e s q u a l i Pourchet on number f i e l d s K t a t i v e l y b e t t e r bounds f o r t h e number ( n o t t h e d e g r e e s ) o f t h e s q u a r e s used t h a n ( i v ) A proofanything known under t h e g e n e r a l c o n d i t i o n s i n ( i i )above. t h e o r e t i c a n a l y s i s o f A r t i n ' s argument by s t a n d a r d metamathematics o f c l a s s i c a l systems ( a c t u a l l y s i m p l i f i e s t h e l a t t e r , and) d e r i v e s q u a n t i t a t i v e l y b e t t e r bounds under c o n d i t i o n s ( i i )t h a n t h e g e n e r a l r e c u r s i v e ones ensured by model

...,

.

,

G. KREISEL, A. MACINTYRE

238

theory. It has a l s o suggested a d i f f e r e n t argument r e l y i n g on t h e g e n e r a l quanti f i e r e l i m i n a t i o n method f o r r e a l c l o s e d f i e l d s . It i s s t i l l open whether improvements i n t h e p r o o f - t h e o r e t i c a n a l y s i s could b e rewarding. They a r e not i f t h e (known) bounds a r e swamped by t h o s e f o r t h e a l g e b r a i c d e s c e n t . ( S p e c i f i c -

l,...,~

:

E K'(x g ) f o r & r o o t a" o f p = CG2 f o r s u i t a b l e n' some polynomial o f odd degree over K' Then p i s a l r e a d y a SOS i n ) , b u t g e n e r a l l y more complicated. The o t h e r c a s e , o f a d j u n c t i o n s K'(x1,..,,x

a l l y , suppose

.

& and f o r a E K' , i s not so complicated.) ( v ) For an o r d i n a r y cons t r u c t i v e s o l u t i o n o f H i l b e r t ' s 1 7 t h problem over such undecidable ground f i e l d s a s R , each summand must b e , roughly speaking, continuous b o t h i n t h e v a r i a b l e s (Here, ' o r d i n a r y ' means: without enrichment of and t h e c o e f f i c i e n t s o f p d a t a . ) More p r e c i s e l y , t h e r e i s a t o p o l o g i c a l v e r s i o n , i n v o l v i n g b o t h a choice o f topology and o f t h e c a t e g o r y of r e p r e s e n t a t i o n s , t h a t i s , maps: p c-+ pi

.

.

( v i ) Stengle's algebraic

p o s i t i v s t e l l e n s a t z i s i n f a c t a normal form 'showing' t h a t t h e summands can b e chosen so t h a t denominators vanish only a t t h e zeroes of p This t a k e s c a r e o f c o n t i n u i t y i n t h e v a r i a b l e s xl, x (vii)

..., .

.

D e l z e l l ' s papers give s o l u t i o n s (and counterexamples) i n a wide c l a s s o f c a s e s f o r t h e proper c a t e g o r i e s of s o l u t i o n s ; semi-algebraic or even r a t i o n a l i n t h e c a s e of q u a d r a t i c polynomials t r e a t e d i n t h i s volume. The methods and n o t i o n s come from mainstream mathematics, I n t e r e s t i n c o n s t r u c t i v e l o g i c h a s , perhaps, been u s e f u l i n drawing a t t e n t i o n t o t h e t o p o l o g i c a l v e r s i o n ; i n c o n t r a s t , i n ( i v ) above, metamathematics o f c l a s s i c a l systems a c t u a l l y provided t h e e a s i e s t constructive solutions. Enriching d a t a and Bishop ( 1 9 6 7 ) ; a t r i v i a l example. The high school r e p r e s e n t 2 a t i o n s o f a x + 2bxy + cy2 by 'completing t h e s q u a r e ' a r e not continuous i n t h e b a r e d a t a ( a . b , c ) ; c f . D e l z e l l ' s paper i n t h i s volume. To be p s d , a+c 2 0, 2 5 ac , and so e i t h e r b2 5 a* o r b 2 5 c 2 ( o r b o t h ) , indexed by 1 , r e s p .

b

2.

Require t h e d a t a t o be e n r i c h e d t o

language:

w i t h t h e d i s c r e t e topology on

(a,b,c,i) i

.

,i

= 1 or

Adding

i = 2 ; i n fancy

i s a genuine enrichment

i

s i n c e it i s not determined continuously nor uniquely by

(a,b,c)

.

For t h e new

d a t a t h e r e i s an obvious t o p o l o g i c a l s o l u t i o n : i f i = 1 : a ( x + y b / a ) 2 + 2 2 (c-b / a ) y 2 , and i f i = 2 : (a-b / c ) x 2 + c ( y + x b / c ) 2 - I n medieval language,

.

t h e s o l u t i o n i s ' i n t e n s i o n a l ' because it depends not o n l y on

on t h e p a r t i c u l a r

i

.

Seen i n t h i s way t h e a d d i t i o n of

(a,b,c)

, but

also

is parallel t o the

i

most s u c c e s s f u l p a r t s of Bishop (1967), a t l e a s t , a s formulated on p. 124 o f Browder ( 1 9 7 6 ) ; viz.:

by e n r i c h i n g t h e d a t a , f u n c t i o n a l equations can b e s o l v e d

by simpler o p e r a t i o n s .

, Bishop

for R

For example, i n t h e c a s e of o r d i n a r y continuous f u n c t i o n s

adds t o t h e graph ( a modulus of c o n t i n u i t y and) a d d i t i o n a l d a t a

which make l o c a l b i s e c t i o n arguments e f f e c t i v e : a map Ix'-xI

< n

-1

and

f(x') = 0

i s decidable.

(x,n)

+

x'

where

This enrichment i s b o t h necessary

f o r i t s purpose and o f t e n a v a i l a b l e i n p r a c t i c e , w h i l e

i

above i s not ' a v a i l -

a b l e , ' i n t h e sense o f not b e i n g r e c u r s i v e i n ( s a y , p r i m i t i v e r e c u r s i v e r e a l s ) a,b,c

.

Besides, by D e l z e l l ' s paper i n t h i s volume, no enrichment a t a l l i s

needed f o r more i n t e l l i g e n t r e p r e s e n t a t i o n s of convex combinations.

-

ax2 + 2bxy + cy

by s u i t a b l e

Other simple enrichments reduce t h e t o p o l o g i c a l v e r s i o n

239

Constructive logic versus algebraization

( v ) t r i v i a l l y t o t h e piece-wise r a t i o n a l s o l u t i o n o f H i l b e r t ' s 1 7 t h problem i n Theorem 2 . 1 o f D e l z e l l ( A ) . B i b l i o g r a p h i c a l remarks. By and l a r g e l o g i c i a n s a r e not aware j u s t how promi n e n t enrichments a r e i n mathematics (and emphasized by b e i n g given a name!). Viewed as an e x p o s i t i o n o f - t h e s t r a t e g y o f - enrichments Bishop (1967) could be most e f f e c t i v e f o r showing how g e n e r a l mathematical s t r a t e g i e s s e r v e trad i t i o n a l l o g i c a l ( h e r e : c o n s t r u c t i v e ) aims. C l e a r l y , t h i s was not t h e i n t e n t i o n , f o r example, o f t h e embarrassing i n t r o d u c t i o n , nor t h e e f f e c t on most l o g i c a l readers. I n s t e a d t h e y looked f o r a 'new' foundation, aping t h e d i s c r e d i t e d analyses o f axiomatic s e t theory (mentioned i n a ( i i ) o f t h e l a s t s e c t i o n ) , and worried, f o r example, whether Bishop (1967)used t h e p r i n c i p l e o f dependent choice! Another, even l e s s p e r c e p t i v e , r e a c t i o n was t o s e e enrichments a s mere padding o r b a l l a s t , not r e c o g n i z i n g t h e e s s e n t i a l knowledge contained i n ( a s k i l l f u l choice o f ) enrichments, even a f t e r t h e l a t t e r had p u t t h a t knowledge before our eyes. (b)

0

If;

Binary diophantine e q u a t i o n s ; c f . t h e s e c t i o n on

-+

113

f o r background,

and t h e Appendix f o r t h e p r i n c i p a l s t e p s i n t h e e v o l u t i o n o f S i e g e l ' s F i n i t e n e s s Theorem.

A t t e n t i o n focuses on methods o f d e r i v i n g

DA

-+

S

, where

S is

S i e g e l ' s c r i t e r i o n , i n terms o f t h e i r genus, f o r e q u a t i o n s which have only f i n i t e l y many i n t e g r a l s o l u t i o n s , and proximation theorem, b o t h

DA

DA

i s a ( s u i t a b l e ) diophantine ap-

S being

and

even from t h e s t r o n g form o f DA due t o Roth Mordell/Weil

If;

.

The c l a s s i c a l p r o o f s

-

used t h e ( s t r u c t u r e ) theorem of

on a f i n i t e b a s i s f o r t h e s e t o f a l l r a t i o n a l s o l u t i o n s o f t h e

equations considered, generators.

-

Now, no known proof o f t h i s theorem provides a l i s t of

But i f only a bound on t h e number o f g e n e r a t o r s i s a v a i l a b l e , t h e

s t a n d a r d proofs of DA

+

S

a r e simply not c o n s t r u c t i v e .

t h e a l g e b r a i c machinery which ( i )g e n e r a l i z e s b o t h number f i e l d s , b u t a l s o r e s t r i c t s

DA

DA

The Appendix e x p l a i n s

and S

t o arbitrary

t o a b e l i a n among a l l a l g e b r a i c v a r i e t i e s ,

t h u s ( i i )e x t r a c t i n g from t h e old p r o o f s c o n s t r u c t i v e i m p l i c a t i o n s : between d i f f e r e n t versions of

DA

o r of

DA

and S

.

(The most ambitious j o b o f t h i s

kind, sketched i n S e r r e ( 1 9 8 0 ) , i n v o l v e s s t i l l more d e l i c a t e a l g e b r a ; c f . A8.2.)

Apart from t h e s t r a t e R i e s i n ( a ) above, t h e work i l l u s t r a t e s a l s o t h e following g e n e r a l p o i n t s which w i l l be sharpened i n A 9 by r e f e r e n c e t o d e t a i l s i n t h e Appendix. One g e n e r a l p o i n t i s t h e l i m i t a t i o n o f f a m i l i a r c l a s s i f i c a t i o n s of problems, f o r example, by a formal q u a n t i f i e r count o r , i n t h e c a s e of ( b i n a r y ) polynomials, by t h e i r degree.

Another concerns t h e r o l e o f l o g i c a l s t r e n g t h f o r t h e success

of a l g e b r a i z a t i o n r a i s e d j u s t b e f o r e ( a ) , l e a d i n g t o a more convincing a n a l y s i s of t h i s success.

More t e c h n i c a l l y , a l g e b r a i c and geometric i n t e r p r e t a t i o n s a r e

used i n t h e Appendix t o choose a b s t r a c t s t r u c t u r e s s u i t e d f o r e f f i c i e n t comput a t i o n , roughly speaking, by p r o v i d i n g s u i t a b l e p r e s e n t a t i o n s o f t h e r e l e v a n t o b j e c t s ; i n c o n t r a s t t o s o c a l l e d t h e o r i e s o f computation i n r e c u r s i o n t h e o r y which assume t h e s t r u c t u r e s t o b e given; c f . t h e end of t h e l a s t s e c t i o n on deceptive terminology.

240

G. KREISEL, A. MACINTYRE

Documentation. ( i ) By t h e f a m i l i a r formal q u a n t i f i e r count, t h e solved by S i e g e l i s more complicated t h a n t h e

1;

$

problem

q u e s t i o n whether a b i n a r y

equation has any i n t e g r a l s o l u t i o n a t a l l ( which i s s t i l l open except f o r genus < 1); 'formal' because, modulo d e r i v a b i l i t y , S i e g e l ' s proof o f course r e d u c e s t h e q u a n t i f i e r complexity. - Reminder. I n c o n s t r u c t i v e l o g i c t h e two formulations o f f i n i t e n e s s , by bounds on number and s i z e , a r e not e q u i v a l e n t . S i e g e l ' s s o l u t i o n a p p l i e s t o t h e former; s i n c e bounds a r e a v a i l a b l e , t h e n a t u r a l formulation i s i n

0

II

form.

( i i ) For t h e problem s o l v e d by S i e g e l , t h e c l a s -

s i f i c a t i o n by degree i s not merely c o a r s e ( t h a t i s , worth r e f i n i n g ) , b u t i n t h e wrong d i r e c t i o n simply because a s u p e r i o r c l a s s i f i c a t i o n c u t s a c r o s s deKree; s p e c i f i c a l l y , by s u i t a b l e b i r a t i o n a l equivalence which p r e s e r v e s f i n i t e n e s s o f t h e s o l u t i o n s e t (and g e n u s ) , and i s incomparable w i t h degree; c f . Theorem 2 on p. 151 of [MI. It i s a s e p a r a t e m a t t e r whether genus i t s e l f i s a p r i n c i p a l s i g n i f i c a n t measure here. For t h e r e c o r d , we f i n d t h e r e s u l t s i n t h e Appendix r e l a t i n g genus and p r o p e r t i e s o f Jacobians p e r s u a s i v e , b u t do not c o n s i d e r ours e l v e s w e l l informed enough t o judge t h e i r s i g n i f i c a n c e ( f o r a choice o f measure). (iii) A i r y - f a i r y s p e c u l a t i o n s on t h e r o l e of l o g i c a l s t r e n g t h a r e o u t d a t e d because t h e r e a r e s t r i k i n g examples a v a i l a b l e f o r i n s p e c t i o n ; i n p a r t i c u l a r , t h o s e s p e c t a c u l a r a l g e b r a i c i n t e r p r e t a t i o n s o f phenomena which on t h e s u r f a c e seem mere c u r i o s i t i e s (as i n Ramanujan's c o n j e c t u r e ) ; b u t a l s o t h e modern a l g e b r a i c p r o o f , f o r example, i n Chapter 5 o f Lang ( 1 9 6 2 ) , o f t h e theorem of Mordell/Weil g i v e s an i d e a o f p o s s i b l e p r o g r e s s ; most d r a m a t i c a l l y , when compared w i t h t h e computational proof - i n c i d e n t a l l y , r e s t r i c t e d t o genus 1 i n Chapter 16 of [MI; f o r a more r e a l i s t i c comparison, c f . t h e proof on pp. 265-266 (and p. 258) o f C a s s e l s (1966), and t h e a n a l y s i s of corresponding 'computational' arguments on pp. 269-272. The choice o f a p p r o p r i a t e a l g e b r a i c s t r u c t u r e s , i n p a r t i c u l a r , a b e l i a n v a r i e t i e s , cohomology groups, i s c r i t i c a l ; not t h e means o f proof used t o e s t a b l i s h t h e i r r e l e v a n t p r o p e r t i e s . Exaggera t i n g very l i t t l e : t h e s t e p s of t h e computational proof ( o r some l i k e them) are used t o e s t a b l i s h and a t t h e same time t h e y a r e i n t e r p r e t e d by - t h o s e algeb r a i c p r o p e r t i e s . Digression on a n a l y t i c number t h e o r y (under t h e slogan: h i s t o r y r e p e a t s i t s e l f ) . I n t h e t w e n t i e s , people toyed w i t h t h e i d e a t h a t i t s s u c c e s s depended on t h e use o f - w h a t we s h o u l d now c a l l - s e t - t h e o r e t i c p r o p e r t i e s of E' A t t h a t time Gadel's incompleteness r e s u l t s could have given some supp o r t t o t h i s idea. For t h e y showed t h a t metamathematical c o n s i s t e n c y propert i e s , q u i t e e v i d e n t from c e r t a i n s e t - t h e o r e t i c a l p r i n c i p l e s , a c t u a l l y r e q u i r e d t h e l a t t e r f o r a formal proof. This w a s unexpected a t t h e time (though n o t 50 y e a r s e a r l i e r when Cantor compared h i g h e r set t h e o r y w i t h a r i t h m e t i c , t a k i n g it f o r g r a n t e d t h a t t h e r e w a s no rewarding r e d u c t i o n o f a r i t h m e t i c e i t h e r ) . I n t h e t h i r t i e s p e r c e p t i v e l o g i c i a n s l i k e Gentzen were s k e p t i c a l of t h e u s e o f s t r o n g p r i n c i p l e s i n f u n c t i o n t h e o r y , and by 1950 h i s impression had been v e r i f i e d beyond any shadow o f a doubt. During t h e l a s t decade l o g i c i a n s have d o t t e d t h e i ' s and c r o s s e d t h e t ' s , w h i t t l i n g down f u r t h e r t h e elementary p r i n c i p l e s which a r e obviously s u f f i c i e n t f o r ( t h e s t a n d a r d p a r t s o f ) f u n c t i o n t h e o r y . Ironi c a l l y , b u t by no means u n t y p i c a l l y , t h i s a c t i v i t y threw up a l o t o f d u s t as it were, obscuring t h e e s s e n t i a l f a c t s w i t h f i l i g r e e pedantry. Some o f t h e s e f a c t s w i l l b e s p e l t o u t i n ( i v ) below. This seems worth doing, e s p e c i a l l y s i n c e alg e b r a i z a t i o n i s t h e n seen t o b e a refinement o f t h e a n a l y t i c s t r a t e g i e s . ( i v ) The 'space' - or s p e c i e s i n i n t u i t i o n i s t i c j a r g o n o f f u n c t i o n s of a complex v a r i a b l e has t u r n e d o u t t o l e n d i t s e l f t o a remarkably manageable t h e o r y . Thus simple p r o p e r t i e s o f a f u n c t i o n i n one r e g i o n o f LC determine a l o t o f i t s behaviour elsewhere; f o r example, t h e l o c a t i o n of p o l e s and t h e maximum modulus o f f t h e r e a l axis determine t h e behaviour ( o f power s e r i e s expansions, z e r o e s , e t c . ) on it. There i s a whole a r s e n a l o f d i f f e r e n t d e s c r i p t i o n s o f t h e same function i n fancy language: r e s t r u c t u r i n g p a r t i c u l a r l y w e l l s u i t e d t o record ( a n d make e v i d e n t ) t h o s e simple p r o p e r t i e s . All t h i s knowledge can now b e brought t o b e a r on number-theoretic problems, e v e r s i n c e Euler s p o t t e d ( a n d o t h e r s e l a b o r a t e d ) how t o 'embed' t h e l a t t e r i n f u n c t i o n t h e o r y . Next, t h e k i n d

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24 1

Constructive logic versus algebraization

of a l g e b r a i z a t i o n expounded i n t h e Appendix e x p l o i t s t h e p a r t i c u l a r l y simple c h a r a c t e r of diophantine e q u a t i o n s . Curves d e f i n e d by them ( t a c i t l y , i n a s u i t a b l e f i e l d ) a r e - t h i s time: l i t e r a l l y - e m b e d d e d i n s u i t a b l y chosen a l g e b r a i c v a r i e t i e s . I n g e n e r a l , t h e y a r e even s i m p l e r and b e t t e r understood o b j e c t s t h a n many o f t h e functions used i n a n a l y t i c number t h e o r y , b e s i d e s b e i n g something new t o t r y out. - Needless t o say, t h e g e n e r a l s t r a t e g i e s of u s i n g f u n c t i o n t h e o r y o r a l g e b r a i z a t i o n l e a v e p l e n t y o f room f o r t h e k i n d of a d d i t i o n a l s k i l l s so promi n e n t i n t h e l a s t s e c t i o n , f o r example, i n t h e choices o f s u i t a b l e embeddings and ' t i g h t ' r e s t r u c t u r i n g . ( a ) R e c a l l from t h e remark a t t h e end o f t h e l a s t s e c t i o n Technical p o i n t s . on deceptive terminology t h a t t h e g e n e r a l s t r a t e g i e s o f mathematics c o n t r i b u t e more t o a proof t h e o r y and a model t h e o r y i n t h e popular sense o f t h e s e words than t h e branches of l o g i c w i t h t h e s e names. Much t h e same a p p l i e s t o t h e p a r t of r e c u r s i o n t h e o r y c a l l e d ' t h e o r y o f computation on a b s t r a c t s t r u c t u r e s . ' The l a t t e r t e l l s us what we g a i n if t h e induced r e l a t i o n s a r e r e c u r s i v e , b u t not o f t e n which s t r u c t u r e s t o choose. The a l g e b r a i c l i t e r a t u r e does t h i s much b e t t e r provided, a s always, i t s r e l e v a n c e i s recognized a t a l l ; c f . 58.2 of t h e Appendix where i n t e r p r e t a t i o n s corning from a l g e b r a i c geometry a r e needed t o + I I; theorems. choose e f f i c i e n t p r e s e n t a t i o n s of Selmer's group. ( B )

$

The d i f f e r e n c e between p r a c t i c a l l y useful and i n t u i t i o n i s t i c meanings o f such i m p l i c a t i o n s i s w e l l i l l u s t r a t e d i n 18.1 o f t h e Appendix. There two p r o o f s a r e considered f o r d e r i v i n g S i e g e l ' s theorem f o r b i n a r y diophantine equations i n one number f i e l d K from Roth's theorem assumed f o r a r b i t r a r y K' ( 3 K) The s t a n d a r d proof shows e x p l i c i t l y how ( i )t h e r e l e v a n t e x t e n s i o n K' and t h e a l g e b r a i c numbers w i t h e x c e p t i o n a l l y c l o s e K ' - r a t i o n a l approximations ( o f l a r g e h e i g h t ) a r e got from ( i i ) K and t h e e q u a t i o n under c o n s i d e r a t i o n . The non-standard p r o o f , r e f e r r e d t o a t t h e end o f Robinson and Roquette (1975), does not. Recursion t h e o r i s t s w i l l recognize h e r e t h e familiar weakness, f o r 'posi t i v e ' r e s u l t s , o f - p r i m i t i v e o r general! - r e l a t i v e r e c u r s i v e n e s s compared t o many-one r e d u c i b i l i t y .

.

A p o s i t i v e c o r o l l a r y t o ( i i i ) above.

If a l g e b r a i c o r t o p o l o g i c a l i n t e r p r e -

t a t i o n s of numerical computations can do m i r a c l e s , why not t r y t h i s s t r a t e g y on formal computations w i t h Heyting's r u l e s ?

This might s e r v e as a crude s a l e s

p i t c h f o r t h e sheaf model i n d u s t r y r e p r e s e n t e d i n t h i s volume. refined, a t various l e v e l s of generality.

I t can even b e

Most t r i v i a l l y , we a l r e a d y have

p l e n t y o f f l a s h y metatheorems, r i p e f o r i n t e r p r e t a t i o n s , and s o t h e r e i s l i t t l e r i s k o f b e i n g l e f t t o t a l l y empty handed.

A s f o r a g e n e r a l a r e a of i n t e r p r e -

t a t i o n , t h e s i n g l e most dramatic use o f Heyting's l o g i c , i n Brouwer's b u s i n e s s of choice sequences, c e n t e r s around c o n t i n u i t y ( o f d i s j u n c t i o n s o r , more gene r a l l y , t h e q u a n t i f i e r combination VIX 3x ).

Bringing i n t h e technology of

sheaves i s doubly c o n s e r v a t i v e ; t e c h n i c a l l y , s i n c e i t s o r i g i n a l f u n c t i o n w a s some k i n d o f continuous p a s t i n g , and t h u s c l o s e t o Brouwer's venerable a i m ; more b r o a d l y , s i n c e sheaves a r e c e r t a i n l y among t h e b l u e chips of t h e mathem a t i c a l market. t o apply

-

True, t h e r e a r e more imaginative t h i n g s t o do w i t h them t h a n

t h e sheaf-theoretic i n t e r p r e t a t i o n s of

-

t h e l o g i c a l operations.

But t h e same could b e , and was, s a i d o f o r d i n a r y model t h e o r y , and t h e o r d i n a r y set-theoretic interpretations.

Yet even i t s e a r l i e s t c o n t r i b u t i o n s , by Malcev,

a r e something we s t i l l want t o know about.

A l l t h i s i s encouraging (though

t h e r e i s a n o t h e r s i d e t o which we r e t u r n i n a moment), and q u i t e enough t o

G. KREISEL, A. MACINTYRE guarantee a coherent theory.

The successes w e know a r e ( f o r us) more d i f f i c u l t

t o judge. Examples. ( i ) Sheaf models have 'brought a i d and comfort' - as t h e Royal Soct o some non-logicians, e s p e c i a l l y , t o p o l o g i s t s who a r e c u r i o u s t o i e t y p u t s it learn independence proofs i n s e t t h e o r y , b u t not t o u s e them. I f t h e y d i d t h e y might b e n e f i t e q u a l l y w e l l from boolean-valued models ( u s e d more widely i n t h e l i t e r a t u r e ) . - Unquestionably, i f ever t h e r e were t o b e a s i m i l a r p a s s i v e i n t e r e s t i n t h e metatheory o f c o n s t r u c t i v e l o g i c , sheaf models would s u r e l y provide n e a t e r e x p o s i t i o n s ( a n d e v e n t u a l l y some new r e s u l t s f o r s p e c i a l i s t s ) . ( i i ) It i s c e r t a i n l y easy t o imagine p l a u s i b l e u s e s , j u s t a s one d i d 1 5 y e a r s ago f o r c o n s t r u c t i v e l o g i c and formal c o n t i n u i t y axioms; c f . 1. 15-20 on p. 1024 of Statman (1979) f o r t h e ( b e g u i l i n g ) p o s s i b i l i t i e s envisaged, and t h e c u r r e n t l i m i t a t i o n s . - Reminder. Here a p p l i c a t i o n s o f sheaf models t o t h e broad cons t r u c t i v e t r a d i t i o n are considered.

-

Nevertheless t h e r e remain nagging doubts connected - perhaps p a r a d o x i c a l l y with t h e comparison above t o e a r l y model t h e o r y .

-

A f t e r having handled o r d i n a r y

models f o r 50 y e a r s , c o u l d n ' t we expect more r a p i d p r o g r e s s w i t h sheaf models? But then:

t h o s e e a r l y uses o f model t h e o r y were made at an e a r l y s t a g e o f

o r d i n a r y axiomatic mathematics which has developed i t s own devices i n t h e meantime, f o r example, b a s e change theorems i n p l a c e o f model-theoretic t r a n s f e r .

Has t h e growth o f o t h e r devices of t h i s s o r t simply 'preempted' t h e p o s s i b i l i t y of easy, s p e c t a c u l a r a p p l i c a t i o n s o f sheaf models?

L a s t b u t not l e a s t , t h e

number o f a r e a s where o r d i n a r y model t h e o r y has had s t r i k i n g u s e s i s q u i t e l i m i t e d , and t h e e x p e c t a t i o n s

-

on t h e o b j e c t i v e evidence a v a i l a b l e

-

i n , say,

1945, 1955, 1965, and 1975 f l u c t u a t e d t o o much f o r any s t a t i s t i c a l l y s i g n i f i c a n t assessment.

In such circumstances, i t seems premature, a s a l r e a d y mentioned i n

t h e I n t r o d u c t i o n , t o have any views on t h e broad p o t e n t i a l o f sheaf models o r o f o t h e r mathematical uses o f c o n s t r u c t i v e l o g i c .

-

For t h e r e c o r d , one o f us

hopes t h a t l o t s o f people w i l l work on sheaf models; j u s t a s it would have been n i c e f o r us, t h e consumers, i f more d i v e r s e v e n t u r e c a p i t a l had gone i n t o n u c l e a r e n e r a ( a t t h e c o s t o f b a n k r u p t c i e s o f some companies and losses t o many stockh o l d e r s ).

Grounds f o r optimism: p o s s i b l e s t r a t e g i c e r r o r s i n c u r r e n t work. Both ( a ) and ( 6 ) below a t t r i b u t e a simple-minded f o u n d a t i o n a l b i a s t o t h a t work. (a) Meticul o u s c a r e i s t a k e n i n choosing between c l a s s i c a l l y , b u t not i n t u i t i o n i s t i c a l l y e q u i v a l e n t d e f i n i t i o n s of t h e same ( b a s i c ) mathematical n o t i o n . But l e s s att e n t i o n seems t o b e given t o t h e q u e s t i o n whether every notion which i s b a s i c f o r ( o r d i n a r y ) mathematics, f o r example, o f r e a l c l o s e d f i e l d s , a l s o has a rewarding s h e a f - t h e o r e t i c a n a l y s i s . The answer would b e a u t o m a t i c a l l y p o s i t i v e i f one thought o f sheaf models as t h e ' t r u e foundation' o f ( w e l l - t e s t e d ) mathematical p r a c t i c e . ( 8 ) A q u i t e extreme form o f s t r a t e g i c e r r o r i s suggested by t h e l a s t paragraph on p. 1010 o f MacLane (1979) about t h e r e l a t e d t o p i c o f topos t h e o r y . A l i t t l e theorem i s p r e s e n t e d as i f it provided t h e f i r s t ' r i g orous' connection between c o n t i n u i t y and i n t u i t i o n i s t i c l o g i c ( a f t e r Brouwer's formal c o n t r a d i c t i o n s between c l a s s i c a l theorems and c o n t i n u i t y p r o p e r t i e s i n t h e i n t u i t i o n i s t i c t h e o r y o f choice sequences!). This s o r t o f r i t u a l o f u s i n g l i t t l e theorems t o ' e s t a b l i s h ' w e l l known p o i n t s i s not only p a i n f u l t o l o g i c a l s e n s i b i l i t i e s , b u t a l s o p r a c t i c a l l y d i s a s t r o u s : it i s bound t o s e t low standa r d s on a p p l i c a t i o n s f o r an obvious reason. No a p p l i c a t i o n can compete i n

Constructive logic versus algebraization

243

i n t e r e s t w i t h t h e s o c a l l e d conceptual p r o g r e s s o f having a t long l a s t given an i n t e l l i g i b l e meaning t o t h e i d e a a t i s s u e , h e r e : t o c o n s t r u c t i v e l o g i c ( i f t h a t i s what sheaf model semantics i s thought t o have done). A t t h e same time an i n t r i n s i c a l l y q u i t e poor a p p l i c a t i o n may b e more t h a n enough f o r t h e pedagogic purpose o f i l l u s t r a t i n g t h a t p r e c i o u s meaning; c f . a l s o a ( i ) o f t h e l a s t s e c t i o n concerning low s t a n d a r d s . - Reminder. S t r a t e g i c e r r o r s c o n s t i t u t e grounds f o r optimism because t h e y may have prevented f u l l e x p l o i t a t i o n o f t h e i n t r i n s i c p o t e n t i a l ( h e r e : of sheaf models). A s f a r as t h e c o n s t r u c t i v e t r a d i t i o n i n mathematics i s con-

Summing up P a r t I.

cerned, ( c o n s t r u c t i v e ) l o g i c has not been, and i s not l i k e l y t o b e , of s i g n i f icance comparable t o t h e g e n e r a l s t r a t e g i e s developed w i t h i n mathematics, d u r i n g t h e same p e r i o d .

More p r e c i s e l y , t h i s a p p l i e s t o t h e l o g i c a l e l a b o r a t i o n s s i n c e

some o f t h e broad i d e a s behind them, f o r example, t h o s e s t r e s s e d i n Brouwer's topology, have been absorbed i n t o t h e g e n e r a l mathematical s t r a t e g i e s t o o . Wherever p a r t s o f l o g i c , more o r l e s s r e l a t e d t o t h e c o n s t r u c t i v e v a r i e t y , were put t o e f f e c t i v e u s e , t h i s r e q u i r e d ' a d d i t i o n a l s k i l l s , ' f o r example, i n f e r r e t i n g out f r u i t f u l problems.

All t h i s i s i n s h a r p c o n f l i c t w i t h t h e e x p e c t a t i o n s cipal attraction foundations:

-

-

and c e r t a i n l y t h e p r i n -

o f t h e p a r t i c u l a r exact methodology a s s o c i a t e d w i t h l o g i c a l

o f s e e i n g i n l o g i c a powerful s y s t e m a t i c t o o l .

even more obvious c o n f l i c t s .

m e r e are other,

Three o f t h e p r i n c i p a l w o r r i e s about c o n s t r u c t i v e

l o g i c , s a y 50 y e a r s ago, concerned:

l a c k o f p r e c i s i o n o r even coherence, ex-

c e s s i v e r e s t r i c t i o n s , and t h e s p e c t r e o f an i n t o l e r a b l e reform of mathematical practice.

Whatever e l s e may b e i n doubt, p r e c i s i o n p r e s e n t e d l i t t l e d i f f i c u l t y ;

i f anything, r e s t r i c t i o n s were i n s u f f i c i e n t , and t h e a c t u a l reform, t h a t i s , change i n mathematical s t y l e , during t h e l a s t h a l f c e n t u r y exceeds anything t h a t was contemplated i n e a r l y debates about c o n s t r u c t i v e l o g i c . For t h e a c t i v i s t

-

concerned w i t h t o o l s , i n p a r t i c u l a r , norms ( t h e most ele-

mentary k i n d o f t o o l ) , f o r r e a s o n i n g

-

t h e s e c o n f l i c t s between e x p e c t a t i o n s and

experience a r e merely r e g r e t t a b l e o v e r s i g h t s .

But f o r t h o s e r e f l e c t i n g on math-

ematics t h e y are o f t h e essence, b e i n g genuine p h i l o s o p h i c a l d i s c o v e r i e s ; now i n t h e popular (no l o n g e r : assessing aims.

Schopenhauer's) s e n s e o f philosophy, concerned w i t h

And exact philosophy o f t h i s k i n d marshals t h e evidence pro-

vided by l a t e r , d e t a i l e d experience.

Work on c o n s t r u c t i v e l o g i c over t h e l a s t

h a l f c e n t u r y has provided remarkably d e c i s i v e o b j e c t l e s s o n s f o r t h i s k i n d o f philosophy.

They w i l l b e t a k e n up i n P a r t 11, t o g e t h e r w i t h aims i n t h e t r a -

d i t i o n o f l o g i c a l foundations f o r which c o n s t r u c t i v e l o g i c remains o f c e n t r a l i n t e r e s t , as a l r e a d y mentioned i n t h e I n t r o d u c t i o n .

244

G. KREISEL, A. MACINTYRE

APPENDIX S i e g e l ' s F i n i t e n e s s Theorem f o r Binary Diophantine Equations This Appendix i s intended f o r r e a d e r s who a r e - w i l l i n g f a m i l i a r with Lang (1962), [La] f o r s h o r t .

t o make themselves-

The r e s u l t of p r i n c i p a l i n t e r e s t i s

i n t h e l a s t chapter but one. It i s an i m p l i c a t i o n between two f i n i t e n e s s theorems, f o r diophantine approximations and diophantine equations (modern v e r s i o n s of Roth's and S i e g e l ' s theorems). more recent S e r r e quantifier-free,

Some s i g n i f i c a n t s i m p l i f i c a t i o n s contained i n t h e

(1980), [ S e ] f o r short, will be g e n e r a l l y reformulated i n so t o speak a l g e b r a i c i n p l a c e of S e r r e ' s i n t u i t i o n i s t i c language

( r e p l e t e with negations l i k e : il n ' e x i s t e p a s d e s u i t e 1.

Aims.

. . . ).

To p o i n t out and e x p l a i n d i s c r e p a n c i e s between ( i )t h e a c t u a l evolution

of- statements and proofs of- t h e theorems mentioned, and (ii)t h e p a r t i c u l a r i d e a l s of systematic a n a l y s i s found i n t h e l o g i c a l t r a d i t i o n .

The most s p e c i f i c

discrepancy concerns p r a c t i c a l l y u s e m i n t e r p r e t a t i o n s of ' c o n s t r u c t i v e ' implications.

This a f f e c t s not only c o n s t r u c t i v i s t s b u t a l s o t h o s e (model t h e o r i s t s

who a r e ) preoccupied with r e l a t i v e g e n e r a l o r , i n f a c t , even r e l a t i v e p r i m i t i v e recursiveness.

More general d i s c r e p a n c i e s concern t h e choice of terms i n which

t h e i d e a s of proofs, and eventually t h e theorems proved a r e formulated.

Thus

t h e r e a r e d e l i c a t e solved and unsolved problems - here: concerning e f f e c t i v i t y formulated i n terms p e r f e c t l y f a m i l i a r from l o g i c a l theory.

But t h e only known

After t h e significance o r 'rele-

general s o l u t i o n s involve a l g e b r a i c geometry.

vance' of t h e l a t t e r i s discovered i n t h i s way, t h e o r i g i n a l problems themselves a r e reformulated i n a l g e b r a i c / geometric terms.

Not only does t h i s a n a l y s i s con-

f l i c t with l o g i c a l i d e a l s , b u t t h e l a t t e r , i f taken l i t e r a l l y , would draw att e n t i o n away from i t s s i g n i f i c a n c e . Remark. Contrary t o a wide spread s u p e r s t i t i o n (among mathematicians) t h e success of t h e general mathematical s t r a t e g i e s used can be made p l a u s i b l e i n remarkably elementary - but of course, not t r a d i t i o n a l l o g i c a l - t e r m s . I n f a c t , many c a s u a l observations i n [ L a ] can b e used f o r t h a t purpose provided only they a r e r e membered where they a r e r e l e v a n t . Two examples of s p e c i a l importance below involve uniformity p r o p e r t i e s of h e i g h t s (even beyond t h o s e s t r e s s e d on t o p of p.50), and t h e e f f e c t of ' c o n d i t i o n s of a m u l t i p l i c a t i v e n a t u r e ' (bottom of p.94) on Roth's theorem; more p r e c i s e l y , i t s optimal form f o r abelian, as opposed t o a r b i t r a r y a l g e b r a i c v a r i e t i e s , though t h i s matter i s not-up i n V I $ 1 0 on pp.115-118 of [La] where, i n our view, it belongs. The m a t e r i a l below i s organized as follows.

52 s t r e s s e s geometric elements i n

t h e statement of S i e g e l ' s theorem ( S ) . $3 r e c a l l s some, s u p e r f i c i a l l y i n s i g n i f i cant a l g e b r a i c t w i s t s on diophantine approximation theorems (DA).

$ 4 f i r s t goes

i n t o ambiguities of ' c o n s t r u c t i v e ' meanings of implications, i n p a r t i c u l a r , (DA) + (S)

or, equivalently, of t h e phrase: from a counterexample t o ( S ) we g e t

Constructive logic versus algebraization

one f o r

8 5,

(DA),

245

and then makes a p r a c t i c a l l y u s e f u l choice between t h e s e meanings.

on p r o j e c t i v e embeddings and heights, s e t s out some of t h e notions needed t o

62,

develop t h e geometric i n s i g h t of t o exploit $ 3 .

and

96,

on a b e l i a n v a r i e t i e s , t h o s e needed

Both 8 6 . 2 and 87 i l l u s t r a t e how standard proofs of implications

between v a r i o u s modern v e r s i o n s of

(DA),

resp. of (DA)

f o r abelian varieties

and ( S ) can be reformulated t o s a t i s f y t h e c o n s t r u c t i v e meaning explained i n $4.4 (with a s t r o n g a d d i t i o n a l hypothesis i n §6.2, a mild one i n 87).

98 r e f e r s t o

more o r l e s s complete sketches i n t h e l i t e r a t u r e , e s p e c i a l l y [Se],

giving con-

s t r u c t i v e p r o o f s of (DA) + ( S )

f o r o t h e r v e r s i o n s of

(DA); s p e c i f i c a l l y , f o r

a l g e b r a i c v a r i e t i e s over a r b i t r a r y number f i e l d s , or t h e p a r t i c u l a r numberfield

99, on l o g i c a l and mathematical c l a s s i f i c a t i o n s , expands $1 and

involved i n ( S ) .

82 by r e f e r e n c e t o t h e preceding m a t e r i a l .

2.

Q u a n t i f i e r complexity, l o g i c a l and mathematical parameters ( c l a s s i f i c a t i o n s ) .

2.1. Logicians a r e most f a m i l i a r with diophantine equations from H i l b e r t ' s 10th Problem; c f . p p . 3 2 3 - 3 7 8 of Browder (1976). of t h e number of ( i n t e g r a l ) v a r i a b l e s .

There most r e s u l t s a r e s t a t e d i n terms

It i s observed t h a t t h e d e c i s i o n problem

f o r 1 v a r i a b l e i s solvable, b u t not f o r 9 v a r i a b l e s , and t h a t t h i s i s t h e l i m i t of c u r r e n t k n o w l e a e (with an a s i d e on t h e d e c i d a b i l i t y of quaclratic equations, Grunewald and Segal (1981) being an up-to-date r e f e r e n c e ) . 2.2. We g e t a d i f f e r e n t p e r s p e c t i v e if we l o o k a t e x i s t i n g knowleke i n t h e case of 2 v a r i a b l e s .

Consider:

(1)

When does

f(x,y) = 0

have a s o l u t i o n i n

(2)

When does

f(x,y) = 0

have i n f i n i t e l y many s o l u t i o n s i n

2 ? 2 ?

S i e g e l (1929) answered ( 2 ) completely, b u t one needs t h e rudiments of algebraic geometry t o understand h i s r e s u l t .

The s u c c i n c t answer t o ( 2 ) i s i n

terms of genus of curves, or, equivalently, t h e dimension of t h e i r Jacobians. c o n t r a s t , (1)i s s t i l l open, though widely conjectured t o be solvable.

In

Be t h a t

a s it may, as f a r as e x i s t i n g knowledge i s concerned, a formal q u a n t i f i e r

i s misleading

s i n c e it makes (1)simpler t h a n ( 2 ) .

2.3. S i e g e l ' s Theorem. If we assume f a b s o l u t e l y i r r e d u c i b l e over Q, f def i n e s a curve. The genus of ( t h e curve defined by) f i s a nonnegative i n t e g e r , which can .be p r i m i t i v e r e c u r s i v e l y computed from t h e c o e f f i c i e n t s o f If t h e genus ----

Siegel

of

f

i s greater than

0, f

has only f i n i t e l y many ----

f.

solutions

(199)a l s o provided d e f i n i t i v e information on t h e c a s e of genus

however does not concern us here.

Then

%

2.

0 , which

246

G. KREISEL, A. MACINTYRE

There i s no w s t e r y i n t h e appearance of geometrical i n v a r i a n t s i n a diophantine problem.

I n p a r t i c u l a r , genus i s a b i r a t i o n a l i n v a r i a n t , and b i r a t i o n a l t r a n s -

formations e s s e n t i a l l y r e s p e c t t h e p r o p e r t y of having i n f i n i t e l y many r a t i o n a l solutions.

It i s , perhaps, more remarkable t h a t genus i s such a useful parameter

f o r t h e problem of i n t e g r a l s o l u t i o n s .

-

One way i n which t h e

-

specific

notion

of genus i s u s e f u l i s t h a t it admits an e x t r a o r d i n a r y number of d e f i n i t i o n s i n

widely d i f f e r e n t contexts; i n p a r t i c u l a r , it connects with s e v e r a l e l a b o r a t e coho-

mology t h e o r i e s .

This kind of t h i n g i s t y p i c a l of mathematical g e n e r a l i t y i n con-

t r a s t t o t h e g e n e r a l i t y of a ' u n i v e r s a l ' l o g i c a l language, as i n model theory, which i s of l i t t l e help i n i d e n t i f y i n g p a r t i c u l a r l y s i g n i f i c a n t s p e c i f i c notions.

3 . Diophantine approximations t o a l g e b r a i c numbers.

Modern r e a d e r s w i l l be

f a m i l i a r with t h e relevance of diophantine approximations t o diophantine equations, f o r example, approximations t o 3 f i

2 - 22

and i n t e g r a l s o l u t i o n s o f

p.1.8 of [ S e ] f o r up-to-date q u a n t i t a t i v e information.

-

=

c ; cf.

Evidently, t h e l o g i c a l

p u r i s t a t t h e time of W h a g o r a s would have been ill at ease with t h i s connection. be r a t i o n a l numbers i n t h e i r lowest terms, and a

3.1. Let p/q

numbers of degree > 2 l i t e r a t u r e takes

DA

over

Q

.

algebraic

Then t h e c l a s s i c a l number-theoretic

i n t h e form:

For a l l such

t h e r e a r e only f i n i t e l y many where t h e best-known i n s t a n c e s of

and p o s i t i v e i n t e g e r s

n

< q -T(n),

q : Ia-p/ql 7:

a

are:

W + IR

( i ) n = degree of a ! , T(n) = n / 2 + 1 (Thue's theorem, a p r e c u r s o r S i e g e l ' s theorem, and adequate for = l),

Of

2 22

(ii) as above, but

T(n) = 2

fi , used

i n S i e g e 1 (1929), and f i n a l l y ,

(iii) Roth's theorem, which i s independent of t h e degree of

~ ( n =) 2 + l / n

.

The theorem would f a i l f o r

take arbitrary positive rationals Bounds on t h e number of exceptional Davenport and Roth (1955).

E

~ ( n =)2 .

i n p l a c e of

a,

not even f o r

q ( i n t h e sense o f (iii)above) a r e given i n

No e f f e c t i v e bounds on t h e s i z e

n=l.

Each of ( i ) -(iii) above i s a $i

with

l/n. )

l a t e r : height (bottom of p.95 of [ L a ] ) - o f t h e exceptional arbitrary

a,

(Equivalently, one can

statement.

q

o r , as we s h a l l say a r e known f o r

247

Constructive logic versus algebraization

3.2. Versions of (DA) incomparable with (iii)above. A t r i v i a l example: r e s t r i c t i n g a: t o be r a t i o n a l when 2 can b e replaced by 1; c f . ( v i ) on p.94 of [La].

On p.95, a m c h more i n t e r e s t i n g v e r s i o n i s s t a t e d as a COROLLARY, giving

s p e c i f i c conditions of a ' m u l t i p l i c a t i v e nature' mentioned e a r l i e r i n t h i s Appendix.

They a r e b e s t s t a t e d by f i r s t formulating Roth's theorem f o r a r b i t r a r y

number f i e l d s

K

i n p l a c e of

sidering approximations of

@,

as i n Theorem 1 on p.93 of [La], and t h e n con-

a: ( a l g e b r a i c over K ) by s p e c i a l elements of

satis-

K

Then 2 can be replaced by 0 .

fying those m u l t i p l i c a t i v e conditions.

3.3. Implications of t h e successive improvements i n 3.1 f o r f i n i t e n e s s theorems ' j a b o u t diophantine equations. p.186,

(ii)Theorem

4

Obvious examples are given i n

on p. 188, and

(iii)p.

( i )Theorem 1 on

189, 1. 5, of Mordell (1969)i n

terms of t h e s y n t a c t i c f o r m of t h e equations, f o r example, i n case (iii),f o r P = Q where

degree

<

zeros of

P i s homogeneous and i r r e d u c i b l e (over

n-2.

The relevant

~ ( 1a,)

= 0

CT

@) of degree

n

2 3,

a r e determined i n an obvious way by

P,

Q of as

or ~ ( a : 1) , = 0.

But f o r o t h e r equations, even those defining curves of genus 1 such as looking a t t h e s y n t a c t i c form j u s t won't help t o f i n d a relevant

y2=d+ k,

This i s

CT.

where t h e new icleas of Siege1 (1929), and eventually t h e formalism of a b e l i a n v a r i e t i e s come i n ( i f one wants t o d e r i v e ( S ) from some (DA) at a l l ;

Baker and

Coates (1970) have a d i f f e r e n t l i n e of a t t a c k ) .

4.

Constructive meanings of: (DA) + ( S ) .

Perhaps t h e f i r s t meaning t h a t comes

t o mind - and c e r t a i n l y t h e only one guaranteed by a proof i n c u r r e n t i n t u i t i o n i s t i c systems

-

i s this.

There i s a c o n s t r u c t i v e procedure ( f u n c t i o n a l ) of passing from a height bounding function(of CT and n) f o r (DA) t o one f o r t h e height of s o l u t i o n s of ( S ) , together with a proof t h a t t h e procedure works.

(Note t h a t t h i s formulation does

not r e s t r i c t i t s e l f t o hypothetical height bounding functions f o r a r e themselves constructive:

this would be weaker.)

meaning i s t h a t t h e r e i s no e f f e c t i v e height bounding function in

(DA)

(DA)

which

The f a t a l d e f e c t of t h i s f o r arbitrary

12

i n sight.

4.1. A much more u s e f u l meaning requires, f o r any given equation f ( x , y ) = 0 i n ( S ) , an e x p l i c i t l i s t of a: (and a number n ) which a r e relevant t o (S), with t h e hope t h a t t h e y f a l l i n t o subclasses may become a v a i l a b l e .

of

CT

f o r which height bounds a r e or

G . KREISEL, A. MACINTYRE

248

4.2. Better s t i l l , one would hope f o r an arsenal of methods ( d i f f e r e n t ) s e t s of

which provide

such t h a t height bounds f o r approximations t o these

01

01

some height bounds f o r solutions of t h e equation considered.

give

4.3. Best of a l l , t h e procedure should be e f f i c i e n t i n one o r both of t w o senses: ( i )the ( r e l a t i v e ) height bounds should not grow too fast,and ( i i ) t h e procedure

should work both ways: not only t o get bounds f o r ( S ) from (DA) applied t o specific

01,

volved i n q,

but t o pass

-

from

exceptionally large solutions

of -

( S ) t o exceptionally close r a t i o n a l approximations

'exceptional' i n t h e sense of

t h e equation in-

5

01

w i t h large

(DA). summarizing 4 . 1 - 4 . 3 above.

4.4. Formal constructive meaning of: (DA) + ( S ) , To give (an arsendl o f ) procedures which assign: (i)

To each binary equation (defining a curve)

ai ( 15 i

of algebraic numbers

C and integer

(ii) To each such

(iii)

To

C , Q, and

i(15 i

an

5 pC )

5 pc)

aw

and an integer

i n t e g r a l point

Q

and

of genus

Q : an integer

and a f r a c t i o n p/q

1 q1 2

c

(x,y)

X,

z

1: a f i n i t e s e t

n (more precisely,

)

n

.

and

C w i t h max(Ixl, l y l ) > X :

on

i n lowest terms such t h a t

lai -p/ql < q-T(n)

.

4.5. Remark on Baker and Coates (1970). Obviously, t h e absolute height 'bound given t h e r e f o r curves of genus 1, s a t i s f i e s 4.4 t r i v i a l l y ( f o r a r b i t r a r y

xc, 01)

since t h e r e simply a r e no i n t e g r a l

+

(x,y)

on

m u ( 1x1, IyI ) > . ( S ) , constructive o r otherwise, implies t h a t of very large, 4.4 may r e t a i n corresponding

X

C which s a t i s f y

This i s i n accordance with t h e f a c t t h a t t h e v a l i d i t y of (DA)

But since

(S).

an i n t e r e s t f o r a limited range of

i s much smaller then

various ' e f f e c t i v e ' height constants i n

XC.

Q

XC

is

provided t h e

(To check t h i s p o s s i b i l i t y t h e

5 5 6-7

below would have t o be computed

explicitly. ) Number t h e o r i s t s who are more competent t h a n we are, might be able t o i l l u s t r a t e t h a t i n t e r e s t by working out an example, perhaps, f o r t h e curves y2

= g+ k .

C defined by

Specifically, t o use t h e ideas of Siege1 (1%9), i n particular,

suitable nonconstant functions on

C and corresponding

01

(of degree

n), t o

exploit 'surprisingly' l a r g e i n t e g r a l points on C f o r t h e purpose s t a t e d i n

4.3 (ii). 4.6. Remark on extending 4.4 t o modern versions of (DA)

(S).

Here t h e r e i s

a p r i c e t o pay for t h e power of t h e modern mechinery which comes from 'looking

Constructive logic versus algebraization C

a t a curve

249

from d i f f e r e n t p o i n t s of view’ ( f o r example, by means of d i f f e r e n t

embeddings i n t h e same p r o j e c t i v e space, o r even i n spaces of d i f f e r e n t dimension). This i s alrea&y c l e a r from t h e r e f e r e n c e i n tacitly, different-heights.

61

t o uniformity p r o p e r t i e s of -

Thus, i n t h e extension of

4.4

b e chosen before we have f i x e d t h e p a r t i c u l a r embeddings of

may have t o

(ii), X

C, which determine

t h e h e i g h t s corresponding t o some of t h e a b s o l u t e v a l u e s occurring i n (iii)of

4.4; c f . t h e Remark i n 86.2 f o r another i l l u s t r a t i o n .

The next two s e c t i o n s

give an admittedly q u i t e s u p e r f i c i a l l i s t of d i v e r s e notions and p o i n t s t o which one must pay a t t e n t i o n .

5.

Geometry: P r o j e c t i v e Enibeddings and Heights.

5.1. By r o u t i n e e f f e c t i v e removal of s i n g u l a r i t i e s , S i e g e l ’ s problem (generalized K ) i s reduced t o t h e following geometric version:

t o a r b i t r a r y number f i e l d s Let C is

C be a p r o j e c t i v e nonsingular curve d e f i n e d over

2

1. Show t h a t

Reminders (i)

C

has only f i n i t e l y many

K.

Suppose t h e genus of

K - i n t e g r a l points.

of u s e f u l p r o p e r t i e s of p r o j e c t i v e spaces and non-singular curves:

p n ( ~ ) i s complete, and, f o r a= completions

a c t u a l l y compact, making l i m i t arguments p o s s i b l e .

of

K~

K,

IP”(K~)

is

.

I n aiming f o r e f f e c t i v i t y ,

we s h a l l have t o f i n i t i z e t h e l i m i t arguments, b u t we would b e l o s t without them. The completeness meant here i s a kind of formal compactness ( c f . Mumford

(1976)

82C, pp.33-36), r e l a t e d t o a s u b t l e r q u a n t i f i e r - e l i m i n a t i o n , i n c i d e n t a l l y somewhat neglected by l o g i c i a n s ; f o r an exception, c f . van den Dries ( A ) . bedding

( i i ) Em-

C i n i t s Jacobian v a r i e t y (about which more i n 66.1 below), r e q u i r e s

C t o be non-singular.

5.2. Height, H(P), t h e b a s i c measure, i s f i r s t defined f o r p o i n t s or

Pn(L)

for

L _3 K

P

i n #(K)

i n terms of c e r t a i n normalized absolute values on

The normalization makes t h e Product Formula hold, and t h i s i n t u r n makes defined;

H

K.

well-

p.44 of [ L a ] , on K-absolute heights, g i v e s a n axiomatic treatment.

The c e n t r a l r o l e of t h e product formula i s p l a u s i b l e enough since, by A r t i n and Whaples

(1945), it a x i o m t i z e s a l g e b r a i c number theory.

For an a r b i t r a r y v a r i e t y

V

,

each morphism V + IPn

defines (constructively) a

height on V.

The r e l e v a n t connections between a l l such h e i g h t s a r e s e t out i n

Chapters 3 and

4 of

[La].

F o r our p r i n c i p a l t o p i c , S i e g e l ’ s theorem and i t s r e l a t i o n t o diophantine

approximation, t h e most obviously u s e f u l p r o p e r t y of h e i g h t s i s t h i s : t h e s e t of p o i n t s i n p.

Pn(K)

49 of [ ~ a l .

of bounded height i s f i n i t e and e f f e c t i v e l y computable, by

G. KREISEL, A. MACINTYRE

250

6.

Algebra:

Abelian Varieties.

Abelian v a r i e t i e s enter i n t o our topic i n at

As mentioned already ( i n $ 5.1)

l e a s t two ways.

, non-singular

curves a r e embed-

ded i n t h e i r Jacobians which a r e abelian v a r i e t i e s , and then, a s i n $3.2, t h e natural version of

(DA)

lowers t h e exponent 2 i n Roth's theorem ( f o r general

-

algebraic v a r i e t i e s ) t o 0 .

C denotes non-singular projective curves over

K.

6.1. The constmction of Jacobians and t h e r a t i o n a l i t y properties which enrich t h e c l a s s i c a l theory of divisors, a r e t o be found i n I1 $2, pp.30-40, of Lang (1959). A s f o r effectiveness, t h e r e a r e primitive recursive procedures

C , n3 on C and any L-rational point

r a t i n g on

on

P

C

n,(C) = A , n2(C) and n3(C, P) a r e embed3ings: A + P n A i s t h e Jacobian of C , and IPn some projective space,

and

A

A + P n a r e defined over

i s equal t o t h e genus of

A

induced by t h e embedding (C, P)

-,Pn . -

L,

and

ope-

, resp. L

C +A,

where

i s an a r b i t r a r y

C.

-

A s i n $5, heights on

C are

A + IPn, or, more precisely, t h e embedding? :

Reminder: uniformity properties of ( r e l a t i o n s between) heights

f o r different choices of 6.2.

over

+A

n2

.

number f i e l d _3 K The fiimension of

c

K,

and

n1

such t h a t

P

w i l l be needed.

On p . l . 9 of [Se], (DA) i s s t a t e d f o r points B on algebraic and abelian

varieties

V

defined over a f i e l d

means of a completion Kv If t h e point

DA 6

. CT

i n V(K ) v

where H

5

H(B)-6

i s a property of K,v,V

DA6

For

K-varieties

For all abelian

V V

V

V.

i s algebraic there

B i n V(K) such t h a t

,

i s t h e height induced by a K-embedSing of

that

defined by

3.

i s the s e t of K-rational points o f

V(K)

a r e only f i n i t e l y many dv(a,B)

and a natural distance

K,

a. -

and of

and a l l algebraic points

i n some P n

Then, f o r a l l

. - Note

K and v,

CT'EV(K~):(V6 > 2 ) D A g ( a : ' ) .

a:

and a l l algebraic points

V

E V(Kv) : (Vp

>

O ) D A ~( a ) .

Formally speaking t h e two statements a r e incomparable because, i n t h e l a t t e r , t h e v a r i e t i e s considered a r e r e s t r i c t e d but t h e range of the quantifier (compared t o

6).

(Theorem 2 on p.

The corresponding socalled

116, resp. Theorem 1 on p.

p

extended

geometric formulations i n [La]

122 )

a r e more convoluted.

-

Though

we have not checked t h e exact r e l a t i o n s between Roth's theorem f o r number f i e l d s and points on algebraic v a r i e t i e s they should be close since t h e co-ordinates of trivially. If 6 5 p then DA6 + D A P

points a r e elements of a numbef f i e l a . NB.

-

Constructive logic versus algebraization

25 1

6.3. If AbDA means DA r e s t r i c t e d t o a b e l i a n V over K, and 6 > (*)

0

(a')+ ( V p > 0 ) Va AbDAp(0)

Va'A'bDA6

6=3

This i s e x p l i c i t on p.1.11 of [ S e ] (where

i s chosen), and i m p l i c i t i n t h e

'geometric' v e r s i o n i n Theorem 1 on pp.l22-l24 of [La].

- Weil

The proof a p p l i e s t h e

theorem f o r a b e l i a n v a r i e t i e s t o V ( K ) / m V ( K ) where, for some 6' > 6 , m must s a t i s f y (m2 - 1)p 2 6' Following pp.l.13-1.14 and 2 . 1 of

weak Mordell

.

[Se], t h e conclusion of (*) w i l l be applied i n 97 t o deduce constructively, i n t h e sense o f 4.4, S i e g e l ' s theorem (S) f o r curves now for V t h e Jacobian v a r i e t y C

those Remark.

C o f genus

2

1, by taking

C , A being a b e l i a n (constructively, f o r

of

A

t h a t a r e given t o g e t h e r with an embedding i n

A,

defined over

K.)

The p a r t i c u l a r proofs o f (*) i n [ S e ] o r [La] e s t a b l i s h (*) constructively-

as they stand, without introducing 'new i d e a s ' - f o r those v a r i e t i e s which some s e t

of r e p r e s e n t a t i v e s of

VK/m

a height bound

Bm

i s a v a i l a b l e (whence a height

V(K)/mV(K)

on t h e elements of bound

B

V over K f o r

with

VK/m

on a s e t of generators f o r V ( K )

can be

e f f e c t i v e l y determined; b u t t h i s i s not used h e r e ) ; i n t h e jargon of t h e track: by enriching t h e d a t a by

B

m

(from which a complete l i s t of a l l t h e elements i n VK/m

can be e f f e c t i v e l y computed).

a

E v ( q ) , we solve

for

a = W ! + x. 1

where

1

Now g i v e n - defining equations f o r

a! also i n 1' H(ni)

5 Bm

-

any point

V(Kv),

and

ni E V K / m .

Roughly speaking, t h e aim i s t o determine

f o r any i n t e g e r

h' : an i n t e g e r

from exceptionally c l o s e g e t an exceptionally c l o s e with

H(B')

To determine

ho

2

h'

p

ho

- approximations 6-approximation

such t h a t

B t o a: with H(B)_> ho we B' t o

one of

the

a;

.

independently of any ( h y p o t h e t i c a l ) p-approximations

B, one

uses t h e strong uniformity of t h e r e l e v a n t Property 4 of heights (on p.68 of [La]); since

p ' w i l l b e chosen t o s a t i s f y

B =a@' + a

f o r some x eA(K)/mA(K),

we know i n advance t h a t

Now, p.1.11 of [ S e ] needs : H(B')m2-1 only

c E (for all

E

>

0)

< H(B) while quasi-equivalence provides

such t h a t for all B '

and

r,

G. KREISEL, A. MACINTYRE

252 2

<

H(B')m Choose

c g H(B)'+',

so s m a l l t h a t

G

m2

2 2 m -1 < (m - l / 2 ) / ( l + y e )

H(B)

1h2 , one c

(h')

f3

C

P.

dv(a,B)

B

B'

and

.

B

p =mB'+ x i

E

V(K)

say.

5

H(B)-'

x.

f o r one o f t h e

a;.

h0

>

-

h2

V(K) h3

delarge

3

U.)

a r e determined by

Such a choice of

C

a,

is

are a l l &ale.

5

hi'

above t h e n

. B'

i s a c l o s e approximation t o

B

I n f a c t , by t h e r e l a t i o n between t h e h e i g h t s of

i s independent of

So, i f

if

and s a t i s f i e s :

dv(a;, B ' ) < C ' H ( B ' ) - 6 ' C'

h'

h-'

(Since t h e

may depend on

dv(a,B)

where

<

( p.1.11 of [ S e ] ) t h e morphisms : B ' , - t m B ' + x .

Now, suppose t h a t

one of t h e

2

, B ' ) < c dv (%B)

i s independent of

possible since

If

so l a r g e t h a t H(B')

To ensure t h a t

and

i

i s chosen a f t e r a, C

h3

ho

(It a l s o remains t o v e r i f y t h a t

=@' + n

a,(";

and

and t h e n

H(B).

The l a s t preliminary s t e p i s t o choose

effectively.)

enough t o e,"sure t h a t i f

where

.

c = h7 c € 1

Cg

u s e s t h e o t h e r d i r e c t i o n of quasiequivalence, making

sufficiently larger than termines

- l / 2 < m2(l+c),

if

<

H(B')m2/(1'E)

t h a t is,

max(h1,h2 dv(";,

B'

.

Since

6'

8' :

and

,

> 6,

C'

<

h6'-6

for all

h

2

h2 '

, h3) B ' ) < H(B')-6

.

6.4. For r e f e r e n c e below: t h e i m p l i c a t i o n (*) i n 6.2 i s uniform i n t h e 3 (suppressed) parameters the field

K, t h e v a l u a t i o n

V assumed t o be defined over K . assume t h e premise f o r tencling

v

and t h e a b e l i a n v a r i e t y

V,

A n a t u r a l formal weakening of (*) i s t o K' 2 K and v a l u a t i o n s v ' on K' ex-

extensions

v ( a n d now p o i n t s i n V(K') a r e considered ; i n 8.2 below we s h a l l have

t o vary a l s o f u l only i f

V).

Since

v

has s e v e r a l extensions

v'

,

t h e weakening i s use-

AbDAB i t s e l f i s r e s t a t e d f o r a r b i t r a r y f i n i t e s e t s

( c f . [Se] p . 2.09):

S

of v a l u a t i o n s

253

Constructive logic versus algebraization

7. Application t o S i e g e l ' s theorem ( S ) .

Recall t h a t

points on a p r o j e c t i v e , non-singular curve

9 6.

explained i n

concerns

(S)

1.

C of genus

[Se], pp. 1.13 - 1.14, and i m p l i c i t i n

The idea, e x p l i c i t i n

Theorem 2, on pp. 124-125, of [La] i s t o d e r i v e (S) from AbDA i s nowthe ( a b e l i a n ) Jacobian v a r i e t y

V

>

p(

C

of

A

,

0) and a l g e b r a i c p o i n t s O i on A a r e determined by (K and) C

A s s t r e s s e d i n 3.3, t h e choice of

(199). - Remark.

already i n S i e g e l

example, f o r

q*

p

and

f o r which t h e

, of

genus

g,

particular

standing

E A(K),

-

lP1 .)

For any points

H* c P

"1

for

where Bo(P)

>

order of

, introduced i n

+ lPl

denotes t h e height on

po,de-

P, on

C ,. for

B

= (xo,

xl,

s t e p s o n p.l.14 i n [Se].

S i e g e l (1929), i s given by

. . ., xg ) ,

(1

) above. (cp" maps Bo(C), C A , i n t o C or, equivalently, on Bo(C) induced by (p"

in

0, a n e f f e c t i v e bound i s obtained f o r t h e height of those i n t e g r a l

on

c

f o r which

IlogIq*(P)l,/log

H*(P)I e

So i n t e g r a l p o i n t s on

(9".

say

c o n s t r u c t i v e proofs of implications involve t h e contra-

(p" : C

q*(P) = xo/

B,

By 6.1, a

K-point,

f o r t h e r e s t of t h i s s e c t i o n .

p o s i t i v e , t h e exposition below r e v e r s e s t h e

7.1. The

may

C in A ;

i s e f f e c t i v e l y a v a i l a b l e from any

Since, by 4.4,

Reminder.

-

C by

will b e assumed below as p a r t of t h e d a t a .

C

and

K

, of

C is

the

K = I, from a r a t i o n a l , not n e c e s s a r i l y i n t e g r a l p o i n t .

example, f o r Fix

Bo

some embedding,

K,

ratio of

p/q i s (uniformly) bounded away from 1.

q

a l s o be thought of as defined on t h e image of

Po

makes t h e

Specifically, f o r

and v = v m , not o n l y t h e number of i n t e g r a l p o i n t s on

K=$

number of d i g i t s i n

f i n e d over

defined on

Nevertheless, [Se], p.1.14,

bounded, b u t a l s o of those r a t i o n a l p o i n t s

(t)

.

i s not a t a l l banal; they come from t h e

O.

apparently novel p o i n t t h a t t h e argument proves more t h a n ( S ) .

some such

where

P

poles and zeroes of a c e r t a i n (non-constant) r a t i o n a l f u n c t i o n

C,

integral

1, and t h e notions

-C

5

c,

C of l a r g e height

There a r e such s i n g u l a r i t i e s of

i s not constant.

t h a t is, f o r

< 1q*(p)lv < ec

H*(P)

2 3( >

e) :

.

can occur only near p o l e s o r zeros of

logIcq*I,

, at

least a t i n f i n i t y , i f

(The proviso i s checked by t h i n k i n g of

(p"

C as a curve i n a . )

G . KREISEL, A. MACINTYRE

254

(I),

CI by Po i n which a r e a l g e b r a i c p o i n t s over A , a r e t h e i' candidates f o r exceptionally c l o s e approximations by elements of A . This i s t h e

Their images,

f i r s t half required, i n 4 . 4 ( i ) , f o r a c o n s t r u c t i v e proof of ( S ) from AbDA ( o r , P -1 l o c . c i t . ) . The next s t e p s determine p equivalently, p = n i l s i n c e T(n) = n

.

from (K and) C

The f a c t s above a r e enough t o explain t h e choice of t h e socalled geometric formul a t i o n s on pp.115-118 of [ ~ a ] .

(p"

7.2. The e f f e c t i v e separation of poles and zeros of

C

i n t e r v a l s on

-

or, as always:

of an upper bound

c*

B (12)-

equivalently, on

)logIq*I

for

Iv

by a f i n i t e number of

, and

o u t s i d e those i n t e r v a l s

t h e computation Ii

around

Cx

i '

i s a r a t i o n a l function; i f one d i s l i k e s t h e empty

only uses t h e f a c t t h a t

q* s e t , one adds: which i s not constant.

Furthermore, t h e c.

, and

ai

at

order of

a l l t h e poles and zeros involved can be computed, say

ci

constants

such t h a t ,

with t h e convention t h a t poles have p o s i t i v e order.

7..3. Evidently, t o apply fl*

an3, t h e height, H

an embedding of

A

,

AbDA, it i s c r i t i c a l t o u s e t h e known r e l a t i o n between

i n some

volved i n t h e statement of

positive

ko

and

%

A (of

on t h e Jacobian

C) induced by (B

lPN (and t h u s a height on

C via

:C + A ,

and)

Bo : C + A )

in-

AbDA:

being obtained e f f e c t i v e l y , and uniformly i n

B.

7.4. To s a t i s f y t h e remaining conditions of 4.4 (without attempting Optimal bounds) t a k e c* from 7.2 f o r c i n 7.1. Then, i f H*(P), a f o r t i o r i , i f klH (B)ko > e , and

then

P

i s l o c a t e d i n j u s t one of t h e i n t e r v a l s

and so, i f

C

= maxlc.

-c*/,

ho

Then, for


l a r g e enough t o absorb i n

H(B)

>

ho

of 7.2.

Also by 7.2,

I ,

dv(ai,B) Choose

Ii

and

p = k0c*/2z

-k c*/E H(B) O k0c*/2 5 a l l the coefficients i n sight.

hO

Constructive logic versus algebraization

X=max(Q, h )

To s a t i s f y 4 . 3 ( i i ) , take

connection between heights i n

.

C and A

255

This i s simple because of the close (but no more so than f o r the c l a s s i c a l

analogues i n 3.3).

4 . 3 ( i i i ) i s s a t i s f i e d because any P t o be considered i s located i n a certain

,

Ii

and t h e ( r a t i o n a l ) approximating element on A

i s provided by Bo(P).

Remarks f o r comparison w i t h t h e corresponding r e s u l t s i n Theorem 2 on p.124 of F i r s t , we have considered only

[La] o r p.2.01 of [Se]. not a s e t

one valuation

v, and

involved i n t h e (usual) statement of ( S ) f o r S-integral points.

S ,

Secondly, we have not made e x p l i c i t t h e dependence of

p

on t h e degree of

K

(over t h e prime f i e l d ) which a r i s e s from t h e exponents i n t h e definition of heights needed t o make t h e product formula hold, but have hidden it i n

k -somewhat l i k e

p.1.14 of [Se] and unlike p.324 of [La]. Reminder. for

6

When knowledge of

> p to

pendent of

>

K

6/p

i s used, a s i n 6.3, the passage from AbDA

AK/m

i s uniform i n

P

K(v and A), and so t h e

Nevertheless, t h e relevant m

K.

affected by (m2 -1)

AbDA

since t h e l a t t e r a f f e c t s

p,

6

m needed i s inde-

f o r deriving ( S ) from AbDA6

is

i s required t o s a t i s f y :

and m

.

8. What can be done without knowing AK/m?

The l i t e r a t u r e , i n particular,

[Se],

contains sketches of two answers, s o t o speak, at opposite ends of t h e scale.

8.1. Siegel's Theorem ( S ) f o r curves V01

AbDA

above.

6

(01)

C over K i s derived, not from

f o r A ( K ) , but f o r a suitable extension K'

Specifically,

A(K')

of

as i n 6.4

K;

has an mth ' r o o t ' of each point on

A(K).

, copied

6.4 from p.2.09 of [Se]. i s determined (from C , K , v ) a s i n 7.4, modified f o r a s e t

valuations, say, place of one

AbDAi

in

> p),

of

S

I n particular S

p

of valuations i n

v.

Now t h e general idea of 6.3 can be applied ( f o r an 6

At

set

t h i s place one needs t h e formulation of Roth's theorem f o r a f i n i t e

AK/m.

but without recourse t o

m

derived from

p

and

Specifically, f i r s t , a s i n $ 7 , (S) i s

derived constructively from t h e conjunction of AbDAi(CXi),

applied t o t h e abelian v a r i e t y

f o r t h e s e t of valuations are obtained as follows. points

0li

on A(K);

algebraic over

v' As in

f o r each

.

extending t h e

07

v

involved i n (S), and the

t h e poles and zeros

ai, one takes

A(K), anct so over

A(K')

A(K').

(m

their

of

(p*

are algebraic

mth roots

i s determined from

0";

C

01;:01.

, K,

1

v

=MZ!

1

via

G. KREISEL, A. MACINTYRE

256

Now, since f o r any B i n A ( K ) , some B ' i n A(K'): B = I@' , B i s d i v i s i b l e by m without remainder, the - f i n i t e l y many - possible

p , as before.) that i s ,

AK/m, used i n 6.3, a r e no longer needed. Roughly speaking, i f p i s a p-close approximation t o Oi then p ' i s a 6-close approximation t o one o f t h e a ; . - Incidentally, t h e r e i s an obvious device f o r avoifling the 'enrichment of data' used i n $7, of presenting C together with an embedding remainders

Po : C +

xi,

ni

E

Since we introduce extensions of K

A.

and use

L'

i n place of

anyway, we might a s well s t a r t

L ( i n which a

C t o form a f i e l d

by adjoining a point on

Po

can be defined),

.

K'

We plan t o give d e t a i l s elsewhere, f i l l y enough t o compare t h e r e s u l t with a weak precursor, s t a t e d a t the end of Robinson and Roquette (1975).

It i s weak; not so

much f o r quantitative reasons ( s i n c e - a s i s usual i n model theory

-

only general

recursiveness i s established), but because of t h e f a t a l defect, explained at t h e beginning of

$4 of

r e l a t i v e recursiveness, primitive o r general; cf. t h e contrast

between Turing and many-one reducibility. - A more formal defect of Robinson and Roquette ( 1 9 5 ) i s t h a t it gives no warning of using the passage t o an extension a t a l l ! I n f a c t , Lemma 7.5 on p.170 loc. c i t . i s false, as confirmed by

K'

Roquette ( i n correspondence), precisely because such a passage i s involved i n t h e proof, but not s t a t e d i n t h e Lemma.

Incidentally, the proof, i n Roquette (1976),

i s purely algebraic, and at l e a s t as long as t h e non-standard argument of Robinson

and Roquette

(1975), not

counting the l e i s u r e l y general exposition of non-standard

arithmetic.

8.2.

The second sketch, on pp.1.12

-stays

in

K,

though

1.13 of [Se], i s much more ambitious: one V. Less knowledge of VK/m i s

-

on the given variety

used than i n 6.3, but an i n t r i c a t e 'cover' b u i l t on t h e f i n i t e l y presented Selmer group S ( m ) ; c f . Cassels (1$6), p.259 and especially p.265. (with each Va end Aa defined over K ) are: an embedding em:V(K)/mV(K) -,S(m) ; f o r each a

not

Au:

necessarily abelian variety

-, V

Vu

For each

V

where, f o r

and

d: (B; , P i )

B ' E Vu(K)

< Co

dv

e S(m) :

(homogeneous space), and a morphism

u = emn, Au(Vu(K)) i s t h e coset of VK/m containing

t h e r e i s a height

for all

V

u

Other a u x i l i a r i e s

H (and distance d:)

m2 -1 and p = A B ' : Hu(B )

(BlrB2)

for

B,B1,B2

Now, as i n t h e Remark of 6.3, f o r any

K,v,

such t h a t

<

H(B),

of large height, say, abelian

TI.

V

2

and a e V (Kv),

ho

.

AbDAp

i s derived constructively from t h e conjunction (valid by Roth's theorem) of

DAg applied t o :

all vu , u e s ( m ) ,

a

= A

This i s a l a r g e r s e t of equations than i n

u

6

(a'0 ). NB. 0

( 0,":

I Ie

VK/m)

. -For

each

p

and

E V(K)

e

with

vu ($)

H(B)

>_

satisfying: ho

6.3, corresponding t o

, look

for

u

E

S(m)

257

Constructive logic versus algebraization

and

B ' c Vo(K)

Such a

p'

s.t.

e x i s t s ; i n f a c t , f o r some u = n 9,

: x B VK/m (n

i s t h e remainder

B a f t e r 'division' by m ) . Because of t h e bound on H ( P I ) , by 5.2 a l l such B ' can be effectively determined. For each p a i r (0,P ' ) , and one of the a; : d: (a;,B ' ) < H(p')-6 , a s required; c f . also t h e FS. of

9.

Logical and mathematical parameters ( c l a s s i f i c a t i o n s ) .

tinues

This f i n a l section con-

$ $ 1-2 of t h i s Appen3ix i n t h e l i g h t of d e t a i l s i n $$3-8.

9.1. The difference between l o g i c a l and mathematical choices of parameters i s i l l u s t r a t e d well by 8.1, specifically, t h e kind of effectiveness of

K , relative t o DA,

number f i e l d

K' of

f o r extensions

K

for the

S ,

established logi-

cally a t t h e end of Robinson and Roquette (1975) and mathematically on pp.2.092.10 of Serre (1980).

The former i s so crude t h a t

some

of i t s weaknesses are

easily s t a t e d i n familiar l o g i c a l terms; f o r example, general versus primitive recursiveness, Turing versus many-one reducibility.

But t h i s gives no idea of t h e

dominant parameters involved, f o r example, t h e features of t h e curve which a r e enough t o determinethe degree of applied t o

a:

K'

obtained by

K' involved i n DA',

, (with similar questions about t h e degree of the - possibly choosing a variant of - (p* i n 5 7 ).

C in

S

short f o r : DA algebraic numbers

A s s o often with

c l a s s i f i c a t i o n s t h e inadequacies of t h e l o g i c a l variety mentioned a r e seen most clearly by looking a t cases which f a l l i n t o t h e 3 class, f o r example, the easy reduction of

S

to

DA'

, and

t h e delicate one of S

resp. pp.1.12 -1.13 of Serre (1980). as much as i n economics, only

after one

language : of the relevant parameters

to

DA

on pp.2.09-2.10,

Without exaggeration: i n mathematics almost

-

has a good idea what t o count

-

i n fancy

can a c l a s s i f i c a t i o n by some 'universal'

complexity measure, such a s r a t e of growth, be rewarding. 9.2. A s f o r axiomatizations, inspection shows t h a t t h e argoments involved i n t h e proof of Siegel's theorem can be formalized i n a r a t h e r weak subsystem of some of them a r e d e c i d e w d i f f i c q l t .

T h i s confirms (also

Ip ,though

i n t h i s instance) general

experience t h a t d i f f i c u l t y i s not correlated with socalled l o g i c a l strength, measured by t h e l o g i c a l complexity of t h e principles and t h e l i k e

-style

-

t h a t a r e used.

- of

induction, comprehension

More significantly, there i s another, mathematical

of axiomatization, which i s highly effective here; specifically, by using

t h e Product Formula t o axiomatize t h e subject of S i e g e l ' s theorem and, indeed, of algebraic number theory i n Lang (1962) pp. 122-U5, resp. Artin & Whaples (1$5), already mentioned i n 5.2.

I n contrast, t h e l i t e r a t u r e on ( f i l i g r e e ciistinctions

concerning) l o g i c a l strength has kept very close t o axiomatizations suggested by

258

G . KREISEL, A. MACINTYRE

Gentzen's o r d i n a l a n a l y s i s .

They always involve some p a r t i a l orderings, such as

f i n i t e l y and i n f i n i t e l y branching, wenfounded and o t h e r t r e e s .

That l i t e r a t u r e

has not even considered t h e imaginative mathematical axiomatizations above or, f o r t h a t matter, t h e most f a m i l i a r ones (of ordered, p-a3ic and o t h e r f i e l d s ) , l e t alone whether they f i t i n t o any of t h e c u r r q t l o g i c a l c l a s s i f i c a t i o n s ; more prec i s e l y , modulo some system regarded as ' b a s i c ' i n t h e t r a d e . Once t h i s formal matter i s decided ( p o s i t i v e l y ) , one could go on t o t h e more i n t e r e s t i n g question: Which s i d e g a i n s what from such equivalences?

Comparable equivalences were popu-

l a r at t h e t u r n of t h e century; f o r example, between Pappus theorem i n p r o j e c t i v e geometry and commutativity i n skew f i e l d s ( i n a l g e b r a ) , with a q u i t e d i s a s t r o u s balance of t r a d e f o r algebra: a s o r t of welfare p r o j e c t f o r t h e kind of f i l i g r e e This i s i n sharp c o n t r a s t t o

axiomatic geometry i n t h e p a r a l l e l considered here.

t h e many c o n t r i b u t i o n s of t h e more c e n t r a l p a r t s of geometry t o o t h e r branches of mathematics, l e t alone physics. PS. summarizing t h e broad conclusions of § § 7-8.

S i e g e l ' s theorem f o r K-integral

C i s d e r i v e d c o n s t r u c t i v e l y from Roth's theorem ( i )f o r

p o i n t s on a curve

p o i n t s on t h e Jacobian A

of

C i n t h e extension

Though t h e a l g e b r a i c numbers r e l e v a n t f o r any given

required by 4.4,

-

described i n 8.1, and

K'

(ii) for K-points on t h e not n e c e s s a r i l y a b e l i a n v a r i e t i e s

8.2.

K'

A (K)

described i n

C are s p e c i f i e d as

it remains t o be seen whether e i t h e r ( i )o r ( i i ) i s p r a c t i c a l l y

u s e f b l along t h e l i n e s of

4.1, t h a t i s , whether height bounds a r e a v a i l a b l e f o r

approximations t o a l g e b r a i c numbers r e l e v a n t t o some

C

of i n t e r e s t .

Concerning 8.2 it should b e noted t h a t , though t h e search procedure starts with E S(m) , and S(m) may contain ($x : x E VK/m) properly, it f i n i s h e s up u i n t h e smaller s e t , s i n c e by p.265 of Cassels (1966) for o t h e r : Vu(K) i s empty, and so could not c o n t a i n f3' . It i s e s s e n t i a l t h a t we need

all

u

with

only decide whether

Vo(K)

-

-

contains

K-rational p o i n t s of height

<

hy(m2-1)

' e s s e n t i a l ' i n t h e sense t h a t no e f f e c t i v e method i s known f o r deciding whether t h e corresponding diophantine equations have K-rational s o l u t i o n s t o u t c o u r t .

;

259:

Constructive logic versus algebraization REFERENCES

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1967, Foundations of c o n s t r u c t i v e a n a l y s i s , Mc G r a w H i l l .

BROWDER, F.E. (ed. )

1976, Proc. Symp. Pure Math. 28.

1967, C a l c u l d i f f g r e n t i e l , Hermann.

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1962, Arithmetic on curves of genus 1. I V . CASSELS, J.W.S. vermutung, J. r e i n e angew. Math. 211, 95-112.

Proof o f t h e Haupt-

CASSELS, J.W.S. 1966, Diophantine e q u a t i o n s w i t h s p e c i a l r e f e r e n c e t o e l l i p t i c curves, J. London Math. SOC. 41, 193-291.

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1955, R a t i o n a l approximations t o a l g e b r a i c numbers,

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GOAD, C.A.

1975, Programs and r e s u l t s i n l o g i c I, Notices A.M.S. 22, 476

1980, Computational u s e s of t h e manipulation o f formal proofs, STAN-

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4,

KREISEL, G. 1965, Mathematical l o g i c , T.L. matics, v o l . 3 , 95-195. Wiley, N.Y.

S a a t y (Ed.) L e c t u r e s on modern mathe-

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G. KREISEL, A. MACINTYRE S. 1959, Abelian Varieties, Interscience.

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MINTS, G.E.

MORDEU, L.J. 1.969, Diophantine Equations, Academic Press, London. MUMFORD, D.

1976, Algebraic Geometry I, Springer Verlag.

ROBINSON, A. and ROQUETTE, P. 1975, On the finiteness theorem of Siege1 and Mahler concerning diophantine equations, Number Theory 7, l21-176. ROQUETTE, P. 1976, On the division fields of an algebraic fbnction field of one variable. An estimate for their degree of irrationality, Houston J. of Math. 2, 251-287.

RGTH, K.F. 1955, Rational approximations to algebraic numbers, Mathematika 2, 1-20.

SCHWICHTENBERG, H. 1977, Proof theory: some applications of cut-elimination; pp. 897-895 in : J. Barwise (Ed.), Handbook of Mathematical Logic, North-Holland. SERRE, J.P. 1980, Autour du Thgorkme de Mordell-Weil, Lectures of 6th and 13th October, Collkge de France. Notes by M. Waldschmidt. SHOEWIELD, J.R. 1967, Mathematical Logic, Addison Wesley. ,I

SIEGFL, C.L. 1929, Uber einige Anwenciungen diophantischer Approximationen, Gesammelte Abhandlungen, 209-266.

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THE LJJ. BROWER CENTENARY SYMFOSIUM A S . lheLrtm ond D. von Dolen (editors) 0 North-Hollond Publishing Compmy. 1982

26 1

SOME EXTENSIONS OF MARKOV'S CONSTRUCTIVE CONTINUUM AND THEIR

APPLICATIONS TO THE THEORY OF CONSTRUCTIVE FUNCTIONS B.A. Kuher Computing Center, Akademia Nauk ul. Babilova 40 117967 MOSKWA, USSR.

1. a,B

Constructive real numbers are rational numbers or pairs of algorithms

(coded in the proper way) such that a

B

and

determine, respectively, the

sequences of rational and natural numbers such that

Markov's continuum in the title of the paper is the totality of constructive real numbers with the arithmetical operations and equality and order relations between them. Modulo technical details it is precisely the model of the continuum proposed by A.A. Markov (cf. for example, C i - 3 J ) for developing mathematical analysis within the framework of the constructive approach to mathematics which he has founded.

This paper is carried out within this framework. Let us introduce some notations and definitions (unexplained concepts and

2.

definitions of constructive analysis can be found in C41). use N ,

R

and P

tive real numbers.

In the sequel we shall

for, respectively, the sets of natural, rational and construcThe letters i,j,k,L,m

(with or without indices) will range

over the natural numbers. Analogously, the letters r,s used,respectively, for the elements of

and

a,b,x,y

will be

R and D.

By "algorithm" we mean normal algorithm in some sufficiently large alphabet. The scheme of a normal algorithm can be coded as a word in a two-letter alphabet. We call this word the encoding for the given algorithm. We shall say that an algorithm a

is of type M ,

-;+

M2,

where M

M2

are two sets of words, if

262

B.A. Ia(P)

(Here

means that a

KUSNER If, in addition, a

P).

is applicable to the word

is applicable to any word in M I , then we shall say that a Let a be an algorithm, P

a word.

By

-ap

is of type M

1

+ M2.

we shall denote the algorithm con-

structed in some definite way such that, for every word

Q,

Following A . A . Markov 1 2 1 , we shall define a constructive function (c.f.) as an algorithm f of type P

+

0 such that if x = y then f(x)

=

f(y)

paper we shall deal only with everywhere defined functions).

By

denote the constructive segment (interval) with endpoints a

and b

(xEaAb

z

asxsb, x

A c.f.

E

(aVb) we

a A b 5 a < x < b).

f will be said to be uniformly continuous on the segment aAb

one can construct an algorithm 6

A c.f.

aAb

(in this

f

of type N

+

N

such that

is said to be pseudo-uniformly continuous on aAb

Let us call

f

if

to be effectively nonuniformly continuous on aAb

construct a rational number

if

r > 0 and algorithms 6 1 , 6 2

of type N

if one can +

P such

that Vn(61(n),62(n)'EaAb

3.

Besides

& 161(n)-62(n)1

< 2-"

&

lf(61(n))-f(62(n))l

2

r).

D, several larger constructive number systems have been consid-

ered in the literature; among these, we think the system of pseudonumbers to be especially interesting. Let us recall that by pseudonumbers we mean rational numbers or encodings of algorithms of type N

+

R

satisfying the "non-effective" Cauchy criterion

The system of pseudonumbers has been thoroughly studied in recent years.

In par-

Extensionsof the continuum

263

ticular, simple descriptive and computable characteristics of pseudonumbers have been found. Namely, pseudonumbers can be interpreted as

A -numbers in terms of 2

the arithmetical hierarchy or (what is constructively more convenient) as #'-computable real numbers (cf. C5-81). the set of pseudonumbers by denoted in that style by

uI

In view of this observation, we shall denote (the constructive continuum

Do).

We shall use the corresponding notation 4.

might have been

(pAq)l

where

p,q

A s is well-known, any c.f. is continuous at any point

Dl

E

x

E

u.

and

p 2 q.

But, in

spite of this, such a function can be unbounded on the unit segment, or bounded and effectively nonuniformly continuous on

(cf. c 4 ; 9-10]).

Oh1

In connection with this important result the question arises whether an effectively nonuniformly continuous c.f.

f has some special properties in connection

with some constructive continuum richer than by new points one may discover that those points.

f

U. Indeed, after an enrichment U

cannot be continuously extended at some of

If this would happen, we would get a new understanding of the char-

acter of the paradoxical properties of tioned system of pseudonumbers

u1

f.

It is natural to take the above men-

as the required extension of

u.

Let us note

that there is another aspect to the questions considered. A s is well-known, there exist two essentially different approaches to a definition of computable functions over the real numbers.

In the first case, an operator which calculates a function

finds approximations to a value of the function using approximations to an argument number.

In the second case, the operator uses a code of an argument number to get

a code of the resulting number.

In the first case we say that the function is

computed by a Kleene operator (The partial recursive operators of Kleene [ I l l are of this kind).

In the second case we speak of Markov operators (cf. definition of

constructive functions).

Kleene's computdbility is nearer to the intuitionistic

interpretation of real functions, while Markov's approach allows one to develop the theory of constructive functions much more closely to the intuition given by classical analysis since one regards functions just as point-to-point mappings. The constructive definition of functions over

u

computable by Kleene may be given

B.A. KUSNER

264

without any difficulties (cf. 16-71).

By a tree we shall mean a set of finite se-

quences of natural numbers which, together with each finite sequence, contains all of its initial segments.

if Vnll3mT(n,m)

and

... *i

*

numbers

i

we have

f(k,ik).

r

A binary predicate

Vnk&r(n,k)

r(n,e))

&

is said to be functional (over 3

k=L). Afinite sequenceof natural

r

is said to be a beginning of

0 < k 5 n,

if, for every

A functional predicate is said to be a path in a tree M

each of its beginnings belongs to M. is an algorithm

IY

N

of type

+

N

T

A path

Let

An algorithm

be the set

T ,

of type T , 7

w

said to be a (I,O)-functional if, for every two finite sequences s I , that

is an initial segment of

s1

We shall say that a (I,O)-functional

NXT,

of this path such that

s

N

-rt

ery k.

and

is defined Qn 5

r.

w(s)

m = Y

on

'Y

r

=

-

1

r

is

such

s2

we have w(s ) = in T ,

An algorithm

n.

W(S*).

and as-

Y

of type

is a (I,O)-functional for ev-

Yk

r

to be defined on

The value of

- .

N

if one cannot fail to find a begin-

is said to be a (I,I)-functional if We shall consider Y

P(n,m)

lw(s)

lw(s ) 2

is defined on a path

w

(written w < T > = n)

sumes on it the value n ning

lw(s,),

and

s2

if

is said to be computable if there

such that Vi(r(i,ct(i))).

of all finite sequences of natural numbers.

N)

if every (I,O)-functional

i s the predicate Y < D

-

k'

such that

The concepts just introduced are constructive versions of

partial recursive functionals and operators over everywhere defined functions. We have taken this approach from C121. Let T2 T -9

be the set of all finite sequences composed of

different from 0

or

Let us fix some single-valued effective enumeration of

1. 1

1

3

3

segments of the form -cA-c + n will be denoted by

T

.

3,

where

for a sequence n aAa + ;(b-a)

=

in T,,,2. ning in T -3

of length

2

I,

E

T -3

2

we set 0

=

I OS*] = a+g(b-a)Ab.

and

We shall say that of this path,

s

i s any integer.

c

We shall assume that

associate with every finite sequence s

s*o

and

be the set of all finite sequences in which only the first component can be

2

0

I's,

and

0's

x

E

0

is defining some x

T

0

2

OA-

=

3

The segment with index and

T~

=

a rational segment 0

$AI.

.

Let us

Namely,

T ~ ,and, if 0 = aAb, then Let x

D and let r be a path

E

r

defines x

.

It is not hard to prove that any computable path

E

(written

r

= x)

D and, conversely, for any

if, for every begin-

x

E

0,

one can effec-

Extensions of the continuum

tively find a computable path in Tm,2 al, f

a c.f.

computes

We shall say that Y

r

every computable path

in T

2

-3

defining x. f

r

such that

265

Let

be

Y

a (1,l)-function-

if, for every x

=

x

E

D and for

is a path in

Y

'm, 2

Y = f(x).

and

Analogously, pseudonumbers can be defined by a limiting-computable or, in (cf. L61). It is also clear that, i n 2 representing numbers in the unit segment, one can limit oneself to paths in T2.

other words, @'-recursive path in T

-9

By virtue of the well-known theorem of G.S.

Ceitin (see, for example, L 4 1 ) , every

c.f. is computed by some (1,l)-functional.

It turns out that extendibility of that

functional to paths defining pseudonumbers guarantees the uniform continuity of the c.f. under consideration. In fact, let u s call a (1,l)-functional to be I-complete on

if it is defined on every @'-computable path in

T2

Theorem I.

A c.f.

is uniformly continuous on

f

T2

exists a (I,I)-functional which is I-complete on

The proof of this theorem can be found in C6l.

D

over

T2'

OAl

if and only if there

and computes f. So a Kleene operator computing

an effectively nonuniformly continuous c.f. always has a singularity in

the continuum P I .

The situation is quite different in the case of Markov's com-

putability. The construction of the corresponding counterexample (theorems 2-3 below) is the principal aim of this paper.

5. An algorithm p,q

P I , from p

E

of type D l

0

= q

+

it follows that

Dl

is said to be a I-c.f. if, for every

0(p)

= O(q)

(the equality relation over

pseudonumbers can be introduced in the obvious way, in fact, if and

algorithm a ,

- B(m)I

Vnll3m(Ia(m)

q

by <

B,

p

is given by an

then p = q means that

2-")).

Let u s note that I-c.f.'s, similarly to c.f.'s,

have properties of continuity (cf. C131). Let

x

D.

E

We shall denote by

of the algorithm of type N x

-

E

D\R. -

+

It is clear that

x1 = x2.

R E

the x

D,

for every

x

Thus we obtain a natural injection of O

extends a c.f.

f

itself if

from the description of

I

I-c.f.

-

x

if

E

D

D

x

x

E

R

and the encoding

(cf. section 1 ) if

and that x 1 = x2 into P I .

implies

We shall say that

266

B.A. K U ~ N E R Vx(xrP

A c.f.

f

3

f (x)=O(r;)).

is said to be I-complete if there exists a I-c.f. which extends f.

Theorem 2.

One can construct a I-complete c.f. which is unbounded on

Theorem 3 .

(cf. c 7 1 ) .

OAl.

One can construct a 1-complete c.f. which is effec-

tively nonuniformly continuous on

OAl.

Let us note that in all earlier examples of effectively nonuniformly continuo u s c.f.’s the property of I-completeness was not obtained.

To prove the theorems 2-3 we shall need the following lemma which is of independent interest. One can construct an algorithm y

Lemma 1.

N

of type

+

R

such that:

OVl;

a)

M(y(k)

b)

for every p

c)

One can construct algorithms 6 1 , 6 2

E

E

P I , one can find a natural number m

of type

P

-+

N

E

such that

N such that,

for every

B2(X)

x

E

D and k

Proof.

2

6,(x),

the condition

Let u s fix an algorithm U

algorithms of type any algorithm a

N

2 2-

of type Ed2

holds. -s+

R

which is universal for

R. More precisely, the following condition holds: for

of type

Let u s further denote by

ly(k)-xl

N

[Ul

R

one can find an

i

E

N

such that

an algorithm such that

Let us construct an algorithm a(’)

such that a(l)(i,O)

r

0 and

26 I

Extensions of the continuum

L,j),j+l) r O

i,j),j+l)

a(')

It i s not d i f f i c u l t t o see ( ( 1 ) - ( 3 ) ) t h a t

7

1.

possesses t h e following prop-

erties:

I)

a(')

2)

f o r every

N2

i s an a l g o r i t h m of type

+

N

and

k k ilj(a(')(i,j)

7

k+l

& lU(i,e).

t=0 Let f u r t h e r

v 1 , 'p2

Let t i o n of

'pl

a(2)

be an a l g o r i t h m such t h a t

b e algorithms of t y p e

i f t h e r e e x i s t s an algorithm

I)

0 5 B(n+l) - B ( n ) 5 1

2)

tflc3e(B(L) = k ) ;

3)

if

B(n) > 0

then

f o r every

@,(n)

By p r o p e r t i e s 1)-2) o f

N2

-+

R;

and, f o r

ti

2) i f

Let now

N

B

R.

We s h a l l say t h a t

of type

N

-+

N

Q2

i s a dilu-

such t h a t :

n;

'pl(B(n)).

a(')

w e have:

i s a n a l g o r i t h m o f type N

a ( 3 ) , a(4),

j 2 k

=

+

I) -+

R,

a(2)

i s an a l g o r i t h m of type

then

be algorithms such t h a t

Liz)

is adilutionof

-

Ui.

268

KUSNER

B.A.

It i s n o t h a r d t o check t h a t

N2

i s an algori thm of t ype

has the following properties:

R;

+

2) i f

ci

1)

N

i s a n a l g o r i t h m of t y p e

+

R

such

that

then

For any p

p

E

(Vn(p(n) = p)

DI

w e s h a l l d e n o t e by

if

p

E

R).

Let

bers

(5)

r

t h e a l g o r i t h m of type

From p r o p e r t i e s 11-2)

follows e a s i l y t h a t , f o r every number of

p

p

E

D,,

i n t h e n u m e r a t i o n g i v e n by

N

+

defining

of t h e a l g o r i t h m

it

one c a n f i n d a n a t u r a l number U)

R

i

(the

such t h a t

be an a l g o r i t h m t r a n s f o r m i n g any f i n i t e s e q u e n c e of r a t i o n a l num-

* ... *rk

i n t o a r a t i o n a l number from

Iri-a(')(r0*

The r e q u i r e d a l g o r i t h m

... *rk) I y

> 2- i - 3

OVI

.

i s now g i v e n by

such t h a t , f o r

0 5 i

5

k

269

Extensions of the continuum

It i s obvious t h a t

y(n)

E

OVI.

b

... * a ( 5 ) ( n , n ) ) .

i s a n a l g o r i t h m of t y p e

y

Let f u r t h e r

L e t u s suppose t h a t

(6)

*

= a(6)(a(5)(o,n) * a ( 5 ) ( l , n )

y(n)

p

0,.

E

We s h a l l f i n d

N i

+

R

and t h a t

such t h a t (4) holds.

i s such t h a t i t holds

Vk(a(5)(i,b)

~ x ( ~ ) ( i , b + k&) l p - a ( 5 ) ( i , l ) /

=

Then, by (5) and t h e d e f i n i t i o n o f

for

y,

n 2 max(i,b)

< 2-i-4 -

).

w e have

which w i t h ( 6 ) g i v e s u s

-

~p

y(n)l >

z-~-~,

and t h i s f i n i s h e s t h e p r o o f o f c l a u s e b ) o f t h e lemma. L e t now type

N

+

R

x

D.

E

a (') ,

Therefore, a(3),

such t h a t t h e algorithm

-U .

is a n a l g o r i t h m of

and it h o l d s t h a t

By p r o p e r t y ( 2 ) of ues.

i

We f i n d

a(')

the algorithm

l ~ ( ( ~ ) ( i ) L. e t

a(5)

and by (7),

a(5)(i,j)

=

Iv(n) -

u(i,i)I

l u ( i , i ) - xl

5

Then, by c o n s t r u c t i o n o f

w e have

lo

n 5 max(i,m),

g e t s a l l n a t u r a l numbers as i t s v a l -

a ( 4 ) ( i ) = m.

U(i,i)

Hence, f o r

a(') i

for

j

for

j 2 m

we have ( c f . > 2-i-3

z-~-~

<

m

(S), ( 8 ) )

a(1)

,

B.A. K U ~ N E R

270

from which we we obtain that that for n 2 max(i,m)

The construction of algorithm

from clause c) of the lemma is now obvious.

62

The lemma has been proved. Lemma 1 is of independent interest from the point of view of a classical theorem of Bolzano-Weierstrass. The algorithm y

provides an example of an effective

sequence of rational numbers from the unit segment which has no limiting points in This result turns out to be precise enough.

Dl.

continuum D2

If we took into consideration the

consisting of $"-computable real numbers (cf. Gal), it would be not

hard to see that every algorithmical, bounded sequence of rational numbers (and even numbers from 0) has a limiting point in 6.

y(i) i and

f

Proof of theorems 2-3. y(j).

It is not difficult to construct an algorithm 6

= $

for

i

f

6(n)

and right endpoints of

-Fn

It is clear that we can assume that, for

into a rational neighbourhood of 8(i)n6(j)

D2.

j.

y(i)

Let us denote by

transforming every

6(i) G O A I ,

such that

rn, sn

i z j,

I6(i)I

<

2-l

respectively the left

be an algorithm such that, for every n,

and let F

is a c.f. and

10

and, in addition,

Fn

for

(xsr

v x r s )

r is linear on

r +s

+s

r A

n

and

As

.

Let us consider the functional series m

In virtue of lemma 1 , this series constructively converges for every x one can construct a c.f. m

g

such that, for every

x

E

D

E

D.

Hence,

Extensions of the continuum r +s Clearly, g ( v )

=

n

and therefore g

27 1

is unbounded on

OAI.

I-completeness

follows without any difficulties from clause b) of lemma 1 which implies

of g

the local uniform continuity of

g.

Theorem 2 has been proved.

Let us remark that

theorem 3 follows from theorem 2 because it is not difficult to see that every c.f. unboundedon the unit segment is effectively nonuniformly continuous. But it is r +s easier to note that g(r ) = 0, g ( v ) = n and that lr - s 1 < 2-". n n

7.

Theorems 2,3establish an essential difference between Markov- and Kleene-

computability over 0 which reveals itself and can be formulated after we have taken into consideration the extension

D,

of

D.

enough to obtain that continuous extendibility of a c.f. antee any properties of uniform continuity of

f.

o2

then f

therefore, f

f onto 0 ,

f has a continuous extension

cannot be effectively nonuniformly continuous on

is a pseudo-uniformly continuous function on

Bibliography

would guar-

But at the same time, from a

remark after the proof of lemma 1 it follows that if onto

D, is not

On the other hand,

OAl.

OAl

and,

BA.K U ~ N E R

212

Editorial note.

Of the items in the bibliography, the nrs. 2, 7, 9, 10, 13 are

available in translation: 2.

A.A. Markov,

7.

B.A. Kugner,

On constructive functions. AMS Transl. 29(1963), 163-195.

On a type of computable real functions, Soviet Math.Dok1. 15(1974), 466-470.

Extensions of the continuum

9.

I.D.

Zaslavski?,

Some properties of contructive real numbers and constructive

functions. AMS Transl. 57(1966), 10. I . D .

273

Zaslavski;, G.S.

1-84.

Ceztin, O n singular coverings and properties of construc-

tive functions connected with them. AMS Transl. 98(1971),

41-89.

13. B . A . Kuxner, Continuity theorems for some types of computable operators. Soviet Math.Dok1. 14(1973),

Nathematical Reviews: ( 4 ) 52, p.7 (#53); (Uspenski;).

(I)

221-225.

16, p.436

(Markov);

(6) 53 p.1477 ( # 10652);

(3) 27, p.684 (#3528); ( 8 ) 80a #03071;

( 1 2 ) 19, p.2

THE L.EJ. B R O W E R CENTENARY SYMPOSIUM A S . lbektm and D.van Dalen (editors) 0North-Holhnd Fubshing Company,1982

215

TWO SHEAF REPRESENTATIONS OF ELEMENTARY TOPOSES.*)

J. Lambek and I. Moerdijk McGill University and University of Amsterdam

We study two sheaf representations of elementary toposes, by drawing two different analogies between topos theory and ring theory. When the Heyting algebra of subobjects of 1 is Boolean, the two representations coincide. In particular, when the topos satisfies the axiom of choice, this sheaf representation comprises Henkin's completeness theorem for type theory, the stalks of the sheaf being his general models.

1. INTRODUCTION A

theorem of fundamental importance in Algebraic Geometry asserts [GD]:

Every commutative ring is isomorphic to the ring of continuous global sections of a sheaf of local rings. While many generalizations of this result to other algebraic systems have been pursued (e.g. [Cl], [W]), there is no unanimity on how to extend it to noncommutative rings. It therefore came as a surprise to the authors when they realized that the result remains valid if "commutative ring" is replaced by "elementary topos". The closest approximation to this result in the literature is the result by Comer [CZ] for cylindrical algebras, an algebraization of first order logic, whereas toposes may be regarded as an algebraization of higher order logic. Let us recall the outlines of the proof for commutative rings. With any commutative ring R one associates a compact topological space Spec(R), whose points are the prime ideals P of R, and whose topology is the so-called StoneZariski or hull-kernel topology, with basic open sets of the form D(r) = IPESpec(R) IrtPI. On Spec(R) one defines a sheaf of rings whose sections over D(r) of the ring R[r-']

obtained from R by adjoining an inverse r-'

are the elements for the ele-

ment r. The stalks of the sheaf are the localizations Rp obtained by localizing R at the prime ideal P. For each element r of the ring R one has a continuous section of the sheaf space, where ;(P) = Lp(r), Lp being the canonical homomorphism R -+ Rp. The correspondence r I+ 1 between the elements o f R and the sections of the sheaf space is one-to-one and surjective.

J. LAMBEK, I. MOERDIJK

216

P

Figure 1 Let us t a k e a c l o s e r look a t t h e l o c a l i z a t i o n tained by i n v e r t i n g a l l elements of 1 Rp = R[X- 1, where X = {seRlsp!PI. and

s

The homomorphism

C.

E

i s c a l l e d t h e component of

0

R/Op

when

at -f

i s von Neumann r e g u l a r :

R

Rp

-f

Rp

at

R

I t i s ob-

P.

P, that is,

r onto r / l .

sends

r/s with

r

E

R

I t s kernel

P;

it plays an important r o l e i n Commutative

Rp

i s f r e q u e n t l y an isomorphism, f o r example,

r

f o r each

E

R

I n t h i s p a r t i c u l a r case one a c t u a l l y has

rsr = r.

of

which a r e not i n

I t s elements have t h e form

L : R P

Algebra.

The monomorphism

R

there exists Op = P

and

s

E

such t h a t

R

Rp = R/P,

and

t h e sheaf i n t h i s s i t u a t i o n has been s t u d i e d by P i e r c e [PI. SPECTRAL SPACES

2.

I n [Ho], Hochster d e f i n e s a category of topological spaces, t h e so-called

s p e c t r a l spaces, and shows t h a t t h e s e a r e p r e c i s e l y t h e spaces of t h e form Spec(R)

f o r some commutative r i n g

They a r e a l s o p r e c i s e l y t h e spaces which

R.

show up a s Stone spaces ( o r s p e c t r a ) o f d i s t r i b u t i v e l a t t i c e s .

In t h i s s e c t i o n

we c o l l e c t some f a c t s about s p e c t r a l spaces t h a t w i l l be needed below.

For more

d e t a i l s , the reader i s r e f e r r e d t o [Ho] and [BD] and t h e references c i t e d t h e r e . Recall t h a t a topological space s e t has a unique generic p o i n t . space i s

To,

but sober and

DEFINITION 2 . 1 (Hochster).

T1

sober i f

is

X

each i r r e d u c i b l e closed sub-

Every Hausdorff space i s sober and every sober a r e incomparable.

A t o p o l o g i c a l space i s s p e c t r a l i f i t

i s compact and

sober and has a b a s i s of compact open s e t s which i s closed under f i n i t e i n t e r sections.

(We use "compact" i n t h e sense t h a t it does n o t imply "Hausdorff".)

H

= (H,h,v,O,l)

A prime f i l t e r i n

x

A

y

E

P,

and

Let

be a d i s t r i b u t i v e l a t t i c e (with l a r g e s t and smallest elements).

H x 2 y

i s a proper f i l t e r E

P

implies

x

E

P 5 H (i.e. P)

0

k

P # !ii,x , y

E

P

with t h e a d d i t i o n a l p r o p e r t y :

implies

Two sheaf representations if Let

Spec(H)

x v y

E

P

then e i t h e r

x

E

277 or

P

H,

be t h e s e t o f prime f i l t e r s of

y

E P.

topologized by taking a s

basic open s e t s t h e s e t s

for

x

for

x,y

E

E

H,

(For t h e t h i r d statement:

if

x $ y,

choose a f i l t e r y 1 P;

i s maximal with r e s p e c t t o t h e property t h a t

second of t h e s e p r o p e r t i e s implies t h a t topology.

H i n BH(x).) Observe t h a t

(Usually, we w i l l omit t h e s u b s c r i p t

H.

The space

Spec(H)

{B(x)IxeH}

i s prime.)

P

H - P

The

i s indeed a b a s i s f o r a

Thus, a subset

i s a prime i d e a l .

P

of

of

H

H. is a

Therefore, t h e space

could a l s o be defined a s t h e s e t of prime i d e a l s of

Spec(H)

x which

containing

i s c a l l e d t h e spectrum (or Stone space)

The dual of a prime f i l t e r i s a prime i d e a l . prime f i l t e r i f and only i f

P

then

H , topologized by

taking a s b a s i c open s e t s t h e s e t s D(x) = {ICHII PROPOSITION 2 . 2

Let

i s a prime i d e a l and

H be a d i s t r i b u t i v e l a t t i c e . i s sober;

(1)

Spec(H)

t h e complete Heyting algebra subsets of

Spec(H)

Heyting algebra

In p a r t i c u l a r ,

Proof:

Spec(Cl)

the s e t s

B(x)

O(Spec(H))

o f open

i s isomprhic t o the complete

Idk(tl)

of i d e a l s o f

H;

a r e a l l compact.

i s a s p e c t r a l space.

The proofs a r e analogous t o t h e proofs of s i m i l a r statements about

i s a commutative r i n g .

Spec@),

where

ideal

t o t h e open subset

I

Then

(2)

(3)

~11).

(3) i s immediate.

R

The isomorphism of ( 2 ) sends an

U{B(X)IXEI} o f

Spec(H).

Using t h i s isomorphism,

J. LAMBEK. I. MOERDIJK

278

The easy proof of t h e following p r o p o s i t i o n i s l e f t t o t h e reader. PROPOSITION 2 . 3

Let

be any sober space and

X

compact open s u b s e t s of (2)

X

(3)

S(X)

Spec(H) X

1

Spec(B(X)).

i s a c o n t r a v a r i a n t functor: cp

H;

f o r some d i s t r i b u t i v e l a t t i c e

i s closed under f i n i t e i n t e r s e c t i o n s

( i . e . , i t i s a l a t t i c e ) and Spec

v - s e m i l a t t i c e of

Then t h e following a r e equivalent:

X.

X

Note t h a t

the

is spectral;

(1)

homomorphism, then

B(X)

if

induces a continuous mapping

cp:

ff

K

-+

is a l a t t i c e

Spec(cp): Spec(K)

+

Spec(t()

by

and Spec(cp)-l(Sff(x)) = BK(cp(x)). I n f a c t , it i s n o t d i f f i c u l t t o s e e t h a t t h e p a i r of f u n c t o r s (Spec,O) d e f i n e s a c o n t r a v a r i a n t adjunction between t h e category of d i s t r i b u t i v e l a t t i c e s and t h e category TopSp

of t o p o l o g i c a l spaces

and continuous mappings. The mappings

have t h e a d d i t i o n a l p r o p e r t y t h a t t h e i n v e r s e image of

Spec(cp)

a compact open subset i s again compact open. Let us w r i t e

mappings.

Then e s s e n t i a l l y t h e same adjunction may a l s o be regarded a s giving a

r e f l e c t i o n from

TopSp

SpecSp

Hochster c a l l s such mappings

spectral.

to

f o r t h e category of s p e c t r a l spaces and s p e c t r a l

SpecSp: t h e functor

SpecoO: TopSp

a d j o i n t t o t h e i n c l u s i o n f u n c t o r . (Of course, t h e category

+

SpecSp

SpecSp

is l e f t

i s dual t o t h e

category of d i s t r i b u t i v e l a t t i c e s . ) Finally, l e t one has a b a s i s

US

say a word about sheaves on s p e c t r a l spaces.

8

f o r a topologic space

s e c t i o n s , then a functor Sets,

8.

+

Sets

extends uniquely t o a sheaf

G : O(X)OP -+

where

i f and only i f of

F: Bop

In g e n e r a l , i f

which i s closed under f i n i t e i n t e r -

X

F

has t h e s h e a f p r o p e r t y with r e s p e c t t o coverings by elements

By t h i s we mean t h a t , given s e t s

B, Bi(ieI)

in

8 such t h a t

B =

U

i61

and a family of elements

x.

F(B~ n B for a l l

i,j

E

I,

E

j

F(B.)

+

which i s compatible i n t h e sense t h a t

Bi)(xi)

t h e r e e x i s t s a unique

=

F(Bi n B x

E

F(B)

j

+

B.)(x.) 1 1

such t h a t

F(Bi+B)(x) = xi

Bi

Two sheaf representations

for all i

E

219

I.

Thus, for spectral spaces X to define a functor

=

Spec(H),

to define a sheaf on X it suffices

having the sheaf property with respect to finite coverings. The same holds for sheaves with values in an arbitrary category (see [GI) having enough inverse limits to define G

from F.

QUOTIENT TOPOSES Before we can transport the above ideas out what corresponds to the construction of multiplicative subset C of R . If E is the set E(1,R) of all arrows 1 + R is a

3.

from rings to toposes, we must find the ring of fractions R[C-l] for a a topos with subobject classifier R, Heyting algebra (in particular, a

distributive lattice), and a multiplicative subset X of E(1,R) is just a filter base on E(1,R). Universally inverting the subobjects K(q) 1 classiwill be seen t o give rise to the construction of quotient toposes. fied by q E C Note that we may just as well assume that I: is a filter; for, if U V 5 1 and U is inverted, then V also becomes an isomorphism. For each filter F on E(1,Q) we shall define a topos E/F and a logical functor L(F): E + E/F. The objects of E/F are the same as the objects of E. The arrows f: A + B in E/F are arrows f: 1 + P(AxB) = nAXB in E such that E f i s an element of F. F o r arrows f,g: A + B the proposition t'xcA3!yEB<~,y> in E/F we write f*=-g to mean that f = g is an element of F. Moreover, we Put

Clearly, E/F is a category; it is not difficult to see that it is a topos and that L(F) is a logical functor. E/F will be called a quotient topes of E. (It is called "filter power" in [Jl].) While we prefer the construction of E/F given here, there are two other constructions that may have more appeal to the cognoscenti. First, one may regard E/F as the category of fractions obtained from E by inverting all those arrows 3!XCAfx = y belongs t o F. To f: A + B of E for which the proposition V YCB explain the second construction, let us first assume that F = (p) = {qlE k p * ql is the principal filter generated by p. Then E/F is equivalent (but not isomorphic) to the "slice" category E/K(p), where K(p) is the subobject of 1 classified by the proposition p: 1 + Q (see [LS3]). In general, E/F is equivalent to the direct limit of all E/K(p), where p ranges over all elements of F

J. LAMBEK, I. MOERDIJK

280

The quotient topos E/F is quite analogous to the notion of a quotient ring. In fact, we have the usual "homomorphism theorem" of algebra, which we state without proof. PROPOSITION 3.1 Every logical morphism L: E F between toposes factors uniquely into E L(F) E/F + F, where F is the filter of all propositions p in E for which L(p).=-T in F. -+

We claim that the arrow E/F

-+

F is uniquely determined, that is, L(F)

is

an epimorphism in the category of toposes and logical morphisms. This is proved like the uniqueness of

GI

in

[L2, Theorem 11.11.

COROLLARY 3.2 L(F): E + E/F is initial among all logical functors E which send all elements of F onto T.

+

F

We also state without proof the analogue of one of the usual "isomorphism theorems" of algebra. PROPOSITION 3.3 There is a one-to-one correspondence between the filters G E/F and the filters G of E containing F, and (E/F)/C 2 E/G.

of

THE FIRST SHEAF REPRESENTATION When R is a ring, one knows immediately what is meant by the underlying set IRI. In the case of a topos E, we shall define IEl as the union, automatically disjoint, of all sets E(l,A), where A ranges over objects of E. In passing from E to IE), we don't lose to much: an arrow f: A + B in E is still represented by the corresponding element of E(l,BA). o r even by the element {<x,y>cAxBlfx=y} of E(l,P(AxB)). What are we to make of the analogue of Figure 1 for toposes? Imitating the 4.

definition of the spectrum of a ring, we define the spectrum of a topos E to be the space whose points are the prime ideals Q of E(1,O) and whose basic open sets are the sets D(p) = IQlplQl. Thus, the spectrum of E is just the space Spec(E(1,n)) described in Section 2. From now on, we shall write Spec(E) f o r this space. Mostly, we shall think of the points of Spec(E) as prime filters P, so that basic open sets have the form B(p) = IPlprPl, for p E E(1,n) (see Sectioh 2). In the case of rings, we saw that the sections over D(r) correspond to the elements of the ring obtained from R by inverting r. It is clear from the description of quotient toposes given in Section 3 what the analogue for toposes is: the sections over B(p) correspond to the elements of the underlying set of the topos E/(p). More precisely, for each object A of E we have a functor A

281

Two sheaf representations defined on t h e b a s i c open s e t s of

Spec(€)

with t h e obvious r e s t r i c t i o n maps

A(B(p))

by

+

A(B(q))

for

B(p) 2 B(q),

that is,

E b q * p.

for

The important p r o p e r t y of t h i s functor i s t h a t it extends t o a sheaf on Spec(E).

In f a c t , i n view of what we s a i d a t t h e end of Section 2 , it s u f f i c e s t o

check t h e sheaf p r o p e r t y f o r f i n i t e covers by b a s i c open s u b s e t s . such t h a t , f o r a l l So suppose B(p) = B(qi) and ai E E/(qi)(l,A)

igl

j,

t h e r e s t r i c t i o n s of

t h e r e i s a unique to

€/(qi)

and

a.

a. t o

I

a E E/(p)(l,A)

According t o t h e c o n s t r u c t i o n of

qj)

agree.

Then we claim t h a t

t h e r e s t r i c t i o n of

i,

E/(qi),

i s an arrow

ai

1

3 ! x c A ~ ~ a i ,henceforth abbreviated a s

E C qi *

that is,

(qi),

A

a

ai.

is

such t h a t t h e p r o p o s i t i o n to

€/(qi

such t h a t , f o r a l l

and

i

(Actually,

3!a,.

ai

PA

-f

3!a..

i s an arrow

in

E

belongs

1 +'P(lxA).)

1

E

We a r e given t h a t

(qi

qj)

A

=+

ai = a j ,

for all

i

and

j.

Now r e c a l l a

theorem of i n t u i t i o n i s t i c type theory, which may be c a l l e d d e f i n i t i o n by c a s e s :

(This holds more g e n e r a l l y i f a r e terms of type

B.)

PA

E b p

=,

-

i s replaced by any type

Eb p

Using t h e f a c t t h a t

n 3!UEPAA (qi

*

i=l

B,

a s long a s t h e

(qlV"*Vqn),

= ail

o r , equivalently, t h a t

where

-

ai

Z

-a: 1

L((p))(ai).

unique arrow

-+

PA

Using d e s c r i p t i o n i n t h e topos in

E/(p)

we obtain a

such t h a t n

€/(PI b

E/(p),

A

(qi

*

- -

a = ail.

i=l

By showing t h a t E k 3!; (as i n t h e argument below), i t i s e a s i l y seen t h a t -a.=.L((p))(a) f o r some arrow a: 1 PA i n €. Hence E b qi * a a for +

each

i.

Since

E b qi

E b p * 3!a. Therefore,

3!ai, a

we have

i s an arrow

=

E C qi * 1

-c

A

in

3!a

f o r each

E/(p).

a.

we i n f e r t h a t

i,

i hence

J. LAMBEK, I. MOERDIJK

282

Obviously, t h e d e f i n i t i o n of

i s functorial in

(simply compose), s o we

A

we have a r r i v e d a t t h e following: PROPOSITION 4 . 1 defined by

For any topos

E,

qE: E

t h e r e i s a functor

8 , where 8(B(p))

nE(A) =

A s i n t h e case of r i n g s , t h e global s e c t i o n s of t h e sheaf

arrows when

in

1+ A

p*=*T, a

E.

E/{T)

* Sh(Spec(E))

= E/(p)(l,A).

1

are just the

This i s obvious from t h e d e f i n i t i o n of

corresponds t o an arrow

a: 1

-+

in

A

E.

A;

for,

A l t e r n a t i v e l y , i f we

p r e f e r t o p r e s e n t t h e sheaf by i t s espace BtalB, we a s s e r t t h a t t h e r e i s a b i j e c t i o n between mapping

a b

functor

E

-+

E(1,A)

a,

where

and t h e s e t of continuous s e c t i o n s of a(P) = L ( P ) ( a ) ,

L(P)

h

given by t h e

being t h e canonical l o g i c a l

E/P.

Spec E

P

Figure 2 Another way of expressing e s s e n t i a l l y t h e same observation a s above is t o say t h a t t h e functor on

Spec(E)

determined by

B(p)

C.E/(p)

gives a sheaf

-

with values i n t h e category of toposes and l o g i c a l morphisms.

on

Spec(E)

is

(see [GI), given arrows

p, q l , . . . q n : 1

+

such t h a t

R

Eb p

S(E) That

(qlv*-*vqn),

we have an e q u a l i z e r diagram of toposes and l o g i c a l f u n c t o r s

n

n

-

(The arrows i n t h i s diagram a r e t h e usual ones a r i s i n g from t h e canonical projections

E/(r)

+

E/(s)

for

E b s

r.)

The f a c t t h a t t h i s i s an e q u a l i z e r i s

again a s t r a i g h t f o r w a r d a p p l i c a t i o n of " d e f i n i t i o n by cases". between t h e sheaf

S(E)

and t h e set-valued sheaves

Observe t h e r e l a t i o n

h:

Having obtained a sheaf, one wants t o know what t h e s t a l k s look l i k e . u s u a l , t h e s t a l k a t a p o i n t (prime f i l t e r )

P

of

Spec(E)

As

i s defined t o be t h e

d i r e c t l i m i t of t h e toposes of s e c t i o n s of t h e sheaf on t h e (basic) neighbourhoods

of

P.

This i s t h e topos

283

Two sheafrepresentations

m-

( F o r the last equality, see Section 3 . ) We will call the topos E/P the ization of E at P. Observe that the global sections of an object A of E in the localization E/P are precisely the elements of the stalk at P of the sheaf A, since

Hence, the relation between the sheaf of toposes S(E) and the sheaves of sets pointed out above also holds at the stalks. Let us call the topos E if it has the disjunction property: for any p,q: 1

+

n,

EC p

v

q implies E b p

or E C q .

This terminology is analogous to that in ring theory: E is local if and only if clearly E/P E(1,Q) has exactly one maximal ideal. For a filter P on E(l,Q), is local if and only if P is prime. Not only is every stalk local, but every local topos arises as a stalk in a sheaf representation. For, if E is local, {TI is a prime filter in E(l,Q), hence E is isomorphic to the stalk of S(E) at {TI. Thus "local" is the characterizing property of stalks. We can now formulate the representation theorem as follows: Every topos is isomorphic to the topos of global sections of a

THEOREM 4 . 2

sheaf of local toposes. Let us briefly consider one particular case of the sheaf representation. We has exactly two elements, call a topos E cosimple if the Heyting algebra E(1,Q) cosemisimple if E(1,Q) is Boolean. The motivation for these terms comes from ring theory. Indeed, E is cosimple if and only if {TI is a maximal filter, and it is cosemisimple if and only if {TI is the intersection of all maximal filters. More generally, in any topos E the intersection of all maximal filters is the set of all propositions q such that E C l T q , which thus might be called the "coradical" of the topos. Still more generally, the intersection of all maximal filters containing a given filter F is the set of all propositions q such that

7.q

E

F.

PROPOSITION 4 . 3 The stalks of the sheaf S(E) the topos E is cosemisimple.

are all cosimple if and only if

J. LAMBEK, I. MOERDIJK

284

Proof: F o r a prime filter P of E(l,Q), P is maximal. But each prime filter of E(1,Q) I _

E(1,Q)

E/P is cosimple if and only if is maximal if and only if

is a Boolean algebra, by the remark preceding the proposition.

The reader will have noticed that there are two analogies with ring theory. According to the first, ideals in rings correspond to ideals in E(l,Q), while, according to the second, they correspond to fiIters in E ( 1 , Q ) . We have used the prefix "co" whenever a topos concept is named in accordance with the second analogy. Some topos theoretical concepts fall under both analogies and so acquire two distinct names. For example, cosimple = simple, coprime = local'. 5.

THE SECOND REPRESENTATION

For a topos E we also have a distributive lattice E(l,Q)OP, which may be considered (as in Section 2). Let us write

Cospec(E)

=

the spectrum of

Spec(E(l,Q)OP).

Using the correspondence between prime filters and prime ideals in any distributive lattice and the duality between E(1,Q) and €(l,Q)op, we may also describe Cospec(E) as the space whose points are the prime filters of E(1,Q) and whose basic open sets are the sets -

Note that

and that the complete Heyting algebra of open subsets of Cospec(E)

is isomorphic

to the complete Heyting algebra of filters of E(1,Q). For any arrow p: 1 .+ Q in E, we shall write

for the set of alternatives of p. Alt(p) is a filter, and if V(p) E: V(q) then so Alt (4) C Alt (p) Therefore, we have a presheaf T of toposes defined on basic open subsets of Cospec(E) by

E C q.* p and

.

the restriction maps being the canonical projections E/Alt(p)

-+

E/Alt(q)

for

Two sheafrepresentations

285

V(P) 5 V(q). In general, T is not a sheaf, as will become clear below. Therefore, we shall consider its associated sheaf T. The crucial point about T is that no new global sections are added.

-

LEMMA 5.1

-

The topos of global sections o f

-T

is the same as that of T, that

is, E. The proof of this lemma again uses a version of "definition by cases" f o r intuitionistic type theory, but this time of a rather unusual form and not as easy to prove as the version we used in the preceding section. (Of course, the two versions are equivalent classically.)

LEMMA 5.2 (New definition by cases). If pl,. . . ,pn are formulas and a are terms of type A, then from al,

...,

(i)

for each i and j ,

pivpjvai=a

j'

and

we may infer a!x,A((x=a

Vp )A-*-A(x=anvpn)). 1 1

Proof: Existence is proved from (i) by induction on n. Clearly, the result holds when n = 1. Assume the result for n-1. Now

hence v p i v /\ai i5n i#n Case 1. Case 2. x = 5,

= a

. .

A a . = an. Then the existence statement is witnessed by x = a ifn pi f o r some i 5 n. Then the existence statement is witnessed by where 5

is chosen, by inductional assumption,

so

that

J. LAMBEK, I. MOERDIJK

286

There i s a l s o a d i r e c t proof of e x i s t e n c e . Uniqueness i s proved from ( i i ) a s follows.

n

A

n (x = a 1. v p 1. )

A

and

a

B

A

(y = ai v p i ) .

i=l

i=l

Then t h e r e e x i s t subsets

Suppose

of

v = {l,

...,n l

such t h a t

~ pi. A x = a i ~A p i ~ , r \ y = a i A ica icv-a 1CB icv-8 If

B

and

a

a r e d i s j o i n t , then

( v - a ) u (v-%) = v ,

A pi,

hence

which

icv contradicts ( i i ) .

Therefore,

s o we have

A

x = a.

y = a,,

Proof of Lemma 5.1:

8

and

a

hence

have an element i n common, say

i,

and

x = y.

Given a f i n i t e covering

V(pl)u--*uV(p )

of

Cospec(E)

by b a s i c open s e t s , we have t o show t h a t any compatible family of s e c t i o n s over tlie

V(pi)

al,...,a

comes from a global s e c t i o n . o f type (1)

(recall that 3!ai (2)

that is, the

ai

PA,

A

To t h i s end, suppose we have terms

E,

being an o b j e c t of

f o r each

E/Alt(pi)

i,

such t h a t

I= 3!cri,

, a b b r e v i a t e s 3!x c A ~ ~ a i ) and for a l l

i

and

a r e compatible.

E/Alt(pivpj) b ai = a j ,

j,

Since t h e

V(pi)

cover

Cospec(E),

we have

Moreover, from (2) it follows t h a t

E C p.

V

p.

1

1

V

a. = a . . 1

Therefore, using Lemma 5.2 and d e s c r i p t i o n i n

and that

a

1

E,

we f i n d a term

a

such t h a t

i s unique up t o e q u a l i t y i n E . To complete t h e proof, we have t o show E b 3 ! a . But from (1) and (4) i t follows t h a t n

287

Two sheaf representations

hence that

Eb

Therefore, by ( 3 ) , REMARK 5.3

3!a.

Lemma 5.2 of course also has a relative version: Given any propo-

sition q. if q v l(p A..-AP,) 1 and, for all i and j, then q v 3!xeA((~

al

=

q V

V

p,)

pi

V

A"'A

p. 3

a.

V

(x

a.

=

I'

1

=

an

V

pn))

However, this does not suffice to prove a generalization of Lemma 5.1 for sections instead of global sections. For, if V(p,) U " - U V(pn) covers V(q), in other words, if E b (plh--.hpn)e=* q , we can in general derive that E k q v .(plh*--Ap ) o?ly if q is such that E b q V 19. Thus, T is a sheaf

over V(q)

only if E ( 1 , f i )

is a Boolean algebra. But this case is of no interest here,

because it is easy to see that the present sheaf representation is isomorphic to the one discussed in the previous section whenever E(1,fi) is Boolean. Let u s now turn to the stalks of the second sheaf representation. For a prime filter P, we write Tp s Ip

In o u r second analogy, Tp

E

E(1,fi)

1xqlpE C p v q}.

corresponds to the component of 0 at P in ring

theory defined by Op

The stalk of the sheaf

{r

E

R13slpr-s= 01.

at the point P may now be calculated as

since

To characterize the stalks of 7 we require an auxiliary concept corresponding to the notion of "regular" element of a commutative ring. Call the

J. LAMBEK, I. MOERDIJK

288

proposition p: 1 -+ s2 superfluous if, for all propositions q, whenever . E b p v q, then already E C q . It is not difficult to prove that p is superhence t o the fluous if and only if it belongs to every maximal ideal of E(l,n), "radical" of E, a fact which we mention only by the way. LEMMA 5 . 4 (1)

-

If F is any topos, the following statements are equivalent: F is isomorphic to a stalk of T(E) for some topos E.

There is a prime filter P in F(l,s2) such that Tp*='{T}. (3) For any n-tuple of propositions p i in F, if F b 7 ( p l ~ - - - )~ p then one of the pi is superfluous. (2)

-

(1) ( 2 ) . If F is a stalk of T(E) at the point P of Proof: Cospec(E), that is, F = E/Tp, then the prime filter P in F(1,Q) corresponding to the prime filter P in E(1,O) has the property that T--=*{T} in P F(l,s2). Conversely, if P is a prime filter in F(1,n) such that Tp-=-{TI, F

-

is clearly isomorphic to the stalk of ?(F) at P. (2) ( 3 ) . Tp-=* IT} if and only if P contains all superfluous propositions. But a prime filter with this property exists if and only if (3) holds. A topos having the equivalent properties of Lemma 5.4 will be called primeval. We may then summarize the above discussion in the following representation theorem. Every topos is isomorphic to the topos of global sections of a THEOREM 5 . 5 sheaf of primeval toposes. There is no need to consider the special case of cosemisimple toposes as in Section 4, as it was remarked above that the two representations coincide for cosemisimple toposes. In' analogy to local rings, we call a topos E colocal if E(1,n) has exactly one maximal filter. It is easily seen that E is colocal if but not both, the and only if, for every proposition p. E b lp or E b 3.p We shall call a topos E cosemilocal if, for maximal filter being IplE b .,PI. p v -,-,p. This is equivalent to saying that E(1,hl) every proposition p. E b . satisfies DeMorgan's Law (see [JZ]): l(p A q)-=-,p V .q. Now a topos E is cosemilocal if and only if all the stalks in its second sheaf representation are colocal. For E/Tp is colocal if and only if, f o r every proposition p, either . p o r -,.p belongs to Tp (but not both). This is the same as saying that there exists q k P such that either E b q v . p or E k q v . , p (but not both). Now this certainly holds if E is cosemilocal, for or q C ,p, as they cannot both belong to P. Conversethen we may take q :..p ly, if all the stalks E/Tp are colocal, then, for any prime filter P in E(l,s2),

Two sheafrepresentations either ,p o r filter, and so

-,,p

289

belongs to Tp C P, hence ,p v ,,p belongs to every prime In particular, we have the following: V , ,p.

E b ,p

Every cosemilocal topos is isomorphic to the topos of global PROPOSITION 5.6 sections of a sheaf of colocal toposes. We remark that unfortunately there is no reason why there should be a unique prime filter containing the non-superfluous propositions in an arbitrary stalk, although this is so when the stalk is colocal. (See [Ll] for a similar situation in ring theory.) It may be worth pointing out that, when E = Sh(X) REMARK 5.7 sheaves on a topological space X, then E is iff X has exactly one point, cosimp1e cosemisimple iff every open set is closed, colocal cosemilocal 6.

iff iff

is the topos of

every nonempty open set is dense, the closure of every open set is open.

HENKIN'S COMPLETENESS THEOREM FOR TYPE THEORY REVISITED

In as much as a higher order language may be regarded as a topos E, a standard model of E is a logical functor into Sets. Henkin [He] observed that, in general, there are not enough standard models, hence one requires other toposes in place of Sets to serve as (targets) of models. Let u s call S a model (topos) if it has the disjunction and existence properties, to wit, (1) if S b p v q then S b p o r S b q , (2) if S k Zx,,(p(x) then S k q(a) for some a: 1 + A.

--

Algebraically, (1) asserts that the terminal object 1 of S is indecomposable and (2) asserts that 1 is projective. If S is Boolean, (2) also implies if, for all a E S(l,A), S b cp(a), then S b Vx,,cp(x). This means that 1 is a generator, that is, that the functor r = S(1,-): S + Sets is faithful. This implies that (3)

and, in particular, that r(PA) Pr(A). So, essentially, S is like Sets, except that it has "smaller" power sets: S is equivalent to a general model in the sense of Henkin [He]. Henkin's completeness theorem asserts that a Boolean topos has enough (general) models, that is, when E is Boolean, E C p if and only if, for all logical functors L: E + S , S being a model, S b L(p). To simplify the argument, he assumes in [He] that E satisfies the axiom of choice. We shall make the same

J. LAMBEK, I. MOERDIJK

290 assumption h e r e .

According t o Lawvere, a topos s a t i s f i e s A . C . i f a l l e p i s s p l i t , o r equivalentl y , i f a l l objects are projective.

f o r some

f

E

E(A,B).

E

If

LEMMA 6.1

Proof:

This means:

s a t i s f i e s A.C.

E/F b p,

Suppose

and

F

where

p

i s a f i l t e r , then 5

VxEA3yEBq(~,y).then

A . C . , t h e r e i s an arrow f : A + E b VxEA(p=* cp(x,fx)), hence E k p VxEAq(x,fx). Since p

satisfies

+

is i n

F,

s a t i s f i e s A.C p

E b VxcA3yEB(p* q ( x , y ) ) . B i n E such t h a t

i s Boolean, by Diaconescu's r e s u l t , hence

VxcAq(x,fx)

E/F

E

F,

F.

E

Now

Since

E

E

it follows t h a t

E/F b VxEAq(x,fx).

that is,

Any topos with A . C . has enough models.

THEOREM 6.2

Proof:

If

L(P): E + E/P

P

i s any prime f i l t e r , t h e canonical l o g i c a l functor

i s a model, because

E/P

has choice, so t h a t

1 is projective in

There a r e enough models, because t h e i n t e r s e c t i o n of a l l prime f i l t e r s i s

E/P. {TI.

Actually, we know more: s e n t a t i o n of

t h e models of

E

a r e t h e s t a l k s of t h e sheaf r e p r e -

E.

In g e n e r a l , f o r an a r b i t r a r y topos, not n e c e s s a r i l y Boolean, one can prove a Henkin-Kripke s t y l e completeness theorem using "general" Kripke models.

The

argument f o r t h i s general case i s somewhat more complicated and w i l l be reserved f o r another paper [LM]. 7.

GEOMETRIC MORPHISMS AND ARBITRARY BASE TOPOSES I t i s p o s s i b l e t o c a r r y out t h e sheaf r e p r e s e n t a t i o n over an a r b i t r a r y base

topos;

but then one cannot d e f i n e t h e spectrum o f a topos a s before, because t h e r e Instead, we may look a t t h e l o c a l e (complete

may not be enough prime f i l t e r s .

Heyting algebra) of i d e a l s ( s e e Section 2 ) . looks l i k e . Let (in

E be any topos and

E ) , SpecE(F),

lattice

f,(OF)

sheaves on

F

f

E

We s h a l l b r i e f l y sketch what t h i s

any topos over

i s now defined t o be t h e l o c a l e

in

SpecE(F),

E.

In

E,

E.

The spectrum of

IdE(f,QF)

we can now form t h e topos

F

of i d e a l s on t h e

ShE(SpecE(F)) of

and we have a l o c a l i c geometric morphism

ShE(SpecE(F)) + E.

(The d i r e c t image i s defined by t a k i n g global s e c t i o n s , t h e i n v e r s e image i s t h e

Two sheafrepresentations

29 1

-

f E always factors The geometric morphism F + E. The easiest way to see this is to use the hyperthrough ShE(SpecE(F)) f connected-localic factorization of [J3] and to write F E as a composition "constant sheaf functor!')

F & ShE(f,nF) In E, we have a frame map V: IdL(f,QF) morphism

and the composite L = kog

+

h

f,QF,

E. which induces a geometric

is the required factorization

It is not difficult to show that L

is the unit of an adjunction SpecE -/

ShE

between toposes over E and the category of "spectral locales" (locales of ideals of distributive lattices) and spectral maps in E. In the case where E is Sets, every spectral locale is spatial (there are enough prime filters), and we have returned to the situation described in Section 4 : the functor . n F defined there is the direct image of the geometric morphism R . The crucial property of R is that its direct image L, preserves disjunction V. This property corresponds to the fact that the stalks of the sheaf representation of Section 4 were all local. Let us therefore call a geometric morphism local if its direct image preserves V. The diagram above says that every geometric morphism factors through a local one. The class of local morphisms is fairly well behaved: every hyperconnected geometric morphism is local, and a geometric morphism is local if and only ipthe localic part of its hyperconnected-localic factorization i s local. A localic geometric morphism ShE(A) + E is local if and only if lA is an indecomposable element of the locale A

in E, that is,

A l s o , the class of local geometric morphisms is closed under composition.

J. LAMBEK, I. MOERDIJK

292 :

APPENDIX

THE INTERNAL LANGUAGE OF A TOPOS

To make t h i s paper e s s e n t i a l l y s e l f - c o n t a i n e d , we remind t h e r e a d e r of some p e r t i n e n t f a c t s about toposes and t h e i r i n t e r n a l language, which happens t o be a

For o u r present purpose we r e q u i r e

version of applied i n t u i t i o n i s t i c type theory.

n e i t h e r a type of n a t u r a l numbers i n t h e language n o r an o b j e c t of n a t u r a l numbers i n t h e topos. A language c o n s i s t s of t h e following d a t a :

e: 1, R ,

(1)

PA

and

assuming

AxB,

B

and

A

t o be t y p e s , and

perhaps o t h e r s .

Terms:

(2) a = a'

{x

PA;

of type a

a

and

o f type

a

Alp(x)I

E

of type

A

and

and

a

PA,

assuming

p(x)

R

B;

x

and

of type

and

P

and

A

o f type

A);

E

of type

a'

a r e a l s o c a l l e d formulas.

AxB,

The symbols

assuming

T , I,

A,

v,

=-,

-,,

[LSZ].

of v a r i a b l e s a deduction r e l a t i o n

X

of type

a

a variable

x

and perhaps o t h e r s .

I I may be defined i n terms of t h e above

F o r each f i n i t e s e t

(3)

assuming

of type

b

Terms of type

Y, 3 , 3 !

P,

(as i s i n d i c a t e d by w r i t i n g

A

of type

* of type 1;

v a r i a b l e s of each type and a l s o t h e following: E

between

f i n i t e s e t s of formulas and s i n g l e formulas, a l l f r e e v a r i a b l e s of which a r e contained i n

X.

The r e l a t i o n s

s a t i s f y t h e "usual" axioms and r u l e s of i n -

ference of i n t u i t i o n i s t i c type theory, but may a l s o s a t i s f y o t h e r s . A topes E i s a Cartesian closed category i n which t h e subobject functor i s r e p r e s e n t a b l e by an o b j e c t a , t h a t i s , Sub E(-,R).

E

The i n t e r n a l language of

has t h e following t y p e s , terms and deduction

relation: (1)

Q and

The types a r e t h e o b j e c t s of t h e subobject c l a s s i f i e r ,

E;

PA = RA

1 i s t h e terminal o b j e c t ,

in particular, and

AxB

i s t h e Cartesian product o f

A

B.

Closed terms of type

(2)

A

i n t e r p r e t e d as indeterm'inates.

a r e arrows

category obtained from

E

where

x: 1

by a d j o i n i n g x,

+ A

in

but variables are x

E[x],

o f type

A

is

t h e Cartesian closed

much a s one a d j o i n s an indeterminate

Moreover:

* i s t h e unique arrow a = a'

E,

For example, t h e v a r i a b l e

i n t e r p r e t e d a s t h e indeterminate arrow t o a ring [L2, L3].

in

1+ A

1 -+ 1;

i s t h e arrow,

i s t h e " c h a r a c t e r i s t i c " morphism of t h e subobject

under t h e correspondence Sub(AxA)

E(AxA,Q) ;

:

A

-+

AxA

Two sheaf representations a

E c1

is the arrow

where

E~

corresponds to

IpA: PA

+

E(PA,P A) {x

E

293

Alp(x) 1

is the arrow 1

+

b

PAxA ---+EA

61,

PA under the correspondence

= E(PAxA,a)l

PA corresponding to the arrow f: A

+ 61

in

E under the correspondence E(A,Q) where

2

E(lxA,Q)

2

E(l,QA),

f is the unique arrow making the diagram

a

*

:f I I

A in E[x] commute; is the arrow 1 + AxB with the same name. (3) To explain p,(X) ,. ,pn(X) q(X), take for example X = {XI, then pi(x)*=*fix and q(x).=-gx in E[x]. We then interpret pl(x), ...,p n(x) I- q(x) to mean: for all objects B of E and for all arrows h: B + A, if T flh,..., f h are all equal to B + 1 0, then so is gh.

..

h

-

When X = {xl,..., x 1. take E[x] = E[xl] ...[xn], which is also isomorphic to E[x], x being a single indeterminate whose type is the product o f the types of the xi. Note, in particular, that p q means that p * q-=-T in E. We usualiy just write this as E b p * q. MLre generally, if p E E(l,R), we write E b p to mean that p*=-T in E. It is not difficult to verify that the "usual" axioms and rules of inference of intuitionistic logic do in fact hold in any topos. There are many ways in which the topos interacts with its internal language. We shall only mention two that play a r6le in this paper. The first is expressed by the slogan "internal equality implies external equality". It asserts that, if f and g are arrows A -P B such that

294

J. LAMBEK, I. MOERDIJK

E C VxcAfx = gx, then f.=-g. Note that we write

-=.

for equality in the matalanguage whenever

it is important to distinguish it from the equality =

in the internal language.

The second is expressed by the slogan "description holds in every topos". asserts that, if r is an arrow 1 + P(AxB) in E such that

It

E b VxEAg!yEB<~,~> 6 r, then there exists a unique arrow f: A

+

B

in E

E b VxEA<x,fxz E

such that

r.

REFERENCES R. Balbes and P. Dwinger, Distributive lattices (University of Michigan Press, 1974) . S.D. Comer, Representation by algebras of sections over Boolean spaces, Pacific J. Math. 38 (1971), 29-38. , A sheaf-theoretic duality theory for cylindrical algebras, Trans. Amer. Math. SOC. 169 (1972), 75-87. J.W. Gray, Sheaves with values in arbitrary categories, Topology 3 (1965), 1-18. A. Grothendieck and J. DieudonnB, El6ments de g6om6trie algarique I, I.H.E.S. Publ. Math. 4 , Paris 1960. L . Henkin, Completeness in the theory of types, J. Symbolic Logic 15

(1950), 81-91; reprinted in: 3 . Hintikka, The Philosophy of Mathematics, Oxford University Press 1969. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. SOC. 142 (1969), 43-60. P.T. Johnstone, Topos theory (Academic Press, London 1977). , Conditions related to DeMorgan's law, in "Applications of sheaves", Proc. L.M.S. Durham Symposium 1977, Lecture Notes in Mathematics 753 (Springer Verlag, 1979), 479-491. , Factorization theorems for geometric morphisms I , Cahier top. et g8om. diff. 22 (1981), 3-17. J. Lambek, On the representation of modules by sheaves of factor modules, Can. Math. Bull. 14 (1971), 359-368. , From types to sets, Advances in Math. 36 (1980), 113-164. , From A-calculus to Cartesian closed categories in: J.P. Seldin and J.R. Hindley (editors), "To H.B. Curry, Essays in combinatory logic, lambda calculus and formalism" (Academic Press, 1980), 375-402.

Two sheafrepresentations

295

J. Lambek and I. Moerdijk, A Henkin-Kripke completeness theorem for intuitionistic type theory, in preparation. J. Lambek and P. Scott, Intuitionist type theory and the free topos, J. Pure and Appl. Algebra 19 (1980), 576-619.

, Intuitionist type theory and foundations, J. Phil. Logic 7 (1980), 1-14.

, Algebraic aspects of

topos theory in "31me colloque sur les

catEgories", Cahiers de Topologie et GGomGtrie DiffGrentielle 22 (1981) 129-140. R.S. Pierce, Modules over commutative regular rings, Memoirs Amer. Math. SOC. 70 (1967).

H. Rasiowa and R. Sikorski, The mathematics of metamathematics, (Warszawa, 1963). A. Wolf, Sheaf representation of arithmetical algebras, in "Recent

*

advances in the representation theory of rings and C -algebras by continuous sections", Memoirs Amer. Math. SOC. 148 (1974), 87-93. FOOTNOTE

"1

One of these sheaf representations was first presented at the Columbia Topos meeting in December 1980. Both authors had support from the National Science and Engineering Research Council of Canada, the senior author also from the Humanities and Social Science Research Council of Canada, moreover he shared a grant from the Quebec Department of Education.

THE L.E.J. B R O W E R CENTENARY SYhfPOSION A S . i?oe[stra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

291

LAWLESSNESS ANU TNUEPENUENCE

U. van Paten, J.S. Loddeh

0. I NTRDDUCTI ON

The theory LS of lawless sequences contains an infinite list of continuity axioms, LS3(n) of the form CP(S~,S~ ,... ,s,,)

A

M C,3ci

iZ 1

+

gxvnET,x(

M nfSi

+

d n , S 2,... ,tn)).

i+l

In ID11 it was shown that LS3(1) PLS3(2), and it seemed plausible that in gene In the present paper we refine the forcing technique of

ral LS3(n) f+LSj(n+l).

I D 1 1 so that models can be constructed for LS3(2) in which LS3(3) fails.

We copy the modelconstruction from [Dll, which was based on the existence o suitable Cohen generic sequences that behave in the Beth model for analysis as lawless sequences. Our refinement consists of the incorporation of a certain dependence of the generic sequences that is reflected in a corresponding dependence of the lawless sequences. The simple minded approach however spoils the LS3(3) together with the LS3(2), so we have introduced infinite clauses in the forcing conditions to save the LS3(2). In the first section we give some results on dependence and independence for generic sets of natural numbers. In particular we look at closure of generic sets under complement, union, intersection and symmetric difference. The latter operation turns out to be useful for the application to lawless sequences, mainly because it has natural 'inverses'. Independence for generic sets is defined by the genericity of a suitable coding, presented by X + Y in definition 1.1. Our models are simplified versions of the models in [ D l ] , because we consider only 0-1-sequences, and also the binary tree a s underlying poset of the Beth model. As a consequence we can straightforwardly adapt Feferman's treatment, cf.1 F ] . We found it convenient to extend the language by quantifiers over generic sets, this could be avoided at the cost of extra details. The construction of the model

BR is by transfinite iteration o f the addition of definable sets, as in

[F].

J.S. LODDER. D. VAN DALEN

298

We c o n s i d e r a c o l l e c t i o n o f g e n e r i c s e t s t h a t i s c l o s e d u n d e r s y m m e t r i c d i f f e r ence, $, t h i s c l o s u r e i s embodied i n t h e f o r c i n g c o n d i t i o n s i n such a way t h a t t h e presence o f bound l a w l e s s v a r i a b l e s i n LS3(2) i s t a k e n c a r e o f , however, a t t h e p r i c e o f some t e c h n i c a l i t i e s .

1.

CLOSURE PROPERTIES OF GENERIC SETS

1.0 As i n [ F ] f o r c i n g c o n d i t i o n s p w i l l c o n s i s t o f f i n i t e c o n s i s t e n t sets o f sentences n E P , m B P .

F o r c i n g i s d e f i n e d i n d u c t i v e l y as i n [ F ] . A s e t G o f

f o r c i n g c o n d i t i o n s i s g e n e r i c i f f o r a l l sentences cp t h e r e i s a p E G such that p

It

cp o r P

Ik TCP.

A g e n e r i c s e t o f f o r c i n g c o n d i t i o n s d e t e r m i n e s a subset o f

subset X o f

IN. C o n v e r s e l y a

IN i s c a l l e d g e n e r i c i f i t i s d e t e r m i n e d by a g e n e r i c s e t o f f o r -

cing conditions.

GIN

1 . 1 D e f i n i t i o n . The g e n e r i c s e t s X,Y

a r e independent i f X + Y

i s g e n e r i c , wh

2n E X + Y - n E X ,

where

2n+l E X + Y -nEY For a c o n d i t i o n p we d e f i n e

p+ = { n

1 (nEP)Ep},

p- = I n

I (nBP)EpI.

The a p p l i c a t i o n s t h a t we have i n mind r e q u i r e c l o s u r e o f t h e c o l l e c t i o n o f gene r i c s e t s under some o p e r a t i o n s , and o f t h e p r e s e r v a t i o n o f indepence.

I n the

Dresent s e c t i o n we work o u t a few cases.

1.2 P r o p o s i t i o n .

PRoob. D e f i n e and

*

I f X i s g e n e r i c then Xc i s generic.

*

a translation

by ( n E P ) * := n B P , (p* := cp f o r a l l o t h e r a t a n s ,

i s a homomorphism f o r t h e c o n n e c t i v e s .

-

+

For c o n d i t i o n s p we p u t (p*)Claim: p

If cp

p

:= n E p .

1- nE P

and d e f i n e p* a c c o r d i n g l y .

p* lk (p*.

I n d u c t i o n on cp.

(i)cp

+

= p

Q

(n E p) E p

- n E p+

n E (p*)-

-

( n F P) E p*

For t h e r e m a i n i n g atoms t h e c l a i m i s t r i v i a l .

(GI

-

pII-cpv$-pItcp

P * I t cp*v+*

(.iiil

P Il-+P

Vq>p

or

pIt-$-p*tt(p*

-

or

vq*>p*

q*

IV (p*

p

*.

( n E P)

p*~k$*-

p * I t (cpv$)*. q IV cp

-

A

p*lt-lCP*

-

p*

It

(+PI*

299

Lawlessness and independence

For unranked sentences t h e c l a i m f o l l o w s i m m e d i a t e l y f r o m ( i ) Now l e t X be g e n e r i c , and g i v e n by c o n d i t i o n s p t h e n Xc

*

and

*

i s g i v e n by c o n d i t i o n s p*,

I f X and Y a r e independent g e n e r i c s e t s , then X U Y i s g e n e r i c .

Pm06. L e t 2 = X + Y

a translation

(v)

I7

and t h e c l a i m y i e l d s t h e g e n e r i c i t y o f X c .

1.3 Proposition.

-

be t h e generic s e t g i v e n by f o r c i n g c o n d i t i o n s i n P. We d e f i n e

by ( n € P ) * := 2 n E P v 2 n + l E P , cp*

:= cp f o r t h e r e m a i n i n g atoms,

i s a homomorphism f o r t h e c o n n e c t i v e s .

For t h e c o n d i t i o n s we p u t (p,)'

:= C m E 2 m

(p*)-

-

:= I m E 2 m

I p+ v 2 m + l E p + } , I p- A 2 m + l E p - ) .

T h i s d e f i n e s p*. Claim: p I k cp*

p * I t cp.

I n d u c t i o n on cp

(i)cp

:= n E p

pll-

-

(~EP)*+=*~II- 2 n E ~ v ( 2 n + l ) E ~ o 2 n E p + o r2 n + l E p + o

n E (p,)+

~kn

p*

e P.

For t h e r e m a i n i n g atoms t h e c l a i m i s t r i v i a l .

ILL1

plt- (cpv$)*-plt

P*Ik c p v i .

(-1

11( 7 cp)* p* Ik 1cp ; p

t tt

p

Vq>p

Observe t h a t q > p

If r>p*

(iv)3xcp(x) , (V)

-

cp*v$*-pll-

Ik

q

i* t* p*It- i

cp* o r p l t

IY (P*

LL vq,>p,

q*

IW cp

tt

o r p*

vr>p,

r

Q

IW cp

-

* q,>p,.

then there e x i s t s a q > p

s.t.

r=q*.

trivial.

3x'lcp(xa)

q$(P($)
A

Q

p*

g$(P($)
Ik

W({X

A

p

I $(X)}))

Ik

CP*({X

Q

P*

I $(X)))

It

Q

3xolQ(xa).

Now f o r each sentence cp t h e r e i s a f o r c i n g c o n d i t i o n p b e l o n g i n g t o 2, such t h a t

PIE v*

o r p I k ~ c p * . Then p* b e l o n g s t o t h e c o n d i t i o n s d e f i n i n g X

c l a i m , p * I k cp o r p,IE

u

Y and by t h e

~ q Hence . X U Y i s g e n e r i c , by v i r t u e o f t h e c o n d i t i o n s p*.

3 00

J.S. LODDER, D. VAN DALEN

1.4 Proposition. If X and Y are independent generic sets, then X n Y is generic. Ph~rooa. Similar to that of 1.3.

0

1.5 Proposition. If X and Y are independent generic sets, then X and $(X,Y).are independent generic sets.

Phood. Let Z = X + Y be the generic set given by forcing conditions in W = X+$(X,Y). We will show that W is generic. We define a translation * by: (nEP)* := 3x (n = 2x A n E p ) v (n = 2x+1 A ((2xEP A 2x+19P) v ((2x9P A 2x+1Ep))). cp* := cp for the remaining atoms, and

*

P,

let

is a homomorphism for the connectives.

Definition. A forcing condition p is symmetric if 2 n p+~ u p2n+1 E p+ u p-

For the remaining atoms the claim is immediate.

( G )cp v $, immediate.

( i i i l (a]Suppose p l t Let q>p*. Hence r IhL

lcp.

Take r>q

UP.

So q

Ilf

such that r is symmetric. Then r*>p, so

UP.

Hence p* II-

7

r*IW

cp.

cp*.

( b l Suppose p* ~k1 (P*. Let q>p. Take r > q such that r i s symmetric. Then r*>p*. Hence r Ilf cp. So q Ibc cp. Hence p* II- 7 w.

So

r*llf cp*.

301

Lawlessness and independence

(iwl j x c p ( x ) (WI

trivial.

~x"v(x')

trivial.

For each sentence cp t h e r e i s a symmetric p b e l o n g i n g t o t h e f o r c i n g c o n d i t i o n s which d e f i n e Z p * k cp or p*

such t h a t pII- cp* o r pII- 7 c p * .

11- 7 c p .

Since (p*)* = p the c l a i m y i e l d s

p* b e l o n g s t o t h e c o n d i t i o n s d e f i n i n g W,

so W i s g e n e r i c . 0

There a r e more p r o c e d u r e s t o o b t a i n new g e n e r i c s e t s f r o m o l d ones, e.g.

the

i n t e r s e c t i o n o f an i n f i n i t e a r i t h m e t i c s e t and a g e n e r i c s e t y i e l d s a g e n e r i c s e t a f t e r an a r i t h m e t i c " s t r e t c h i n g "

n

sider X {n

{Zn

I

o f the arithmetic onto

W . As an example, con-

n E I N } , then the s t r e t c h i n g y i e l d s the ' f i r s t p r o j e c t i o n '

I ZnEX}, which i s generic. We can c o n s i d e r g e n e r a l i s a t i o n s t o o p e r a t i o n s w i t h more arguments.

For o u r

purpose i t s u f f i c e s t o c o n s i d e r a r i t h m e t i c a l o p e r a t i o n s w i t h i n v e r s e s . Example: t h e t e r n a r y symmetric d i f f e r e n c e ,

g3(X,Y,Z)

and X =

= $(X,$(Y,Z))

$ ($ (X,Y,Z),Y,Z), 3 3

etc.

For convenience we w i l l use c h a r a c t e r i s t i c f u n c t i o n s i n s t e a d o f s e t s .

f : {0,lIk

+

{O,l} together w i t h inverses

fi : { 0 , l I k

+

{O,l}

We c o n s i d e r o p e r a t i o n s

( i = 0 ,...,k - 1 )

such t h a t (TI,

,..., f ,...,n k - l )

o

( n o ,..., f i

. . ,f , . . . , n k - l )

o

(

and ( n o , .

n o , .. . , f

,..., nk-l) ,. . . ,

= id,

T ~ - ~= )

id.

These o p e r a t i o n s may n o t be t h e most g e n e r a l ones, b u t t h e y s u f f i c e f o r t h e a p p l i cations

.

1.6 Proposition.

I f Xo,...,Xk-l

xo ,. . . ,f (x,,

. .. ,xk-l ) ,. .. ,xk-l.

-

i s an abuse of n o t a t i o n , we p u t

Here f(Xo,...,Xk-l)

,...,Xkml)

nEf(XO where Ki

a r e independent g e n e r i c s e t s , t h e n so a r e

f(Ko(n)

,...,K k - l ( n ) )

= 1

i s t h e c h a r a c t e r i s t i c f u n c t i o n o f X..

+...

L e t Z := X o + . . . + X k - l , W := X f ( X o ,..., Xk-l)...+Xk-l. 0 t h a t W i s g e n e r i c . We w i l l i n t r o d u c e a second abuse o f n o t a t i o n :

Pkoob.

nEf(Xo...Xk-l) 3y0.

. .yk-l

-

We w i l l show i n t h e language,

i s an a b b r e v i a t i o n f o r :

M I

((yi = 1

knwiEP)

A

(yi = 0

i.+ km+iFP)

A

f(yo...yk-l)

= 1).

3 02

J.S. LODDER, D. VAN DALEN

Define a t r a n s l a t i o n

*

by

( n E P ) * = 3x(n = kx+i

,..., Xk-,))

xEf(X,,

A

v Vx(n#kx+i

A

nEP).

A c o n d i t i o n p i s k-symmetric i f f o r a l l j < k

*

k n + j E p + U p-

@
kn+QEp+ U p-.

We p u t K (n) = 1

0 n E p + ; K (n) = 0 Q n E p - . P P k-symmetric c o n d i t i o n s a r e d e f i n e d by:

= {n I3m(n = km+i

(p,)'

vVm(n#km+i

A

A

( p * ) - = {n I3m(n = km+i vVm(n#km+i

A A

( p * ) + = {n I3m(n = km+i vVm(n#km+i

vVm(n#krn+i

A

= (p*),

Note t h a t (p,)* Let n E ((p*),)+,

(i]Vm

I-&] n

A A

...

f ( K (km) K (km+k-1)) P P K (n) = 1 ) ) .

*

and

*

= 1)

P f ( K (km) K (km+k-l)) = 0) P P K (n) = 0)).

...

P

A

( p * ) - = { n I3m(n = km+i

The t r a n s l a t i o n s

...K P (km+l-1))

= 1)

...KP (km+k-1))

= 0)

f i ( K (km) P K (n) = 1 ) ) . P fi(K

P

(km)

K (n) = 0 ) ) . P

= p.

then t h e r e a r e two cases.

n#km+i : Then n E ((p*),)+

0nE(p*)+

t=$

nEp+.

K *(km+k-l)) = 1 Then f ( K *(km)... P P (km+i) = f i ( K (km) K (km+k-1)) i t f o l l o w s t h a t P* P P (km+k-1)) = 1 , f (K ,(km). . f i (Kp(km). .K (km+k-1)) .K P P P* So K (km+i) = 1 ( s i n c e f i i s t h e i n v e r s e o f f ) . P The r e s t o f t h e p r o o f i s s i m i l a r . = km+i.

...

Since K

.

Claim. -

.

,. .

For k-symmetric c o n d i t i o n s p : p 11- IP

li] cp

-

p*

11- @.

:= n E P .

a) n = km+i. Then pII- n E P

*

f ( K (km)

P

t* t=)

b) Vm

* nEp+-

K (km+i) = 1 P

,... f i (Kp(km) ...KP (km+k-1)) ,...KP (km+k-l))

f ( K *(km)...K (km+i) P P* p*Ik (nEp)*.

,... K

P

,(km+k-l))

= 1

n#km+i.

Then p k n E P

Q

p*II-

(nEP)*.

For t h e remaining atoms t h e c l a i m i s t r i v i a l .

(2) cp v

$;immediate.

= 1

f o r the

3 03

Lawlessness and independence

1 x 1 (a)

Suppose p

11-

1 cp.

L e t q>p*.

Take r > q

So q

IV

v*..

Hence p*

IF

( b ) Suppose p* L e t q>p.

( i w ] gxcp(x)

7

I+

@.

i

cp*.

Take r > q

such t h a t r i s k-symmetric.

So r*Iv (p*.

Then r * > p * . Hence r

such t h a t r i s k-symmetric.

so r * Ilf cp. Hence r Ilf cp*.

Then r * > p ,

Ilf cp.

So q

IF cp.

Hence p*

11-

cp.

trivial.

I W I 3xacp(x")

trivial.

For each sentence cp t h e r e i s a k-symmetric p b e l o n g i n g t o t h e c o n d i t i o n s d e f i n i n g Z such t h a t p

If

(p* o r p

11-

2.

Ik cp

(p*. The c l a i m y i e l d s p*

belongs t o t h e c o n d i t i o n s d e f i n i n g W ,

o r p*

Ik

cp. S i n c e p*

0

t h i s implies t h a t W i s generic.

CONSTRUCTION OF A MODEL

We w i l l e x h i b i t a model of t h e t h e o r y of l a w l e s s sequences i n w h i c h LS3 i s v a l i d f o r 2 parameters. We u s e t h e symmetric d i f f e r e n c e o p e r a t o r on independent g e n e r i c s e t s i n o r d e r t o o b t a i n a model

i n w h i c h LS3(2) h o l d s b u t LS3(3) f a i l s . The t e c h n i q u e o f model-

c o n s t r u c t i o n i s adopted f r o m [ D l ] . approach,

For convenience we have chosen a s i m p l i f i e d

i n w h i c h we o n l y c o n s i d e r 0-1-sequences.

T h i s has t h e advantage t h a t we

can use f o r c i n g f o r a n a l y s i s w i t h s e t s , as i n Feferman's [ F ] . We w i l l use t h e framework o f [ 011, s e c t i o n

6,

m o d i f i e d i n accordance w i t h [ F ] .

2.1 We c a r r y o u t o u r f o r c i n g i n c l a s s i c a l second-order a r i t h m e t i c w i t h two s o r t s o f set variables. The g e n e r i c ones a r e i n d i c a t e d by s c r i p t c a p i t a l s X,Y,Z by X , Y , Z

...

...,

and s e t s i n g e n e r a l

The c o n d i t i o n s c o n t a i n more t h a n j u s t t h e t r a d i t i o n a l c l a u s e s ,

o r d e r t o h a n d l e t h e dependencies. The language c o n t a i n s a symbol m e t r i c d i f f e r e n c e s , and u n a r y p r e d i c a t e s P,,Pz, f i n i t e c o n s i s t e n t s e t o f t h e form p = { n E P where P,Q,R,

..., P ' , Q ' , R '

... . A

4

forcing condition i s a

,...,m e Q,...,P '

= q(Q',R')

,... 1,

are variables f o r the P.'s.

We p u t p+ = I n

i

I (ncPi)Ep v

...

in

f o r t h e sym-

v [ ( n ~ p . ) ~ Ap ( n e P k ) E p A $ ( p j , p k ) = P ~ E P I J v [ ( n E p . ) E p A ( n g P k ) E p A $(Pi,P.) = PkEpl} J J

J.S. LODDER, D. VAN DALEN

3 04

p f = I n I ( n B P i ) E p v [ ( n E p . ) E p A ( n E P k ) E p A $ ( P ~ , P ~= ) p i E p I J v v [ ( n g ~ . ) E pA ( n B P k ) E p A $(pj,Pk) = P i E p l } . J

...

F o r c i n g i s d e f i n e d as u s u a l : p t n E P i The c l a u s e s f o r v , - , 3 x , 3 X

PII- ~ X W ( X ) i f 3pi

nEpt.

if

a r e s t a n d a r d , f o r t h e g e n e r i c q u a n t i f i e r s we p u t

p k dpi).

A g e n e r i c sequence i s a sequence o f c o n d i t i o n s (P,)

(i]f o r

each sentence cp we have 3n

ILL) V i , j 3kn

($(Pi,Pj)

= P),

p

It

cp

or

such t h a t p It

1

cp.

E pn.

F o r c i n g has t h e f a m i l i a r p r o p e r t i e s , as one can e a s i l y check.

I n p a r t i c u l a r we

have t h e T r u t h Lemma. L e t (p ) be a g e n e r i c sequence and, ill2 the c o r r e s p o n d i n g model t h e n

D !

cp

e=*

p I t cp, f o r sentences cp.

3n

2.2 C o n s t r u c t i o n o f t h e g e n e r i c sequence. S i n c e we have r e s t r i c t e d o u r s e l v e s t o b i n a r y sequences, we can do w i t h o u t s p e c i a l measures t o g u a r a n t e e t h e i n i t i a l segment c o n d i t i o n s o f LS1. So a l l t h a t i s l e f t concerns t h e d e c i d i n g o f sentences and t h e " c l o s u r e under L e t cp1,cp2,cp3,...

$".

be an e n u m e r a t i o n o f t h e sentences o f second-order a r i t h m e t i c

w i t h symbols f o r t h e P i ' s . Step k . We have c o n s t r u c t e d c o n d i t i o n Pk-l. q 11- cpk o r q 11-

Claim:

(P,)

cpk.

As u s u a l t h e r e i s a q>pk-l

Choose any such q f o r p

such t h a t

k'

i s generic.

We o n l y have t o check c l o s u r e under $. Suppose t h a t no pn f o r c e s 3X $(P.,P.) = X t h e n f o r some I J Pmlk -3x $ ( P i ' P . ) = p, J so Vq>pm VPk q I v $ ( P i , P j ) = Pk, w h i c h i s f a l s e .

x,

A s i n 1011 we use a Beth-model on t h e b i n a r y t r e e t o i n t e r p r e t e sequences as

growing f a m i l i e s o f 0-1-functions namely t h e s t a n d a r d model.

(i.e.

s e t s ) . The f i r s t - o r d e r domain i s c o n s t a n t ,

L a w l i k e sequences a r e t o t a l i n t h e b o t t o m node. The

l a w l e s s sequences a r e d e f i n e d as f o 1 ow5

2.3 Definition. €Pi

Lawlessness and independence

:= 3p

Observe t h a t

2.4

11-

p I F VX(XEPi +=+xEP.) J

i s used f o r two d i s t i n c t purposes.

e. LS(1)

Pnood.

305

holds. i n t h e model.

L e t u be a f i n i t e 0 , l

For each c1 w i t h I t h

Then t h e r e i s a p

c1 =

sequence w i t h I t h u = n.

n we w i l l show nIk 35

S E u . Suppose f o r a l l p

such t h a t

which i s n o t t h e case. Hence t h e r e i s a p

such t h a t

M

P m k 3x

biEX

f-t

u . = 1).

OQ i
2.5 P e r m u t a t i o n s . Define

if

~ ! ( p = ~ p )i J

k = j

pj if k = i pk e l s e .

D e f i n e a s y n t a c t i c a l p e r m u t a t i o n by .rr!(nEPk) = ( ( n E P i J v (nEPk

A A

k = j) v ( n € P . A k = i ) J k # i A k f j)).

2.6 P e r m u t a t i o n lemma. p 11- cp

Pnood.

* n p 11-

mp.

I n d u c t i o n on cp.

(i)For

t h e a t o m i c case,

(close the

i t s u f f i c e s t o look a t t h e

$ - f r e e p a r t under

-

8-free part o f p

4).

(Lii)The s t a n d a r d c o n n e c t i v e s a r e r o u t i n e , we w i l l j u s t check 3XwX p

Il- 3 X w ( x )

-

( i n d . hyp.) TIP

II-

3Pi

p

3Pi

npll- vw(Pi)

3X?Icp(X).

0

dPi)

J.S. LODDER, D. VAN DALEN

306 Forcing i n t h e Beth-model

W

in

i s straight-forwardly translated i n t o satisfaction

i n e x a c t l y t h e same manner as [ D l ] ,

Result: f o r each cp t h e r e i s a cp*

6.5.

such t h a t

cxlt

cp

* !IT b

cp*(a).

Observe t h a t LS2 holds t r i v i a l l y i n t h e model.

Lemma.

2.7

L S 3 ( 2 ) holds i n t h e model.

Ptlood. We have t o show t h a t $ (6,,Ck)

h

v(cn.Ck)

*

3 x [ c n E x h v c E x ( f (595,)

-f

'P(E,ck)l

holds, where a l l lawless parameters a r e shown.

I t t u r n s o u t t o be t e c h n i c a l l y more convenient t o consider i n i t i a l segments o f lawless sequences f o r t h e purpose o f v a l i d a t i n g LS3(2). auxilliary fact: c1

IF t i f

sj

-

aI t

3x

Ti ( X I

#

We need t h e f o l l o w i n g

sj( x ) .

a I t ti f 5 and consider a 1- 3x Ti (x) f T . ( x ) . j J This reduces t o V f € a 3y T ( y ) 11- Ti (y) # (y) * 3y Vf E a i ( y ) 11- Ti ( y ) # rj ( y ) (by the f a n theorem) t=, 9JI I=3y V f E a Vi(i(Y)) fFj i ( y ) ) . Pa006 06 6act. L e t

rj

We use the Pi,P. as c h a r a c t e r i s t i c f u n c t i o n s . The above i n e q u a l i t y merely J s t a t e s t h a t t h e sets P. and P . d i f f e r i n one o f t h e (coded) i n i t i a l segments o f J f(Y). Suppose now t h a t p l k 1 3 y V f E a vq>p VY 3 r > q i .e.

V q Z p vy 3 r > q

3f

r

Fi(-f(y))#Fj(T(y)),

then

s IV P i ( f ( y ) ) + F j ( i ( y ) ) ,

3 f Vs>r

1-

Pi(i(y))

= Fj(f(y)).

Now look back a t t h e Beth model and determine t h e longest node t h a t occurs i n the c o n d i t i o n p, say w i t h l e n g t h y o . Extend p t o q such t h a t P i and P . d i f f e r i n J a l l nodes w i t h l e n g t h y o + l . Then f o r such a q

r 11- P i ( i ( y o + t ) )

i t i s impossible t o f i n d r > q

and an f such t h a t

= Fj(?(y0+1)).

C o n t r a d i c t i o n , hence t h e r e e x i s t s a p such t h a t

PIC 3y V f E a P i ( i ( Y ) ) f P j ( i ( Y ) ) a I- 3x Ti (x) # (x)

and t h e r e f o r e

sj .

The converse i s c l e a r . L e t all- cp(<

,c

) , then by t h e t r a n s l a t i o n lemna LQ +q*(a,Pn,Pk). So by t h e t r u t h n k lemma t h e r e i s a p such t h a t p k rp*(a,Pn,P,). L e t p contain, a p a r t from Pn,Pk a l s o

P

...P9 .

ml

307

Lawlessness and independence

We want a l l t h e P ' s t h a t o c c u r i n p t o be d i s t i n c t ,

b u t i n a s t r o n g form; we

extend each o f them such t h a t t h e y d i f f e r on a b a r i n t h e model,

B

There i s a b a r

for

Let B l t

k

,P

. .. P

ml

rn, (h)

3n 611- M T i ( n ) # r . ( n ) . L e t B E E , J p r e s c r i b e s a t B d i f f e r e n t i n i t i a l seg-

a such t h a t f o r a l l BEB

then we may e x t e n d p t o p B such t h a t p ments o f Pn,P

i.e.

B

o f t h e same l e n g t h h. Then p B IC v*(B,Pn,Pk). m!2

rn(h).

I n o r d e r t o d e f i n e t h e p e r m u t a t i o n we have t o c o n s i d e r t h e $ - p a r t s o f t h e c o n d i t i o n pB. For convenience we i n t r o d u c e t h e n o t a t i o n (n1,n2,n3) $(P, 1

. The

) = Pn

,Pn

for

$ - p a r t s o f pB can have t h e f o l l o w i n g form:

3

2

(1)

(n.k,mi 1

) ;

The p e r m u t a t i o n t a k e s n t o n ' and keeps k f i x e d . The m. i s d i s t i n c t from the '1 t o m.' , which i s $ ( n ' , k ) . I f mi m i s s i n case ( 2 ) , ( 3 ) . The p e r m u t a t i o n t a k e s m. '1 '1 = P ( n , k ) and s u b s t i t u t e : (mi,mj,9(n,k)), which o c c u r s i n case ( 4 ) , we w r i t e m i1 i s e q u i v a l e n t t o (n,$(k,m.) , m . ). We add t h e l a t t e r t o t h e 1 i s t (2). We proceed I J t o l i s t (2).

) t o (n',m ,mi' ) . '3 i2 3 We l o o k f o r o c c u r r e n c e s o f mi ( i n t h e l i s t s (3) and We permute (n,m.

,m.

'2

3 i n l i s t ( 2 ) o n l y once). We w r i t e m i r e p l a c e (mi,m.,m.

= $(n,mi

3

)I,

) by (mi,m.,&(n,mi

'3 J We add t h e l a t t e r t o t h e l i s t ( 2 ) .

(4)) n o t e t h a t m. can o c c u r '3

) and proceed t o e l i m i n a t e m

3

2 which i s e q u i v a l e n t t o (n,$(mi

2

.

i '

,mi),mj).

2

When we come t o t h e new a d d i t i o n s t o l i s t (2), we permute as p r e s c r i b e d by t h e preceding steps, e.g.

(n,$(k,mi),mj)

-t

(n',8(k,mj),mj'),

where m ' = $ ( n ' . $ ( k , m i ) ) .

A f t e r f i n i s h i n g l i s t (Z), we s t o p s i n c e t h e r e m a i n i n g $ - p a r t s

j

i n t h e l i s t s (3)

and ( 4 ) a r e n o t i n f l u e n c e d by t h e above s t e p s . We o b s e r v e t h e f o l l o w i n g f a c t : The t r i p l e s c o n t a i n i n g $ - s i g n s can be t r a n s f o r m e d i n t o $ - f r e e t r i p l e s .

-

Example: ( n ' , , % ( k , m i ) , m . ' ) c-t ($(n',k),mi,m.' ) (mi',mi,m,'). J J 1 J i s j u s t the permutation o f the Observe t h a t t h e r e s u l t i n g t r i p l e (mi',mi,m.') J i n case ( 3 ) o r ( 4 ) . o r i g i n a l t r i p l e (m ,mi,m.) i1 J

1

J.S. LODDER, D. VAN DALEN

308

So f a r we have permuted complete P i 1 s among themselves, but we have t o consider the $-f ree parts o f p

B' but f i x e s t h e i n i t i a l segment o f Pn t h a t

The permutation takes Pn t o Pn,,

occurs i n $ - f r e e c o n d i t i o n s i n pB. From t h e c o n s t r u c t i o n we see t h a t t h e perm u t a t i o n f i x e s a l l i n i t i a l segments o f t h e same a p p r o p r i a t e l e n g t h . The cond i t i o n s t h a t r e s u l t from the permutation belong t o t h e g e n e r i c sequence, s i n c e t h e c o n d i t i o n s i n v o l v i n g Pn a r e taken t o c o n d i t i o n s on Pnl by d e - f i n i t i o n . I n the stepwise d e f i n i t i o n o f IT on t h e remaining Pmls we have obeyed t h e $ - c o n d i t i o n s , and we permute t h e $ - f r e e c o n d i t i o n s a c c o r d i n g l y . F i n a l l y we observe t h a t t h e permuted c o n d i t i o n s Pml original P

do n o t occur i n t h e

B'

An a p p l i c a t i o n o f t h e permutation lemma y i e l d s ~ ( p , ) IC

cp*(B,Pn,P,)

so

nbB)

IF cp*(B,P,',

Pk)

Since ~ ( )p i s contained i n our g e n e r i c sequence we may apply t h e t r u t h

B

1 emma :

rn

II- (dC(B,Pn,Pk)

and hence

B (I- d E n , Lk). Now we can f i n d t h e r e q u i r e d i n i t i a l segment o f f o r a l l B E E , aIk 3x V E E x

Hence LS3(2) holds i n our model.

2.8

Lemma.

since

BI- VEErn(h)

d5,Sk)

0

LS3(3) f a i l s i n t h e model.

Ph006. There i s a

However ( )

5:,

cp(c,c,).

Y 3x

5,

such t h a t our model ( )

c i E x A VEEx

5 =

$(c,,c,).

IF Ei = $ ( 5 , , t 2 ) . 0

2.9 C o r o l l a r y . LS1, LS2, LS3(2) Y LS3(3) Contrary t o t h e i n i t i a l e x p e c t a t i o n o f t h e authors t h e above r e s u l t does n o t immediately c a r r y over t o t h e n-parameter case. We s t i l l c o n j e c t u r e t h a t LS3(n) lf

LS3(n+l),

g e n e r i c sets.

but presumably one needs more s o p h i s t i c a t e d o p e r a t i o n s on

Lawlessness and independence LITERATURE

[Dl]

D . van Dalen. An i n t e r p r e t a t i o n o f I n t u i t i o n i s t i c A n a l y s i s .

Ann. Math. L o g i c 1 3 ( 1 9 7 8 ) , pp. 1-43.

-

(D21

The c r e a t i v e s u b j e c t and H e y t i n g ' s A r i t h m e t i c ( t o appear i n ' A l g e b r a and i t s A p p l i c a t i o n s ' , Banach C e n t r e P u b l i c a t i o n s ) . T h i s paper c o n t a i n s c o r r e c t i o n s t o [ D l ] .

[F]

S.

Feferman. Some a p p l i c a t i o n s o f t h e n o t i o n o f f o r c i n g and g e n e r i c s e t s . Fund. Math. 56(1965),

pp. 325-345.

309

THE LXJ. BROUWER CENTENARY SYLWOSIUvl A S . Troelstra and D. van Dalen (editors) 0 North-HollandPublishing Company, 1982

311

CREATIVE SUBJECT AND BAR THEOREM Enrico Martino Seminario matematico Padova University V. Belzoni 35100 PADOVA (Italia) In the present article a reasonably precise description of Brouwer's notion of "creative subject" is proposed and an axiom is introduced which is conceptually equivalent to the Bar theorem.

1.

THE CREATIVE SUBJ€CT. The idea of the creative subject occurs in various writings of Brouwer in a

somewhat vague manner.

Several logicians, among them Kreisel, Troelstra and

Dummett, have recently tried to analyse this concept by proposing some axioms, which are however rather controversial, as Dumett's discussion shows. The creative subject, which we will call I, carries out its mathematical activity in time, which we assume divided in w

distinct states of knowledge.

The

main feature of the theory of the creative subject consists in using the fact that knowledge increases in time by explicitly referring to the stage in which E to know a certain proposition. operator

t.

If A

gets

Such a reference is expressed by the propositional

is a formula,

F A

stands for:"at stage n .I has evidence

for A". The kind of axioms for

we put forward essentially depends on our ideali-

zation of t. were pro?A the strict one, according to which I can at any stage know only a finite

In the literature two essentially different conceptions of posed:

number of propositions, and the wider one, according to which at any stage .I is allowed to know infinitely many propositions. The wider conception is proposed, among others things, by Troelstra [TI as a possible solution of a paradox of diagonalization in the strict theory (see § 3 ) , which he pointed out. We begin with some critical observations on the wider conception. The following axioms are usually assumed both in the strict theory and in the wider one:

E. MARTIN0

312

These axioms are not sufficient to yield a precise conception of the creative subject, but they seem to be minimal requirements:

they are implicitly used by

Brouwer himself in the well-known constructions of counterexamples for some classical theorems. 1.1

is usually justified in the following manner:

decidable since at the stage n X.

the predicate

%A

knows whether he has evidence for A

is or not.

We will shortly discuss such a justification. 1 . 2 says that the knowledge of

1

is cumulative:

at every stage he knows

again (or he maintains the knowledge of) what he knew at the preceding stages. 1.3,

from left to right, says that E

reasons in an intuitionistically cor-

rect manner, that is he knows only intuitionistically true propositions. From right t o left it seems to express the "solipsistic" conception of intuitionistic mathematics: subject.

the true propositions are only those which are known to the creative

If one does not want to commit oneself to this last statement, 1 . 3 can

be weakened by replacing it by 3nkA

1.3'

A

+

and 1 .3"

1.3"

A

-f

113nkA

(called by Kreisel the Principle of Christian Charity) says that if A

true, it cannot be excluded that, in some stage, X In the wider conception it is allowed that, if ate consequence" of A,

kB.

then

is

comes to know it. %A

and if

B

is a "immedi-

Apart from the difficulty of generally char-

acterising in a sufficiently precise manner the vague notion of immediate consequence, it seems that the following axioms for it are hardly objectionable (see [Dul). 1.4

t;;VXP(X)

1.5

% P(i5)

I .6

t;;(AvB)

-t

3xP(x)

+

-f

vxt;; P(x)

%A

v

%B

In 1.6 it is understood that A

and

B

do not contain choice parameters nor

any information which is not available at stage n .Now, we see at once that from 1.1-1.6

(see 1 2 ) .

we can deduce some intuitionistically

incorrect propositions. In fact, let P(x) such that Vx(P(x)

be a decidable predicate (without choice parameters),

V lP(x)).

From 1.3 it follows that, for some n,

3 13

Creative subject and bar theorem

whence, by 1 . 4 and 1.6, VX(kP(X) It follows that, if by 1 . 5 ,

k3xP(x)

Ik-lP(x)).

3xP(x),

.

then there exists a k

such that

kP(c)

and

so,

Therefore the following equivalence holds

3xP(x) Thus, since, by 1 . 1 ,

v

tf

k3xP(x).

k3xP(x) v lk3xP(x),

we have

3xP(x)

V

13xP(x).

But this

cannot be intuitionistically acceptable for an arbitrary P(x).

In my opinion this incoherence arises from the fact that 1 . 1 is acceptable only if, in order to have of A.

%A,

it is required that

Now, the evidence of VxP(x)

Z

is aware of having evidence

does not involve (unless we greatly stress

the idealisation of Z) the conscious evidence of all single instances P(k). Thus 1.4 holds only if evidence is meant as implicit, not necessarily conscious, evidence.

Hence the incompatibility of 1 . 1 and 1 . 4 results.

Since X so

is no real subject but an idealized subject, we could idealize him

that the evidence of VxP(x)

P(k).

involves the conscious evidence of each single

But such an idealization would attribute to

Z

superhuman powers which

would allow him to know some classically but not intuitionistically true propositions.

Since intuitionistic truths are to be humanly knowable, we must be very

cautious in idealizing Z. Therefore, since 1.4 seems t o be a minimal requirement of the wider conception, we hold that it forces the axiom 1 . 1 to be abandonned. Then, at the same stage, Z

can implicitly know infinitely many propositions.

he knows the soundness of the axioms and of the in-

If, fo'r instance, at stage n

ference rules of a certain system F, evidence of all theorems of

F

then at the same stage n

he has implicit

(but he i s not necessarily able to decide whether

a given formula is a theorem, as 1 . 1 requires).

Indeed one could even assume that

every stage is closed with respect to "analytic" deductions and that the passage from one stage to another is characterized by an increase of external information (see Posy (1977)). be read:

But we think that in that case

"At the stage n

$A

t can have evidence of A".

should be more properly The potential interpreta-

tion serves the purpose of avoiding that superhuman powers (as that one of deducing actually all what is deducible from the available information) are attributed to Z.

The closure with respect to the analytic deductions seems to be in agreement

with Grzegorczyk's semantics, (cf. [GI). Nevertheless it seems to me that such a conception has not the effect of taking the idea of the creative subject seriously, but rather of replacing it by other notions, such as "deducibility (in an intuitive sense) from certain given information", (cf. [Dal).

On the contrary we believe that the originality and the strength

314

E. MARTIN0

of the theory of the creative subject consists just in the possibility of exploiting the concept of conscious evidence, which Brouwer has already partially exploited by his implicit assumption of 1.1.

We hold that new interesting results are obtain-

able not just by suppressing 1 . 1 but rather by adding to it new axioms intended to exploit the concept more deeply. Therefore we will abandon the wider conception in favour of the strict one, according to which the request of the conscious evidence i s certainly plausible. This request can be made even more explicit by stating, for instance, that, in order to have

$A,

X

has to

write

a proof of

A

at the stage n.

We here obviously

pass over the inadequacy of natural language to express an intuitive proof; our statement serves only the purpose of clarifying metaphorically the meaning of the adjectif "conscious".

What is of importance is that, proving only finitely many

propositions at each stage, .I has the possibility of explicitly directing his attention at each one of them. .

The axiom 1 . 2 results to be obvious. Without committing ourselves to the so-

lipsistic conception, we will accept 1.3 with the following justification: since A

is intuitionistically true, if it has been proved by someone, we suppose that as

soon as a proof of

A

is found (by someone), it is communicated to .I who includes

it among the propositions of which he has evidence (at the stage he is in at the moment of communication). We now intend to propose a modification of the strict conception in order to obviate, at least in part, the drawback of the invalidity of 1.4 and of other axioms which seem to hold only in the large conception. We replace 1.4 with the scheme 1.4'

Vn(kVxP(x) If 1 . 4 '

+

k~(ii)).

is understood in the sense that all its instances should be simultane-

ously true, then it is certainly not acceptable in the strict conception, since it presents the same difficulties as 1 . 4 .

Therefore we propose to understand the val-

idity of an axiom scheme in the following sense:

for each instance, it is possible

"to programme" 1 at the stage 0 so that the instance in question is true. "to programme E

"

we mean "to instruct E

favourable conditions, he performs certain deductions which interest us". sense 1.4' is thoroughly acceptable: whenever we have fixed P and struct 1 so that, if he deduces VxP(x)

P(k)

By

so that if he happens to be in certain k,

In this we can in-

in some stage, he takes care of deducing

at the same stage. Similarly for 1 . 5 :

3xP(x)

whenever

as soon as he has deduced

P

is fixed, 1 can be instructed to deduce

P(m)

for some m.

necessary restrictions on the choice parameters).

In general we admit the following principle:

Likewise for 1.6 (with the

Creative subject and bar theorem 1 . 7 Whenever, for a fixed proposition

state of knowledge

s

A,

315

recognize that, if we were in a certain

we could prove A,

we can suppose that whenever 1

is in

the state s, he actually deduces A. Some other axioms which turn out to be sound are the following:

On the contrary we observe that 1.8'

(A-tB)

+

Vn ( k A

-f

I ; B )

does not hold, since until we know a proof of A instruct 1 to deduce B

-f

B

(or of l(A+B))

we cannot

from A.

But from 1.8 and 1 . 3 we get: 1.8''

(A-tB)

-t

3n( \;;A

+

kB).

We observe that a formula A(x)

with a free variable x must be considered

true if it is possible to programme t neously true. from it V x k x

=

x

t

that all instances A(k)

k x = x

is not true:

are simulta-

in fact we deduce

which would involve the evidence, at the stage n ,

finitely many propositions k = k. true:

so

For example, the formula

can be instructed

so

of the in-

On the other hand the formula V x t x = x

is

that, at every stage n, he knows n = n.

Finally we observe that, according to our concept of programming, t can be given only positive instructions, i.e. instructions for performing (under favourable conditions) certain deductions, and not negative instructions, i.e. instructions for not performing certain deductions. At every stage t must be free to perform at will some other deductions, in addition to the ones which are imposed on him. if A

This guarantees that two programmings are always compatible. Therefore, and

B

are instances of two sound axiom schemata, there is a programing

which makes them simultaneously true.

It follows that since only a finite number

of formulas occur in a formal deduction, every formula which is deduced by use of some sound axiom schemata is true with respect to a suitable programming. Thus the usual axioms and inference rules are in accordance with our concept o f soundness of a scheme.

This would not be the case if we would admit negative instructions.

Then, for instance, both formulas vx ( *=x)

and

Vx(lFx=x)

would be sound, since t could be instructed both to prove n = n for every n

and not to prove n = n

inconsistency would follow.

at stage n

for all

n.

at stage n

Hence syntactic

E. MARTIN0

316

7ffE CREATIVE SUBJECT ANU EXlSTENTlAL STATEMENTS.

2.

The axiom 2. I '

Vn (

t-, 3mP (m)

+

3ml;P

(m) )

is usually accepted with the following justification: 3mp(rn)

~(i;)

is a proof of some instance

an intuitionistic proof of

P(x).

of

But we observe that it is possible to have evidence for 3mP(m) proof of an instance of

P(x)

even if a

is not available, but only a method for constructing

such a proof is available. Therefore if, at the stage n, t has evidence for 3mP(m),

it is the case that at stage n

an instance of

P(x).

t has a procedure

The execution of

TI

for determining

TI

must be feasible in a finite number of

stages but not necessarily at the same stage n,

since the execution can require

some information which can be obtained only at future stages. Thus what holds in general is not the axiom 2 . 1 ' 2. I

Vn(k3mP(m) For instance, let

but the weaker axiom

-t

be a free sequence such that the value

p

has been chosen (and known to k3xp(m)

3p3m tP(m)). P

=

at stage n.

Z )

x

p(n)

of

p

Then, if m > n

but not

3x

I-

p(m)

=

x.

Observe that in this example, even though t cannot determine an instance of

P(x),

at stage n, yet, at stage n, he knows at which stage (at the stage m) the execution of the procedure

TI

will be completed.

But one sees at once that neither is this the general case, i.e. not always k3xP(x) F o r example, if

+

P(x)

is

3y(1;3x

l-P(x)). Y

p(p(m))

=

only at which stage (at the stage m) m

and

p(m))

at which

IT

x

(and

n < m),

at the stage n 1 knows

he can determine the stage (the maximum of

will be completed.

In general, it seems reasonable to admit as implicity involved in the evidence (of the fact) that the procedure is that if

k3xP(x),

1

TI

must terminate in a finite number of stages

should at least be able to point out,at the stage n,

stage in which he can obtain an important piece of information concerning a stage'at which

II

IT,

For a precise statement of this idea, let us consider, for every sentence 3xP(x)".

the species S We define

i.e.

is reduced to a procedure IT', more elementary with respect

to the complexity of the required information. 3xP(x),

a

S

of the stages in which inductively:

"1 has inductive evidence of

317

Creative subject and bar theorem

(i)

if for some m l;P(m),

(ii)

if for some m I-m

then n then n

S,

E

S;

E E

S.

Then the "principle of the inductive evidence" which we intend to propose can be stated in the following way: PIE:

For every sentence of the type Therefore PIE

"If E of

3xP(x),

S3xp(x) = {n k3xP(x)j.

says:

has, at the stage n, evidence of' 3xP(x),

then he has inductive evidence

3xP(x)". In particular, the above considerations can be extended to the axiom

EQUUIVALENCE OF PIE AND BIM.

3.

++ be variables for choice sequences, u,v variables for finite se+ quences (nodes of the universal tree). The length of a sequence u =

Let

is L(:)

a,B

=

n+l, where the length of the empty sequence is L ( < >) = 0. Initial

segments of choice sequences are denoted by a(n) = < a ( O ) , ...,a(n-I)>. We write + + E u for i ( L ( : ) ) = U. For a species of nodes R we define inductively'theF-

a

closure RF

R by:

of

A species of nodes now'state

Theorem Proof.

+

++ u*vcR).

We

the monotone bar theorem in the form:

If R

Bh:

+++

R will be called monotone if VuVv(u~ R

is a monotone species of nodes and Va3xi(x) BIN

++

E

R,

then < >

E

R

.

PIE

Let u s assume BIN.

For the sake of simplicity we consider stage 0 and

suppose that

We assume that it is possible to represent the information, on which the procedure TI

for determining an instance of

that, for each n, 1 at the stage n

-

o(n)

R

execution of

rests, by means of a sequence u

such

is a codification of the amount of information known to

and, for every

possible (i.e. at the stage n) Let

P(x)

k, o(n)

* is a value which is a priori

for o(n+l).

+ be the species of the pieces of information u sufficient for the II.

Then

E. MARTIN0

318 Vadxa(x)

E

R. -f

By induction on the construction of can programme t so that'if I)

if

E

performs 2)

E

+ u

E

S:

E

a,

and gets the required instance of

E

R'

one

at the stage L(:) hence L ( c )

P(x),

he by (i);

S

E

if Vk(z* E RJ), by the induction hypothesis one can instruct 1 s o that + if u* E a, then e(:)+l t S . Then we can instruct 1 so that if E 0 also L(G)+l

-L(:)+l Since R

t takes note of it at the stage l ? ( G ) ,

and

S

E E

whence L(;)

S,

is obviously monotone,

Va3xu(x)

E

R.

%3xi;(x)

E

R

p(n)

E

R

R

of

S, that

i)

If for some m

3xp(x)

S c_ { n l p ( n )

of

p,

induction on m-n, ii) If,for some m, 0

E

S.

after having introduced a free sequence

R,

then if n

2

m,

p(n)

it must hold that Va(b(n)

L S,

E

E

R

X

has in-

because of the monoton-

*a(m-n)

E

RF),

whence, by

RF.

then, by the induction hypothesis, 6(m)

and so < >

S

E

Let u s prove, by induction on the construction

as i n i), we recognize that p(n) By PIE

0

then, since at the stage 1 knows only the initial

p(n)

t-, m

and so

R 1. E

R; if n < m

segment 6(n)

BIM

is chosen at the stage n+l, we can (get)

R.

E

E

t, p ( m )

by

be a monotone species such that

t o be the species of the stages in which

S

and we put

ductive evidence of

icity of

< >

Let

We programme t as follows:

with the property that

i.e.

by (ii).

S

E

Conversely let u s assume PIE. p

then L(:)

RF,

then we instruct t so that if

R,

u

let u s prove that, for each

RF,

E

E

E

RF

and,

RF.

RF.

BIBLIOGRAPHY. I.

L.E.J. Brouwer [Bl, Collected Works, vol.1: Philosophy and Foundations of Mathematics, a cura di A. Heyting, Amsterdam 1975.

2.

M. Dummett [Dul, Elements of Intuitioeism, Oxford 1977.

3.

A. Grzegorczyk [GI,

4.

G. Kreisel [K],

5.

C.J.

A Philosophicalty Plausibte Formal I n t e r p r e t a t i o n o f I n t u i t i o n i s t i c Logic, "Indagationes Mathematicae 26(1964), pp.596-601.

Informal Rigour and Completeness Proofs, in Problems in the Philosophy of Mathematics, a cura di I. Lakatos, Amsterdam 1967, pp. 138-186.

Posy [ P I , Brower's Constructivism, "Synthese" 27(1974), pp.125-159. Va'erieties of Indeteminacy in the Theory of Genera2 Choice sequences, "Journal of Philosophical Logic" 5(1976), pp.91-132. The Theory of Enrperical Sequences, "Journal of Philosophical Logic" 6(1977), pp.47-81.

6. A . Troelstra [TI, 7.

Principles of Intuitionism, Berlin 1969.

D. van Dalen [Dal, An I n t e r p r e t a t i o n of I n t u i t i o n i s t i c Analysis, "Annals of Mathematical Logic" 13(1978), pp.1-43.

THE L.EJ. BROUWER CENTENARY SYMPOSIW A.S. Doehtra and D. van Dalen (editors) 0 North-HolkmdPublishing Company, 1982

319

THE INTRODUCTION O F N O N - R E C U R S I V E METHODS INTO MATHEMATICS* George M e t a k i d e s * and A n i l Nerode U n i v e r s i t y o f P a t r a s , Greece Cornell University,

Ithaca,

New Y o r k

USA

The a u t h o r s s u m m a r i z e t h e u s e o f r e c u r s v e f u n c t i o n t h e o r y as a t o o l t o u n d e r s t a n d t h e t r a n s i t i o n from constructive t o non-constructive mathematics i n the 1 9 t h century. Certain classic a l theorems o f a l g e b r a and a n a l y s i s a r e analyzed f o r recursive content. 1.

INTRODUCTION

As t h e n i n e t e e n t h c e n t u r y b e g a n , v i r t u a l l y a l l m a t h e m a t i c a l r e s e a r c h was o f a c o n c r e t e , c o n s t r u c t i v e , a l g o r i t h m i c c h a r a c t e r . By t h e e n d o f t h e n i n e t e e n t h c e n t u r y much a b s t r a c t , n o n - c o n s t r u c t i v e , n o n a l g o r i t h m i c m a t h e m a t i c s was u n d e r d e v e l o p m e n t . W h a t h a p p e n e d , how d i d i t h a p p e n , a n d w h y ? We h a v e s o u g h t t o u n d e r s t a n d t h i s s p e c t a c u l a r t r a n s i t i o n by u s i n g t h e t o o l s and i n s i g h t s o f r e c u r s i v e function theory systematically. We h a v e u s e d t h e n i n e t e e n t h c e n t u r y sources e x t e n s i v e l y t o s t u d y these changes. H e r e we g i v e a b r i e f a c c o u n t o f some o f t h e s e c h a n g e s i n a l g e b r a a n d a n a l y s i s . We a t t e m p t t o e x p l i c a t e them i n a few cases, u s i n g r e c u r s i v e f u n c t i o n t h e o r y . T h i s w i l l be o n l y a s k e t c h f o r a l o n g e r c a r e f u l l y d o c u m e n t e d s t u d y , a n d we h o p e t h a t b r e v i t y d o e s r r o t l e a d t o s e r i o u s e r r o r s i n e m p h a s i s or historical fact. B r o u w e r was a m o s t i l l u s t r i o u s c r i t i c o f t h e i n t r o d u c t i o n o f n o n - c o n s t r u c t i v e methods, s o i t i s f i t t i n g t o examine t h i s i s s u e f o r t h i s symposium. M i d d l e n i n e t e e n t h c e n t u r y and e a r l y t w e n t i e t h century mathematics l o o k q u i t e d i f f e r e n t . I n a d d i t i o n t o t h e g r o w t h o f new s u b j e c t s , t h e r e i s a g r o w i n g p r e f e r e n c e f o r s h o r t c o n c e p t u a l noncomputational proofs (often i n d i r e c t ) over long computational proofs (usually direct). T h e s e c h a n g e s h a v e b e e n l o o k e d a t f r o m many d i f f e r e n t p o i n t s o f v i e w . B e s i d e s B r o u w e r , s u c h g r e a t names a s K r o n . e c k e r , P o i n c a r 6 , C l e b s c h , G o r d a n , E. B o r e 1 h a d r e s e r v a t i o n s a b o u t t h e n o n computational methods. B u t o n l y a few t r i e d t h e i r hand a t s y s t e m a t i c development o f mathematics from a c o n s t r u c t i v e p o i n t o f view. I n a l g e b r a , K r o n e c k e r d e v e l o p e d much o f a l g e b r a a n d a l g e b r a i c n u m b e r t h e o r y as a s u b j e c t d e a l i n g w i t h f i n i t e m a n i p u l a t i o n s o f f i n i t e expressions. I n a n a l y s i s , Brouwer e s t a b l i s h e d systematic c o n s t r u c t i v e s u b s t i t u t e s f o r t h e theorems o f t h e Weierstrassian foundations o f real analysis. Unfortunately, the general mathematical p u b l i c tends

*Research s p o n s o r e d i n p a r t by US-Greece I n t e r n a t i o n a l Development P r o g r a m o f NSF-Y.E.E.T. M C S - 8 0 - 1 7 0 7 3 , i n P a r t . u n d e r NSF g r a n t n o . MCS-80-03016.

320

G . METAKIDES, A. NERODE

to remember Kronecker for his attacks on Cantor, and Brouwer for his attacks on classical analysis. In this paper, when we view the changes from algorithmic to nonalgorithmic mathematics, we do s o wholly within the domain of classical mathematics, and use its full power. This is made possible by the existence of recursive function theory as a developed branch of classical mathematics. This tool was not available to Kronecker or Brouwer. Its significance is that it removes the question as to what can or cannot be computed from a fuzzy informal intuition to the status of an exact subject. The key advance was the formulation of a sufficiently general notion of algorithm so as to cover all possible computations. The principal difference between the use of recursive function theory as an approach and the methods of either Kronecker or Brouwer is that in proofs that computations terminate (or don't terminate), any sort of argument at all from classical mathematics is acceptable. This includes higher set theory and indirect proofs. The classical attitude behind this is that we are interested in whether computation procedures exist in principle to do certain jobs, not in how they are proved to work. Thus our investigations can be understood and accepted by any classically trained mathematician, and require no understanding of or commitment to any constructivist point of view. (It is very difficult to read constructivists like Brouwer, Shanin, Kronecker, or Bishop without such a commitment.) In summary, our program is to determine the recursive content o f mathematical constructions, and to use this to shed light on the changes alluded to above. We have just delineated,the difference between the recursive function point of view and constructive points of view. There are very strong common features as well. Whenever a constructivist exhibits the construction of a mathematical object from certain initial data, it is usually trivial to convert this to a recursive procedure which will yield a recursive presentation of the object from a recursive presentation of the data. It is usually an uninteresting exercise. On the other hand, suppose a constructivist produces a Brouwerian intuitionistic proof that the assumption that there is a procedure (constructing an object from certain data) leads to an absurdity. Then often a recursion theorist can use this as a hint to begin a laborious priority construction o f a recursive presentation of a model in which the data are recursively presented, but the object asserted to exist by the classical construction cannot be recursively presented. (This wil.1 be made clear by examples later.) The use of the priority method for determining the recursive content of algebraic constructions was introduced by u s (1974) and now a whole literature exists. We have only recently succeeded in adapting the same methods to analysis. In Section 2 we recount a little of the history o f the theory o f fields. We also indicate how this is related to the recursive content of theorems of field theory. In Section 3 we recount some of the history o f ordinary and variational calculus. In Section 4 we use recursively presented metric spaces as a language to describe the effective content of ordinary and variational analysis. In Section 5 we give a preliminary account of the recursive content of the Hahn-Banach theorem. In Section 6 we give concluding remarks.

Non-recursive methods

2.

321

F I E L D THEORY

Lagrange i n 1772 d e v e l o p e d a G a l o i s t h e o r y f o r t h e g e n e r a l e q u a t i o n o f degree n, b u t d i d n o t r e s o l v e t h e q u e s t i o n o f s o l v a b i l i t y by radicals f o r n 5 t h a t had been w o r k e d o n f o r t h e 200 y e a r s s i n c e t h e successes i n degrees 3 and 4 i n Cardano's time. After inc o m p l e t e i m p o s s i b i l i t y p r o o f s b y R u f f i n i i n 1799 and 1813, Abel gave t h e f i r s t i m p o s s i b i l i t y p r o o f i n 1824, and t h e n G a l o i s gave t h e gene r a l G a l o i s t h e o r y o f normal extensions o f f i n i t e degree (published i n 1846). T h e a p p r o a c h t a k e n b y a l l was w h o l l y a l g o r i t h m i c , a l l o w i n g a t each stage t h e computation ( i n p r i n c i p l e ) o f everything, i n c l u d i n g the f i n a l group t a b l e o f t h e Galois group. Gauss d e v e l o p e d i n 1801 a t h e o r y o f b i n a r y q u a d r a t i c f o r m s w h i c h g i v e s e s s e n t i a l l y an e q u i v a l e n t o f t h e t h e o r y o f i d e a l s i n q u a d r a t i c number f i e l d s , i n w h o l l y a l g o r i t h m i c form. I n 1 8 4 5 , Kummer d e v e l o p e d a t h e o r y o f i d e a l p r i m e s i n c y c l o t o m i c number f i e l d s , b o t h f o r t h e s t u d y o f h i g h e r r e c i p r o c i t y laws and f o r F e r m a t ' s l a s t theorem. The d e f i n i t i o n s were e n t i r e l y algorithmic. An i d e a l p r i m e i s a c e r t a i n s y s t e m o f c o n g r u e n c e r e l a t i o n s , t w o i d e a l p r i m e s a r e e q u a l if and o n l y i f a c e r t a i n c o m p u t a b l e r e l a t i o n h o l d s between t h e congruence r e l a t i o n systems. Kummer's d i s c i p l e K r o n e c k e r w e n t o n i n 1 8 5 9 t o g e n e r a l i z e t h e Kummer t h e o r y t o a r b i t r a r y number f i e l d s , and a l o t o f f u n c t i o n f i e l d s , and i n d o i n g s o r e v e a l e d much a l g e b r a i c g e o m e t r y . T h e k e y i d e a f o r K r o n e c k e r was t h a t o f an i d e a l prime as a f o r m a l l i n e a r combination o f elements His theory o f ideal w i t h indeterminate c o e f f i c i e n t s (formal GCD). p r i m e s was w h o l l y a l g o r i t h m i c , a n d t h e c r i t e r i a f o r t w o s u c h f o r m a l l i n e a r c o m b i n a t i o n s t o g i v e t h e same p r i m e w e r e c o m p u t a t i o n a l . He delayed p u b l i c a t i o n a l o n g time, and i n t h e meantime Dedekind developed f r o m Kummer's c y c l o t o m i c t h e o r y a h i g h l y c o m p u t a t i o n a l t h e o r y o f i d e a l p r i m e s i n number f i e l d s . B u t t h i s i s n o t what Dedekind publ i s h e d f i r s t ( i n 1871). He d i d n o t l i k e t h e d e f i n i t i o n o f i d e a l prime i n computational terms w i t h an equivalence r e l a t i o n which t e l l s when t w o s u c h a r e e q u a l . He w a n t e d a m o r e i n t r i n s i c , s e t - t h e o r e t i c a n s w e r t o t h e q u e s t i o n " W h a t i s a n i d e a l p r i m e ? " He d e r i v e d t h e t e x t b o o k d e f i n i t i o n t h a t we now u s e a n d w i t h some e f f o r t u s e d i t t o d e v e l o p a much l e s s c o m p u t a t i o n a l v e r s i o n o f t h e t h e o r y . This information about Dedekind i s from a paper w r i t t e n by him i n response t o t h e Russian Z o l a t a r e f f ' s p u b l i c a t i o n o f an incomplete rediscovery o f Dedekind's e a r l y c o n s t r u c t i v e approach. Complete v e r s i o n s had been p u b l i s h e d b y b o t h K r o n e c k e r and Dedekind by 1882. D e d e k i n d ' s work a c q u i r e d a g r e a t f o l l o w i n g due t o b e a u t y and s t y l e , w h i l e K r o n e c k e r ' s cramped s t y l e a t t r a c t e d o n l y a few readers. B u t a t l e a s t e v e r y o n e knew t h a t K r o n e c k e r h a d o b t a i n e d t h e main e l e m e n t a r y theorems on p o l y n o m i a l i d e a l s i n p o l y n o m i a l domains by computation. I t i s a t t h i s p o i n t t h a t we s e e a s p l i t i n t h e d e v e l o p m e n t o f f i e l d t h e o r y i n t o a c o n s t r u c t i v e t r a d i t i o n f o l l o w i n g Kronecker and a l e s s c o n s t r u c t i v e s e t - t h e o r e t i c t r a d i t i o n f o l l o w i n g Dedekind.

Dedekind i n t r o L e t us l o o k b r i e f l y a t t h e Dedekind t r a d i t i o n f i r s t . duced f i e l d s as s u b f i e l d s o f t h e complex numbers. H i s co-worker, Weber i n t r o d u c e d a b s t r a c t f i e l d s ( 1 8 9 3 ) . I n response t o examples as d i v e r s e as t h e r e a l f i e l d and t h e p - a d i c f i e l d s o f Hensel, S t e i n i t z (1910) d e v e l o p e d o u r p r e s e n t t h e o r y o f a r b i t r a r y f i e l d s ( a s opposed t o t h e f i n i t e dimensional t h e o r y of Kronecker). Among S t e i n i t z ' s theorems were t h e e x i s t e n c e and uniqueness o f a l g e b r a i c c l o s u r e , t h e o r e m s t h a t we w i l l d i s c u s s b e l o w . Then A r t i n and S c h r e i e r d e v e l oped t h e t h e o r y o f r e a l c l o s e d f i e l d s (1927) and K r u l l d e v e l o p e d

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(following a h i n t of Dedekind) t h e Galois theory of i n f i n i t e extensions (1928). N e x t , l e t us e x a m i n e t h e t r a d i t i o n s t e m m i n g f r o m K r o n e c k e r . T h i s was h i g h l y a l g o r i t h m i c a n d , a s a w h o l e , c o n f i n e d by c h o i c e t o t h e f i n i t e d i m e n s i o n a l c a s e . His s o - c a l l e d e l i m i n a t i o n t h e o r y ( a n d m o r e ) was e x p o s i t e d i n a 500 p a g e book by G . KSnig ( 1 9 0 3 ) b u t t h e r e w e r e e r r o r s . A s t u d e n t . H e n z e l t , t u r n e d i n a d i s s e r t a t i o n i n 1915 which p u r p o r t e d l y d i d c o n s t r u c t i v e i d e a l t h e p r y o f p o l y n o m i a l i d e a l s , b u t he d i e d i n World War I . E . N o e t h e r e d i t e d H e n z e l t ' s w o r k t o e l i m i n a t e many c o m p l e x i t i e s ( H e n z e l t , 1 9 2 2 ) . N o e t h e r ' s s t u d e n t G r e t a Hermann ( 1 9 2 5 ) p u b l i s h e d a p a p e r w h i c h was s u p p o s e d t o c a r r y o u t a t l e a s t p o l y n o m i a l ideal theory over f i n i t e dimensional f i e l d s of c h a r a c t e r i s t i c zero i n a f i n i t e number o f s t e p s . T h e r e w e r e m i n o r e r r o r s , a n d a n e x p o s i t i o n c o r r e c t i n a l l c h a r a c t e r i s t i c s was p u b l i s e h d by S e i d e n b e r g G r e t a Her( 1 9 7 4 ) . B u t t h e r e was a m a j o r c o n c e p t u a l e r r o r a s w e l l . mann a l s o s p e a k s o f f i e l d s w h i c h a r e e x p l i c i t i n t h e s e n s e t h a t t h e r e a r e a l g o r i t h m s ( i n f o r m a l l y , or i n t u i t i v e l y ) f o r f i e l d o p e r a t i o n s and f o r t e s t i n g e q u a l i t y o f e l e m e n t s . Now K r o n e c k e r showed t h a t e v e r y f i n i t e d i m e n s i o n a l f i e l d c a n b e r e a l i z e d a s a n e x p l i c i t f i e l d . He B u t what a b o u t showed t h a t e v e r y s u c h h a s a " s p l i t t i n g a l g o r i t h m " . o t h e r e x p l i c i t f i e l d s w h i c h a r e c o u n t a b l e : Do t h e y n e c e s s a r i l y h a v e a l g o r i t h m s w h i c h d e t e r m i n e i n a f i n i t e number o f s t e p s w h e t h e r o r n o t a g i v e n p o l y n o m i a l o v e r them s p l i t s ? H e r m a n n ' s e r r o r h a s t h e f o l l o w i n g r o o t . Think o f a c o u n t a b l e e x p l i c i t f i e l d a s l i s t e d a s a s e q u e n c e , t a k e t h e s u b f i e l d F n g e n e r a t e d by t h e f i r s t n e l e m e n t s l i s t e d , r e p r e s e n t the whole e x p l i c i t f i e l d a s t h e union of t h e c h a i n Fn. S i n c e e a c h F n h a s a s p l i t t i n g a l g o r i t h m , s h e a s s e r t s t h a t so d o e s t h e w h o l e e x p l i c i t f i e l d . B u t c o n s i d e r a n y p a r t i c u l a r polynomWe h a v e no i a l o v e r t h e f i e l d , whose c o e f f i c i e n t s l i e i n a n F n . i d e a w h e t h e r some F m , m > n , e x i s t s w h e r e t h e p o l y n o m i a l s p l i t s . T h i s g a p was o b s e r v e d by Van Der Waerden ( 1 9 3 0 ) . He was w e l l a c q u a i n t e d w i t h i n t u i t i o n i s m i n Amsterdam, a n d a t t e m p t e d t o g i v e a kind of counterexample. He h a s no e x a c t d e f i n i t i o n o f w h a t a n e x p l i c i t f i e l d i s , a n d makes no a t t e m p t t o p r o d u c e a s p e c i f i c o n e without a s p l i t t i n g algorithm. I n s t e a d he s u p p o s e s t h a t t h e r e i s a g e n e r a l p r o c e d u r e l e a d i n g f r o m t h e d a t a f o r an e x p l i c i t f i e l d ( o p e r a t i o n s and e q u a l i t y ) and a p o l y n o m i a l o v e r i t , p , t o a 1 i f p i s He s h o w s t h a t t h i s l e a d s t o a n a b s u r d irreducible, a 0 i f not. r e s u l t i n t u i t i o n i s t i c a l l y . The a b s u r d r e s u l t i s t h a t f r o m an e n u m e r ation f ( O ) , f(l),:.. o f an a r b i t r a r y s e t A o f i n t e g e r s , t h e r e would t h e n be a g e n e r a l p r o c e d u r e l e a d i n g f o r a n y i n t e g e r n t o a d e c i s i o n 1 i f n i s i n A , 0 o t h e r w i s e . To s e e t h i s e n c o d e t h e = O,l, ...I enumeration of A a s t h e e x p l i c i t f i e l d F = Q [ f i f ( , ) l n where pn i s the n t h p o s i t i v e prime. A g e n e r a l s p l i t t i n q a l g o r i t h m , a p p l i e d t o t h i s f i e l d and x 2 - p n , d e c i d e s w h e t h e r n i s i n A o r not. A c l a s s i c a l l y minded m a t h e m a t i c i a n w i l l endow b o t h t h e d a t a ( f i e l d

o p e r a t i o n s , e q u a l i t y , polynomial p ) w i t h a p r o d u c t of d i s c r e t e t o p o l o g i e s , and s i m i l a r l y { O , l } , w i t h t h e d i s c r e t e t o p o l o g y , and i n t e r p r e t t h i s a s s a y i n g t h e r e i s no c o n t i n u o u s f u n c t i o n c o n s t r u c t (Products of d i s c r e t e t o p o l o g i e s a r e ing the o b j e c t from t h e d a t a . t h e r u l e i n i n t e r p r e t i n g a l g e b r a ; i n a n a l y s i s , the usual t o p o l o g i e s of a n a l y s i s a r e used f o r the c o r r e s p o n d i n g purpose of g i v i n g a c l a s s i c a l meaning t o i n t u i t i o n i s t i c c o u n t e r e x a m p l e s . )

Non-recursive methods

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F r o h l i c h a n d S h e p h e r d s o n ( 1 9 5 5 ) saw t h a t t h e n o t i o n o f r e c u r s i v e f u n c t i o n c o u l d b e u s e d t o make t h e p r e v i o u s l y i n t u i t i v e n o t i o n o f a n explicit f i e l d into a purely classical definition. Namely, f o r them an e x p l i c i t f i e l d i s a r e c u r s i v e s e t w i t h r e c u r s i v e o p e r a t i o n s and If r e c u r s i v e e q u a l i t y (we c a l l t h e m r e c u r s i v e l y p r e s e n t e d f i e l d s ) . t h e A above i s r e c u r s i v e l y enumerable and n o t r e c u r s i v e , t h e f i e l d F o b t a i n e d i s r e c u r s i v e l y p r e s e n t e d and has no s p l i t t i n g a l g o r i t h m . R a b i n ( 1 9 6 0 ) showed t h a t e a c h r e c u r s i v e l y p r e s e n t e d f i e l d has a t l e a s t one r e c u r s i v e l y p r e s e n t e d a l g e b r a i c c l o s u r e . The o r i g i n a l S t e i n i t z p r o o f does n o t r e v e a l t h i s , s i n c e i t depends on t a k i n g i r r e d u c i b l e polynomials o v e r t h e base f i e l d and adding r o o t s t h e r e o f , which requires a s p l i t t i n g a l g o r i t h m (and Kronecker's algorithms) t o be c a r r i e d o u t r e c u r s i v e l y . I n s t e a d , R a b i n used a c o n s t r u c t i o n due t o A r t i n where t h e a l g e b r a i c c l o s u r e i s a q u o t i e n t modulo a r e c u r s i v e m a x i m a l i d e a l o f a p o l y n o m i a l d o m a i n i n i n f i n i t e l y many v a r i a b l e s over t h e base f i e l d . T h i s a v o i d s assuming a s p l i t t i n g a l g o r i t h m . Now S t e i n i t z p r o v e d n o t o n l y e x i s t e n c e , b u t a l s o u n i q u e n e s s o f a l g e braic closure. I n case t h e base f i e l d F has a s p l i t t i n g a l g o r i t h m , h i s u n i q u e n e s s p r o o f shows t h a t a n y t w o r e c u r s i v e l y p r e s e n t e d a l g e b r a i c c l o s u r e s d i f f e r by a r e c u r s i v e F-isomorphism. But what about t h e c a s e when t h e r e i s n o s p l i t t i n g a l g o r i t h m a s i n t h e e x a m p l e ment i o n e d a b o v e ? T h i s i s a t y p i c a l c a s e w h e r e we a s k f o r t h e r e c u r s i v e c o n t e n t o f a theorem o f t h e classical-non-constructive f i e l d t h e o r y . Theorem. (Metakides-Nerode, 1977). L e t F be a r e c u r s i v e l y p r e sented f i e l d . Then any two r e c u r s i v e l y p r e s e n t e d a l g e b r a i c c l o s u r e s o f F d i f f e r b y a r e c u r s i v e F - i s o m o r p h i s m i f and o n l y i f F has a s p l i t t i n g algorithm. Our o r i g i n a l m e t h o d o f p r o o f c o m b i n e d t h e p r i o r i t y m e t h o d w i t h a d e t a i l e d a p p l i c a t i o n o f t h e Noether-Henzelt a l g o r i t h m s on polynomial ideals. T y p i c a l a l g o r i t h m s needed a r e those f o r t e l l i n g whether a p o l y n o m i a l i s i n t h e i d e a l g e n e r a t e d b y o t h e r p o l y n o m i a l s , and t h o s e f o r f i n d i n g g e n e r a t i n g s e t s f o r p r i m a r y and p r i m e i d e a l s c o n t a i n i n g a given ideal. T h e w h o l e t h i n g c a n now b e d o n e b y u s i n g R o b i n s o n s t y l e model t h e o r y i n an e f f e c t i v e v e r s i o n ; and i n f a c t t h e N o e t h e r H e n z e l t a l g o r i t h m s w e r e o b t a i n e d i n c r u d e f o r m t h i s way t h e m s e l v e s b y Van d e r D r i e s a n d L a s c a r . The b e s t c o n t e m p o r a r y v e r s i o n o f t h e N o e t h e r - H e n z e l t a 1 g o r i thins i s S e i d e n b e r g ( 19 7 4 ) .

O f c o u r s e t h e germ o f t h e above i s t h a t t h e r e i s no c o n t i n u o u s f u n c from certain t i o n w h i c h w i l l c o n s t r u c t an o b j e c t (F-isomorphism d a t a ( e q u a l i t y and f i e l d o p e r a t i o n s f o r F and i t s two a l g e b r a i c See closures). T h i s i s always t h e i n i t i a l i n t u i t i o n i s t c h i n t . Metakides-Nerode (1979) f o r d e t a i l s . 3 . A L I T T L E HISTORY OF FOUNDATIONS OF ANALYSIS I n E u c l i d ' s elements the l e v e l o f r i g o r reached i n proofs i n v o l v i n g a r e a a n d v o l u m e was v e r y h i g h . The m e t h o d o f e x h a u s t i o n a t t r i b u t e d t o E u c l i d a n d e x p o s i t e d i n E u c l i d g a v e a way o f h a n d l i n g a r e a a n d volume problems b y l i m i t s o f f i n i t e a p p r o x i m a t i o n s . I t was v e r y p r e c i s e , b u t l a c k e d t h e n o t a t i o n a l f a c i l i t y o f t h e modern t h e o r y o f limits. I n a w a y i t was t h e e q u i v a l e n t o f t h e t h e o r y o f l i m i t s p r o v i d e d by w o r k i n g i n an a r c h i m e d i a n o r d e r e d f i e l d , b u t w i t h o u t t h e e x p l i c i t e x i s t e n c e assumption provided by t h e axiom o f completeness. The d e v e l o p m e n t o f c a l c u l u s i n t h e s e v e n t e e n t h c e n t u r y o u t g r e w t h e s e p r o o f procedures, and t h e e i g h t e e n t h c e n t u r y development o f b o t h o r d i n a r y a n d v a r i a t i o n a l c a l c u l u s was a l m o s t e n t i r e l y f o r m a l a n d a l g o r i t h m i c . T h e f i r s t a t t e m p t we know o f t o p u t a n a l y s i s o n a f i r m

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f o u n d a t i o n was B o l z a n o ( 1 8 1 7 ) . He d e f i n e d w h a t we c a l l Cauchy s e q u e n c e s , more o r l e s s d e r i v e d t h e p r i n c i p l e t h a t Cauchy s e q u e n c e s c o n v e r g e f r o m t h e commonly a c c e p t e d p r i n c i p l e t h a t g e o m e t r i c s e r i e s converge, defined t h e notion of a continuous f u n c t i o n r a t h e r e x a c t l y , and d e r i v e d t h e i n t e r m e d i a t e v a l u e t h e o r e m . He d i d n o t t h e r e r a i s e t h e e x i s t e n c e q u e s t i o n "what i s a r e a l number?" t h a t l a t e r a u t h o r s r a i s e d , but did t r y t o c o n s t r u c t t h e r e a l s i n a paper not published f o r 1 4 0 y e a r s ( B o l z a n o , 1 9 7 5 ) . I n t h e l a t t e r he a t t e m p t e d t o c o n v e r t i n f i n i t e number s y m b o l s ( s a y s e r i e s ) i n t o a s y s t e m w i t h a n e q u a l i t y s u c h t h a t a s u b s y s t e m c a n be s i n g l e d o u t w h i c h c a n b e i d e n t i f i e d w i t h the reals. (He i s p e r h a p s t h e f i r s t t o t r y t o make u p a n o n s t a n d a r d number s y s t e m w i t h i n f i n i t e l y s m a l l a n d i n f i n i t e l y l a r g e q u a n t i t i e s , and t o e x t r a c t t h e o r d i n a r y r e a l s somehow f r o m t h i s . ) I n a n y c a s e , he was v e r y much a w a r e o f t h e e x i s t e n c e q u e s t i o n , a n d t h e q u e s t i o n o f how t o c o n s t r u c t t h e r e a l n u m b e r s . I n 1 8 2 1 and 1 8 2 3 , Cauchy p u b l i s h e d f o r h i s c o u r s e s two t e x t s w h i c h c o n s t i t u t e t h e f i r s t s y s t e m a t i c a t t e m p t s t o e r e c t a n a l y s i s b a s e d on t h e Cauchy c o m p l e t e n e s s o f t h e number s y s t e m . (Modern t e x t b o o k s s i m p l y f i l l t h e g a p s i n Cauchy and e x t e n d t h e r e s u l t s t o c o n t a i n l a t e r d e v e l o p m e n t s . ) He d i d n o t attempt t o j u s t i f y completeness t h e r e . The q u e s t i o n o f d e f i n i n g t h e r e a l n u m b e r s f r o m s i m p l e r n o t i o n s s o a s t o be a b l e t o p r o v e t h e c o m p l e t e n e s s p r o p e r t y s e e m s t o h a v e b e e n t r e a t e d s a t i s f a c t o r i l y f i r s t by W e i e r s t r a s s i n h i s B e r l i n l e c t u r e s i n t h e e a r l y 1 8 6 0 ' s . I n o n e s e t o f n o t e s he d e f i n e s a p o s i t i v e r e a l a s a s e t o f u n i t f r a c t i o n s l / n , whose s e t o f f i n i t e sums i s b o u n d e d a b o v e . He t h e n d e f i n e s t h e a p p r o p r i a t e e q u i v a l e n c e r e l a t i o n on t h e s e s e t s , g e t s t h e p o s i t i v e r e a l s , and e x t e n d s t o g e t a l l r e a l s . I t i s s t r i k i n g l y s i m i l a r t o p a r t of Bolzano's long-unpublished attempt. T h i s c o u r s e o f B e r l i n l e c t u r e s was f a m o u s ; H e i n e a n d C a n t o r w e r e s t u d e n t s t h e r e , a n d D e d e k i n d was o f t e n i n B e r l i n . W e i e r s t r a s s t h u s p u t a f o u n d a t i o n u n d e r r e a l a n a l y s i s . The c h a r a c t e r i s t i c f e a t u r e i s t h e B o l z a n o - W e i e r s t r a s s p r o c e d u r e . D i v i d e an i n t e r v a l c o n t a i n i n g an i n f i n i t e s e t o f p o i n t s i n h a l f . One h a l f c o n t a i n s i n f i n i t e l y many p o i n t s of t h e s e t . Take t h e l e f t m o s t h a l f - i n t e r v a l w i t h t h i s propC o n t i n u e o n w a r d . The l e f t e n d p o i n t s a r e a erty, divide i t in half. Cauchy s e q u e n c e c o n v e r g i n g t o a l i m i t p o i n t o f t h e o r i g i n a l s e t . T h i s r i g o r i z a t i o n was n o t t a k e n s e r i o u s l y u n t i l H e i n e a n d C a n t o r a t H a l l e needed f o u n d a t i o n s f o r t h e t h e o r y of s e t s of u n i q u e n e s s i n t r i g o n o m e t r i c s e r i e s . They b o t h p u b l i s h e d a c c o u n t s of r e a l s a s e q u i v a l e n c e c l a s s e s o f C a u c h y s e q u e n c e s i n 1 8 7 2 , a n d D e d e k i n d immed i a t e l y p u b l i s h e d his v e r s i o n v i a c u t s . Dedekind's d e f i n i t i o n of t h e r e a l s i s s t r i k i n g l y s i m i l a r t o his d e f i n i t i o n of i d e a l . In each c a s e a notion with computational o r i g i n s ( p r i m e i d e a l a s a f i n i t e s e t of c o n g r u e n c e s , r e a l number a s a Cauchy s e q u e n c e ) i s d i s c a r d e d b e c a u s e an e q u i v a l e n c e r e l a t i o n must be taken t o g e t t h e r i g h t o b j e c t (prime i d e a l , r e a l number). A s e t t h e o r e t i c n o t i o n i s s u b s t i t u t e d which i s u n i q u e l y c o r r e l a t e d w i t h t h e d e s i r e d o b j e c t i n s t e a d . This c h a n g e s t h e n a t u r a l o r d e r f o r t h e d e d u c The e x p o s i t i o n becomes c l e a n e r b u t t i o n of elementary consequences. l e s s e a s y t o f o l l o w and o f t e n l e s s c o n s t r u c t i v e . I n summary, o r d i n a r y c a l c u l u s o f a r e a l v a r i a b l e had a f o u n d a t i o n p u t u n d e r i t by W e i e r s t r a s s . B u t t h e s i t u a t i o n was n o t t h e same f o r t h e v e r y s i m i l a r c a l c u l u s o f v a r i a t i o n s . A f t e r L e i b n i z had p u b l i s h e d means o f s o l v i n g o r d i n a r y maxima a n d minima p r o b l e m s . f o r r e a l f u n c t i o n s of a r e a l v a r i a b l e , t h e B e r n o u l l i b r o t h e r s and o t h e r s e s t a b l i s h e d s i m i l a r computation r o u t i n e s f o r r e a l valued f u n c t i o n s of a

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325

function variable (calculus o f variations). Much o f e l e m e n t a r y p h y s i c s can be r e c a s t as problems o f f i n d i n g a f u n c t i o n i n a f u n c t i o n space f o r w h i c h a n o n - n e g a t i v e r e a l v a l u e d f u n c t i o n a l t a k e s on a minimum. P r i o r t o W e i e r s t r a s s and h i s d e r i v a t i o n o f o r d i n a r y c a l c u l u s f r o m t h e c o m p l e t e n e s s p r i n c i p l e , t h e p r o b l e m f o r f u n c t i o n a l s was n o t regarded as any d i f f e r e n t from t h a t f o r functions. This led t o an h i s t o r i c a l l y i m p o r t a n t m i s t a k e and c o r r e c t i o n . We w i s h t o e x a m i n e t h i s , a n d t o a s k w h y t h e f u n c t i o n a l q u e s t i o n was h a r d e r t h a n the function question. Green (1828) used p h y s i c a l arguments based on d i s t r i b u t i o n o f e l e c t r i c a l c h a r g e on c o n d u c t i n g s u r f a c e s t o p r o v e t h e e x i s t e n c e o f p o t e n t i a l f u n c t i o n s w i t h p r e s c r i b e d boundary values on surfaces. Gauss ( 1 8 3 9 ) attempted t o s u b s t i t u t e a mathematical p r o o f by observing t h a t the p o t e n t i a l s d e s i r e d were minima f o r a r e a l valued f u n c t i o n a l d e f i n e d on a n a p p r o p r i a t e f u n c t i o n s p a c e . D i r i c h l e t learned t h i s from him and used t h i s k i n d o f argument i n h i s p h y s i c s c o u r s e s o f 1847-9. Riemann p i c k e d up t h i s p r i n c i p l e f r o m t h e s e t w o a u t h o r i t i e s . This p r i n c i p l e , t h a t a non-negative f u n c t i o n a l on a f u n c t i o n space a t t a i n s i t s m i n i m u m , was u s e d b y R e i m a n n i n 1 8 5 1 t o p r o d u c e p o t e n t i a l s w i t h p r e s c r i b e d behaviour, and t o found h i s geometric complex v a r i a b l e theory. T h e e x i s t e n c e p a r t o f h i s t h e o r y o f a l g e b r a i c f u n c t i o n s was t h u s based on t h i s D i r i c h l e t p r i n c i p l e . B u t a f t e r Weierstrass had f o u n d o u t why c o n t i n u o u s f u n c t i o n s o n a c l o s e d b o u n d e d s e t a t t a i n a m i n i m u m , h e was a b l e t o e s t a b l i s h n o c o r r e s p o n d i n g p r i n c i p l e f o r f u n c t i o n spaces. T h u s t h e r e was n o f o u n d a t i o n f o r R i e m a n n ' s t h e o r y of algebraic functions. Weierstrass p u b l i s h e d a s i m p l e example o f a f u n c t i o n a l and f u n c t i o n space f o r which t h e p r i n c i p l e f a i l s , i n 1869, b u t w e l l a f t e r h i s c o l l e a g u e R i e m a n n ' s d e a t h . Our e x a m i n a t i o n o f W e i e r s t r a s s ' s own a t t e m p t s a t c a l c u l u s o f v a r i a t i o n s m a k e s i t seem t o u s t h a t h e a t t e m p t e d t o r e d u c e t h e p r o b l e m t o a f i n i t e d i m e n s i o n a l one i n E u c l i d e a n space. He n e v e r s u c c e e d e d i n g r a p p l i n g w i t h t h e e s s e n t i a l i n f i n i t e d i m e n s i o n a l i t y o f f u n c t i o n spaces. Once t h e f o u n d a t i o n s w e r e r e m o v e d , t w o g r o u p s a t t e m p t e d t o r e f o u n d Riemann's t h e o r y o f a l g e b r a i c f u n c t i o n s . One ( C l e b s c h - G o r d a n , 1 8 6 6 ) A second group c o n s i s t i n g o f used a p u r e l y a l g e b r a i c approach. S c h w a r t z i n 1 8 6 9 - 7 2 a n d Neumann ( 1 8 6 5 ) c o n s t r u c t e d t h e d e s i r e d p o t e n t i a l s d i r e c t l y w i t h o u t use o f v a r i a t i o n a l methods. An e a r l y h i g h p o i n t i n t h e p o t e n t i a l t h e o r y a p p r o a c h was P o i n c a r e ' s m e t h o d o f Balayage (Poincart?, 1890). The d i r e c t method o f u s i n g v a r i a t i o n a l m i n i m a was e s t a b l i s h e d b y H i l b e r t ( 1 9 0 0 ) . B o t h use a p r o o f t h a t proceeds by stages t h r o u g h a c o u n t a b l e dense s e t ( s e p a r a b i l i t y ) . T h i s would n o t have been t h i n k a b l e b e f o r e C a n t o r ' s development o f Hilbert's justification of t h e t h e o r y o f c o u n t a b l e s e t s i n 1873. t h e d i r e c t method f o r p r o v i n g t h e e x i s t e n c e o f minima f o r c a l c u l u s o f v a r i a t i o n s p r o b l e m s p r o c e e d s b y an i n f i n i t e sequence o f a p p l i c a t i o n s o f t h e B o l z a n o - W e i e r s t r a s s theorem t h r o u g h a c o u n t a b l e dense set. T h i s k i n d o f p r o o f i s more c o m p l i c a t e d t h a n t h e s i m p l e f i n i t e sequence o f a p p l i c a t i o n s o f t h e B o l z a n o - W e i e r s t r a s s p r o c e s s c h a r a c t e r i s t i c of W e i e r s t r a s s ' s f i n i t e dimensional Euclidean space work. I t goes back t o A r z e l a ( 1 8 8 9 ) f o r f u n c t i o n spaces. I t was a l s o u s e d by Peano ( 1 8 9 0 ) i n h i s second p a p e r o n t h e e x i s t e n c e t h e o r e m t h a t y' = f(x,y) w i t h f c o n t i n u o u s on a c l o s e d r e c t a n g l e has l o c a l s o l u t i o n s through any p o i n t . A r z e l a (1895) t r i e d t o apply t h i s t o v a r i a t i o n a l problems w i t h m i l d success. H i l b e r t succeeded.

I n summary, i n l i g h t o f H i l b e r t ' s p r o o f , t h e r e i s n o d i f f i c u l t y i n j u s t i f y i n g Riemann's t h e o r y o f a l g e b r a i c f u n c t i o n s by appeal t o a

326

G . METAKIDES, A. NERODE

v a r i a t i o n a l minimum p r i n c i p l e . But the c o r r e c t generalization o f W e i e r s t r a s s ' s method t o prove compactness o f subsets o f f u n c t i o n s p a c e s ( t h e A r z e l a - A s c o l i l e m m a ) was c l e a r l y v e r y h a r d t o s e e . Even though Weierstrass h i m s e l f proved t h e approximation theorem which clives l o t s o f c o u n t a b l e dense subsets o f f u n c t i o n soaces. t h e e x t e n s i o n o f t h e Bolzano-Weierstrass compactness i d e a t o . i n f i n t e d i m e n s i o n a l f u n c t i o n s p a c e s seems t o h a v e e l u d e d h i m . 4.

R E C U R S I V E METRIC S P A C E S

T h e t h e o r y o f m e t r i c s p a c e s was d e v e l o p e d b y F r i c h e t ( 1 9 0 6 ) t o c o v e r i n common t h e W e i e r s t r a s s s c h o o l ' s f i n i t e d i m e n s i o n a l a n a l y s i s a n d t h e n e w l v emercled t h e o r y o f f u n c t i o n a l s o n i n f i n i t e d i m e n s i o n a f u n c t i o n spaces. f t s h o u l d - t h e r e f o r e n o t be s u r p r i s i n g t h a t t h e c o n t e x t o f complete m e t r i c spaces which a r e r e c u r s i v e l y presented forms a convenient language f o r f o r m u l a t i n g t h e r e c u r s i v e c o n t e n t o f c l a s s i cal analysis. T h i s i s n o t t h e p l a c e t o t i e down e x a c t d e f i n i t i o n s . We w i l l s a y t h a t a l m o s t a l l t h e u s u a l s e p a r a b l e c o m p l e t e m e t r i c spaces f r o m t h e textbooks have easy n a t u r a l r e c u r s i v e p r e s e n t a t i o n s . For these t h e r e i s a dense subset enumerated w i t h o u t r e p e t i t i o n s such t h a t t h e standard system o f neighborhoods ( c o n s i s t i n g o f spheres w i t h r a t i o n a l r a d i i , c e n t e r s i n t h e dense s e t ) i s so e x p l i c i t l y g i v e n t h a t e l e m e n t a r y c o m p u t a t i o n s on such n e i g h b o r h o o d s can be c a r r i e d out. ( T h e c o u n t a b l e d e n s e s e t s we h a v e i n m i n d a r e f o r e x a m p l e t h e r a t i o n a l p o i n t s i n n - d i m e n s i o n a l E u c l i d e a n space, t h e f i n i t e l y nonzero r a t i o n a l sequences i n H i l b e r t space, t h e t r i g o n o m e t r i c p o l y nomials w i t h r a t i o n a l c o e f f i c i e n t s i n t h e space o f continuous funct i o n s on an i n t e r v a l , e t c . ) B u t u n l i k e many c o n s t r u c t i v i s t s w h o d o n o t a c c e p t t h e c l a s s i c a l c o n t i n u u m , we a r e i n t e r e s t e d i n t h e w h o l e c o m p l e t e m e t r i c s p a c e , n o t merely i t s r e c u r s i v e p o i n t s (see d e f i n i t i o n below). For example, i n d i s c u s s i n g f u n c t i o n s o n t h e r e a l i n t e r v a l , we a r e i n t e r e s t e d i n r e c u r s i v e continuous f u n c t i o n s d e f i n e d on t h e whole i n t e r v a l . The f u n c t i o n s a r e r e c u r s i v e , b u t t h e y o p e r a t e on a r b i t r a r y p o i n t s . To t h e r e c u r s i o n t h e o r i s t t h i s should be n a t u r a l , since t h a t i s what t r u t h t a b l e T u r i n g r e d u c t i o n p r o c e d u r e s do. Our m o t i v a t i o n i n choosi n g t h i s course i s n o t by analogy w i t h recursion theory, b u t r a t h e r b y g e n e r a l i z a t i o n o f common e x p e r i e n c e i n n u m e r i c a l a n a l y s i s , w h i c h was a g r e a t s o u r c e o f i n s p i r a t i o n i n t h e s e t t i n g u p o f c l a s s i c a l a n a l y s i s anyway. T h i n k o f a p h y s i c a l p r o c e s s w i t h measurements as i n p u t s and outputs,, d e s c r i b e d by a f u n c t i o n on r e a l s t o r e a l s . One h a s n o i d e a w h a t i n p u t s w i l l come i n ; t h e y a r e n o t r e c u r s i v e r e a l numbers, b u t a r e g i v e n b y n a t u r e , and t h e c o m p u t a t i o n p r o c e e d s b y u s i n g a p p r o x i m a t i o n s t o t h e i n p u t number t o g e t a p p r o x i m a t i o n s t o t h e o u t p u t number. There a r e no r e s t r i c t i o n s on n a t u r e as t o t h e i n p u t numbers n a t u r e p r o v i d e s , t h e b e t t e r t h e a p p r o x i m a t i o n t o t h e i n p u t , the better the approximation t o the output. This i s the notion o f real function that the particular d e f i n i t i o n o f recursively continuo u s f u n c t i o n we a d o p t r e f l e c t s ( i t d o e s n o t o r i g i n a t e w i t h u s ) . Assume we a r e i n r e c u r s i v e l y p r e s e n t e d c o m p l e t e m e t r i c s p a c e s . How i s a p o i n t p d e s c r i b e d ? The answer i s , b y a d e s c e n d i n g sequence o f s t a n d a r d b a s i c n e i g h b o r h o o d s Ui 5 Ui+l such t h a t each

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a r e c u r s i v e l y p r e s e n t e d m e t r i c s p a c e i s s i m p l y one wh'ich i s d e s c r i b e d b y a r e c u r s i v e sequence o f s t a n d a r d n e i g h b o r h o o d s o f t h e above f o r m . When a p p l i e d t o t h e r e a l n u m b e r s , t h i s y i e l d s t h e u s u a l r e c u r s i v e

Non-recursive methods

327

r e a l n u m b e r s ( n u m b e r s w i t h r e c u r s i v e d e c i m a l e x p a n s i o n s , or n u m b e r s w i t h r e c u r s i v e D e d e k i n d c u t s ) . When a p p l i e d t o t h e s p a c e o f c o n t i n uous f u n c t i o n s on t h e i n t e r v a l i n t h e u n i f o r m t o p o l o g y , t h e r e c u r s i v e p o i n t s a r e t h e n t h e r e c u r s i v e c o n t i n u o u s f u n c t i o n s on t h e u n i t i n t e r v a l . Although e l e g a n t and e a s i l y remembered, t h i s does n o t r e v e a l t h a t t h e s e a r e t h e r e a l v a l u e d f u n c t i o n s f on t h e u n i t i n t e r v a l w h i c h c a n be c o m p u t e d e v e r y w h e r e by a p p r o x i m a t i o n by a T u r i n g m a c h i n e . Such an f i s r e c u r s i v e i f and o n l y i f t h e r e i s a n i n p u t - o u t p u t T u r i n g machine such t h a t f ( x ) y i f and o n l y i f : whenever a s e q u e n c e o f n e i g h b o r h o o d s i s p r i n t e d o n t h e i n p u t t a p e which d e s c r i b e s x , t h e o u t p u t t a p e e n d s u p w i t h a s e q u e n c e of n e i g h b o r h o o d s which d e s c r i b e s y . When a p p l i e d t o t h e s p a c e o f c o n t i n u o u s f u n c t i o n s on Z W t o ZW, t h e r e c u r s i v e p o i n t s ( o r r e c u r s i v e l y c o n t i n u o u s f u n c t i o n s ) a r e p r e c i s e l y t h e truth t a b l e reductions (Rogers, 1967). For t h e rec u r s i o n t h e o r i s t , t h e more g e n e r a l T u r i n g r e d u c t i o n s a r e s i m p l y c o n t i n u o u s f u n c t i o n s d e f i n e d on c e r t a i n G 6 ' s i n Z W . A n a l y s t s do n o t i n g e n e r a l have any i n t e r e s t i n f u n c t i o n s w i t h t h a t s o r t o f domain, a l t h o u g h many o f t h e i r t h e o r e m s may be e x t e n d e d t o s u c h d o m a i n s . We now h a v e s k e t c h e d e n o u g h a p p a r a t u s t o l o o k a t t h e r e c u r s i v e c o n t e n t of a l i t t l e a n a l y s i s . F o r s i m p l i c i t y , l e t us s t a r t w i t h t h a t b a s i c f a c t of W e i e r s t r a s s i a n a n a l y s i s , t h e theorem t h a t a continuous r e a l v a l u e d f u n c t i o n f on t h e u n i t i n t e r v a l a t t a i n s i t s minimum a t some p o i n t i n t h e i n t e r v a l . The c l a s s i c a l p r o o f g i v e s an a l g o r i t h m which l e a d s from a r e c u r s i v e c o n t i n u o u s f u n c t i o n f t o a c o m p u t a t i o n I s t h e r e a c o n s t r u c t i v e procedure f o r f i n d o f i t s maximum v a l u e M . i n g a p o i n t a t w h i c h t h e maximum i s a t t a i n e d ? A B r o u w e r i a n c o u n t e r e x a m p l e c a n be e a s i l y t r a n s f o r m e d i n t o t h e f o l l o w i n g c l a s s i c a l phenomemon. T h e r e i s no c o n t i n u o u s f u n c t i o n G w h i c h , a p p l i e d t o a n y c o n t i n u o u s f d e f i n e d on t h e u n i t i n t e r v a l , y i e l d s a r e a l n u m b e r G ( f ) i n t h e u n i t i n t e r v a l a t w h i c h t h e maximum v a l u e o f f i s a t t a i n e d . ( T h e t o p o l o g y i n t e n d e d on t h e domain i s t h e u n i f o r m t o p o l o g y . ) B u t t h e s i m p l e f a c t t h a t t h e r e i s no c o n t i n u o u s G w i t h t h i s property does n o t e x h i b i t a p a r t i c u l a r r e c u r s i v e continuous f s u c h t h a t no r e c u r s i v e y h a s f ( y ) t h e maximum o f f . Such an e x a m p l e was c o n s t r u c t e d by S p e c k e r ( 1 9 5 9 ) . Huang a n d N e r o d e h a v e shown i n 1 9 8 2 t h a t f o r a n y bounded A: s e t , there i s a recursive c o n t i n u o u s f on t h e u n i t i n t e r v a l s u c h t h a t t h e d e g r e e s o f t h e r e a l set. y w i t h f ( y ) a maximum a r e t h e d e g r e e s o f t h e s e t s i n t h e A; The t h e o r y o f J o c k u s c h - S o a r e ( 1 9 7 2 ) t h e n a p p l i e s , and f o r e x a m p l e shows t h a t t h e r e i s a r e c u r s i v e c o n t i n u o u s f u n c t i o n on t h e u n i t i n t e r v a l such t h a t t h e r e i s a continuum o f such y which have a l l mutually incomparable Turing degrees. (The T u r i n g d e g r e e of a r e a l i s t h e Turing degree of i t s decimal expansion.) There a r e s i m i l a r r e s u l t s f o r many t h e o r e m s w h i c h e s s e n t i a l l y d e p e n d on t h e B o l z a n o W e i e r s t r a s s method. F o r many t h e o r e m s d e p e n d i n g on A r z e l a - A s c o l i , unbounded I: classes c a n b e made t o p l a y t h e r o l e t h a t b o u n d e d o n e s do f o r B o l z a n o But W e i e r s t r a s s . P u b l i s h e d examples do n o t y e t e x h i b i t t h i s . n o t e w o r t h y was A b e r t h ( 1 9 7 1 ) who g a v e a r e c u r s i v e c o n t i n u o u s f ( x , y ) i n t h e u n i t s q u a r e s u c h t h a t y ' = f ( x , y ) h a s no l o c a l r e c u r s i v e c o n t i n u o u s s o l u t i o n t o y ( x ) i n any i n t e r v a l c o n t a i n i n g (0,O). I t i s e a s y t o compound t h i s t o make o n e s u c h t h a t t h e e q u a t i o n h a s no l o c a l r e c u r s i v e s o l u t i o n s a t a l l , a n d t h i s was d o n e by R e t z l a f f i n 1 9 7 7 ; a n d i n d e p e n d e n t l y o f R e t z l a f f a n d A b e r t h a l s o by P o u r - E l and R i c h a r d s ( 1 9 7 9 ) .

328

G . METAKIDES, A. NERODE

Define a r e c u r s i v e l y t o t a l l y bounded s e t t o be one such t h a t f o r a r a t i o n a l E one can r e c u r s i v e l y o b t a i n a f i n i t e E - n e t c o n s i s t i n g of r e c u r s i v e p o i n t s of t h e s e t . C a l l a s e t r e c u r s i v e l y compact i f i t i s complete and r e c u r s i v e l y t o t a l l y bounded. Then a r e c u r s i v e l y c o n t i n u o u s f u n c t i o n on a r e c u r s i v e l y compact s e t a t t a i n s a r e c u r s i v e maximum v a l u e ( b u t n o t n e c e s s a r i l y a t a r e c u r s i v e p o i n t ) . T h i s i s a n a t u r a l e f f e c t i v i z a t i o n of t h e theorem of t h e maximum and w i l l - be used i n S e c t i o n 5. The proof i s s t r a i g h t f o r w a r d . 5.

THE H A H N - B A N A C H T H E O R E M

I n f i n i t e d i m e n s i o n a l f u n c t i o n s p a c e s t u r n e d u p i n many l a t e n i n e t e e n t h c e n t u r y c o n t e x t s , i n c l u d i n g i n t e g r a l e q u a t i o n s and c a l c u l u s of v a r i a t i o n s . These i n f l u e n c e d g r e a t l y t h e s e t t i n g u p of f u n c t i o n a l a n a l y s i s . B o t h H a h n ( i n 1 9 2 7 ) and Banach ( i n 1929) i n v e n t e d , i n d e p e n d e n t l y , complete normed l i n e a r s p a c e s (Banach s p a c e s ) . T h e i r theorem a s s e r t s t h a t a r e a l v a l u e d bounded l i n e a r f u n c t i o n a l f on a c l o s e d s u b s p a c e has a t l e a s t one e x t e n s i o n t o a bounded l i n e a r f u n c t i o n a l g on t h e whole s p a c e such t h a t 11 f ( l = 1) glI. The proof f o r a r b i t r a r y Banach s p a c e s u s e s Z o r n ' s lemma o r w e l l - o r d e r i n g , b u t t h i s i s u n n e c e s s a r y f o r s e p a r a b l e s p a c e s . However, even i n t h i s c a s e f o r t h e s t a n d a r d s p a c e s t h e proof i s n o t e n t i r e l y c o n s t r u c t i v e . Bishop (1967) gave a c o n s t r u c t i v e v e r s i o n , i n which t h e c o n c l u s i o n i s weakened t o t h e a s s e r t i o n t h a t f o r any p o s i t i v e E , t h e r e i s a g w i t h 11 g l l < ( l + c ) l l fll. From h i s p o i n t of view t h e E i s n e c e s s a r y , a s he gave an i n t u i t i o n i s t i c c o u n t e r e x a m p l e t o t h e theorem w i t h o u t t h e E. We r e p r o d u c e t h e c l a s s i c a l c o u n t e r p a r t of t h i s example o f h i s below. a s an a s s e r t i o n t h a t no c o n t i n u o u s c h o i c e of q from f i s p o s s i b l e w i t h 11 f l l = 11 g l l . His p o s i t i v e r e s u l t h o l d s i n t h e r e c u s i v e c a s e w i t h o u t c h a n g e , once t h e r e c u r s i v e f u n c t i o n t h e o r y d e f i n t i o n s of t h e w o r d s i n v o l v e d a r e u n d e r s t o o d . As i n e a r l i e r c a s e s b e f o r e , t h e r e i s s t i l l t h e q u e s t i o n of an a c t u a l r e c u r s i v e c o u n t e r e x a m p l e - - a n d . t h i s was o b t a i n e d by M e t a k i d e s , Nerode, and S h o r e i n 1981. We remind t h e r e a d e r of t h e u s u a l

( n o n - s e p a r a t i o n theorem) d i r e c t proof o f t h e Hahn-Banach theorem f o r s e p a r a b l e s p a c e s First, div o u t b v t h e k e r n e l of f , s o t h a t f i s 1-1 and def ned on a 1 d i m e n i i o n a l s p a c e , g e n e r a t e d by x l . Let x1 , x 2 , . . . be a c o u n t a b l e d e n s e s e t . L e t S n be t h e s u b s p a c e g e n e r a t e d by x, . . ' X n . Assume a 1-1 l i n e a r f u n c t i o n a l f l d e f i n e d o n S1 and t r y t o o b t a i n from f n ( d e f i n e d o n S n ) an f n t 1 ( d e f i n e d on S n t l ) with s t a n d a r d com[I f n I I = 11 f n t l I I , and such t h a t f n + l e x t e n d s f n . p u t a t i o n shows t h a t d e f i n i n g f n t l ( x n + l ) = y n t l works i f and o n l y i s in the interval i f Y = Ynt1

.

We c a l l such a y a d m i s s i b l e . The r o l e of Z o r n ' s lemma v a n i s h e s by t a k i n g always ( s a y ) t h e l e f t e n d p o i n t o f t h i s i n t e r v a l a s y n + l . B u t t h i s c a u s e s t h e c o n s t r u c t i v i s t problems. Can t h e o b j e c t y n + l b e chosen i n t h i s i n t e r v a l c o n t i n u o u s l y from d a t a f n , S n , x n ? To make

Non-recursive methods

3 29

t h i s m e a n i n g f u l , n o t e t h a t a n y n d i m e n s i o n a l Banach s p a c e i s t o p o l o g i c a l l y n d i m e n s i o n a l E u c l i d e a n s p a c e , b u t w i t h a new norm. S o we may g i v e a t o p o l o g y t o t h e p o s s i b l e norms o f n d i m e n s i o n a l E u c l i d e a n s p a c e . The t o p o l o g y i s t h a t o b t a i n e d by r e g a r d i n g e a c h norm a s a c o n t i n u o u s f u n c t i o n on ( s a y ) t h e E u c l i d e a n s p h e r e , and u s i n g t h e u n i f o r m t o p o l o g y f o r t h e s e bounded r e a l v a l u e d f u n c t i o n s . DO t h e same t h i n g f o r t h e l i n e a r f u n c t i o n a l s i n v o l v e d . Now we show This t h a t f 2 c a n n o t be o b t a i n e d c o n t i n u o u s l y f r o m f l , S 1 , x l . w i l l be done w i t h S1 f i x e d a s t h e x - a x i s o f t h e p l a n e , S 2 a s t h e plane, f l as f l ( x , O ) = x. The p r o b l e m i s t h e n , c a n f 2 be c h o s e n c o n t i n u o u s l y f r o m t h e norm N o f S2? The a n s w e r i s n o , t h e e x a m p l e i s t h e c l a s s i c a l c o u n t e r p a r t of B i s h o p ' s . Suppose f 2 ( N ) were such a continuous f u n c t i o n . There a r e t h r e e f i g u r e s l a b e l l e d 0 , 1 , 2. Let N i be t h e norm on S 2 g i v e n by t a k i n g t h e i n d i c a t e d p a r a l l e l l o g r a m o f f i g u r e i a s u n i t s p h e r e . N o t e t h a t f l h a s norm 1 , w h i l e f o r s u f f i c i e n t l y s m a l l p o s i t i v e a n d n e g a t i v e a , t h e norms N 1 and N 2 c a n b e made a r b i t r a r i l y c l o s e t o N o . There i s o n l y one p o s s i b l e v a l u e f o r f 2 ( N 1 ) , n a m e l y t h e e x t e n s i o n o f f l w h i c h maps t h e l i n e L1 c o n n e c t i n g ( 0 , - 1 ) and ( 1 , O ) t o 1 . T h e r e i s o n l y one p o s s i b l e v a l u e f o r f 2 ( N 2 ) , namely t h e e x t e n s i o n of f l which maps t h e l i n e L 2 c o n n e c t i n g ( 0 , l ) and ( 1 , O ) t o 1 . T h e r e i s a whole tension passing sphere" through through

gamut o f c o r r e s p o n d i n g c h o i c e s f o r f 2 ( N 0 ) , n a m e l y a n y e x o f f l s u c h t h a t t h e l i n e L 2 mapped t o 1 i s a l i n e through ( l , O ) , b u t not through t h e i n t e r i o r of the " u n i t s q u a r e . Were f 2 ( N ) c o n t i n u o u s , a s a a p p r o a c h e s 0 p o s i t i v e v a l u e s we g e t L o = L 1 , a s a a p p r o a c h e s 0 n e g a t i v e v a l u e s we g e t L o = L 2 .

N o w l e t us l o o k a t t h e r e c u r s i v e l y p r e s e n t e d c a s e .

The n o t i o n o f a r e c u r s i v e l y p r e s e n t e d B a n a c h s p a c e i s e a s y : J u s t make i t a r e c u r s i v e l y p r e s e n t e d c o m p l e t e m e t r i c s p a c e and r e q u i r e t h a t t h e o p e r a t i o n s and norm a r e r e c u r s i v e l y c o n t i n u o u s . A ( r e c u r s i v e l y ) l o c a t e d s e t A i s o n e s u c h t h a t t h e f u n c t i o n a s s i g n i n g t o a n y x i n t h e Banach s p a c e i t s d i s t a n c e t o A i s r e c u r s i v e l y continuous. T h i s i s simply t h e o b v i o u s r e c u r s i v e v e r s i o n o f B r o u w e r ' s n o t i o n a s u s e d by B i s h o p . We c a n p r o v e by i n d u c t i o n on d i m e n s i o n t h a t a f i n i t e d i m e n s i o n a l s u b s p a c e i s l o c a t e d i f and o n l y i f i t has a r e c u r s i v e b a s i s , and t h i s i s used i m p l i c i t l y below. ( A l t h o u g h a s i m p l e f a c t , i t shows f o r e x a m p l e t h a t t h e t h e o r y o f b e s t a p p r o x i m a t i o n s t o e l e m e n t s o f a Banach s p a c e u s i n g a f i n i t e dimensional s u b s p a c e i s r e c u r s i v e i n t h e a n a l y s t ' s s t a n d a r d c a s e s because there a r e obvious r e c u r s i v e b a s e s . ) Now s u p p o s e f i s a b o u n d e d r e c u r s i v e l i n e a r f u n c t i o n a l w i t h l o c a t e d k e r n e l on a c l o s e d s u b s p a c e o f a r e c u r s i v e l y p r e s e n t e d s p a c e . D i v i d e by t h e k e r n e l , g e t t i n g ( d u e t o l o c a t i o n ) a r e c u r s i v e l y p r e s e n t e d q u o t i e n t w i t h f = f l now a s s u m e d t o b e 1 - 1 a n d d e f i n e d on t h e Let o n e - d i m e n s i o n a l s u b s p a c e S1 g e n e r a t e d by r e c u r s i v e x l . x 1 , x 2 . . . . be a r e c u r s i v e l y e n u m e r a b l e d e n s e s e q u e n c e o f r e c u r s i v e

330

G. METAKIDES, A. NERODE

a =

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F I G U R E S 0 , l ,2.

Non-recursive methods

33 1

d e f i n e d on Sn g e n e r a t e d b y xl, . . . , x n , w h a t i s n for t h e a d m i s s i b l e i n t e r v a l y = ynt g i v e n a b o v e ? N o t e t h a t t h e l e t hand e n d p o i n t i s f n + l ( x n + l ) = Yn+ a r e a l whose l o w e r D e d e k i n d c u t i s r e c u r s i v e l y e n u m e r a b l e ( p r o v i d e d points.

With

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The r i g h t e n d p o i n t i s a r e a l whose u p p e r c u t i s

r e c u r s i v e l y enumerable. I f t h e l e f t and r i g h t endpoints c o i n c i d e , we g e t a u n i q u e r e c u r s i v e r e a l y a s a d m i s s i b l e f(xn+l). I f not, any r e c u r s i v e y i n s i d e t h e i n t e r v a l ( s a y a r a t i o n a l y) w i l l do. T h i s i t s e l f shows t h a t i f t h e o r i g i n a l Banach s p a c e i s f i n i t e d i m e n s i o n a l , t h e n t h e f u n c t i o n a l can be e x t e n d e d t o a bounded r e c u r s i v e f u n c t i o n a l o n t h e w h o l e s p a c e w i t h t h e same n o r m . T h a t i s , t h e HahnBanach t h e o r e m i n t h e f i n i t e d i m e n s i o n a l r e c u r s i v e l y p r e s e n t e d c a s e r e q u i r e s no r e v i s i o n . So e a c h o f t h e t h r e e t w o - d i m e n s i o n a l r e c u r s i v e l y p r e s e n t e d Banach spaces g i v e n i n t h e example above has t h e Hahn-Banach t h e o r e m t r u e f o r i t r e c u r s i v e l y . But, the extension p r o c e s s as g i v e n above i s n o t u n i f o r m . I t d o e s o t g i v e fn+l uniformly from on t h e e x t e n s i o n

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0. By n t h e t h e o r e m o f t h e maximum m e n t i o n e d i n S e c t i o n 4 , we g e t r e c u r s i v e endpoints, and a v a l u e f o r can t h e n be computed. Thus w i t h inf

need be t a k e n o n l y o v e r a s p h e r e o f r a d i u s

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a r e c u r s i v e sequence

E g i v e n we r e c u r s i v e l y g e t n ( l + ~ ~ ) f lnl + l ( ( . O f c o u r s e , i f o n e s t a r t s w i t h a n E > 0, o n e c a n c o m p u t e E~ w i t h I I ( ~ + E ~ )< 1 + ~ ,a n d a l l I / fn+lII 5 ( l + ~ ~ ) l l n W i t h a l i t t l e d e t a i l added, t h i s g i v e s a p r o o f o f B i s h o p ' s theorem

11

frill 5

frill.

with E present. We l i k e t h i s p r o o f b e c a u s e i t e x p l a i n s w h a t happens g o i n g up one d i m e n s i o n a t a t i m e . Theorem. (Metakides-Nerode-Shore). There e x i s t s a r e c u r s i v e l y pres e n t e d Banach space and a bounded r e c u r s i v e l i n e a r f u n c t i o n a l on a c l o s e d subspace w i t h l o c a t e d k e r n e l , such t h a t e v e r y e x t e n s i o n t o t h e w h o l e s p a c e w i t h t h e same p r o p e r t i e s h a s l a r g e r n o r m . T h e B a n a c h s p a c e f o r t h i s p r o o f i s a H i l b e r t s p a c e d i r e c t sum o f i n f i n i t e l y many t w o - d i m e n s i o n a l s p a c e s e a c h o f w h i c h l o o k s s o m e w h a t l i k e one o f t h e t w o - d i m e n s i o n a l examples i n t h e B i s h o p argument. The f u n c t i o n a l w i t h o u t a good extension, i t s l o c a t e d k e r n e l , and t h e norm f o r t h e whole space, a r e s i m u l t a n e o u s l y c o n s t r u c t e d by a p r i o r i t y argument. We h a v e s a i d i n t h e p r e v i o u s a l g e b r a i c d i s c u s s i o n t h a t c a r r y i n g o u t these p r i o r i t y arguments tends t o use a l l t h e p o s i t i v e r e c u r s i v e f a c t s known i n o r d e r t o s e e t h a t t h e s t a g e s o f t h e c o n s t r u c t i o n can be r e c u r s i v e l y c a r r i e d o u t . T h i s i s no exception. The r e c u r s i v e ( l + E ) - H a h n - B a n a c h t h e o r e m a l l u d e d t o above i s used, a s w e l l a s t h e maximum t h e o r e m m e n t i o n e d a b o v e .

332

G. METAKIDES, A. NERODE

We n o t e t h a t i n t h e c a s e o f r e c u r s i v e l y p r e s e n t e d normed v e c t o r spaces over recursively presented ordered f i e l d s (everything counta b l e ) , Kalantari (Crossley [1981], 128-146) has given a counterexample t o t h e s e p a r a t i o n theorem and c o u l d have a s e a s i l y g i v e n one f o r Hahn-Banach. T h i s c o u n t a b l e a l g e b r a c a s e p o i n t s t o w a r d , b u t d o e s n o t y i e l d , the uncountable analysis case. 6.

CONCLUDING REMARKS

A good b i b l i o g r a p h y o f r e c u r s i v e a l g e b r a u p t o 1 9 8 0 o c c u r s i n C r o s s l e y , 1 9 8 0 . Among t h e r e c e n t c o n t r i b u t o r s a r e J . B a l d w i n , J . B o h o r q u e z , J . N . C r o s s l e y , R . Downey, E . E i s e n b e r g , J u . L . E r s h o v , 0 . G u i c h a r d , I . K a l a n t a r i , P . L a R o c h e , C . L i n , G . M e t a k i d e s , A . N e r o d e , J . Remmel, A . R e t z l a f f , A . S e i d e n b e r g , L . S c h r i e b e r , a n d R . S m i t h . The s i m p l e r r e c u r s i v e a l g e b r a r e s u l t s a r e b e i n g a b s o r b e d i n t o r e c u r s i v e model t h e o r y . Among t h e c o n t r i b u t o r s h e r e a r e C . Ash, J u . L . E r s h o v , S . Gonchorov, M . Manasse, T . M i l l a r , M. Morley, A . T . N u r t a z i n , A . N e r o d e , M . G . P e r e y a t k i n , J . Remmel.

R e c u r s i v e a n a l y s i s h a s had many c o n t r i b u t o r s s i n c e t h e 1 9 3 0 ' s b u t i n c o n s i s t e n t n o t a t i o n s and r e p e t i t i o n c h a r a c t e r i z e t h e l i t e r a t u r e . T h e r e i s no good b i b l i o g r a p h y t h a t we know o f . Some o f t h e a u t h o r s h a v e been 0. A b e r t h , S . Banach a n d S . M a z u r , G.S. C e i t i n , R . L . Goods t e i n , A . G r z e g o r c z y k , D . K l a u a , D . Lacombe, Y . M o s c h o v a k i s , V . P . O r e k o v , M . B . P o u r - E l , I . R i c h a r d s , N . A . S h a n i n , I.D. Z a s l a v s k i i . There i s a s y e t n o - s y n t h e s i s of counterexamples i n r e c u r s i v e a n a l y s i s c o r r e s p o n d i n g t o t h a t a c h i e v e d by r e c u r s i v e model t h e o r y f o r r e c u r s i v e a l g e b r a . However, N e r o d e and Remmel h a v e made some p r o g r e s s i n 1982 i n t h i s d i r e c t i o n u s i n g t h e n o t i o n o f r e c u r s i v e g e n e r i c p o i n t i n a complete r e c u r s i v e l y presented m e t r i c space. We h a v e a l s o n e g l e c t e d t h e f o r m a l s y s t e m s d e s i g n e d t o c o v e r c o n s t r u c t i v e o r r e c u r s i v e mathematics. Some a r e d u e t o S . F e f e r m a n , H . Friedman, V . L i f s c h i t z , J . Myhill, N . A . Shanin. T h e r e i s a l s o a l i t e r a t u r e on c o n t i n u o u s f u n c t i o n a l i n t e r p r e t a t i o n s and r e c u r s i v e r e a l i z a b i l i t y i n t e r p r e t a t i o n s o f s u c h s y s t e m s , f o r e x a m p l e t h e work o f D . S c o t t . Our o m i s s i o n s a r e d u e t o l a c k o f s p a c e , not i n t e r e s t .

~

The a u t h o r s a r e g r a t e f u l t o E . Constructive Analysis (1981).

B i s h o p f o r i n f o r m a t i v e d i s c u s s i o n s on

Non-recursive methods

333

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H e n z e l t, K . [1922] ( B e a r b e i t e t von E. N o e t h e r ) , f u r t h e o r i e den p o l y n o m i d e a l e und R e s u l t a n t e n , M a t h . Ann. 88 53-79. Hermann, G. [1926] Die Frage der e n d l i c h v i e l e n s c h r i t t e i n d e r t h e o r i e d e r P o l y n o m i d e a l e , M a t h . Ann. 95 736-788. H i l b e r t , 0. [1900] Ueber das D i r i c h l e t ' s c h e P r i n z i p , J a h r e s b . Deut. Math. v e r . 8 184-188. Jockusch C.G. and Soare, R . I . [ 1 9 7 2 ] D e g r e e s o f m e m b e r s o f T: c l a s s e s , P a c . J . o f M a t h . 4 0 6 0 5 616. [1972]

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c l a s s e s and degrees o f t h e o r i e s ,

T r a n s . Amer.

Math.

SOC.

173 33-56. K l a u a , D. [1961] K o n s t r u c k t i v e A n a l y s i s , Math. Forsch. X I , D e u t s c h e r V e r l a g der Wiss. B e r l i n . hKcnig, G . [1903] E i n l e i t u n g i n d i e A l l g e m e i n e T h e o r i e d e r A l g e b r a i s c h e n GroBen, ( L e i p z i g ) . K r o n e c k e r , L. [1882] Grundzuge e i i i e r A r i t h m e t i s c h e n t h e o r i e d e r A l g e b r a i s c h e n , Grb'aen, J . f . M a t h 92 1 - 1 2 2 . K r u l l , W. [1928] G a l o i s c h e t'heorie d e r u n e n d l i c h e n A l g e b r a i s c h e n E r w e i t e r u n g e n , M a t h Ann. 100- 687-698. M a s l o v , S. Yu., M a t i y a s e v i c h Yu., e t a l . [1980] N i k o l a i A l e k s a n d r o v i c h Shanin, R u s s i a n Math. S u r v e y s 35 277-282. Metakides, G., and Nerode, A. [1974] R e c u r s i o n Theory and A l g e b r a , A l g e b r a and L o g i c , S p r i n g e r V e r l a g l e c t u r e n o t e s 45 209-219. [ 1 9 7 9 ] E f f e c t i v e c o n t e n t o f f i e l d t h e o r y , Ann. M a t h . L o g i c 1 7 289-320. Neumann, C . [ 1 8 6 5 ] V o r l e s u n g e n u e b e r Riemann ( T h e o r i e d e r Abe s c h e n I n t e g t a l e ) (2nd ed. 1 8 8 4 ) . Peano, G. [1890] D G m o n s t r a t i o n de l i n t e g r a b i l i t ; des 6 q u a t ons d i f f e r e n t i e l l e s o r d i n a i r e s . Math. A n n a l e n 37 182-228. P o i n c a r e , H. r 1 8 9 0 1 S u r l e s e q u a t i o n s aux d e r i v e e s p a r t i e l l e s de l a p h y s i q u e m a t h e m a t i d u e . Am. J. M a t h . 1 2 211-294. P o u r - E l , M . B . , a n d R i c h a r d s , I. [1979] A computable o r d i n a r y d i f f e r e n t i a l equation which posesses no c o m p u t a b l e s o l u t i o n , Ann. M a t h . L o g i c 1 7 6 1 - 9 0 . [ 1 9 8 l ] The w a v e e q u a t i o n w i t h c o m p u t a b l e i n i t i a l d a t a s u c h t h a t i t s u n i q u e s o l u t i o n i s n o t computable, Advances i n Mathematics 39 215-239. Rabin, M.O. [ 1 9 6 0 ] C o m p u t a b l e a l g e b r a , T r a n s . Amer. Math, SOC. 95 341-360. Riemann, B. [1902] Gesammelte m a t h e m a t i s c h e Werke, 2nd ed., ( l e i p z i g T e u b n e r ) . Rogers, H . J r . [1967] Theory o f r e c u r i s v e f u n c t i o n s and e f f e c t i v e c o m p u t a b i l i t y , M c G r a w - H i l l C o . . New Y o r k . Seidenberg, A. [ 1 9 7 4 1 C o n s t r u c t i o n s i n a l q e b r a T r a n s . Amer. Math, SOC. 197 273-31 3.

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S p e c k e r , E. [ 1 9 5 9 ] D e r S a t z vom M a x i m u m i n d e r R e c u r s i v e A n a l y s i s , C o n s t r u c t i v i t y i n M a t h e m a t i c s ( P r o c . C o l l . Amsterdam 1 9 5 7 ) , N o r t h H o l l a n d , Amsterdam 254-265. Steinitz, E. [1910] A1gebraisch.e t h e o r i e d e r KGrper, J . f . Math 137 167-309. Van d e r W a e r d e n , B . L . [1930] E i n e Bemerkung Qber d i e U n z e r l e g b a r k e i t von Polynom, M a t h . Ann 1 0 2 7 3 8 - 7 3 9 . Weber, N. [1893] Untersuchungen i b e r d i e allgemeinen Grundlagen d e r G a l o i s ' s c h e n G l e i c h u n g s t h e o r i e , M a t h . Ann X L I I I 5 2 1 - 5 4 4 .

THE L.E.J. B R O W E R CENTENARY SYMPOSIUM A.S. Tmebtra and D. van Dalen (editors) 0North-Holland Publishing Company, 1982

337

ALGEBRAIC NUMBER THEORY, A SURVEY Ray Mines Department of Mathematical Sciences New Mexico State University Las Cruces, New Mexico 88001 U.S.A.

This

paper gives an exposition of the constructive development of

algebraic number theory IMR31,

[MR41, and

as

work

carried still

out

being

in

[JMRI,

developed.

IMR11, The

[MR21,

setting is

Bishop's constructive ma.thematics [Bishop, 19671. Number

theory

is

a good test for constructive mathematics as it

applies both discrete and continuous constructions, the development

at all apparent. facie

constructive

brings to light constructive difficulties which were not Indeed,

constructive,

algebraic

and

it

number

theory

appears

prima

is common for authors to give routines

for the construction of the objects which occur in the subject. of

the

see

best

their

because

excellent

of

One

examples of this is [Borevich and Schaferevich, 19661, the

discussion

of

the

unit

theorem.

However,

mix of classical and constructive methods it is not

at all clear that these routines are finite. As

there

is a fairly complete bibliography of the literature on

constructive number theory [Crossley and Miranda, 19801, we will

not

include an exhaustive list. Kronecker exposition

was

the

first

to

of the rational number

field

results

routine

is

a

coefficients

Another

finite in

irreducible fields are

give

a

systematic

an

[Kronecker,

algebraic

for

18821.

factoring

number

field

into

his

many

polynomial with a

product

of

factorial. early

infinitely

are factorial. activity

Among

any

factors. That is, Kronecker showed that algebraic number work

is

[Hermann, 19261.

this work contains several errors, one of which that

constructive

of the algebraic number fields, that is finite extensions

in

generated

extensions

Hermann's mistakes constructive

have

algebra;

in

It is unfortunate that was

the

assumption

of the rational number field spurred

a

particular

great the

deal

of

search for

conditions which insure that a field is factorial. Another

approach

that

has

been carried out in the direction of

R. MINES

338

developing algebraic number theory recursive

function

"erode

19601,

development

and

which

Metakides,

are

constructively

of

by

means

of

.

19751

The

results

in

this

interest to us are negative in that they

show the nonexistence of certain algorithms. obtained

is

[Frohlich and Sheperdson, 19551, [Rabin,

theory

The

positive

results

tend to be suspect because of the use of classical logic in

recursive function theory. The

first investigation of valuation theory from the constructive

point of view was in interested

in

[Staples,

valuations

per

19711.

However,

se

so

and

subject beyond the basic definitions. the

problem

of

There is also

a

extending

using

Staples

was

to

not

did not develop the

For example, he did not

valuations

development

he

study

finite extension fields.

recursive

function

theory

in

"Smith, 19811. The table of contents for this paper is:

I.

Algebraic number fields Basic definitions Discreteness of algebraic number fields

11.

Factoring Polynomials

111.

Valuation Theory Definitions Locally precompact fields Pseudo factorial fields Hensel's lemma Extension of valuations

IV.

Unit theorem and finiteness of class group Dedekind domains The unit theorem

The material in the first three soon

appear.

The

material

sections

has

appeared

or

will

in the last section, with exception of

Dedekind domains, is an anouncement of work in progress.

I. Basic definitions. with

ALGEBRAIC NUMBER FIELDS

An axiomatic

definition

of

an

abstract

field

equality and inequality modeled on Bishop's complex numbers was

given in [JMRI. makes

the

Conventional notation was used in that paper,

subject

appear

which

similar to classical developments, while

retaining the constructive finitistic interpretation. This

definition

uses

the

category

of

sets

equipped with an

Algebraic number theory

339

equality relation and an inequality relation satisfying: i)

x = x

ii)

x = y

implies

iii)

x = y

and

iv)

A

x

y

L

y = x

y = z

implies

y

x

#

z

v)

x = y

and

y

L

z

implies

vi)

x = y

and

x

L

y

is impossible.

set is discrete if for any pair

y.

x = z

implies

Inequality

to

is

be

x

x

and

thought

of

L

y

x = y

either

as

a

positive

or

x

#

notion

of

both

the

is

the

distinctness rather than denial of equality. A function

f

from one set

to

another

must

respect

equality and the inequality in the sense that: i)

x = y

ii)

f(x)

implies

X

The product of t w o sets Cartesian

X

product

i)

A

x

L

Y

y.

Y

in

X

of

X

on

x

and

Y

x

inequality are defined

ii)

f(x) = f(y)

implies f(y)

L

this

Y,

and

category

where equality and

by:

(x,y) = (u,v)

if and only if

x = u

and

(x,y)

if and only if

x

or

y 0

field is

L

u

a set

k

with distinguished elements

and

*

from

+

two functions

(u,v)

L

k x k

k

to

y = v v.

L

and

which make

1

and

into

k

a

commutative ring with identity and satisfy the further restrictions:

For each

i) ii)

iii) Condition

0

L

ii)

X

the

notion

L

0

there is

b

ab = 1.

with

for some integer

n

then

a = 0.

1. is needed because we do not assume that

is an

L

This is in contrast t o IHeyting, 1971, page

apartness relation. where

a

an = 0

If

511

of a field is defined using an apartness relation

satisfying vii) viii)

x X y is absurd implies

implies that for all

z X x

or

in addition t o our equality

condition

field vii)

given may

z

either

z X y .

conditions

and inequality.

axioms of a

x = y

x # y

through

i)

Condition above

be false:

viii)

vi)

for

sets

with

is a consequence of the

[JMRI, while,

as

we

shall

see,

the residue class field of a field

with a valuation gives a n example of a field which does

not

satisfy

340

R.MINES

vii) ,

while it satisfies if it is absurd that x = y

vii') One from

x z y

to

k or

a z 0

no

nonzero

interpreted

is the following.

to

b

be a function

0 = 0 + 0,

#

then

0. Another observation is that a local ring

#

nilpotent

mean

a + b

If

IMR21.

+

of the consequences of the requirement that k x k

either with

then

-

a

elements

b

is

is a unit.

a

field

if

a s b

is

Thus theorems about fields,

in this sense, may be interpreted as theorems about such rings. maximal

The

ideal of such rings consists of those elements which are not

different

from

zero.

constructivist's

This

notion

to be different from

gives

classical

insight

into

the

of numbers that are not known to be zero or

zero,

and

clarifies

the

problems

that

the

constructivist faces in proving theorems about fields. A prime field

integers

is a field with no proper subfields.

localized

at

a

prime

p

Every field has a unique

field generated by

and

k

If

is

a

field

characteristic of the infimum is positive not

be

sequence of be

the

-,

a

the

Thus a

and

subfield: the

discrete one

the

prime inf (n

point field

subfield

>

positive

compactification of

integer

rational or

with at most one

1's

(anpn : n = 1 , 2,

of

. . . 1.

rational

integers

Let

be

k

characteristic of

of

k

the

where

of

the

numbers

arbitrary

-:

1.

Let

Let

(a

pl,

has field be a

)

p2,

P

Let

generated

by

the

(x

if

R.

of

is neither a positive integer nor

algebraic number fields.

The

whether a polynomial is irreducible or has a

set

-

y)

E

P.

Then the

-.

ability

nonconstant

to

decide

factor

is

repeatedly in classical expositions of algebraic number theory.

Even authors who are being careful about the constructive content the

...

be the

the field of quotients of the

Z with equality defined by x = y k be the field of quotients

let

Discreteness

Z

then

0 : n.1 = 0 )

increasing enumeration of the prime numbers.

discrete ring

used

prime

and the characteristic of of an

either 0's

ideal of the ring

Finally,

in

integers.

characteristic need

with

taken

of

1.

is defined to be

k

ring

is an example of a prime field

that is not discrete. 0

The

subject

Van der Waerden subfield

of

factored into 19301.

The

of

assume that every polynomial has an irreducible factor. was

the

the

first

complex

irreducible following

is

to

numbers

observe

polynomials a

that

there

exists

a

and a polynomial which cannot be over

this

field

[Waerden,

slight variation of van der Waerden's

Algebraic number theory

field.

Let

{an)

the

polynomial

the

field

discrete

be a n increasing sequence of x2 + 1

cannot

W = U Q(ani). .k

field

341

and

0's

1's. Then

be factored into irreducibles over

Following

factorial if

[Hermann,

19261

we

every polynomial in

call

k[xl

a

can be

written as a product of irreducible polynomials. Even

though

one

cannot

factor

irreducible polynomials one c a n , Euclidean

algorithm

to

a

polynomial into a product of

over

produce

a

discrete

GCD

the

field,

of

any

use

two

the

nonzero

polynomials.

[JMR,

Lemma 1.1. subfield,

Lemma

f(a)g(a) = 0,

f(a) = 0

then

Proof. By

If

the g(a) = 0

Hence

0.

The

Euclidean

E

Let

be a field,

f,

g E k[xl

or

g(a) = 0.

So

0.

f

a discrete and

algorithm

either

f ( a ) = 0,

or

k

( f , g )= 1

and

Euclidean

s(a)f(a) + t(a)g(a) ='1 #

3.31.

a E E.

and

s(a)f(a)

write or

0

f

t(a)g(a)

respectively. 0

algorithm also allows us t o obtain the next result

which replaces unique factorization into

a

product

of

irreducible

polynomials.

IJMR, Lemma 3.51.

Theorem 1.2. nonconstant

a

polynomial

Let

in

k

be a discrete field

kCxl.

Then

relatively prime polynomials of the form positive

integer,

q

characteristic of

k,

derivative

f'.

Proof.

deg g = 1

If

then

0,

>

1

h

and

1

or is

g'

g(x) = h(xp)

f

0

then compute the G C D

power

If

by

induction

and

g2

in the desired form.

write

g1

these

factors

under

induction t o the

(g,g').

on the degrees of

is to

a

in

k[xl. If

deg

(g,g') = 1 g. and

Now close the

its g' =

1 and

g.

If

Write g2

set

By g we g

we can of

all

the taking of GCD's and the result of applying

case

g' = 0

when

ever

such

a

polynomial

is

fl

A s an application of t h i s useful result w e shall

algebraic

>

h

g1

into

m

prime

deg g

for some

g

g

of the finite

g 1 = (9.9')is a proper factor of

Then

discovered.

a

where

relatively

the result is clear. and

w e can factor

fm(xq)

can be written as desired; hence so can

are done. Otherwise

= g1g2.

either f E k[xl

and

k = p

char

induction

is

and

numbers

form

However, because of the

prove

that

the

a discrete subfield of the complex numbers. need

to

know

a

similar

result

for

the

R. MINES

342 algebraic

elements

in

the

p-adic number field, we shall state the

result in its general form.

[JMR,

Theorem 1.3.

Theorem

E.

discrete subfield of

a = p and

F

Let

a

part

k

k,

a

then

be the product of a monic polynomial satisfied by

monic ii)

f(xq)

be a field and

are algebraic over

asp.

or

Proof.

E

Let

3.61.

a, p E E

If

polynomial

8.

satisfied by

Applying Theorem 1 . 2 and

of the definition of a field we can

q = pn

where

for

some

prime

a

p

assume

F(x) =

that

(f,f') = 1.

and

Using

Taylor series, f(y) = (y

-

pq)f'(pq) + (y - 1 3 ' ) ~ K ( y )

0 = (aq - pq)[fo(pq) + (aq - pq)K(aq)l.

( f , f ' )= 1

Since

f(gq) = F ( p ) = 0, we have

and

f ( p q ) = tf ( p q )

0

I

+ (aq

-

pq)K(aq)I + [-(aq

-

pq)K(aq)1.

So either

(aq Thus 0

so

-

either

(a

-

pq)K(aq) z 0, or 0

f

Condition 1 + a)

-

g q ) = (a - p I q

p b q = 0, and by

field, a - p = 0

order

(aq

for

ii) this

with

inequlaity

so

f'(pq) + (aq - pq)K(aq)

a = p-

part

on

of

a

to be true:

multiplication

relation

F

of

p,

f

the

or

aq

-

definition

8'

=

of

a

0

in the definition theorem

a

giving

ii)

0.

f

defined by

x

f

field

necessary

a2 = 0.

by

y

is

F be the ring

Let if

x

-

y

in

(0,1, a,

Define

is a unit in

an

F.

F satisfies all the axioms of a field except ii). The field F is an algebraic extension of the 2 element field: a satisfies the polynomial equation x 2 = 0. However, F is not discrete. The next result is proven by using the above results and others in [JMRI. We shall not give a proof. Then

Theorem 1.4. field, that is

[JMR. a

Theorem

finitely

4.31. generated

contained in the complex numbers.

Let

k

be an algebraic number

extension

of

the

Then

is discrete.

a)

k

b)

k = Q(a)

c)

k is factorial.

d)

k is a finite dimensional vector space over

for some complex number

a. Q.

rationals

343

Algebraic number theory

e)

K is a detachable subfield of the algebraic numbers. 11.

FACTORING POLYNOMIALS

Kronecker was the first to investigate the constructive of

polynomials.

He

gave

a

finite

routine

for

Van

der

Waerden

extended

1882,

a

finite

separable

' a l s o factorial.

editions

of

extension

well

this

fact

in

known book [Waerden, 19531.

the norm of an element and it is unclear if the

that

of a factorial field then

He included a proof of

his

van

the

if

K

K

is

earlier

This proof uses

der

Waerden

means

norm defined in terms of a splitting field and the conjugates of

an element therein, as defined on page 178, or the terms

of

the

field

polynomial

norm

defined

as defined on page 132.

all

obvious from the second definition. out

is nonconstructive [Seidenberg, 19781.

and

then

repeats

van

der

not

he

properly

Because of these

difficulties Seidenberg redefines the norm using a generic field,

are

Seidenberg has read van

der Waerden's proof as using the first definition which points

in

The proof

also uses several properties of the norm which, while true, at

any

section

Hronecker's result to fields other

than extensions of the rational number field by showing is

factoring

factoring

polynomial over an algebraic number field [Kronecker, 41.

0

Waerden's proof.

splitting

A proof which

avoids the norm map altogether is given in [MRII.

of

Examples

inseparable extensions of factorial fields which are

not factorial have been given by [Seidenberg, 19701 [Frohlich

and

Shepherdson,

recursive function point of view 'explicit' algebraic

the

19551

field

and

k

an

with

'explicit' algebraic extension

K

algebraic

so that

of

and

[MRlI.

In

problem is studied from the example

is

splitting

given

algorithm

of

an

and an

k

corresponding to a simple K does not have a splitting algorithm. They also show that if K is an 'explicit' field having a splitting algorithm and K the 'explicit' extension of k given

by

the

extension

elements

algebraic

over

of

k,

a l ,. . . , an k,

then

where

K

each

ai

is

separable

has a splitting algorithm.

'explicit' field can be translated in our terminology

as

and

The term the

field

is a countable discrete field. In [ M R l I

the

relationship

polynomials

was

studied. A

field

separability

and

factoring

A polynomial is separable if it is equal

to a product of polynomials derivatives.

between

is

which

are

relatively

separably factorial if

prime each

to

their

separable

R. MINES

344

polynomial is a product of irreducible separably

factorial

To prove this result and avoid

Theorem 3.91. norm

map

polynomials.

which

both

van

der

following result was proven.

Waerden

[ M R l , Theorem 3.21.

the

E,

polynomial. such

Let

over

5

k.

a

be

--.

A i E k [ x l , . . . ,xnl

Define

the the

we

construct

be a discrete subfield

n-1 + .a h ( x ) = x n + a n- lX i s n) be a set of algebraically

Let

tr, : 1

k

of

ktxl.

in

monic

q ( x ) E ktxl

h

be

of

(separable)

Then there exists a (separable) polynomial

+

Proof.

h

h ( x ) E ktxl

let

qCx).

roots of

of

used

problem

the coefficients of any monic factor of

that

class

Seidenburg

As in [Seidenberg. 19781

Theorem 2.1.

and

the

and

a generic splitting field for a polynomial

field

The

is closed under finite extensions [ M R l ,

fields

Etxl

in

in

ard

k[xl.

independent

Let

elements

H ( x ) E ktAo , . . . , An-l,XI

and

by ~ ( x =) H(ri) = 0

As over

for each

-

polynomials

- r i 1 = xn + ~ i,

Define

a

ai.

+

Q(x1

of

Let

are

ri.

As the elements

in

all

are

symmetric

algebraic

H

goes t o

IWaerden,

Ai

are the and

and

the

the ri

1953,

k[AO,...,An-l,xI h,

by

symmetric

r.

the

see

homomorphism from

Under this m a p

~

polynomials in all the

q

is

separable

be a countable subfield of

K

Let

Corollary

ri

-

to

it page ktxl q(x1

image

is the desired polynomial.

ko h.

the

polynomials

ring

x+ .A~

Q ( x ) E ktr l,...,rn,xl

Define

It remains t o show that

of

it follows that the

Q ( x ) E k[AO,...,An-l.xI,

follows that

Ai

Q

of

- + ~

~

ranges over all the elementary

1

subsets

symmetric

coefficients

811.

where

X I

in

elementary

of

(x

k[AO,...,An-ll.

Q(x1 = i l (x

by

n

be

3.91.

a

splitting

field q

The polynomial

are combinations of the roots of any polynomial which splits in This result is improved in polynomial polynomials

in

ktxl g

coefficients of

and g

and

k

for

splits in

h.

K

if

But

h

is

separable.

containing the coefficients

K/k

h

over

K

is separable

and

is separable [ M R l , Lemma 3.11.

tMR2l where it

is

shown

that

of

q so

0

if

a

factors into a product of t h e relatively prime h

in h

an

extension

are separable over

field k

E

then

the

t M R 2 , Lemma 4.61.

The trick of constructing the countable subfield generated by coefficients

,

t JMR

ko,

as the roots of

the a set of polynomials a s in the proof of Theorem 2.1

Algebraic number theory

is often used adjoin

345

This is because over countable fields one can

roots

of

polynomials,

form

splitting

fields,

always

or form an

algebraic closure [JMRI. The

discrete

k

field

has

a

root test for (separable) f(x) E k [ x l

polynomials if given a (separable) polynomial there

a E k

exists

or

f(a) = 0

with

f(a)

either

Thus the rational root test is a root test for the field of numbers.

In

[Frohlich

and

a

splitting

algorithm

an

'explicit'

construction

algorithm.

for

turning

However,

by

field

k

if and only if it has a root algorithm.

Unfortunately, their proofs are not constructive: a

ratlonal

1955, Theorem 4 . 4 3 1 and

Shepherdson,

[Rabin, 1 9 6 0 , Theorem 8 1 it is proven that has

a E k.

for all

0

#

the

using

root

they do

algorithm

Theorem

we

1

not

give

into a splitting

can

obtain

such

a

construction. 3.41.

[MRl, Theorem

Theorem 2.2.

(separably) factorial

if

and

A

only

discrete

if

field

has

k

a

is

k

root test for

(separable) polynomials. Proof.

k

If

is (separably) factorial, then we can determine the

linear factors of prove

the

f.

any factor of

use

f

Theorem

2.1

is

clearly

k[xl

in

q.

are roots of k

can determine the finite set of elements of

q,

necessary,.

to construct

is separable) polynomial so

f

monic (separable if

of

so the condition

converse

To

q(x) E k [ x l that

the

a

roots

By the root text we

which

are

roots

of

and thus we can obtain a finite set of polynomials containing all

the honic factors of

f

Corollary 2.3.

field

The

in

k[xl. of

0

rational

numbers

form

a factorial

field. The rational root test is a root test.

Proof.

The characterization of (separably) which

have

a

root

test

of

how

we

can

factorial

fields

as

fields

for (separable) polynomials allows one to

prove the following theorem. example

0

We shall give a proof as it is

another

restrict our attention to countable fields

generated by the coefficients of a polynomial. Theorem 2.4.

[MRL, Theorem

3.91.

a separably factorial subfield, and

F = kla) k

is separably factorial.

is factorial, then

F

Let a E K

K

be a discrete field, algebraic over

Moreover, if

is factorial.

a

k.

k

Then

is separable and

R. MINES

346 Proof.

Let

f E k[xl

be a separable polynomial.

Icountable) separable closure, within the

coefficient

of

the

k,

irreducible

polynomials over follows

by

ko

Theorem 1.2,

Fo

f

a

Fo

in

that

ko

is separably closed in

[JMR, Corollary 3.91.

L.

By

gi E Fo[xl ri E Fo

[MRl, Lemma

Let

3.01

occur

in

is a separably factorial field. f

that

kla),

lie in k l a ) = F 0' 0 and the coefficients of

prove

we

have

found

L

of

rl,...,rn be the roots

f f

of

we can find irreducible polynomials

that are satisfied by the and

linear, then To

a,

is countable, we can construct a splitting field

Fo

over

which

by

by

Fo.

lie in As

satisfied

f.

Thus it suffices to find the roots of because

be the

As we can determine whether separable it have roots in k, and hence in kO'

of

coefficients

kQ

of the field generated

polynomial

together with the coefficients of all powers of the

Let

a

If

ri.

root of

gi

f

in

f has no roots in FO. the second statement replace

closure instead of the separable closure,

is

ko

and

linear,

Fo.

If no

then

gi

is

by the algebraic

apply

Corollary

1.3

rather than Lemma 3.8 of [MRlI. 0 Corollary 2.5. factorial

IMR1,

Corollary

field, and let

3.101.

f E k[xl

there is a separably factorial field f

is

a

k

over

product

of

K

f.

k

K[xl

a

extension

field

of

[Rabin, 19601.

Recall that a subset in

Theorem 2.6. factorial

Let

field

B

Kp

K

be

a

k

of

finite

is detachable in

is

f

of

finite

Kp

is 0

K.

form

that

is generated K

are

a

in

f(xq)

condition

in

K.

A

or

of a set b

Z !

B

is

detachable

A.

dimensional

extension

characteristic

p.

of

Then

the K

is

K. To show the sufficiency assume that

polynomial then either By Theorem 1.2 we may

where

characteristic

order

a factorial field be factorial

By Theorem 2.2 we need to show that

nonconstant

f o r all

the

the finite

a

K

is detachable in

The necessity is clear.

E K[xl

g(a)

b E A

either

factorial if and only if Proof.

Then

such

Moreover, any two such fields

if for each

b

separably

0

k.

finite

a

k,

and

The following gives necessary and sufficient that

be

containing

linear factors in

by the roots of

isomorphic over

Let

be a separable polynomial.

of

lf,f') = 1 k.

As

k

and is

g

if

g

has a root or

assume

that

q = pm where factorial

p

K

g is is

Algebraic number theory separably

factorial

f

roots of

f.

As

K.

Kp

Theorem

K.

in

If

a

Thus

2.4.

is a root of

K

is detachable in

347 we

g

can determine all the

then

is a

'a

root

we can find all the roots of

of

g

in

0 111.

Definitions. function

VALUATION THEORY

Staples defined a rank one

v

from

a

field

k

to

general valuation to the

be

a

nonnegative real numbers

satisfying: i) ii)

vx

f

iii)

B

if and only if

to

be

k

is called a bounding constant

able

valuation

for

1x1

valuations

in

k

is a

I

function

I

clear

2.

that

0

f

if and only if

Ix

+

yl

and

1x1

5

0

L

lyl.

+

B

a valuation is a general valuation if we take

O n the other hand,

constructive

x

lxyl = IxlIyI

ii')

is

v.

t o the nonneqative real numbers satisfying

iii') It

for

Staples's

to distinguish them from the definition given in

i')

to be

B.

5

[MR21 where a rank one valuation on a field from

0

#

such that if

v(l + x )

then

1,

5

We use the term general order

x

B

There is a constant vx

The constant

0

v(xy) = (vx)(vy)

shows

[Weiss,

that

1963,

Proposition 1-1-81

B = 2

a general valuation with

is is a

valuation. A

general valuation is nontrivial if

nonarchimedean if vn

>

1

for

<

vn

1

some integer

n.

IMR21,

nonarchimedian or equivalent if

we

is equivalent t o

cannot

archmedian

they

induce

v2

>

1

x

for some

n,

and

to

deny

assert

that

that

IMR21.

Two

a

the

valuation

general

valuations

the same uniform structure, that is,

if f o r each

> <

c

0

there is

d

>

0

such

< implies v x < E , and v x d implies v x C E. 2 2 2 We cannot tell if a general valuation is nontrivial. Let

a

nonnegative

number

I

and

I

Because

conditions

of for

this two

problem valuations

the to

theorem be

are v 1 that

r

be

be ordinary absolute value.

Define the valuation on the rational numbers by 0.

is

valuation is either

v x 1

real

k,

in

archimedean if

Although t o assert that the valuation

is nonarchimeadean is the same as archimedean

vx

for some integer

vx = (xir for

x

f

which gives equivalent

equivalent

takes

on

the

348

R.MINES

following form. Theorem 3.1.

IMR2, Theorem

Let

1.11.

v1

and

v2

be

general

valuations and consider the following three conditions: a)

There is a positive real number

b)

v x = (v x)r for all nonzero 1 2 v1 is equivalent to v

c)

vlx

Then

a)

v x < 1 2

implies

implies

b)

then

implies

c).

implies

c)

nontriviality

< 1

Moreover,

a).

requirement "v1x

the condition reads

x

for all

Thus

a

if

v

in Theorem 1 is essential even if

if and only

< 1".

v 2x

if

wit,hout the nontrivility condition we can prove neither b)

nor If

b) k

implies is

a

a)

field

In

c)

fact

implies

[MR21. with

a

valuation

valuation to the rational function field be

1

nontrivial general

0

valuation is equivalent to a valuation. The

such that

2'

< 1

nontrivial,

r

x,

then

k(x)

we can extend the

1x1

by defining

to

a

positive real number and then the valuation of a polynomial is i 1 1 aix I = sup ( lai I IxIi : 0 L i 5 n > where n is the given by degree of the polynomial (MR2, Theorem 4.21. Locally precompact valuations. occur

in

number

positive integer of

k

theory N

1x1

Such a subset is called is

the

and positive

such that if

valuation

The fields

have

called

< N an

k

with

following there is

E

Ix - yl <

then

valuations

a

finite

subset

y

for some

E

6-approximation to the N-ball,

locally precompact.

not assumed to be a subset of the locally precompact valuation,

that

F o r each

property.

in and

Y

Note that the set

N-ball. If then the

Y Y.

the is

k is a field with a valuation is either

archimeadean or nonarchmedean [MR2, Theorem 2.11. The

residue class field of

valuation Ix

is

- yl < 1

the

x

and

L

L

y,

k

the

while

residue

class

the absurdity of

with

a nonarchimedean

where Note, field

x

L

x = y f

y

if

in contrast to

x = y

is

does not imply

x = y.

that The and

x

field

-

y

[Heyting, 19711, that in absurd. then

a

(x E k : 1x1 5 1) if Ix yl = 1.

set

p

element

residue

class field may not be a field in our sense:

be two real numbers with field.

sup (a,B) = 1.

Define a valuation on

klx,y)

by

Let

Ix

k

Let

be the

= a

and

a

two

lyl

349

Algebraic number theory

= 8.

I x + yl = 1

Then

x + y

and so x

field yet we cannot assert that

If the value group is a discrete field

L

y

group

a field and is also discrete.

is

in

0

or

0

L

the

residue

class

residue

class

0.

f

then

the

We c a n obtain an example of

a residue class field which is a field and which is not defining

a valuation on

k(x)

number less than or equal t o valuation

A

such that if

y

following

on

k

gives

1x1

discrete if

t o be a positive real

there is an

lyl = J x l n f o r

a

by

1.

is

then

0,

L

by taking

discrete

some

characterization

x

of

in

0

L

integer

n.

those

k

The

nontrivial

nonarchimedean valuations which are locally precompact. [MR2,

Theorem 3.2.

Theorem

Let

2.21.

nontrivial nonarchimedean valuation. locally

precompact

k

be

a

field

with a

Then the valuation o n

is

k

if and only if the valuation is discrete and the

residue class field is finite. Proof.

Suppose

k

is

locally

nontrivial we can find

z

Iz I-approximation

...,

Choose for

x

y

k

such that

x

to

0

the

< lzl < 1.

N-ball

for

Choose N

2

a

121-'.

1 , . . . , x n of largest value less than 1. Then among m' L 0 there is an integer m such that lyl = Ixil. Now

choose a finite Y

in

*

i

any

of

x 1,

A s the valuation is

precompact.

Y

1-approximation

t o the

1-ball.

The

elements

form a system of representatives for the residue class field,

which is finite since it is discrete: the value group is discrete. If

the

valuation

finite, let finite

N

and

then

(zlm <

<

1

is

and

E

<

for

E

I z I - ~2 N

k[xl.

factorial

of

that

some

A

field

residue class field is and

A

let

in

Y

be

Chose

m

precisely

l's,

and

a

Given

such that if

Y.

k

1x1

so that

with

a

valuation

when

it

is

K =

k

is

pseudofactorial under the trivial

To obtain a n example of

and let

is

exists for each polynomial

The reason for this terminology is that a field

is not pseudofactorial let

0's

y

inf ( If (a1 I : a E k )

valuation, Theorem 2.2.

K

the group

and let

Pseudofactorial fields. in

and

we must find a finite set

E

Ix -yl

pseudofactorial if

f

discrete

generate the value

system of representatives for the residue class field.

positive

s N

(21

(an)

a

factorial

field

be a n increasing sequence

U Qca nil. A

If

k

is

field

with a valuation w e may form the completion . k

R. MINES

350 k

of

in

the

t h e same w a y t h e r e a l numbers are [Bishop, 19671

rationals

,.

naturally to

The

obtained

valuation

by

completing

on

k

extends

A

and

k,

is a field where

k

x

y

f

-

Ix

means

yI

L

A

0.

If

and

only

is the natural map from

@

1x1 = 0.

if

and

k

say

to

k

If

k

is nonarchimedean,

k

and

E

have the

o(x) = 0

then

k.

is

then

isomorphism. o(k)

We

k

complete if

E

and

same

if

is

0

an

is a field between

residue

class

fields

and value groups. If if

k

is a discrete field with a pseudofactorial

f

a

is

separable polynomial in

k[xl,

valuation,

then either

f

and has a

A

root in

k,

or

)f(a)\ is bounded away from

for all

0

a

in

k

IMR2, Corollary 3.21. The next result gives conditions

that

show

how

pseudofactorial

f ie Ids arise .

Theorem 3 . 3 . either

[MR2, Theorem 3.41.

If

a factorial residue class field, then The

is a

k

discrete

field

with

locally precompact valuation, o r a discrete valuation with

a

fields

which

occur

k

is pseudofactorial.

in number theory are pseudofactorial as

they either have a finite residue class field subfields

of

the

0

or

else

are

located

complex number field with ordinary absolute value

as the valuation, and hence are locally precompact. The importance of psuedofactorial fields is the following. CMR2, Corollary 3.51. If

Theorem 3.4. a

pseudofactorial

of

k

in its completion

Proof.

The

algebraic

discrete, Theorem 1.3. is

dense

in f

is a discrete field

with

K.

is separably factorial.

k

closure K of k in k is K is pseudofactorial since k is a separable polynomial in K t x l ,

(separable) Moreover,

Hence if

f A

then either

k

valuation, then the algebraic (separable) closure

*

K = k or \f(a)1 is bounded away K. But any root of f in k is in K ,

has a root in

n

from so

K

0

€or

all

a

in

is separably factorial, by Theorem 2.2.

Hensel's lemma.

The

version

of

Hensel's

0

lemma

that we use is a

modi'fication of the somewhat elaborate version that is used by [.Artin, 1967, Page 2 9 1 . the coefficients of a introduce

the

Artin

Because we are unable to determine which of

polynomial

are

nonzero,

we

are

forced

to

concept of the formal degree of a polynomial which is

defined as follows.

Algebraic number theory

i)

351

The formal degree of a constant is The formal degree of

ii)

iii)

x

is

0,

1,

The formal degree of a sum is the maximum of the formal degrees of the terms,

iv)

The formal degree of a product is the sum of the formal degrees of the factors.

Note

that the formal degree is an operation in the sense of [Bishop.

19671 rather than a function

-

that is,

the

formal

degree

of

two

equal polynomials may differ. The following definltion is the conclusion of given in [Artin, 1967, Theorem 51 when Definition. say

C

that

Let k

k

be a field with a nonarchimedean valuation.

and

where

d

M

is in

k,

then

we

f,

u,

0 ,

h,

B

A,

We and

is at least as large as the maximum of

s =

( O ( x )

B(x)h(x)

and

C(x)h(x);

l/lanl

@(x)

construct polynomials

can

as

such that

the formal degrees of and

lemma

s = 1.

is Henselian if when ever

klx ,

are in

Hensel's

and

Y ( x )

in

ktxl

such

that vi 1 vii 1 viii) ix ) X)

Hensel's Lemma.

tMR21

If

k

nonarcherndian valuation, then Proof. If

k

is

a

complete

field

under

a

is Henselian.

0

CMR21.

is a discrete field with a nonarchimedean valuation, then

k

we define the Henselization

k"

of

k

t o be

the

separable

closure

A

of

k

in

k. This

terminology

is

justified

discrete field with a nonarchimedean valuation is

by the fact that a Henselian

exactly

R. MINES

352 when

it

separably

is

c l o s e d i n i t s c o m p l e t i o n [MR2, Theorem 4 . 7 1 .

The r e a s o n f o r r e p l a c i n g t h e c o m p l e t i o n of of

by

k

the

Henselization

t h a t t h e H e n s e l i z a t i o n of a d i s c r e t e f i e l d i s a d i s c r e t e

is

k

f i e l d , Theorem 1 . 3 . One

consequence

of

the

above

characterization

H e n s e l i z a t i o n f i e l d s is t h a t t h e a p p a r e n t 1x1

disappears.

k(x)

is given with

Extension

of to

e

e

Let

be

algebraic

field

and

is f i n i t e dimensional over

k(B)

E

a

finite

Let

dimensional

extends uniquely t o

k

Proof.

The p r o o f of t , h i s

result

in

the

constructive

is

extension

result

1963,

[O'Meara,

separable

extend

k

the when

is

discrete of

Henselian Then t h e

k.

If t h e value

group

of

k

E.

follows

Theorem

the

proof

of

the

same

w i t h s e v e r a l changes t o

14:11

subtleties.

then

a

extension

E.

That

k.

[JMR, Theorem 3 . 7 1 .

k

be

k

i s d i s c r e t e , so i s t h e v a l u e g r o u p of

overcome

over t h e f i e l d

w e must i n g e n e r a l r e s t r i c t o u r s e l v e s t o

that

[MRZ, Theorem 5 . 3 1 .

v a l u a t i o n on

value

c a n a l w a y s assume t h a t t h e v a l u a t i o n on

s a t i s f i e s an i r r e d u c i b l e polynomial over

Theorem 3.5.

discrete the

I t i s shown i n [ M R Z I t h a t i n o r d e r t o

k[e1

t h e case where

of on

1x1 = 1.

valuations.

with a valuation. valuation

we

Thus

dependence

One

shows

that

if

e

t h e v a l u e of a n e l e m e n t

the

in

E

is

g i v e n by l/n

l e i = INte) 1 where

is t h e f i e l d norm f r o m

N

E

of

over

an element

e

If

k

p

and

k

n

is t h e

q = pm

where

and

VE

of

that

E

there

k

is a d i s c r e t e is

an i n t e g e r

group

and

that

t h e f a c t o r group

t h e r e a s o n i n g u s e d i n [ B o r e v i c h and

E

of

subset

m

p o s i t i v e real m (VE) C Vk.

such t h a t

Schaferevich,

19661

is

VE

is f i n i t e .

VE/Vk

a

This is

when

they

t h a t t h e e x t e n d e d v a l u a t i o n is a g a i n d i s c r e t e a n d t o d e f i n e t h e

r a m i f i c a t i o n index it

the

0

t h e n it i s e a s y t o s e e t h a t

U s i n g t h i s and c l a s s i c a l r e a s o n i n g i t is e a s y t o see t h a t

show

is

i s a f i e l d w i t h a d i s c r e t e n o n a r c h i m e d e a n v a l u a t i o n and

t h e value group

cyclic

dimension

is t h e nonzero c h a r a c t e r i s t i c .

a f i n i t e d i m e n s i o n a l e x t e n s i o n of

numbers,

to

l e i = leqll/q

i s g i v e n by

d e g r e e of t h e e x t e n s i o n a n d

is

E

F o r a p u r e l y i n s e p a r a b l e e x t e n s i o n t h e v a l u a t i o n of

k.

is

difficult

e ( E / k ) = order ( V /V ) . E k f i n d t h e generator

However,

to

C l a s s i c a l l y , t h e r e s i d u e class deqree

f(E/k)

of

constructively

t h e new v a l u e g r o u p . is defined t o

be

the

Algebraic number theory degree

of

the

residue

Constructively, unless a

basis

-

E

for

class

-

k

-

over

introduce

a

notion

P

if

field

-

extension

E

due

-

k.

construct

The next theorem gives conditions under Before we state this result

to

every

over

is factorial, it is not easy t o

k.

which these numbers exist.

condition

353

A

Seidenberg.

finitely

we

field

generated

need

k

to

satisfies

purely

inseparable

extension field is finite dimensional, [Richman, 19811. lMR2, Theorem 5.51.

Theorem 3.6. field

with

a

discrete

separable extension of satisfies

k,

P.

condition

groups has finite

-

extension

E

k

be

Let

a

be

Then the factor group

-

k

e(E/k),

has

finite

the

P.

satisfies condition

for

which

one

field

E

class

is

n = and

0

cannot

assert

extension

that the factor group

class

Henselian

field

can

not

be entirely removed.

is a field such that whenever

field

with

residue

class

dimensional separable extension field of field from

5

-

k

is

finite

k

is and

that the

k,

dimensional over

k

then

of

on

is the

It is actually

k

-

field

E

VE/Vk

residue

-

field

discrete,

There is also an example t o show that the condition

k

k

and

finite.

shown that if

-

of the value

f(E/k),

I n lMR21 an example is given of an inseparable

k

class

residue

dimension

Henselian

an n-dimensional

VE/Vk

e ( E / k ) f ( E / k ) . I n particular, the valuation on

E

discrete

E

and suppose the residue

cardinality

over

Let

valuation.

a

discrete

E

a finite

residue

cp

class

is detachable

( M R 2 , Corollary 6.31.

Applying

Theorem

3.6

it

is

not t o o hard t o prove the next' two

results. Theorem 3.7.

Theorem 6.41.

[MR2,

Let

k

be a discrete field with

a valuation such that:

If

E

E

a)

The valuation is discrete.

b)

The Henselization is separably factorial.

c )

The residue class field satisfies condition

is a finite separable extension of

extending the one o n

Theorem 3.8. valuation

[MR2,

k

satisfies

6.51.

Theorem

satisfying

the

of

E

Let

b), k

valuation

and

be

c).

a

field

hypotheses of Theorem 3.6.

n-dimensional separable extension of Henselizations

k , then any

a),

under

the

k.

Let

valuations

El, on

P.

..., E

Let

Es

on

0 with

E

a

be an be

the

extending the

354

R. MINES

v a l u a t i o n on

Proof. n =

k.

Then

by Theorem 3 . 6

and

U N I T THEORCM AND FINITENESS O F CLASS GROUP

IV. While

: el

0

tMR2, Theorem 5 . 5 1 .

m a t e r i a l i n t h i s s e c t i o n h a s been worked t h r o u g h i t is

the

not y e t i n

= tEi

e ( g i / c ) f l g i / c ) = ni

We have n . by

final

Whaples,19451,

form.

follow

We

the

approach

in

[Artin

and

u s i n g p r o d u c t f o r m u l a f i e l d s , t o prove t h a t t h e c l a s s

g r o u p i s f i n i t e and t o p r o v e t h e u n i t t h e o r e m . Let on

be a d i s c r e t e f i e l d and

k

Then

k.

a

f o r each

i)

a d i s c r e t e s e t of v a l u a t i o n s

M

is a p r o d u c t f o r m u l a f i e l d i f

(k,M)

0

f

in

f i n i t e number of

\alp = 1

k

f o r a l l but a

p E M

n PEM l a l p = 1.

ii)

T h e r e e x i s t s one v a l u a t i o n

iii)

I

in

6

M

which i s

e i t h e r d i s c r e t e with a f i n i t e r e s i d u e class f i e l d o r Archimedean w i t h c o m p l e t i o n i s o m o r p h i c e i t h e r t o t h e f i e l d of

r e a l numbers o r t h e f i e l d of

complex numbers. Examples

of

product

numbers w i t h t h e s e t of absolute

value,

and

formula

the

together

f i n i t e dimensional

t h e f i e l d of r a t i o n a l

of r a t i o n a l f u n t i o n s

field

f i n i t e f i e l d with t h e p ( t ) - a d i c polynomial,

are

fields

a l l p-adic valuations t o g e t h e r with

with

valuations,

the

extensions

p(t)

an

these

fields

are

over a

irreducible

l/t.

v a l u a t i o n g i v e n by of

ordinary

k ( t )

Separable

also

product

formula f i e l d s . Dedekind d o m a i n s .

In

developed

the

using

i) ii)

Each v a l u a t i o n i n For each

iii)

If

a

q

and

q'

S

theory

of

theoretic

)a

-

A

i s a Dedekind

p

in

nonempty

set

if:

in

llq

<

k

with

E

and

v a l u a t i o n s are i n e q u i v a l e n t ) .

T

S

so

of

S\T.

a r e d i s t i n c t v a l u a t i o n s of a

was

domains

is d i s c r e t e . '

f o r each

then there e x i s t s and s u c h t h a t

k

Dedekind approach.

t h e r e is a f i n i t e subset

K

in

\alp s 1

that

the

valuation

of v a l u a t i o n s on a f i e l d

S

discrete s e t

[MR31

S

lalp c 1

<

E

and for all

E

>

0,

p E S

(so d i s t i n c t

355

Algebraic number theory

Let

(k,M)

be a p r o d u c t f o r m u l a f i e l d

S

valuations

of

Dedekind domain i s a s u b r i n g set S

R

of

a field

:

la1

a

subset

of

discrete

S :M .

valuations if having

k

a

A

Dedekind

of v a l u a t i o n s s u c h t h a t R = (a E k

The

then

i s a Dedekind s e t of

M

field

5

P

1 for a l l p E S ) .

i s t h e q u o t i e n t f i e l d of

k

R.

t h e Dedekind domain

are

r i n g s o f a l g e b r a i c i n t e g e r s i n a l g e b r a i c number f i e l d s

The

Dedekind

domains.

R

Let

be a Dedekind domain w i t h q u o t i e n t

Dedekind

set

of

S.

valuations

f i n i t e l y g e n e r a t e d R-submodule

Then

field

a

having

k

the

fractional ideal is

I t i s shown i n IMR31 t h a t i f

of k .

a A

is a f r a c t i o n a l i d e a l then

I A I p = max(lal

P

: p E S)

i s d e f i n e d and t h a t

A = (a E k : la1 Using

this

that if k

so

description

A

P

s lAlp

f o r a l l p E S).

t h e f r a c t i o n a l i d e a l s i t i s easy t o show

of

A

that

the

is

T h i s d e s c r i p t i o n also shows t h a t t h e s e t

(a E k

C

P

of

principal

la1

:

< 1)

P

The

set

I/P

i s c a l l e d t h e i d e a l c l a s s qroup.

Theorem 4.1.

Let

and

f r a c t i o n a l i d e a l g e n e r a t e d by

I

of

form a f r e e a b e l i a n g r o u p w i t h b a s i s t h e s e t of :

a

fractional

in b.

ideals

p E S). The f a c t o r g r o u p

be a p r o d u c t f o r m u l a f i e l d and l e t

(k,Ml

b and

fractional ideals

i d e a l s form a s u b g r o u p .

be a Dedekind s e t of v a l u a t i o n s .

Then t h e c l a s s

I/P

group

S C M

is

a

be

a

0

f i n i t e group.

Corollary 4 . 2 . fractional

a

is a f r a c t i o n a l ideal then there e x i s t s

Let

k

ideal.

and

S

A

Then

be a s i n Theorem 4.1.

Let

A

is

or

A

either

principal

is n o t

principal. Proof.

Check

to

c l a s s group

I/P.

The u n i t -

-

theorem.

be a s u b s e t of

M.

see

if

A

is t h e i d e n t i t y element i n t h e f i n i t e

0 Let

ik,M) be ~

Then t h e s e t of

u(S1 = (a E k

: la1

P

a product formula f i e l d . S-units

= 1

is t h e set

for all

p E S).

Let

S

R. MINES

356 The group

u(M)

Theorem 4.3.

is the group of absolute units of k.

The

group of absolute units is a finite cyclic group,

and consists of the roots of unity in Proof.

It

k.

is first shown that there is an integer

D > 0

which is

a bound for the number of absolute units and hence for the orders the

units.

As

k

xD1 - 1

polynomial

of

is a product formula field it is factorial.

The

can be factored

and

into

irreducible

factors

the set of linear factors gives the set of absolute units.

[7

The unit theorem can now be stated. Theorem 4.4.

Let

(k,M)

be a product formula field and let

a set of discrete valuations contained in is

finite

with

s L 1

elements.

is a direct product of the abelian group of rank Proof. is

-

group

Then the group of of

argument

independent

in

the

absolute

units

elements

in

M\S

S-units

u(S)

with

free

a

1.

basis

[Artin

be

S

such that the set

The main problem with the constructive proof of this

constructing

the

s

M

for the free factor.

and

Whaples,

u(S1.

set t o a basis for the free factor.

19451

to

theorem

We roughly follow obtain

s

-

The problem is then t o change this 0

REFERENCES Artin, E. 1967

Alqebraic numbers and algebraic functions, Gordon and Breach, New York.

Artin, E. and G . Whaples 1945

=.

Axiomatic characterization of fields by the product formula for valuations, Bulletin &. Math. 51. 469-492.

Borevich, 2 . I., and I. R. Schaferevich 1966

Number Theory, Academic Press, New York.

Bishop, E. 1967

Foundations York.

of

constructive analysis, McGraw-Hill, New

Crossley J. N. and S. Miranda 1981

1

A bibliography of effective algebra, Aspects of Effective Algebra (Proceedings of a conference at Mona.sh University Australia 1 - 4 August, 1979). Upside Down A Book Company, Yarra Glen, 251-290.

357

Algebraic number theory

Frohlich, A . and J . C. Shepherdson 1955

Effective procedures in field theory, Philos. Trans. Roy. Ser. A , 248. 407-432.

SOC. London, Hermann, G . 1926

Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 9 5 , 736-788.

Heyting, A . 1971

Intuitionism, & Introduction, 3rd revised edition, North Holland, Amsterdam.

Julian, W . , R . Mines and F . Richman 1978 [JMRI Algebraic numbers, a constructive development, Math., 7 4 , 91-102.

m. 2.

Kronecker, A. 1882

Grundzuge einer arithmerischen Theorie der algebraischen Grossen, Journal fur die reine und anqewandte Mathematik 9 2 , 1-122.

Metakides. G. 1982

Proceeding of t h e Brouwer Centenary.

Mines, R., and F. Richman 1981 [MR31 Dedekind domains, Constructive Mathematics, (Proceedings, New Mexico, 1 9 8 0 ) , Springer Lecture Notes 873, Springer-Verlag, Berlin, 16-30. 1982 [MRlI Separability and factoring polynomials, Rocky Mtn. ( t o appear).

2.

m.

1982 rMR21 Valuation theory, a constructive view, ( t o appear). 1982 [MR41 Archimedean valuations, ( t o appear). O'Meara, 0. T. 1963

Introduction

quadratic forms, Springer-Verlag. Berlin.

Nerode, A , and G. Metakides 1975

Recursion theory and algebra, Algebra and loqic (Fourteenth Summer Res. Inst. Austral. Math. S O C . , Monash Univ.,

R. MINES

358

Clayton 1974), Springer Lecture Notes 450, Springer-Verlag, Berlin, 209-219. Rabin, M. 0 . 1960

=.

Computable algebra, general theory and theory of computable 95, 341-360. fields, Trans. Am. Math.

Richman, F. 1981

Seidenberg's condition P , Constructive mathematics, (Proceedings, New Mexico, 1980). Springer Lecture Notes 873, Springer-Verlag, Berlin, 1-11.

Seidenberg, A. 1970

Construction o f the integral closure of a finite integral domain, Rend. Mat. g . Milano 40, 100-120.

1978

Constructions in a polynomial ring over the ring of 2. Math. 100, 685-703. integers, +.

e.

Smith, R. L. 1981

Effective valuation theory, Aspects of Effective Algebra (Proceedings of a Conference at Monash University Australia 1 - 4 August, 1979). Upside Down A Book Company, Yarra Glen, 232-245.

Staples, J. 1971

On constructive fields, Proc. London Math. 753-768.

E. 13) 23,

Waerden. 8. L. van der 1930

Eine Bemerkung uber die unzerlegbarkeit von Polynomen, 102, 738-739.

Math.Ann. 1953

Modern Algebra, Ungar, New York.

Weiss., E. 1963 Alqebraic number theory, McGraw-Hill, New York.

THE L.E.J. BROUWER CENTENARY SYMPOSIUM A S . Troelstra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

359

GLUEING TOPOI AND HIGHER ORDER DISJUNCTION AND EXISTENCE Ieke Moerdijk University of Amscerdam The Netherlands

During the past few years, the technique of glueing topoi been applied

KO

(IGl,[Wl)

has

prove several kinds of derived rules for intuitionistic higher or-

der logic (the logic of topoi, HAH), such as the continuity rule, the bar induction rule (see IFJI),

and Church's rule (Hyland).

We will be concerned here with

two rules of a slightly different nature, viz. the disjunction rule ( o r disjunction property, DP)

DP:

i f

I-wv$J, then e i t h e r

bc? or

I-$

and the numerical existence rule (existence property, EP)

if

EP: -

The CLS21)

DP

then f o r some nwneraZ

I-3x : NQ(X),

and

EP

2, I-VJ(?).

for HAH have been established by Lambek and Scott

by using a Kleene-Friedman type of realizability argument.

that the DP

and the EP

Freyd noted

could be proved in a much simpler way by glueing topoi

along the global-sections functor to S e t s

(CLSII, 16).

Freyd's method can be con-

sidered as a generalization of Smorynski's proof which uses Kripke-models and shares many of its properties

DP -T' EP -T'

. .

if T

([SI),

([MI).

If, however, one is interested in proving the in the language of

([LSII,

DP

and

EP

for a theory T

HAH,

I-

cpv$

then e i t h e r

if T I- 3x : NQ(x)

T I-Q

or

T I- $

then f o r some numeral

2, T I-

cp(n)

Freyd-glueing is insufficient for kinds of theories that tend to introduce a higher degree of decidability in the logic, such as theories which include versions of Kripke's Axiom (the higher order analogue of Kripke's schema) and choice-principles.

360

I. MOERDIJK

In a nutshell, the reason for this is that in the glued topos (the so-called Freyd-cover of the original topos), the subobject classifier does not have any

non-trivial complemented elements (elements which have a complement and not only a pseudo-complement). The aim of this paper is to present an alternative to the Freyd-cover that meets this shortcoming.

In the first section, we will describe the new cover. ED], where an ana-

The idea behind its definition occurred to us while reading

logue of Smorynski's construction for Kripke-models is presented for Beth-models.

In the second section, we will give a rather detailed description of the toposstructure of the cover as a category of fractions. This will be useful in the

DP and EP

final section, where we use the cover to prove the

for several intu-

itionistic higher order theories. I would like to thank Josje Lodder for stimulating discussions; she also Further, I am

proved the preservation of Bar Induction (proposition 3 . 4 . below).

grateful to Peter Freyd for pointing out a mistake in an earlier version of this paper.

1.

Construction of the cover. One respect in which our cover differs from Freyd's is that we do not cover a

single topos, but a sequence of topoi:

let ( E / n e w ) be a sequence of elementary

topoi, and let SetsW be the category of diagrams on the natural numbers with their usual order.

(Objects of

L:

!En

-

+

a topos. object of

(k)

L((An),(k'),

the comma-category

+

X m

'Xn,,

of

W e then define a functor

for m t n . )

SetsW

by setting L((An)n) L((An)n)(k)

SetsW will be written as sequences

.Im: Xn

sets, with restrictions

=

nn>k

€,(I

,An),

for ksk',

nn E n @ L SetsW

and letting the restriction maps be the projections.

((Setsw

r)

a sequence of objects An

is left-exact, so

in the notation of

Its objects will be written as triples SetsW, A

L

(X,A,(p), of

with

En, and

cp

CJ1)

is again

X = (Xn)n

an

a sequence of

36 1

Glueing topoi

functions (pn:Xn and

A

En(l,An).

+

the boundary,

X

For such an object, we will call

the core.

Recall

([Wl)

that we have an open inclusion

nn En

-f

(nnEn eL Setsw)

with

as direct image part the cofree coalgebra functor, and as inverse image part the forgetful functor U,

(nnEn eL Setsw) classifier in

Q*

=

which is logical. The subobject classifier of

is the object Q

En,

and

I

(Q*,(Qn)n,K),

ikE { O , l } ,

V k ' 2k(ik, = I

and

pkc Ek(l,Qk),

Im = (ik,pk)ka,

is defined by letting En : Q:

+

jn((ik,Pk)k2n)

ii

=

on Q

-%

with

jl,

if

10,

otherwise.

3&

2

is the subobject

ik

I,

=

then

L((Q~)~)

(nn E n 63L

Setsw): j

(jn)n: Q*

is the pair

defined by

Q*

where and V&

k(iL=l)

II: a*

be the projection.

in and

(ii,Pk)km,

=

and if

for m z n .

kgnEk(l,Qk)

We now define a topology j id),

Qn

pk,= tme)},

with restrictions (ik,~k)k2n

((jn)n,

where

is the diagram given by

Q*

i(i k,pk)k2n

=

2

k(p&=true)

j is indeed a topology. We are now ready for the defini-

It is easy to see that tion of the cover. 1.1.

DEFINITION. L e t

(En)n be a sequence of t o p o i .

d e f i n e d to be the topos Note that

j

Shj (nn En eL S e t s w ) , where

Then the cover XnEn

j and

does not "affect the core components".

ful functor XnEn +

nnEn

L

is

are as above.

Therefore, the forget-

will be logical. This is, in fact, also a by-product

of the explicit description of XEn

to be given below.

The subobject classifier of XnEn is the object

a. = J

((fij*n)n,

II),

where Q

*

jn

=

{(ik,pk)k>n

E

Cl:

I

(3k'tk-ik, = 1

&

Vk'2k.pk,= true)

We conclude this section with the following lemma. ward, and omitted.

*

ik = 1 )

Its proof is straight for-

362

I. MOERDIJK

Let

1 . 2 . LEMMA.

(X,A,w)

(nnEnQL S e t s w ) ,

be a moYphism i n

(Y,B,$)

(T’ci)>

a T

:X

L+Y,

(r,a)

(i) lim +n

-f

Bn,

is one-to-one.

--tn lirn Y

is aZmost e p i c iff each

(T,CY)

: limnxn

2. ZE -n

An

is almost manic iff each an is a monomorphism, and

T~ : %nXn

(ii)

with

(a ) n n

ci =

+

lim Y +n n

is an epimorphism, and

an

is onto.

as a category of fractions.

I n this section we will take a closer look at the cover defined above.

This

will be of great use i n section 3 , when we turn to applications to higher-order logic. Recall that X E n = S h . ( n E Q S e t s w ) can be described as a category of fracJ n n L tions

([Jl, 50.1,

(nn En QL

5 3 . 4 , CGZ])

having the same objects as the comma-category

S e t s w ) , but a more general notion of arrow.

In the particular case we

are dealing with, these ‘generalized’ arrows l o o k very nice: 2.1.

(i) A = (A,),

tions

XEn

LEMMA.

i s ( e q u i v a l e n t t o ) t h e category which has

as i t s o b j e c t s , t h e t r i p Z e s a sequence of o b j e c t s

(pn:Xn

+

+

and

X

an o b j e c t o f

(0 = ((pn)n

Setsw,

a sequence o f func-

€n(l,An),

(ii) a s i t s morphisms f : lim X h n n

En,

of

An

with

(X,A,(p)

lim Yn’

+n

and

(X,A,(p)

a

=

+

(an)n

the pairs

(f,a),

where

is a sequence o f morphisms

*,:An

(Y,B,$)

+

Bn,

such t h a t t h e following diagram commutes.

lim

+n (*)

xn

If

lim Y +n n

(where (p x

limnnk>_n~k(~,%)

-

.I

-L%nnk2n

= ---*1 lim ((cp ) k r n ) ,

E,( I , B ~ )

( (pikzn

( ‘ ~ ~ ( x l k ) ) ~ ~similarZy ~; for

defined in t h e obvious way.

: xn

;)I

+

,Sn and

Ek(l

,%)

being t h e f u n c t i o n

a = lh&,nk2,,(txko-).)

Composition is

363

Glueing topoi

Convention:

in the sequel, we will refer

discussion as " t h e

PROOF.

to

any diagram of the form

(*) under

(*)-diagram".

The statement about the objects is trivial, of course. As for (ii), for

an arrow

(X,A,(p)

(Y,B,JI) in the category of fractions,that is, an equivalence

+

class represented by a diagram (Y,B,JI)

(Z,[,;:,y

(X,A,Q) with

bidense (cf. lemma l . Z ) ,

(T,Y)

we can define an arrow

(X,A,(p)

(Y,B,JI)

-1

of the form as described in the lemma by letting anoyn , and f

=

(lim u ) o (&F~T~)-'. Then -n

(u,B)

and

=

{(x,y)

with restrictions 2

E

XnxYn +

(Y,B,JI) as described in the lemma, let

(x,A,(~) JQJ-+

be defined as follows:

Zn

does not depend on the particular pairs

that we chose to represent the equivalence-class.

(~,y)

Conversely, given (Z,C,p)

(f,a)

I

Cn

An,

=

and

f(Cxl) = Cyl, and

Zn+l inherited from X

Vk>n-$k(ylk) and

Y.

=

a,o(p,(xlk)}

The transitionmap

is

p

defined by

zn

P,:

+

En(I.Cn),

We then define maps letting

and

T

o

(T,Y)

P,(X,Y)

= Qn(X).

: (Z,C,p)

+

(X,A,(p) and

be the projections, y

(o,B) : (Z,C,p)

+

(Y,B,$)

by

@ = a . Note that

the identity, and

is bidense.

(T,Y)

It is straight forward to check that this connection establishes an isomorphism between the arrows in the category of fractions and the arrows as described

0

in the lemma.

From now on, I € will denote the category described in the preceding lemma. Let u s have a look at its topos-structure. We start with finite limits. The t e r -

minu2 is

(I,(

where

the terminal in where

XxY

I

is the constant diagram n

En. Products are given by

is the product of

X

and Y

(X,A,(p)

x

in S e t s w ,

+-f

{ * I , and In

is

(Y,B,JI) = (XxY, A x B , q x $ ) ( A x B ) ~= An" Bn,

and

I. MOERDIJK

364

((Pi),: X n x Y n A s for

+

is defined by

En(l,AnxBn)

puZZbacks, take

(P, AxCB,

and

)

(X,A,(p) u ( Z , C ,

Their pullback is given by

(x,y)

(cpx$),

where

(pxp$),

=

(cpn(x),

(Y,B,$)

qn(y))

.

U ( Z , C , p ) .

( A x ~ B )=~ AnxC

is the

Bn

n

En,

pullback in

and

P

is defined by

I

Pn = {(x,y) € X n x Y n

f([xl)

and V k 2 n * a k ~ ~ kIk) ( ~ = Bkoqk(y Ik)}

g(Cyl),

=

with restrictions inherited from X

Y.

and

( Q x ~ $ ) ~ :+ P ~E n ( I , A x

asso-

Bn)

'n

ciates with

the arrow

( x , y ) ~Pn

cpn(x) : 1 + A n

Jln(y) : 1

and

+

1

uniquely determined by

A n x CnBn

+

Bn.

Monomorphisms, epimorphisms, and images are easily described: 2.2.

An arrow

LEMMA.

En,

is mono (epi) in

(X,A,Q)

(Y,B,$)

and

+

f :%Xn

PROOF. We prove the mono-case. For "only if", assume

(f,a)

Then, first of all, each n2

= AnxBnAn

_+

(n2)n f

Also,

f(Cx1)

=

is

fCyl),

k

=@

+

and

Cyl.

Uk={*)

V = I (the terminal of k k

arrows (U,V,id)

([XI, +

(cpk(xlk))k)

(X,A,(p)

Using Z.Z.,

(id,

(f,a).

nz)

from this object to

Hence, for all

Choose n

C >'".

Ek) and

show that

if

and

which

kzn,

Then let

l a x n with

E

and

(U,V,id)

by setting

V = O (the initial in Ek) k

and

k

id: *

(Cyl, ((Pk(ylk))k)

H

(Ik+Ik).

if

k
Then the two

defined in the obvious way

(f,a) cannot be mono, contradiction.

images are constructed as follows:

Bn.

(X,A,(p),

rxl, Cyl

so large that x,ycXn,

Then define an object

if k > n ,

B

Y An

We

(cP,cP)).

An.

and first take the epi-mono-factorization of a =

=

n,

for if not, then there would be

1-1,

[XI #

k
if

nl

must be mono, since

a

Vk2n*ako(pk(xlk) = ako(pk(ylk). U

(onto).

1-1

is mono.

(id,IIl) and

: A n x BnAn

an

For epis, the argument is similar. "If" is clear.

have the same composition with =

is

limy

-+n

A n XB n A n An To see this, make a new object ( X , (AnxB , ) A n

An.

then have two arrows

(nl)n

t En is mono ( e p i ) iff each

in

let

in En,

(X,A,(p)

(f,a),( Y , B , J l ) ,

a say A -% Bn =

0

365

Glueing topoi

= I Y E Y.

Z

n

and define pn: Zn nition of

3x~X~(f[xl=ryl, and En(l,Cn)

-+

by

Vk>n-ako@k(xIk)

pn(y)=yno(pn(x),

(note that the choice of

Zn

x

where

(Z,C,p) : (& 6 ) (Y,B,Jl),

(X,A,(p) (h )

=

x

with

Jlk(ylk))l

is as in the defi-

is irrelevant), Zn

and we have a factorization (X,A,(p) -@&

tions from Yn, =

I

inherits restric-

(Y,B,$) =

h(Cx1) =f([xl),

and

g

the inclu-

sion-functor.

*

as described in ) , (Qn),,II) n n be a mono. We may assume that each

((aj

The subobject classifier is the object section 1 . Yn

c_

Xn,

Let

(Y,B,Jl) >-

and that for

YE

(X,A,q)

Yn, cpn(y) =anoJln(y).

-

xn : An

struct the classifying morphism, let

-n

=

Sl

n

+

true)krk 1.

= [(I,

wards, then h([x])

(h,y)

an

is mono. Bn

be the classifier of

as follows: if “EX, lim il? Jn [(O,~~o(p~(xIk))~~~l, and if X E X and

and define h: lim X then h([xl)

+

A l s o , each

and

To cona

5 An’

Vkrn.xlkdYk,

xlkeYk

from ko on-

is a well-defined morphism,

(i.e.

0

the relevant

(*)-diagram commutes) and its pullback along true is isomorphic to

(Y,B,$). Later on, we will also need an explicit description of exponents objects

(Y,B,+)

and

(X,A,q?)

(Y,B,$)(x’A’(p) where

A An (B )n = En

=

.

For two

the exponent is given by

(U, BA, IIz),

is the exponent in

I

Um= {(f,(~~)~~,)

f : %Xn

+

En,

%Yn,

and

U

is defined by

u k c E,(l,Bkqi),

and below commutes)

(*)

lim -n

x

lf

(*)

lim IIk h E k ( I , \ )

----tn

-

nkZn (,E

lim Y -n where

u

is the

----tn lim IIk>n(-)

assign to a section 1 -&

4,

I ,B~)

of the functions Ek(l,A,) of

4,

the composition

1

+

Ek(l,Bk),

gxuk

%XBkAk

kzn, that B k’

(evl ,evz) The evaluation-morphism is (Y ,B,$) x (X,A,q) ’ (Y,B,$), A where (evz)n: Bn n x A + Bn is the evaluation in En, and

I. MOERDIJK

366

Note that this is well-defined: the relevant (*)-diagram comutes. The transposed of a morphism (Z,C,p)

x

of Bn

in En,

(X,A,Q)

E(L-21,

(-

a,

+ n

=

( g , l z l , g,Czl) (Z,C,p) A

1 . .

(h,y) by taking -+

a

E(Cz1)

U =

: CnxAn

-+

is the transposed

Bn

is given by

~glczl~~xl,

[XI) =

And conversely, given

6 : &Zn

(?),=gn

where

(Y,B,$),

and g:limZ X G ~ X + limyn

where we write g(Cz1)

transposed

(p,B). (Y,B,$I)(~'~'~) is

(Z,C,p)

(Y,B,$),

(X,A,Q)

x

.

"(f,(uk)k2n)".

=

we construct its

.

is the transposed in En, and

(y),=y,

is given by

n

(~lCzl, h,[zI).

with

with n large enough for pCz]

GI

and

c,

defined by

to be defined.

Finally, let us look at the naturaZ number object. If in En, then the n.n.0. numeral embedding k

+-P

in EEn

(k: I

-+

is

N,).

(IN,( Nn)n,

limn: N

Nn -+

is the n.n.0.

En(i, Nn)

is the

The successor- and zero-arrows are the ob-

vious ones.

3.

Applications to higher order 1oe;ic. Now that we have developed some feel for the categorical structure of E E n ,

we turn to logic. We assume that the reader is familiar with the standard interpretation of HAH

in topoi which possess a natural number object, and with the

completeness theorem for this interpretation (see e.g. [ B J ] ) . First of all, we need the following lemma, which says that, when dealing with topoi as models of HAH, we may restrict ourselves to topoi in which the terminal object is projective. It is an analogue of the technique used in classical model-

367

Glueing topoi

theory of adding constants which witness (true) existential formulas, and will be

X (propositions 3.3-3.5).

very useful when we turn to preservation properties of

E

3 ^E

such t h a t

language o f jective i n

and

E,

T I$ cp,

E

[=

IF],

+ 1

Lya : Ey =

+

-f

q~ be a sentence, i n the

E

such t h a t

is pro-

I

The proof of (i) is almost identical to the proof

E

inhabited if it has global support (i.e., the

is epic).

and

is the canonical embedding, if

Ea/Lon(Aa),

= (-)

Ea

E =%B
XLoa(Aa) oLya*

say as

and logical functors and for

y <

a,

is a limit; as for successors, Let

E c1 .

E'=lim -
Now repeat the construction w-many times, i.e. let structed above, and let

E,

Well-order the inhabited objects of

E g ( a < B ) as follows: E = E ;

Ea

i s logical and f a i t h f u l .

.

(Aala<~), and construct asequence of topoi LaB : Ea

F

and a functor

but for completeness sake we will sketch the argument. of

Call an object A unique arrow A

and

there e x i s t s a topos

T, E If cp

(ii) follows from (i).

of theorem 3.21 in

^E

T be a s e t o f sentences, and

HAH. Then i f

5

there e x i s t s a topos

i s projective i n

I

(ii) Let

PROOF.

E

(i) For. every topos

3.1. LEMMA.

!I

=E,

En+1 = E l :

as con-

.

E=lim E --tn

0

n'

Let us now explain how the cover is used for proving the DP higher order theories. Recall that the forgetful functor l E n

+

so all computations proceed on the cores as they do in the topoi

and

TIEn

En

.

EP

for

i s logical,

Therefore,

we may concentrate on the boundaries. Consider the subobject classifier 2 = ((Rj*n)n, (Rn),,n) of l E n as dej has as its boundary-component scribed in section 1 . Disjunction R. x n . 2 R J J j the of the maps lim (R.* xR * ) -t %Rj*n --tn J I I j n

%

jnov n where

(n E~ Q~

jn

: aj *n X

R *

j n

+Q.*

JII

comes from the topology

j, and

V

is like the disjunction in

S e t s w ); more precisely, v ~ ( ( ~ ~ ~ (ii.ei)krn) P ~ ) ~ ~ ~= ,(ik v ii, pk v

I. MOERDIJK

368 with

(Pk,Pi) and pkVpk = - 1 Rk x Rk 2 nk' = max(ik,i,'), k A s for the numerical existential quantifier gN : QN + a , first recall from i vi;

section 2 that

N

=((Vn)n,

(Rn

Vn = I(f,(ak)k2n)I

Nn

)n,Il),

where a k e Ek(l,n k Nk),

f: N +soj*,,

and the (*)-diagram commutes) The boundary-component

-t

c (f,(a,)ktn)l (3N)k:Rk

where

lim V --mn

Nk

+

Rk

gN

is given by

[(~,true~)~,~l, if for some n c m , fn=C(l,truek)ktnl C(0,(3N)ko ak)ktnl,

otherwise,

is the existential quantifier in is preserved by

E

is projective in

En

We say that a theory T

En such that

of topoi

of

+

1

if vhenever we have a sequence

En C

and

T, also t E n C- T.

Using the descriptions of disjunction and existence just given, we can easily prove the following

t. Then PROOF.

Let

E

T be a theory in the language of HAH, which is preserved by

Let

3.2. THEOREM.

T has the disjunction property and the numerical existence property. be such a theory, and suppose T k

Let

T

and

E' be models of T such that E H

assume that in

E

and in

E2n = E , E2n+l=El. [I$] : 1

Write [ c p ] ,

*

and

*

I-+

C

tEn I= cpvJ1, it follows from the description of disjunction

cp for all

etk,

either or

ik = 1

El I= J1

or

i'= k 1.

Hence for some k ,

for all e t k ,

functor is logical), contradicting the choice of

E and E'.

The case of numerical existence is similar:

suppose T

for all m e N ,

T I$ cp(g)

of topoi, each with

where

3

given above that for some k ,

Ee I--

by 3.1. (ii), we may

R. with boundary components given by, say,

c-+ [ (ik,pk)k2nl

respectively. Since

either

E If J1;

preserves T, EEn I= QV$.

Since E +

cp,

T I$ J1.

is projective. Now construct EEn,

I

E',

T I$ cp,

cpVJ1,

1

(m

the numeral of m).

projective, such that

E

(since the forgetful

k

3x:Ncp(x),

Now choose a sequence

I= T

while 'En'n

for each n, while for

369

Glueing topoi

each m

En I# cp(m)

we have

I € I= 3x : Ncp(x),

for infinitely many

Since X E

n.

we find, (using the description of

so

fact that the forgetful functor is logical) m,kcm

gN

I= T,

given above and the

such that

€L

I=

for all

cp(m)

L > k, again a contradiction.

0

One can now start playing the seemingly endless game of proving preservationproperties of X, section 2.

using the detailed description of I:€

as presented in

We will give three examples, viz. Kripke's Axiom, Bar Induction, and

Choice Principles. Some more preservation-properties will be worked o u t elsewhere. 3 . 3 . PROPOSITION.

x

preserves Kripke's axiom

Vp : n 3a : NN(p

KA:

++

3m:Nam = O ) .

First note that whenever an a

PROOF.

it is a decreasing 0-I-sequence, i.e. Nn that NN = (U,(Nn )n, If)

where

KA

In general, validity of ++

3mam =

is epic, where up NN Ix(3mum=O~ >-

NN x

on

++ x

R,

*

Vn,m(nSm

=

:IN +IN,

Un=I(f,(5k)k2n)If

up

U

satisfying KA

(Un)n

ukttk(l,NkNk)

in a topos E

Recall from section 2

for k h ,

means that

>-

3mam = 01

is the pullback of

where NN -Q 3mam=o

XEn,

flx Ci

along

1

A N

along NN

[am=O] >+ x

N Z

N

.

As a general national convention,

for a formula cp with free variables x l ,..., x of

n

classifies the image of

We will now compute these things in IEn.

...,A

amSan
with

N L N N , [am=Oj being the pullback of

the objects A ] ,

occurs, we may assume that

whose s o r t s are interpreted by

we will write the subobject [cpl >+

IIA. a s a

triple ((ucpnn)n,

(ucpniEn)n,

...) .

(Since the forgetful functor is logical, U c p l

indeed is the interpretation of

En

cp in €n.)

I. MOERDIJK

3 70 For the image of [ a =On >(([3niarn=Oln)n, (@mam=OB

t(f,(uk)k,n) Let

En

un I

E

NNx N L N N in ZEn,

)n, I$),

where [3mam=O]I n E U n is

%eN(fm=O

&

W > n evo(m,uk) = I - = . ~ N ~ ) } .

be the classifier of this subobject (see § 2 ) ,

(h,X)

being the classifier of [3m a m = 01 En >+ ipt+ 3 m a m = 0 ]

(n

Nn Nn

so

that x

=

(xnIn,

n'

in En.

is then given by

>-fix"

(E, p t+

we get

3mam =

01 En)n, II) , where E~ = t((ik,pklkzn,

(f,o)) =

Assuming that each E

I= KA

E

oj*,xun

h(C(f,o)l),

I

[(ik,Pk)k>n~

=

and Vklk>n-pk = x k o a k } .

and using lemma 2 . 2 . it is now sufficient to

show that the projection

n:*En is

a

(f,o)

*aj*,

-f

surjection; that is, for each

E

there is an n S n and an

i2j*n

such that

U

E

(ik,pk)kzn

"0

(i) [(ik,pk)k>kol

= h(Cf,ul),

for some

KO

(ii) Vk>n0 Pk = Xkouk'

-

So

I

choose (ik,pk)k>n K

Since 1

E

in Ek

N P

is projectiv'e i n

like to take

T~

:=

Since Ek I= KA,

Slj*,.

such that pk o != x k o r

T~'S

Case. ik '0, value 1. have

r o s as

T ~ .

for each k.

E

r is decreasing).

+

Since pk = Xk 0

Tk,

(ii) would then be satis-

IN such that (i) is satisfied, we have to

a bit, however. Consider the following two cases: Now take for f : N

If we would take u = k

(f,(uk)k>n)

(, and

Ek, K has a global section I 5 K. We would now

fied. In order to find an f : N change these

we easily find a diagram

T ~ ,

+

IN

the function with constant

then (i) would be satisfied, but we don't

Un, i.e. the following does not hold:

To get (iii), we take for uk

the sequence

with a sequence of k

1's

put

371

Glueing topoi in front:

ak

= (

I,. ..,I

)

*

1 + N Nk

Tk:

k

denotes concatenation).

(*

that for

= I for some ko. Then C(ik,pk)kI = c(1, We may assume k0 'k0 k 2 k o , T~ is the constant function with value 0, since =true

for k ? k

.

Case 2 .

i

k'

f(n)

f:W

Now let

t

=

TN

-f

1

if

n < k

0

if

n 2 k

be defined by

We now turn to bar induction. Before we can formulate the axiom, we have to describe the object of finite sequences and introduce two operations on sequences We will code a finite sequence ( n l , ...,\ (n + I I

,..,nk+1,0,0..).

Vk> n-a(k)

=

O)]

as the infinite sequence

)

Thus, N 0 &

of

NN.

The boundary W = (Wn)n

as i n the proof

is given by: (Un

of 3 . 3 . ) Wn = {(f,(ak)krn)

1

Un

E

3n(Vm/mm-f(m)

0

=

factors through [Vm
T(.)

a "cut off" map N
-+

N
: NN

(a,x)

+

a*

N

-f

N
(x)

tt


B(p)

N for V a : N 3 x : N p(ax)

M(p)

for Va : N
-f

a(n),

Suppose that each

I[B(p) AM(p) hI(p)

+

p(())l

&

each a k

11.

it is straightforward to define and a concatenation map

,

-f

p(a))

and write "p

is a bar"

"p

is monotone"

"p is inductive".

of monotone bar induction is

3 . 4 . PROPOSITION. t preserves

PROOF.

N,

p(adx)))

~ ( p ) for Va : N < (~ ~ :x N p(a*(x)) The axiom BIM

# 0

.

be a variable of sort CIN

Now let p

V m < n.f(m)

Vm
&

Using the decidability of the ordering on

&

BIM.

En I= BIM.
We will show that in EEn,

I. MOERDIJK

312 Write

QN
=

(an

((Vn)n,

Nn
so it suffices to know that if ‘k

‘k\ fl,Qp
(g,a) but

E

(g,o) d Up(0)Bn.

E

sequence

+

#

UVyp((y))B,(g,u)

E

I

Vn

through UVyp(0)

#

p(0)

(g,o) E

p(O)lEkI.

extends

1+ 1

Now write transpose of an(k) : 1

+

si

N

such that

#

g(s,)

+

N,

61

*

)

or

=

Wn

Since

g(sm) = I ,

(g,o)

and E

we then find a

such that

for all

nk,

factors

Repeating this argument, we

1.

si has length i,

i.

( T ~ ) ~ ~ ~and ) , define

f(k)

Tk

that

1

i

where

such that

#

and

holds in En,

ElM

g(si) # I ,

(*)

NN = T ~ ,for each k c n . mEN

p(0)

( s ~ )of ~ elements of

k

there must be an

truek}

=

fP(())Bn(h,(Tk)km,

5

it follows from

((no,..,ni-j),

(f,(an)n):

I(p)],,

j n this latter possibility is ex-

p(())jn

-f

Using once more that

si, and =

A

1.

s , < W n of length I ,

s.

=

(the top element of

k I= BIN,

nVyp((y))nn(h(Tk)ktn)

1.

[B(p) AM(p)

and Vkzn*rk(ok)

g(0) # I

Since E

E

then

is given by

= I,

UI(p)Iln G [VYP((Y))

find an infinite sequence and

h(0)

means that either

cluded, so we find

{(h,(Tk)k,n)

1

Vn

(Ip(0)J

(g,o)

with

k k>n

c_ Vn,

(B(p) AM(p) AI(p)],

E

a = (a )

as above), and

Qj*n(Wn

-t

(g,o)

~ ~ ( # truek 0 ~ for ) some k 2 n .

=

n

The boundary of

(g,a) 4 Up(O)l/,

Now since

We may restrict ourselves to the boundaries,

lie argue by contradiction: suppose

up(())nn=I(h,(Tk)ktn) so

g:W

are such that

.

[p(O)Bn

n).

Q

:I

an : Nn

[B(p)nn

contradicting

+

-f

c_

NN

Nn

to be the is such that

13xp(F(x))Bn,

(*),

0

Finally, let us consider some choice principles. (In the following proposition, AC

stands for ‘Axiom of Choice’, RDC

‘from numbers to numbers’, NF

for

for ‘from numbers to (numerical) functions’.

3.5. PROPOSITION. 1 preserves PROOF.

for ‘relative dependent choice’, NN

AC-NN, AC-NF, RDC-NN, and

RDC-NF.

The proofs follows essentially the same pattern in all four cases, except

that notational difficulties increase exponentially. Therefore, we will only describe the case of vp :

x

N

AC-NN:

(Vx:N3y:Np(x,y)

-f

N 3a:N Vx:Np(x,a(x))).

Glueing topoi

En in which

Take a sequence of topoi

E I= AC-NN.

is projective, and suppose that each

1

We have to show that, in IEn,

is contained in the subobject [3aVxp(x,a(x)ll RNXN

=

aries.

Nn Nn '

(an

((PJn'

The boundary of

Q,

3 73

= {

[Vx3yp(x,y)l,

u

6

k

I

Pn

Q N x N . Write

>+

Again, we may restrict ourselves to the bound-

)n, E ) .

(f,(uk)k2n)

say Q = (Qn)n,

is given by

Vm 3m' (f (m,m' ) = !( I , t r : i e k ) k 2 n l ,

o (m,m') = truek

large enough), and Vkflcrn,u

for k

--

factors through IVx3yp(x,y)] while the boundary of Rn

=

[3c(Vxp(x,cr(x))]l,

{(f,(~~)~>,) EP, each m,

I

say

R = (Rn)n,

there exists a = [(I,

f(m,g(m))

t ~ u e ~ ) and ~ ] ,

as in the proof of 3 . 3 ) .

k

€k

I,

is given by

(g,(Tk)kflcrn)E Un

Uvxp(x,a(x))l (Un

NxN

the subobject [Vx3yp(x,y)] >-+R

such that for

( u ~ , T ~ )factors

€k

nk

>-

We will show that Q c R . Choose n- n

through

N k XNk

Nk} Nk

(f,(uk)krdcQn,

in particular, (I)

Vm3m' 3kmVk2km-oko (IJ,~') = truek,

We may without loss assume that the sequence (2)

for each k

(km)m

+

N

for ( I ) ,

satisfying ( I ) .

We want to construct a sequence

sense that

T ~ ) ~ E ~Un, ~ )

(3)

Vn 31n VL L Ln

with k > k m*

i.e.

g(m) = m r

(

~

T

) that ~

for some m r

fits ~ ~ g

ev o (?,-re)

=

+

g(n).

Qk of

uk,

and let

ui

be the transposed

A O ( S ~ , n n Z , [ ~ = ~ + y = g O ] ) : N k ~ N k + Q k ~ Q k + Q k ,where m

such that k,,,>k

-

is theleast

this is possible because of (2).

Then we still have that

Ek k AC-NN, :u

in the

i.e.

Consider the transposed Sk:NkxNk of

=rtr%el.

f(m,m')

is strictly increasing

there are only finitely many m

Now pick any choice-function g : N

(g, (

and (hence)

u;

factors through [Vx3yp(x,y)l

also factors through [3aVxp(x,a(x))l

diagram on the next page.

hence, since

*

Ek'

say by Ek'

q

as in the

314

I. MOERDIJK

Nk 'k

i

p

=

be the pullback of

Il2 o iopk.

uvx P(x,a(x))n

x

Nk Nk

A

we find a global section T~

Nk

pb

L

Let

x

q as in the diagram. I

A

L

Then, evidently,

Ek'

of

L.

Let

Using projectivity of

I,

p k = p o f, and let

( u ~ , T ~ ) factors

through

and furthermore, (3) is satisfied, since

VmVk2km evo ( g , ~=~p0 ) so we indeed have

0

( g , ( ~ ~ ) ~ E> ~Un. )

Collecting the results of this section, we obtain

3.6. THEOREM. Let

T 2 HAH be any theory whose axioms form a subset of

{KA, BIM, AC-NN, AC-NF, RDC-NN, RDC-NF}.

Then

T

has the

DP and t h e

EP.

U

REFERENCES. [BJl CD

1

A. Boileau & A. Joyal, La logique des topos, J S L s ( 1 9 8 1 ) .

D. van Dalen, Another semantics for intuitionistic logic, and some metamathematical uses, preprint ( 1 9 7 9 ) .

[FJl [F

1

[GZI CG

I

M. Fourman & A. Joyal, Metamathematical applications of sheaf theory, t o appear. P. Freyd, Aspects of topoi, Bull.Austr.Math.Soc. 7(1972). P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springer ( I 9 6 7 ) . A . Grothendieck, Thgorie de t o p o s et cohomologie 6tale des schgmas (SGA4), SLN 2 6 9 ( 1 9 7 2 ) , exp. IV.

[J 1

P. Johnstone, Topos theory, Academic Press ( 1 9 7 7 ) .

CLSll

J. Lambek 8 P.J. Scott, JPAA 1980)

s(

.

Intuitionistic type theory and the free topos,

Glueing topoi

CLS21

J. Lambek

&

P.J. Scott,

375

Independence of premisses and the free topos,

to appear.

[M

I

I. Moerdijk,

[S

1

C. Smoryfiski, Applications of Kripke-models, Ch.V in Metamathematical investigation of intuitionistic arithmetic and analysis (ed. Troelstra), SLN 3 4 4 , ( 1 9 7 3 ) .

[W 1

G. Wraith,

On the Freyd cover of a topos,

Artin Glueing, JPAA i ( 1 9 7 4 ) .

to appear.

THE L.E.J. BROWER CENTENARY SYMPOSIUM A S . Troelstra and D. van Dalen (editors) 0North-Holkznd fiblishing Company.1982

317

SMOOTH FUNCTORS AND SYNTHETIC CALCULUS Ngo van Que and Gonzalo E. Reyes* Departement de mathematiques e t de s t a t i s t i q u e U n i v e r s i t e de Montreal Montreal , Quebec Canada

INTRODUCTION T h i s paper d e a l s m a i n l y w i t h models f o r S y n t h e t i c D i f f e r e n t i a l Geometry (SDG, f o r short).

The aim o f t h i s t h e o r y i s t o g i v e an i n t r i n s i c , na'fve a x i o m a t i z a t i o n o f

D i f f e r e n t i a l Geometry as a f o u n d a t i o n f o r s y n t h e t i c r e a s o n i n g i n t h i s f i e l d . b a s i c n o t i o n s a r e t h o s e o f a commutative r i n g w i t h

1

Its

, R ( " t h e l i n e " ) and i t s

subset D o f elements of square 0 ( " i n f i n i t e s i m a l s o f f i r s t o r d e r " ) . The fundamental assumption, t h e Kock-Lawvere axiom, a s s e r t s t h a t D i s b i g enough t o make t h e map a : R x R + R D

i n v e r t i b l e , where

a(a,b)(d)

= a t b - d ,Qd

E

D

.

( " I n t h e i n f i n i t e l y s m a l l , any

curve i s a l i n e " ) . Since t h e axiom i s i n c o n s i s t e n t w i t h c l a s s i c a l l o g i c , no s e t - t h e o r e t i c a l models e x i s t f o r t h i s t h e o r y ( e s s e n t i a l l y because t h e r e a r e non d i f f e r e n t i a b l e f u n c t i o n s i n sets).

On t h e o t h e r hand, t o p o s - t h e o r e t i c a l models have been c o n s t r u c t e d ,

showing, i n p a r t i c u l a r , t h e c o m p a t i b i l i t y o f

SDG

with intuitionistic logic

(see Kock 1981 f o r f u r t h e r i n f o r m a t i o n ) . I n t h i s paper we s h a l l c o n s t r u c t c a t e g o r i e s o f s e t - v a l u e d f u n c t o r s o f t h e form SetsA which a r e models o f t h e Kock-Lawvere axiom as w e l l as t h e axiom o f i n t e g r a t i o n o f Kock-Reyes 1981.

These c a t e g o r i e s c o n t a i n t h e c a t e g o r y

paracompact m a n i f o l d s and smooth maps f u l l y and f a i t h f u l l y .

M o f smooth

Following the lead

o f Lawvere 1979 and Dubuc 1979, 1981a, 1981b, who f i r s t c o n s t r u c t e d such c a t e g o r i e s , our c o n s t r u c t i o n i s based on t h e a l g e b r a i c t h e o r y t h e numerical spaces Rn smooth maps (i.e., Rn i n t o R" Cm

*

into

S

.

A

and whose morphisms

Cm

whose o b j e c t s a r e

Cm (Rn, Rm) a r e

m-tuples o f

maps h a v i n g c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f a l l o r d e r s ) from Cm-algebra

i s a ( c o v a r i a n t ) p r o d u c t p r e s e r v i n g f u n c t o r from

, t h e c a t e g o r y of s e t s .

A morphism between Cm-algebras i s j u s t a

Research p a r t i a l l y supported by t h e N a t u r a l Sciences an E n g i n e e r i n g Research Council o f Canada and t h e M i n i t e r e de l ' E d u c a t i o n , Gouvernement du Quebec.

318

G.E. REYES, N. VAN QUE

n a t u r a l t r a n s f o r m a t i o n between t h e f u n c t o r s . r e f e r t o the objects o f of

SA

,

A

where

The smooth f u n c t o r s o f t h e t i t l e

i s t h e f u l l sub-category o f t h e category

Cm-algebras where o b j e c t s a r e o f t h e form

Rn

,I

on

U).

an a r b i t r a r y i d e a l o f t h e

Cw(U)/I

R-algebra

C"(U)

an open subset o f

(U

o f smooth numerical functions

Results o f Oubuc ( l o c . c i t . ) and Kock-Reyes 1981 a r e extended t o t h i s

context.

I n particular,

SA

i s a model f o r s y n t h e t i c d i f f e r e n t i a l and i n t e g r a l

c a l c u l u s (Theorem 2, Section 111).

These r e s u l t s f o l l o w s from a theorem on i d e a l s

o f f l a t smooth f u n c t i o n s (Theorem 1, Section 111). The category

make a v a i l a b l e t o us a very

(and some sub-categories t h e r e o f )

SA

general n o t i o n o f smooth space (and smooth map) which i n c l u d e s usual manifolds, space o f paths, i n f i n i t e s i m a l spaces, a l g e b r a i c spaces w i t h s i n g u l a r i t i e s , e t c . Section I V shows how t o account f o r t h e n o t i o n o f Schwartz d i s t r i b u t i o n w i t h compact support i n t h i s context. To make t h i s paper accessible t o a wider p u b l i c we r e l e g a t e d t h e a p p l i c a t i o n s of o u r , r e s u l t s t o "smooth toposes" t o remarks, which may be skipped by non-specialists.

Modulo some (modest) knowledge o f category t h e o r y and a n a l y s i s , t h i s paper

i s s e l f contained. The second author would l i k e t o thank L. B C l a i r , E. Oubuc and J . Penon f o r valuable conversations.

I - EXAMPLES OF

1)

C--ALGEBRAS

Let

maps as morphisms. a

M(q)

-f

i n t o Rn

M

: M(Rn)

Any o b j e c t

M : Cm

(?-algebra

maps from

M be t h e category o f smooth paracompact manifolds w i t h smooth

+

If 4 : M + N

E

S

.

.

M

M can be considered, i n a n a t u r a l way, as

of

Indeed, l e t

M(Rn) = Cm(M,Rn)

Furthermore, i f

cp

Cm(Rn,Rm)

E

M(Rm) be d e f i n e d by M(cp)(f) = cp

0

f

,

,

t h e s e t o f smooth

let

.

M , we have, by composition, a map ( f o r each Rn) N(R")

+

M(R")

f t t f o c p

t h a t i s a n a t u r a l t r a n s f o r m a t i o n between t h e

(?-algebras

Conversely, any n a t u r a l t r a n s f o r m a t i o n between t h e d e f i n e s a morphism o f R-algebras N(R) = C"(N,R)

+

d e f i n e d by

Cm-algebras

M(R) = Cm(M,R)

M

M and and

N

N

.

Smooth functors Since map cp

,N

M

from

379

a r e paracompact, such a morphism comes ( b y c o m p o s i t i o n ) f r o m a smooth

.

N

into

M

The g i v e n n a t u r a l t r a n s f o r m a t i o n i s t h e one induced by

9.

I n o t h e r words, we have a f u l l embedding o f t h e c a t e g o r y c a t e g o r y o f a1 1

M

i n t o t h e dual o f t h e

Cm-algebras (Dubuc 1979).

2 ) More g e n e r a l l y , t h e same c o n s t r u c t i o n g i v e s an embedding o f t h e c a t e g o r y o f smooth paracompact m a n i f o l d s w i t h c o r n e r s i n t o t h e d u a l o f t h e c a t e g o r y o f Cm-algebras.

k

A

Let

3)

R

be t h e R - a l g e b r a

f u n c t i o n s on

U

A

as a

Cm

Cm-algebra, by d e f i n i n g

(fl mod I,...,fn mod 1 ) I f = (fl ,...,fn)

{?

=

on t h e o b j e c t s o f we d e f i n e

U i s an open subset of of a l l smooth numerical

C"(U)

.

We may c o n s i d e r

A(Rn)

, where

Cm(U)/I

I i s any i d e a l o f t h e R - a l g e b r a

and

.

A(cp) : A(Rn)

I f cp i s a morphism o f A(Rn) by A(cp)(f) =

Cm

.

+

,

i.e.,

E

Cm(UJln)l cp E Cm(Rn, Rm),

if

The f a c t t h a t

w e l l d e f i n e d f o l l o w s f r o m t h e f o l l o w i n g Hadamard's lemma: I f 1 f(y) - f ( x ) (y-X) * f'(xt(y-x)t)dt 10 f o r a l l x,y E R

f

E

A(cp)

is

, then

Cm(R)

.

From now on, we s h a l l o n l y c o n s i d e r f u l l s u b c a t e g o r i e s o f t h e c a t e g o r y of Cm-algebras where o b j e c t s a r e o f t h e f o r m Remark.

Let

Mc

Cm(U)/I

.

be t h e c a t e g o r y o f (paracompact) smooth m a n i f o l d s w i t h corners.

W h i t n e y ' s embedding theorem o f an o b j e c t M o f Mc i n t o some Rn i m p l i e s t h a t C"(M) = Cm(Rn)/I f o r some i d e a l I (Ii s even c l o s e d i n t h e Whitney t o p o l o g y o f u n i f o r m convergence on compacts o f f u n c t i o n s and t h e i r d e r i v a t i v e s ) , We o b t a i n , i n t h i s way, an embedding Mc

M

+

AoP

+ c~(M)

cm(Rn)/r

which p r e s e r v e s p r o d u c t s and t r a n s v e r s a l p u l l - b a l ks i n t h e f o l l o w i n g sense: R e c a l l t h a t if M of

M

i s a m a n i f o l d w i t h c o r n e r s , we have t h e n a t u r a l s t r a t i f i c a t i o n

,

-

M = M,

where

ik i s tne

3

Mn-,

3

...

3

c l o s u r e o f t h e smooth m a n i f o l d

fl, = Mo Mk

, dlm Mk

= k

I

Mn

i s an

380

G.E. REYES, N. VAN QUI?

open s e t o f

M

and

t h e boundary o f

.

M

L e t now

M,N,B

E

IM,I

and

$ = f x g : M x N + B x B c M c we say t h a t

I$

(X*Y)

E

(MxN)

I$(x,Y)

E

A n(BxB)

and

I$

A in B

i s transversal t o the diagonal

It i s r e a d i l y seen t h a t

I$(x,y)

1 F = 9- ( A )

and we have i n t h e c a t e g o r y

Mc

c

,

iff for a l l

P

i s a smooth mapping f r o m a neighborhood o f

t r a n s v e r s a l i n t h e usual sense a t

B

x

9 (x,y)

in

(MxN)

t o t h e sub-manifold

P

into

(BxB)

A n(BxB)

q

q '

i s a smooth m a n i f o l d w i t h c o r n e r s ,

MxN

a p u l l - b a c k square F-M

-

which g i v e s by t h e p r e c e d i n g embedding a push-out square i n Cm(B)

A :

C"(N)

The p r o o f o f t h i s statement i s about t h e same as t h e one g i v e n w i t h much d e t a i l s by Dubuc 1979 i n t h e case o f smooth m a n i f o l d s w i t h o u t boundary.

- Remark made apropos i f one wants t o show t h a t t h e " c l a s s i c a l " d i f f e r e n t i a l and i n t e g r a l c a l c u l u s on m a n i f o l d s agrees w i t h t h e s y n t h e t i c one i n t h e smooth toposes.

I1

-

-

IDEALS OF FLAT FUNCTIONS

To f o r m u l a t e t h e main r e s u l t o f t h i s s e c t i o n , l e t us i n t r o d u c e f i r s t some notations. If

X

Let

,

i s a c l o s e d subset o f Rn

i n Cm(Rn) which a r e f l a t on TI

.

be t h e c a n o n i c a l p r o j e c t i o n : Rn x Rm -t Rn

T

and I be an i d e a l i n by

the ideal o f a l l d i f f e r e n t i a b l e functions

w i l l be denoted by MY

X

IfonlfcIl

Cm(Rn)

,I

0

TI

w i l l be t h e i d e a l i n C"(RnxRm)

generated

Smooth functors

THEOREM 1.

X

Let

,

C”(RnxRm)

and

381

.

be closed subsets r e s p e c t i v e l y o f Rn and Rm

Y

In

we have t h e f o l l o w i n g e q u a l i t y o f i d e a l s :

~i~~ = MY nl ,uy n2 co

t

0

nl

where

and

respectively

n2 are t h e canonical p r o j e c t i o n s o f Rn x R m on Rn and Rm

,

.

For the p r o o f o f t h i s theorem, we need t h e Pollowing r e s u l t s . There are constants

LEMMA 1.

X

i s a compact subset o f

rentiable function

aE

Rn

which depend o n l y on

Cr

,

and

r

E

Nn

such t h a t :

If

a p o s i t i v e constant, t h e r e i s a d i f f e -

E

Cco(Rn) s a t i s f y i n g

E

i) O c - a s 1 aE

z 0 on a neighborhood o f X

aE(x) = 1

f o r every

ii)

we have of

aE

f o r every x

E

d(x,X)

L E >

0

and

r

x

Rn

E

X

Rn w i t h an E u c l i d i a n distance t o

E

where

...

,...,r n )

Nn , r = ( r l

IDraE(x)I
, Irl = rlt tr n i s the p a r t i a l derivative

DraE

.

This lemma i s due t o J.-C.

Tougeron 1972 (lemna IV-3.3,

page 77).

.

L e t X be a compact subset o f R” We s h a l l denote by Li(Rn) , t h e s e t o f l o c a l l y bounded f u n c t i o n s on Rn which are f l a t on X i n t h e f o l l o w i n g sense: VS

wnere

x

E

Let

Let

,p

E

and

Y

,

( I f ( x ) Id(x,X)-’<E)

=>

for

9

E

Cm(Rn)

and

$I

E

(XSY)

Let

such t h a t

s = q2

0

5

p

2

, and E q , there

1 is

d((x,y),XxY) we have

X

.

i n Rn

.

Rm

.

Then there

Cm(Rm) f l a t and vanishing o n l y on

, *0 , whenever + (x,y) E Rn

93). p

to

N , be a sequence o f f u n c t i o n s i n LiXy(RnxRm)

x

R”

PROOF. (This i s an adaptation o f an idea due t o J.-C. =page

x

r e s p e c t i v e l y , such t h a t f

VP

, (d(x,X)<6)

be compact subsets r e s p e c t i v e l y o f Rn and

Y

are p o s i t i v e f u n c t i o n s

X

> 0

i s t h e E u c l i d i a n distance o f

and

X

, 36

> 0

, d(x,X)

Rn

LEMMA 2. (fp)

N , VE

E

.

, d((x,y),XxY)

>

0

< 6(q)

0

Tougeron 1972, lemma V

By d e f i n i t i o n o f

6(q)

-+

f

P

E

, such t h a t i f

-

LiXy(Rnam)

,

382

G.E. REYES, N. VAN QUI?

2 l f p ( x , Y ) / *d((x,y),XxY)-q p : N

Define then a f u n c t i o n

-f

1

.

N , i n c r e a s i n g and g o i n g t o

m

such t h a t

ai

E

and a l s o a f u n c t i o n S : IiEN,isp(l)}

+

N

,

such t h a t S(i) = q

iff

p(q)

.

i < p(qt1)

5

Using t h e n t h e lemma 1, we c o n s t r u c t a sequence o f f u n c t i o n s

B~ of

,iE

C"(Rm)

E

X

, B.1

E 0

N

,0 s

r

Vx

E

Nn

E

Rn

'p

=

8. 5 1 with 1

5

if

or

,

r

d(x,X)

-, 1

t

ai

Cw(Rn)

and

z 0 i n t h e neighborhood

Y

i n t h e neighborhood o f

a.(x) = 1 and w i t h

,0

s 1

ai

~ ~ ( =y 1) i f

2l +1

d(y,Y)

2

1 i+ l 2

Nm , we have

E

IDrai(x)I

s 2jIr1Cr

and

Vy

E

,

Rm

IDrBi(y)I

5

ZiIr'Cr

Let

,it

E(;)is(i)ai

i t i s t h e n immediate t h a t

and

cp

f l a t and v a n i s h i n g o n l y on

~ ( 1 ) and

X

and

are positive

j,

Y

,

To prove t h e l a s t p a r t of t h e l e m a , l e t q

E

N

such t h a t

p

5

q

and 0

E >

*

1

,it

$ :Z(;)is(i)6i

p(1) and Rm ,

Cm-functions on Rn

respectively

.

d((x,y),XxY)

p

E

N

and

E >

0

be given.

There i s

Assume t h a t

s

1

2"(4)+1

hence f o r such (x,y) g i v e n , assuming, w i t h o u t l o s t g e n e r a l i t y , d(y,Y) i.e. d((x,y),XxY) s 2d(x,X) we have

5

d(x,X)

Smooth functors d((x,y),XxY)

s 2

2d(x,X)

9

383

7

2

z7Gm

5

71

we have Ifp(X'Y Hence

PROOF OF THEOREM 1. f

E

MYxy

,

(i)L e t assume f i r s t t h a t

r

we c o n s i d e r w i t h any

Nntm

E

X

and

E

a r e compact.

For an

N

1 k

gr,k (x,Y) = IDrf(x,y)I N x Nntm

we have so f o r any enumeration o f gr,k

X

and

Vr

and

Vk

Nntm

,

Y

a sequence o f f u n c t i o n s

LYxy(RnXRm)

E

By lemma 2, t h e r e a r e p o s i t i v e f u n c t i o n s

v a n i s h i n g o n l y on E

Y

and

k

Cm(Rn)

ip E

and

$ E Cm(Rn)

f l a t and

r e s p e c t i v e l y , such t h a t

N ,

gr, k ( x ,Y) 0

-t

ipX+$Y

whenever

d ( (x,y) ,XxY)

-f

0

Let then h(x,y) = 0

As any p a r t i a l d e r i v a t i v e o f

if (x,y)

h(x,y)

X x Y

E

i n v o l v e s a f i n i t e summation o f terms o f t h e

form Drf(x3Y)Ds((Pt$)(xlY) (ip(X)t$(Y) It i s immediate t o see t h a t

f(x,y)

w i t h r,s

N"+"

k

and

N

Ik h

i s f l a t on

C"(RnxRm)

E

= h(x,y)(ip(x)t$(y))

with

h

E

X x Y ; that i s Cm(Rn Rm)

The theorem i s proved i n t h a t case.

(ii) I n t h e case where consider l o c a l l y f i n i t e covering: and

Y

j

,

compact s e t s o f Rn and Rm

as b e f o r e ; f o r any

X

X = uX.

(i,j) , we have

and

and

1

respectively. E

j

1

Cm(Rn)

$i,j

E

J

Cm(Rm)

and

Let

LX.xY.(R$Rm)

ryk

E

a r e n o t compact, we

m

g

are positive functions yi,j

Y

Y = UY

j

E

N

with

Xi

g t o be d e f i n e d r,k and by lemma 2, t h e r e

384

G.E. REYES, N. VAN QUB

f l a t and vanishing o n l y on

vr

, vk

N""'

N ,

on

j

E

,

N

,

r e s p e c t i v e l y , such t h a t

g r ,k ( x ,Y)

there i s

,

'pi

0 whenever

-f

q , j ( x ) + $ i ,j(Y)

Again by lemna 2,in ca'se m = 0 f i x e d and

Y. J

and

Xi

f o r t h e sequence o f

qi,j

E

J

Lxi (Rn)

+

0

with

i

, p o s i t i v e , f l a t and vanishing o n l y

Cm(Rn)

E

d((x,y),XixY.)

such t h a t

Xi

Similarly, f o r

$i

. , we have $j

Cm(Rm) p o s i t i v e , f l a t and vanishing o n l y

E

1.J

on

I t i s imnediate t h a t we have again

Y.

J '

gr,k(x

3Y)

+

++$Ij(Y)

whenever

0

d((x,y),XixY.)

0

-f

J

L e t us then consider a l o c a l l y f i n i t e covering o f Rn ( r e s p e c t i v e l y o f Rm) by open sets: Rn=UuUi

U n X = @

and

V n Y = $ L e t us a l s o consider p o s i t i v e f u n c t i o n s

e

x,x

e,ei

Ui

3

Xi

vj

3

Y. J

E

u , supp. ei

ui

E Cm(Rm) w i t h j supp. x = v , supp. xj =

vj

=

j

E

N

Cm(Rn) w i t h support =

supp. and p o s i t i v e f u n c t i o n s

i and

Rm=VuV j

with

Then

are positive Y

Remark.

E

t

c ei

$I

x

t

c

"pi

xj . q j

and Rm

, f l a t and vanishing o n l y on X and

It i s again immediate t o see t h a t

f(x,h) h

e

Cm-functions on Rn

, respectively.

with

cp E

= h(x,y)

(q(X)+$I(Y))

CW(RnxRm) ./. The p a r t i c u l a r case o f Theorem 1, w i t h

has been proved by A.P. Calder6n.

X =

I01

c

Rn

and

Y = I01

c

m ,

R

We a r r i v e d a t our r e s u l t by studying h i s proof,

which he has k i n d l y comunicated t o us.

Smooth functors

COROLLARY 1.

For

,

Y = Rm

385

we have

q o n = M"XxRm My o

In particular,

i s a closed i d e a l ( i n t h e Whitney topology o f

n

But we do n o t know i n general i f Let

COROLLARY 2. have i n

If

Y

U

and

V

Indeed l e t

Let

vanishes on

f

U and V

U x V ; we can a l s o

m

L

m

MUxV = M i nl + MV n2

E

0

M i and M ;

0

c

(theorem 1 )

.

./.

M!

X and Y one can see e a s i l y t h a t

X = I01

and

Y = Rm

,

t o he compact

SMOOTH FUNCTORS be t h e category o f

A

where

U

u

5 R"

,

Cm-algebras whose o b j e c t s are o f t h e form C"(U)/I

Rn and

i s an open subset o f

A

we can describe t h e o b j e c t s o f any

, .we

M i x y , i.e. vanishing i n a neighborhood o f

are compact, t h e r e are compact neighborhoods

C o r o l l a r y 2 i s f a l s e i f we do n o t suppose

Remark.

and Rm

t o be t h e c l o s u r e o f t h e i r i n t e r i o r :

We obviously have M i

-

are compact sets r e s p e c t i v e l y i n Rn

, r e s p e c t i v e l y , such t h a t

Y

f

I11

the ideal o f functions i n

the following equality o f ideals

and

X

and

choose

Y

be a f u n c t i o n i n

f

. Since

X

and

X

C"(RnxRm)

Indeed, l e t of

Mi

We s h a l l denote by

, which vanish i n a neighborhood o f X , then

Cm(Rn)

X x Y

.

be a subset o f Rn

X

C"(R%Rm))

i s closed, whenever I i s closed.

Io n

I an i d e a l o f

as those o f t h e form

C"(U)

.

Alternatively,

.

Cm(Rn)/I

Indeed, f o r

open, Cm(U) = Cm(Rn+' ) / ( t q ( x ) - l )

where

(tcp(x)-l)

i s t h e p r i n c i p a l i d e a l generated by

characteristic function o f Letting and

AoP

U

,

i.e.

~ ( x t) 0

tcp(x) - 1

and ~ ( x ) i s a

, and Q ( ~ t) 0 iffx

SA be t h e Yoneda embedding, we n o t i c e t h a t , for

B = Cm(Rm)/J

, Y(A)xY(B) = hAxhB = h

e

,

u

.

A = Cm(Rn)/I

386

G.E. REYES, N. VAN QUB

where A gB = C"(R%Rm)/(

I 0 r 1

tJnn2) A h (C) = A(A,C)

and hA i s the representable f u n c t o r defined by

.

I n SA , " t h e l i n e " i s t h e r i n g - o b j e c t

.

R = hC"(R) Indeed, we have n a t u r a l transformations

induced ( v i a Yoneda) by t h e (usual) smooth maps

I*}

0

O(*)

= 1 -R

f

*2

1(*) = 1

2

(a,b) (a,b)

by composition.

+

C"(R)/(t2)

THEOREM 2 - i .

atb w

a

-

b

The o b j e c t o f " i n f i n i t e s i m a l s o f f i r s t order" i n SA

by D = hC"(R)/(t2) monomorphism 6 : D C"(R)

= 0

.

, +

R

where t i s t h e i d e n t i t y f u n c t i o n on R We have a induced ( v i a Yoneda) by t h e canonical epimorphism

in A

Let

i s given

RxR

.

5 RD

be t h e exponential a d j o i n t of the canonical

morphism .i,'

where

T

i

t

( 6 03~)

i s the projection f o

.T'

: RxRxD + R

RxRxD

into i s

E

SA

ith component.

Then c1 i s an isomorphism (axiom o f Kock-Lawvere). PROOF.

We have t o e s t a b l i s h a b i j e c t i o n between morphisms f : hA

-f

RD

and morphl'sms (a,b) such t h a t

f =

~

1

(a,b) 0

, for each A

: hA + R x R

= C"(Rn)/I.

.

By d e f i n l t i o n o f exponential, f may be thought o f as a morphism f : hA x D A h x D = hB f o r B = C"(Rn~R)/(I~.i,',t2) Hence f i s

On t h e other hand,

.

+

?7

Smooth functors

387

defined by a d i f f e r e n t i a b l e f u n c t i o n

F : RnxR + R f = F

More p r e c i s e l y ,

.

mod (Ion' , t 2 )

By Hadamard's lemma,

F(x,t) = F(x,O) L e t now a

and

b : hA

F;(x,O)

2

t G(x,t)

t

be defined by d i f f e r e n t i a b l e f u n c t i o n

R

+

-

t

t

F(x,O)

and

, r e s p e c t i v e l y . I t i s now easy t o check t h a t ( a , b ) depends o n l y on t h e morphism f , i.e. on t h e c l a s s o f F modC1.a 1 ,t2 1 , and the Fi(x,O)

on Rn

correspondence from THEOREM 2 - i i .

C"(R)

Let

to

f

i s one-to-one and such t h a t :

(a,b)

RZO = hc"(R)/Mt

, which vanish on

Mt

where

.

Rt = {teR,trO}

f = a0 (a,b) ./.

i s the ideal o f functions i n

Then

i s a sub-object o f

RO ,

R

,

~

and i t d e f i n e s a pre-order algebraic structure.

on t h e r i n g - o b j e c t

2

, compatible w i t h i t s

R

Moreover, a l l n i l p o t e n t "elements" o f

R

are

z 0

( i n the

sense t o be explained below). PROOF.

The monomorphism o f

Rto '

-

i s d e f i n e d by t h e n a t u r a l epimorphism:

R

A R E S By a 2 I) , we mean t h a t we have a commutative diagram

( i ) a : hA i.e.

into

RO ,

.

+

a takes value i n

hA 3 R

Then t h e pre-order i s compatible w i t h t h e r i n g - s t r u c t u r e o f (a)

the constant morphism

(b)

f o r a l l morphism a : hA

(c)

if

a t 0

then

and

b r O ,

a t b t O

and

1 : 1 +

R

-f

61

,

,a - a

a:hA+R

is

t

0

R

i n t h e sense t h a t :

.

0

t

and

b:hB+R,

a . b t O

where

a I n o t h e r words, l e t

t

b

A = Cm(Rn)/I

,a .b

,

defined by a d i f f e r e n t i a b l e f u n c t i o n

: hA@ + R

.

a morphism a : hCm(Rn)/l

f : Rn

+

R ;a t 0

iff

And we have o n l y t o v e r i f y these c o n d i t i o n s ( a ) , (b), ( c ) f o r B = C"(Rm)/J

.

The v e r i f i c a t i o n o f ( a ) and ( b ) i s immediate.

+ R i s then

.

V$ E M+ , $I f E I A = Cm(Rn)/I , 0

388

G.E. REYES, N. VAN QUe

Let

a : hA

R

,

and

g : Rm +I?

,

+

b : hB

R ,

-f

respectively.

be defined by d i f f e r e n t i a b l e f u n c t i o n s By d e f i n i t i o n , t h e morphism

Cm(RnxRm)/Ion'tJo.irz a t 6 : ExF = h

f : Rn + R

+

i s defined by the d i f f e r e n t i a b l e f u n c t i o n f t g : Rn x R m + R (X9Y) If a

0 and 6

2

,

0

2

$

Mt

E

,

Mt

E

, @o(ftg) E

vanishes on Rt xRt

Ion'

+

:

R

R

x

R

-f

and

J,

in $

with

c p o f c

I , as

.

(f+g)(u,v) a > 0

= h(u,v)cp

and

J,og E

a

,

2

t

as

t Jon

0

and

k(u,v)$o g ( v ) 6 2 0 ;

2

.

b

2

that i s

, a.

0

b z 0

-).

A = Cm(Rn)/I ; t h e morphism R ; for

$(utv)

k(u,v)$(v)

+

f(u)

1.71'

hA

-+

Jon2

R i s n i l p o t e n t i f f f o r some i n t e g e r p , up 0-sub-object o f R i . e . we have a commutative diagram

A morphism a : hA

takes i t s value i n t h e

For

0

J

E

I n the same way, one can see t h a t if a

F : Rn

i.e.

Hence

@o(ftg)

(ii)

t

,

0

2

R x R ; by Theorem 1,

Mt 0

*

: (u,v)

@(u+v) = h(u,v)cp(u) cp

a t 6

the f u n c t i o n

e

with

.

f ( x ) + g(Y)

we have t o prove t h a t

V@ If

++

,

a

+R

i s then defined by a d i f f e r e n t i a b l e f u n c t i o n

t o be n i l p o t e n t means t h a t

Fp

E

I

.

L e t then

$

E

Mt

we have by Hadamard's lemma: x cRn

, $oF(x)

= +(O)tF(x)$'(O)t...

tFp-'(x)$(p-')(0)tFp(x)G(x)

= FP(x)G(x)

Hence,

$ O F E I , V$

E

Mt

i.e.

a 2 0

./.

We want t o show here how t h e axiom o f Kock-Lawvere gives r i s e t o t h e n o t i o n o f "synthetic d i f f e r e n t i a t i o n " i n

8

(i)0 i s a sub-object o f

R , c o n t a i n i n g - t h e 0-element

,

Smooth functors

where

0

389

i s t h e monomorphism defined by t h e canonical epimorphism

Hence, i f

E

and

a r e two o b j e c t s of

F

,

SA

f o r any morphism

f : FxD+E

we have a morphism

g : F

+

, which i s the composition o f

E F

cI

F x (1 l ( i 2 3 0 ) ~ x Df E

I n o t h e r words, we have a n a t u r a l epimorphism

n : E ~ + E such t h a t

g

,

n o f

when

formal vector d i e l d on X : E+ED

,

(ii)

such t h a t Let

k

i s considered as a morphism o f

f

i s , by d e f i n i t i o n , a s e c t i o n o f

E

ITOX = idE

be a

.

R-valued function on

Then we can d e f i n e the d i f f e r e n t i a l o f

k

E

i.e.

n

F

into

, i.e.

a morphism

ED.

A

a morphism

k: E

along t h e formal vector f i e l d

X

+

R

.

Indeed, we have t h e n a t u r a l commutative diagram

+n

tX

where

+n k

E

k D i s t h e canonical prolongation o f

R k

Then kDoX: E + R D L R x R where i t i s easy t o check t h a t i 0 k o X = (k,X.k)

The morphism k

X - k :

E

+

: E

+

RxR

R i s , by d e f i n i t i o n , t h e d i f f e r e n t i a l o f the f u n c t i o n

along t h e formal v e c t o r f i e l d

X

.-

( i i i ) More generally, we have a l s o t h e n o t i o n o f " p a r t i a l d e r i v a t i v e " i n t h e s y n t h e t i c sense. L e t us r e c a l l f i r s t t h a t t h e i d e n t i t y morphism Id. : EF + EF , where E and F a r e o b j e c t s of SA , defines, by adjunction, the e v a l u a t i o n morphism ev : Then t h e composition

E ~ X F +

E

.

.

390

gives a morphism +

E~XF'XD

E

and hence, by adjunction, a natural morphism, which generalises the preceding notion of prolongation:

D p : EF

-f

(ED)F

Let E be the line R and let X be a formal vector field on F X : F-fF'

Then we have a morphism (Ro)F'

+

( R ~ ) F (R~R)F

RF x R F

x* the isomorphism being given by the Kock-Lawvere axiom. The composition X,

0

p : RF

4

RF x RF

is readily seen to be (id.,,yX) ; the morphism Ox : RF

+

RF

is, by definition, the partial derivative along the formal vector field X In the particular case where F = C0,ll

=

.

hA where

is the ideal of functions in Cm(R) which vanish on the interval C0,ll . On the object 10,11 of SA , we have a canonical formal vector field 6 : 6 : 10,ll x

D

-f

C0,ll

which is defined by the morphism

induced by the canonical differentiable function

R x R + R : (x,t)

t+

xtt

.

Hence we have a canonical partial derivative along the formal vector field 6 : 9 : RcoS11 R[091] -f

Again as an application of our theorem 1 , we are going to prove the existence of

44 1

Independenceof the fan theorem

Proof.

By Lemma 1, )'(V

Theorem 4.

inherits pertinent principles.

Assume RDC, V a 38-cont. and

BT

ZFI +RDCConSt. + ( ~a 38-cont.)const* +BT +FT Proof.

are consistent with ZFI

.

.

Then:

By Lemma 1, as above.

Remark. Compare Theorem 4 with ZFI+RDC+ Va 3B-cont. F B T , which was shown in [Be], pp. 46-50 by a realizability method. We show BT If FT, but also

,_

,(Q) (KSb)const. for any fragment KSb of KS consistent with whereas no interesting fragments of KS are realizable.

Finally, notice that in all above theorems we really have the Heine-Bore1 theorem.

V a 3B-cont.,

independence of

References M.J. Beeson: Continuity in intuitionistic set theories, in: Logic Colloquium 78 (ed. by M. Boffa, D. van Dalen, K. MacAloon), NorthHolland, Amsterdam, 1979, pp. 1-52. L.E.J. Brouwer: Collected Works, Vol. I, Philosophy and Foundations of Mathematics, ed. by A. Heyting, North-Holland, Amsterdam, 1975. M.A.E. Dummett:

Elements of Intuitionism, Clarendon Press, Oxford, 1.977.

M.P. Fourman, J.M.E. Hyland: Sheaf models for analysis, in: Applications of Sheaves (ed. by M.P. Fourman, C.J. Mulvey and D.S. Scott), Springer LNM 753,1979, pp. 280-301. M.P. Fourman, D.S. Scott: Sheaves and logic, in: Applications of Sheaves (ed. by M.P. Fourman, C.J. Mulvey and D.S. Scott), Springer LNM 753,1979, pp. 302-401. R.J. Grayson: Heyting-valued models for intuitionistic set theory, in: Applications of Sheaves (ed. by M.P. Fourman, C.J. Mulvey and D.S. Scott), Springer LNM 753,1979, pp. 402-414.

R. J . Grayson: Concepts of general topology in constructive mathematics and in sheaves, Annals of Math. Logic 0 (1981), pp. 1-41. W. A. Howard, G. Kreisel: Transfinite induction and bar induction of types zero and one and the role of continuity in intuitionistic analysis, J S L 31 (1966), pp. 325-358.

S. C. Kleene, R.E. Vesley: The Foundations of Intuitionistic Mathematics, North-Holland, Amsterdam, 1965. J. Myhill: Notes toward an axiomatization of intuitionistic analysis, Logique et Analyse, 2 (1967), pp. 280-297.

C. Rousseau: Topos Theory and Complex Analysis, in: Applications of Sheaves (ed. by M.P. Fourman, C.J. Mulvey and D.S. Scott), Springer L N M m , 1979, pp. 623-659.

442

A. SCEDROV

[S]

A . SEedrov:

[TI

A . S . Troelstra:

[Z]

I . D . Zaslavskir, G.S. Ceitin: On singular coverings and properties of constructive functions connected with them, AMS Translations (2) 2 (1971), pp. 41-89 = English translation of Trudy Mat. Inst. Steklov 67 (1962), pp. 458-502.

Consistency and Independence Results in Intuitionistic Set Theory, in: Proceedings of the New Mexico Conference on Constructive N M m , 1981, pp. 54-86. Mathematics (ed. by F. Richman), Springer L Choice Sequences, A Chapter in Intuitionistic Mathematics, Clarendon Press, Oxford, 1977.

THE LXJ. BROUWER CENTENARY SYM?'OSIUM A S . Troelstra and D. van Dalen (editors) 0North-Holland Publishing Company,1982

443

A NOTE ON THE FRIEDMAN SLASH AND FREYD COVERS

Andrej h d r o v Department of Mathematics University of Michigan Ann Arbor, Michigan, USA

0.

and

Philip J. Scott Department of Mathematics McGill University Montreal, Quebec, Canada

Introduction

Kleene's method [K] for proving the disjunction and existence properties (DP, EP) for HA was further developed and extended to intuitionistic type theory (HAH, for: higher-order HA) by Friedman [F] (cf. also Troelstra [ T I , chapter 3), and to intuitionistic Zermelo-Fraenkel set theory by Myhill [My]. More recently, Freyd [Fyl independently gave an elegant proof of DP and EP for HAH, avoiding explicit syntactic constructions of [F] but relying instead on general topos-theoretic facts on glueing of toposes. A detailed exposition of both methods is in the paper of Lambek-Scott [LSlI, which this paper continues in a sense. ' ! e show that both methods are essentially the same, by defining an elementary topos (cf. [BJI, [J]) which embodies Friedman's notion of realizability [F] and then showing that this topos F* is a subtopos of the topos (SetsCr) obtained by gluein the free topos F and the topos of Sets along the global section funcSets, as considered by Freyd. Thus, on the one hand, we show (to a tor F logician) that Friedman's method is a semantical interpretation in (a suitable generalized notion [BJ] of) a model, which was not clear in the papers of Friedman and Myhill; and on the other hand, we prove (for a category-theorist) that F* can be described in purely categorical terms from the Freyd cover f = (SetsCr), from which one can then reconstruct Friedman's method. Thus Freyd's use of retracts and Friedman's impredicative assignment of indices turn out to be one and the same process, a fact that was called a "remarkable resemblance" in rLS7.1 p. 11. We refer to [LSl] for details of particular formulations of HAH and of basic notions of topos theory. We will work with a slightly simpler version of HAH which discards R as a type but uses P(1) instead, otherwise as in [LSlI. A word about terminology. In o u r opinion, the term "realizability" is best reserved for various modifications of Kleene's recursive realizability. We speak of the "Friedman slash" instead. At this Conference, work of I. Moerdijk [Mol, a student of Troelstra, has shown how several other notions of "glueing/slash" considered by Aczel, Smorynski, Van Dalen, etc. are all instances of topos glueing. Finally, since this conference, recent work of Lambek and Scott [LS31 on the internal logic of Freyd covers has shown further connections between the work here and other notions of "Aczel slash". Acknowledgements: Most of this work was done during the first author's visit to Montreal in March 1981 and was subsequently presented in detail by the second author in a series of lectures at the Universitg de MontrEal and at the Brouwer Conference. We wish to thank G.Reyes and J. Lambek for suggestions and comments, the organizers of the Brouwer Conference, Profs. Troelstra and Van Dalen, for inviting us and the referee for careful comments. This research was supported in part by grants from the Quebec Department of Education and the Natural Sciences and Engineering Research Council of Canada.

A. S(5EDROV. P.J. SCOTT

444

1.

The Friedman Slash: a syntactic approach We l o o k at the formulation of HAH in the language

only allows comprehension terms

C

{x~Al$(x)}

F

4

([LSl], p.222),

in which

$(x)

no comprehension terms and has no free variables other than

Friedman slash is defined for the extended language '1 terms:

*,

L

is in

L+

is defined recursively as follows:

V C [A],

where the set

[AxB] consists of all pairs

(iv)

[P(A)]

is the set of all

V E [A]

(a)

if

a

E

V,

if

a

E

V

a-

(i)

* m

(iv)

C

$

$

a

- a',

then a'

w-

for example

(C*)

in

L

E

E

[A] and

b

E

[B].

obtained by erasing all indices (if

*

C ; and

5

V.

-

is the equivalence relation on

-*

-n

iff m

- - Cu iff V

n

Cv $

*

iff = U

a

- a'

and CC

4

and b =

- b'

C

*. L+

Then the Friedman slash is defined for sentences of denotes

a

is a comprehension term in

then C$(a-). and

[A] defined recursively as follows: (ii)

in

*.

with

Cv where

is the term of type A

there are any):

(iii)

[A] of closed terms of type A

satisfies the following Friedman conditions:

(b)

Here

which allows as closed

is the set of all numerals (or natural numbers).

[N]

(iii) and

and

[l] is the set consisting of the symbol

(i) (ii)

L

itself contains The notion of

numerals, ordered pairs, and indexed comprehension terms Ci, where

C

4

X.

which

I$

and I-$-,

occurring in

being obtained from $

$-

as follows, where

I[-$

by erasing all indices in terms

$.

1 .

1.

IT;

2.

Iaccv

3.

I$A*

4.

I$v$

iff

5.

I$++

iff

6.

7.

not

iff acv

$

iff

I$

.

and

I$ .

or

ICJ,

.

It$

implies

IJI .

IVxxeA$(x)

iff

for all

ac[A], [$(a)

lXxcA$(x)

iff

For a comprehension term set of all ae[A] indices, one has

with

there exists ac[A] C9

in

IC$(a).

. with

L , let (C+)+

II-$(a)

.

, C)'(' where V($) is the 4 Using this impredicative assignment of Set5

445

Friedman slash and Freyd covers

Soundness Theorem ([F], [LSl]).

I++,

$+

where

+ (C*) .

Let

is obtained from

+

be a sentence of

$

L.

Then b +

implies

by replacing each CJI occurring in $

by

From the Soundness Theorem we immediately obtain the required existence and disjunction properties. 2.

Freyd's Method: a categorical approach A more conceptual proof of EP and DP was introduced by Freyd [Fy] as pre-

sented in [LSl]. "free topos"

F.

Consider the closed term model of

as a topos

F are the "sets" in L

The objects of

Morphisms in F

type PA for arbitrary types A.

L

relations", i.e. closed terms of type P(AxB)

-

-

the so-called

i.e. closed terms of

are "provably functional

which are provably (in L) the

graphs of functions between objects.

F

is initial in the category of topoi and (strict) logical functors. In

-

other words, the canonical interpretation of higher-order logic in any topos E F E (c.f. [BJ], [LSl]). Freyd's induces a unique strict logical functor F idea was to let E = -+T,

the Freyd cover of

F

+ C +

(X,c,C ) morphisms and

where X

is a set, C

*

(X,E,C+) + (Y,q,C )

4

CJI is a morphism in

F

so

F

F).

F are triples

and

5: X

-t

r(C+)

(f,y) where

f: X

+

Y

an object of

are pairs

-

(denoted

We recall (see [LSl] for details) that the objects of

a function;

is a function

that the following diagram commutes:

c

There is a strict logical functor G:

F

+

F which "erases all the =-structure".

Formally, on objects G(X,<,C+) = C+ and on morphisms G(f,y)_= y. It follows G F , with GF = id. by freeness that F is a retract of its Freyd cover: F F [For the record, Freyd's original proof (c.f. [LSl]) goes as follows: DP and EP state that the terminal object 1 in algebraic property is true in

F

an indecomposable projective in 3.

F is an indecomposable projective. This

and i s preserved under retracts; hence 1 is

F itself.]

The Friedman Slash Topos: a resolution

-

We now compare the Freyd and Friedman proofs. The main results of this F A F* of the unique logical functor section state that the,*

F

F

(= the

canonical interpretation of higher-order logic in F) admits two descriptions (up

446

A. SCEDROV. P.J. SCOTT

to isomorphism):

F* - as a syntactic topos - is constructible directly from the Friedman (i) Slash (see Definition 3.1).

-

-

F* - as a subtopos of F - corresponds to the "definable" objects of (ii) (see theorem 3 . 6 ) .

F

Indeed it will be shown (Theorem 3 . 4 ) that Friedman's index operations ( ) - , yield logical functors F

( )+

-

F F

of

4 F*.

It follows that the image factorization

( )+

-

F

(c.f. (ii) above) becomes

That is, ( ) -

and ( )+

-

G F

Ft \.

F

are exactly (the restrictions of) the functors

-

G

Fi in Freyd's proof.

F

F

So in a very strong sense, the Freyd and Friedman proofs are

virtually identical.

F*

We begin with a direct syntactic construction of

F*

above, show it has the desired "image" properties. topos constructed in the language

(L+, IC).

Definition 3.1 (The Friedman Slash Topos). are indexed comprehension terms Cv

v Cw Cu )

triples

(C

la = a'

iff a

-*

such that

$ Ik"C::

Cv $

F*

Let

L+

of

be a category whose objects U Cv -t C* are

and whose morphisms +

$

is a function", (i.e. since

Cu *

a'):

t VxeAVyeB(8 (x,Y)-t$(x)A$(y))

(1) (2)

$ ' 8'

and, motivated by (ii) is essentially the free

(3)

~vx~A($(x)~~!Y~BB(x,Y)) For all ae[A], bc[B], eW implies acV and

(4)

For all

aa[A],

bcU

there exists be[B] (unique up to -) such that

cW. (In other words, C:

wl-:

Vl-+

UI-

F*

is a morphism in

iff Ce

Equality of objects and equality of morphisms in relation

-.

where W

is the set of all

cw

Cv

0

The identity morphism

C:

Cu

CT

X

+

with

CY

a *A

is a morphism in

F and

is a function for --equivalence classes).

is defined by

Cv

4

Ci

F*

is given by the equivalence W

is given by {<x,~>EAxA~x=~A$(x)},

aiV. The composition of

LT X'

where

u(x,z)

is the formula

Friedman slash and Freyd covers

%EB(e(x,y)AT(y,z)) EW and

and

cY.

Z

c o n s i s t s of a l l

z

Clearly

< a , c > such t h a t f o r some

F*

Theorem 3 .2

beU,

s a t i s f i e s t h e Friedman c o n d i t i o n s , and

Co

composition i s well-defined and a s s o c i a t i v e .

Proof:

447

F*

Thus

i s a category.

i s a topos w i t h c a n o n i c a l s u b o b j e c t s .

F* which w i l l make

We g i v e a topos s t r u c t u r e t o

F*- i s indeed a topos. Terminal Object i n F* i s given by

i t more o r l e s s

obvious that

1.

-.

{xcllx=x}['].

The t e r m i n a l

W

v 'e

C+

morphism W

i s given by l e t t i n g

be t h e s e t c o n s i s t i n g of

2.

The p r o d u c t

+(x)A+(y),

and

CI

T

w i t h

F*

in

ClxC;

B(x,y)

be

and l e t t i n g

aeV.

CT x ( x , y ) being t h e formula X' of s e t s . The first p r o j e c t i o n

i s given by

t h e C a r t e s i a n product

VxU

B ( x , y , x ' ) :x(x,y)Ax

i s then given by

*,

@(x)Ay=

=

and t h e set

x'

The second p r o j e c t i o n i s defined s i m i l a r l y . w2

pairing

of morphisms

aEV, bcU. w 1 w2 CT X

C:

given by

3.

T ( X , Y , Z ): B1(z,x)A82(z,y)

where

C:,

> with

C i

C:

EW1

and

and

and

2

c o n s i s t s of

<*,Cpl>.

i s d e f i n e d by

Q

A

Q

F*

in

aeV

The k e r n e l

with

is

i s t h e s e t of a l l

i s given by

i s given by CX

Cu

vj

$(x) E X t c P ( l ) ( e ( x , t )

t h e s e t of a l l

Ci

Cs

{ x c P ( l ) I x = x }[ P ( U

CW

,

The

eW2.

The subobject c l a s s i f i e r

and t h e morphism " t r u e "

of a l l

W

<,a> with



*

A

E

and

=

*

A

of a morphism

Cv

E t),

W,

e(x,y) E x

x(x,y) E x = y

A

*

y

and

W

c:

Cv

@-n

$(x)

i s t h e set of a l l

X

E

1

and

is

U

with

Cu and Cx f o l l o w from t h e assumption t h a t $ X Moreover, one e a s i l y sees t h a t Cx i s mono. The CY x cw Cv i s given c h a r a c t e r i s t i c morphism Cv A Q of a monomorphism Cu '+ vj c CZ by f i r s t d e f i n i n g a "canonical" monomorphism Cx A C,: where X C : {xEA(XyEBB(y,x)} and X i s t h e set of a l l a s V w i t h eW f o r some X be[B], T ( X , Y )E x = y A ~ ( x ) , 2 i s t h e s e t of a l l < a , a > w i t h aeU and then

aEU.

C:

The Friedman c o n d i t i o n s f o r

defining

o ( x , t ) : (*

such t h a t and

Y Cu

be[Bl. if

F*.

i s a morphism i n

CC

++

E

x(a-)

t

and

a r e not obvious. Thus

++

ke(b-,a-),

and

Y

*

iff

acU.

E

S

Re ( a ) f o r so

l-x(a-).

S

sY, then i n p a r t i c u l a r

Re (b) f o r

Y,

if


s1 >eY,

Z1

S a s t h e s e t of a l l E V

x(x))

Cx X' (b)

Cx X suppose acV, i . e . cW f o r some Y i s obvious f o r X. Re ( a ) f o r C ,

kZ++x(a-), al

-

a2,

[P(l)]

x

The Friedman c o n d i t i o n s f o r

and

s -

-

k*ECz++X(a-), i . e . k o ( a ,(Cz) ) .

so

s1

CZ1

- Csz22 ,

then s i n c e

S

1

=

S

2'

A. SCEDROV. P.J. SCOTT

448

-

-

V P(C )

The power-object

a ( u ) :VxcA(x

Q(x))

E U +

set of all so

x

tf

u)

E

w x <, C >

is the set of all Cw kVx

Q(x)

A

{ueP(A) la(u)IU,

+

U

E

A(+(x)

Vy W aeV, C A

*

with

X

kx c+

that

: C

t

E ((* E

is given by

0

and

V lkVx E A(x E Cw + x E C ) , i.e. 6 @ V evaluation morphism C ~8(x,u,t)

E Y . One can readily check that 2’ Z2

F*.

is a morphism in 4.

Y Ca

s2

/-a, = a2, and kZltfZ2, we have

Q(x))

-+

E

A(y

E

U

and

[P(A)]

and

W

-+

Q(y))),

so

that

u

E

where

E

*

such that

V.

Then the

and

2

S

iff

E

is the aEW, and

The Friedman conditions are then verified as above, and

$(a-).

F*.

is obviously a morphism in

The transposition CT

P(Ci)

of a

CP morphism

CTxC;

(*

x

and

E

t

+-+

E

is defined by

s2

u)

p(y,x,t)

A

Z(y,u)

W

+

R

is the set of all with bET, C

6

definition of Power-Objects) such that IkZ(b,C!), S a E [A] there exists C E [P(l)] such that

bx

Jl(a-).

U

A(x

E

(where

E u -+

S

X

*

and for each if

S

E

Q(x)),

is given in the

U

kZ(b-,C+),

EP,

The natural number object in F*

6.

Canonical monomorphisms in

monomorphisms {x V, kVx,y

E

F*,

is clearly a morphism in

5.

:c

Al+(x)Iu

E

A(B(x,y)

x=y

f-f

F*

a

E

W, and

Ix

A

$(x))

since Cp is. P

is given by

{x

Nix

E

=

XI“],

(c.f. [LSl], p. 2 4 2 ) are given e.g. E

where kVx

A\@(x)}”,

W

and

I

~ E u .

E

P(1)

E

Vx

f

Only the Friedman condition (b) is not immediate, but it follows as

above. Also, C;

U

A3t

E

i.e.

X

+-+

Vx

E

~ ( y )A ~ ( u ) where n(u)

A

We want to show that the assignment ( )

+

E

A(+(x)

is the set of all

+

as

Q(x)),

with

defined in the Soundness Theorem

is the free logical functor from the free topos

F

to

F*. L1

topos was defined in [LSl] with respect to the language

However, the free which allows nested

comprehension terms, so one needs: Lemma 3 . 3 :

Let

terms C

Q

in

L

F ( L ) (F(L1), (in

0

*

(CQ, C,, C ) such that: (i) (ii) in

L

Cvxe AVY

E

kVx~A($(x) (in

L1,

B(e(x,y)

* 3!y

resp.);

-+

E

$(x)

A

Ce

C

are triples

+(Y))

BB(x,y))

equality of objects and equality of morphisms being the

provable equality of terms in composition of relations.

-*

resp.) be a category whose objects are closed

L1, resp.) and whose morphisms C

L

Then

(in

F(L)

L1, resp.); and

F(L1)

composition being as for are isomorphic categories.

449

Friedman slash and Freyd covers

Proof:

F(L)

Certainly

F(L1)

i s a s u b c a t e g o r y of

L1

b e t h e t r a n s l a t i o n of

L0

a b i l i t y of n e s t e d comprehension t e r m s ) , where

Lo.

over

L1 b

Also

$

F

We are t h u s a l l o w e d t o w r i t e

F

is t h e f r e e topos

Theorem 3 . 4 :

G

L

F

( )-

.(

F(L)

for both

( [ L S l ] , pp.236-237,

The f u n c t o r s

L1.

L e t ( )T

L1

C

L1

and

(elimin-

conservative

t h e Lemma e a s i l y f o l l o w s .

$T;

tf

L

since

L O , e . g . as g i v e n i n [ L S l ] , pp.221-222

into

).

F(L1).

and

In particular,

242-246).

F*

d e f i n e d by

(C )+ = C;")

where

$

( )+

V($)

{a

=

I

[A] I k $ ( a ) } ,

E

subobjects.

and

F

Moreover,

Proof: -

( )-

C$,

=

i s a s u b t o p o s of

are l o g i c a l and p r e s e r v e c a n o n i c a l

F*

by

is obviously l o g i c a l .

CT : { u E P(A)lVx

A(x

E

( P ( C g ) ) + = C:(n).

E

1l-V~

E

A(x

Cw

E

C o r o l l a r y 3.5:

+

jl

Let

x

C;(@)),

E

i.e.

+.

By t h e Soundness Theorem,

E($,V): V / - +

E

L.

A(x

r(C )

F.

0

iff

F*

E P(Ci(')

C:

*

C"

E

[P(A)]

*

F* b

Then

cp

in

I-$ i f f I!-$+. On t h e o t h e r h a n d , b$ i f f F* C $+. F as f o l l o w s : l e t F* (Cv ) = (V/-,t($,V),C+)

i s g i v e n by

i s t h o u g h t o f as a morphism i n

x = a

E

r(Cg)

S($,V)([a])

=

.k

t

F:

W e )

~

e

t

C$,

a global

~

C,

r ( C ) 3 y = b-

JI

c o m m u t a t i v i t y b e i n g t h e c o n s e q u e n c e of t h e c o n d i t i o n s ( 1 )

F*.

F*

such

( i n t h e s e n s e of

The Friedman c o n d i t i o n s on Cv e n s u r e t h a t t h i s 0 * v w u $ d e f i n i t i o n i s l e g i t i m a t e . On morphisms, F (C , C , C ) = (W/-, C e ) , where C

in

W C + $ ( x ) ) , h e n c e U=V(a).

E

-

Fb $ W e d e f i n e a f u n c t o r F*

in

is

F

i s given i n

0

i s t h e set o f a l l

lbVx

be a s e n t e n c e of

$

By Theorem 2 ,

s e c t i o n of

+

* $ ( x ) ) ( c . f . [ L S l ] , p.236 and Lemma 1 h e r e ) .

u

[ B J ] , o r [ J ] , Chap. 5 ) i f f

where

P(C )

On t h e o t h e r h a n d , t h e power-object

w a s computed i n t h e p r o o f o f Theorem 1: U

Proof: F C $.

)+ = i d . ,

The Soundness Theorem i m p l i e s t h a t ( )

o b j e c t s , t h e rest i s l e f t a s a s t r a i g h t f o r w a r d e x e r c i s e .

that

)-a(

We c h e c k f o r example t h a t i t p r e s e r v e s power-

and t h e n a t u r a l number o b j e c t .

by

(

It o b v i o u s l y p r e s e r v e s p r o d u c t s , t e r m i n a l o b j e c t , s u b o b j e c t c l a s s i f i e r ,

a functor.

Then

and

( )+

F*.

F i s a retract of

i.e.

(Cy)-

-

( 4 ) on morphisms

A. SCEDROV. P.J. SCOTT

450

F*

Theorem 3.6.

* F F

F.

is a logical inclusion. Moreover, F*

morphic to) the topos of subobjects of pure types in

Proof:

-

It suffices to show that

F.

subcategory of pure types of

F* is an isomorphism between

and the full

Before we do this by induction on pure types, FA is faithful and injec-

let u s notice that the Friedman conditions ensure that tive on objects.

F*

is (iso-

In the course of the following induction we will prove all other 1

claims of the theorem. Pure types A certain triples to A.

(S

Obviously

F were described in [LSl], p.

in

We check that F*({xcA/x=x)[*~) is isomorphic A ,A A ,{xcA(x=x)). [1]/- = S1, “I/- = SN, and X1 and AN are maps given by

obvious conditions. For the induction step, if the claim is true for A

B, it is surely true for AxB B

=

P(A).

247 as

Remember that

ri

as well.

[P(A)]

and for

The only problem then is with the case

is the set of all C” satisfying the Friedman

d

-

is (bijective t o ) the set of all canonical P (A) all (X,C;,C ) such that (by induction hypothesis)

conditions. On the other hand, S subobjects of

A

in

F, i.e.

r(cd) where

i

i

___f

r(c8)

+

F,

We know that

=

d

C

4

‘e

>-

3

x = a-

{xcAlx=x?

is a canonical

and c[a] = {clxAly=*Ax=a-].

C4 :{xcAl$(x)}

so

F*(Cv)

3

r({xcAlx=x})

is an inclusion of sets and

monomorphism in

d

sA

( V k , C;($,V),C4)

in

is a canonical subobject of

F‘.

On the other hand, given such a canonical subobject V

=

c[a]

(X,C;,Cd), let We need only check Friedman’s condition (a) : if

{ac[A] I [a]cX}.

as given above is a global section of

* v

F , k$(a-).

acV, then

C 4 , so by definition of morphism in

Notice that then F ( C ) = (X,C;,C ) . Again,

erases superscripts. ‘P (A) @ 4 We have seen so far how the Friedman conditions are actually conditions on

-

canonical subobjects of pure types in we check that (Y,u,C$)

F*

k

objects in

F.

To finish the proof of the theorem,

is a full subcategory in

F‘.

be canonical subobjects, let

F*, and let

be a morphism in F.

Let W

be the set of all

Let

(X,C;,C4) k

*

C” Cu

4’

and

be the corresponding

with

f[a]

=

[b].

we need to check only Friedman’s condition (a), (b) being obvious. So let

Again

45 1

Friedman slash and Freyd covers

cW, i.e. f[a] = [b]. the square gives FO(a-,b-).

Again, F*(C:)

+ ) .

o(

= =

observe that the commutativity of

(y=b-) as global sections of

C+

in

F,

hence

(f ,C,).

The free logical functor

Corollary 3.7 :

* F

To show Ce(a-,b-),

C,o(x=a-)

F

F

---+

F-

factors through

F*

as

Thus the functors

are exactly the restrictions of

Remark.

The reader acquainted with [HJP] will have noticed that our definition

of the Friedman slash topos topos of Q-sets

(c

F*

proceeds differently from the construction of a

being the Heyting algebra of global sections of

Q

in

F*,

which surely is a "model of second-order propositional logic" in the sense of [HJP], p.214) which is suggested by analogy with section 2 of [HJP].

Nevertheless,

for arithmetical sentences, the structure of Heyting algebras of subobjects of Nn (n 2 1) coincides with the Heyting-valued presentation of the Kleene slash given

in the appendix of [D]. It is also worth remarking that the Friedman slash in [LSl] was extended to

..

include parameters of type P(A) the Freyd cover

([LSl], p.230).

*

F and then restricting to F

By adjoining indeterminates to (c.f. [LSL], p.249-250) we include

this more general situation here. References [BJI

A. Boileau, A . Joyal: La logique des Topos, JSL 46(1981),

[Dl

A.G. Dragalin: Matematiceskii Intuicionizm: Teorija Dokazatel'stv,

[FYI

P. Freyd: On proving that 1 is an indecomposable projective in varlous

pp.6-16.

Moscow, 1979 (in Russian). free categories, manuscript 1978. [Fl

H. Friedman:

Some applications of Kleene's methods for intuitionistic

systems, in:

Cambridge Summer School in Mathematical Logic (ed. by

A.R.D. Mathias and H. Rogers), Springer LNM 337, Berlin 1973, pp. 113-170. [HJP]

J.M.E.

Hyland, P.T. Johnstone, A.M. Pitts: Tripos Theory, Math. Proc.

Cambridge Phil. SOC. 88(1980),

[Jl

pp.205-232.

P.T. Johnstone: Topos Theory, Academic Press, London, 1977.

A. SCEDROV. P.J. SCOTT

452 [K]

S.C. Kleene:

Disjunction and existence under implication in elementary

intuitionistic formalisms, JSL 27(1962), JSL 28(1963),

pp. 11-18; an Addendum,

pp. 154-156.

[LSl]

J. Lambek, P.J. Scott: Intuitionist type theory and the free topos,

[LS2]

J. Lambek, P.J. Scott: Independence of Premisses and the Free Topos,

[LS3]

J. Lambek, P.J. Scott: New Proofs of some Intuitionistic Principles

[Mo]

I. Moerdijk:

J. Pure Applied Algebra 19(1980),

pp. 215-257.

Symposium on Constructive Mathematics, Las Cruces, Springer LNM 873. (to appear). On the Freyd cover of a topos, preprint no. 81-15,

University of Amsterdam (Department of Mathematics). [My]

J. Myhill: Some properties of intuitionistic Zermelo-Fraenkel set theory, in: Cambridge Summer School in Mathematical Logic (ed. by A.R.D. Mathias, H. Rogers), Springer LNM 337, Berlin 1973, pp. 206-231.

[TI

A.S. Troelstra:

Metamathematical investigations of intuitionistic

arithmetic and analysis, Springer LNM 344, Berlin 1973.

THE L.E.J. BROUWER CENTENARY SWOSIUiU A S . Troelsm and D. van Dalen (editors) 0North-Holhnd Atblishing Company,1982

453

COMPLEXITY OF NORMALIZATION IN THE PURE TYPED LAMBDA

-

CALCULUS

By t h e p u r e typed A - c a l c u l u s we mean as u s u a l t h e system o f terms b u i l t up from typed v a r i a b l e s xT,yT, c a t i o n (t'+Tsu)

...

and maybe t y p e d c o n s t a n t s aT,bT,

and A - a b s t r a c t i o n

(Ax'tT)u*T.

by means o f a p p l i -

Here t h e t y p e s T , U , . .

t i v e l y g e n e r a t e d f r o m a ground t y p e 0 by means o f ( u * T ) . e.g.

...

.

a r e induc-

I t i s well-known

(cf.

[TI) t h a t any such t e r m has a u n i q u e l y d e t e r m i n e d normal f o r m w i t h

Troelstra

respect t o so-called B-reductions

... ( A x t ) s ... +...t x [ s l ..., and

that t h i s nor-

mal form w i l l e v e n t u a l l y be reached no m a t t e r w h i c h sequence o f r e d u c t i o n s t e p s 1) one chooses

.

I n t h i s paper we w i l l be concerned w i t h e s t i m a t e s f o r t h e number o f r e d u c t i o n s t e p s necessary t o r e a c h t h e normal form. We w i l l g i v e an G4 lower bound i n 5 1 by w r i t i n g down terms tn o f l e n g t h ')

3n and showing t h a t i t t a k e s a t l e a s t Zn-'-n

r e d u c t i o n s t e p s ( w i t h Z 0 := 1, 2n+1 :=;")

I n 52

t o b r i n g tn i n t o i t s normal form.

we d e s c r i b e a p a r t i c u l a r n o r m a l i z a t i o n p r o c e d u r e and g i v e an E4 upper bound ( i n terms o f m a x ( l h ( t ) ,

L ( t ) ) , where I h ( t ) denotes t h e l e n g t h o f t and L ( t ) denotes

the inner type level o f t , i.e.

t h e maximum t y p e l e v e l

3,

o f a subterm o f t ) f o r

t h e number o f r e d u c t i o n s t e p s t h i s p r o c e d u r e w i l l c a r r y o u t . The r e s u l t o f I 1 a l s o f o l l o w s f r o m Statman

[S],

where i t i s shown more gene-

r a l l y t h a t t h e problem whether two terms tl and t2 have t h e same normal f o r m i s n o t e l e m e n t a r y r e c u r s i v e . However, f o r t h e more s p e c i f i c q u e s t i o n we a r e i n t e r e s t e d i n h e r e i t i s p o s s i b l e t o g i v e o u r much s i m p l e r p r o o f . A l s o ,

t h e mere r e s u l t

o f 52, namely t h a t f o r some s p e c i f i c n o r m a l i z a t i o n p r o c e d u r e t h e r e i s an

t4

upper

"We make use h e r e o f t h e f o l l o w i n g c o n v e n t i o n s . (1) Type s u p e r s c r i p t s w i l l be o m i t t e d whenever t h e y a r e c l e a r f r o m t h e c o n t e x t o r i n e s s e n t i a l . ( 2 ) Terms t h a t d i f f e r o n l y i n t h e bound v a r i a b l e s used a r e i d e n t i f i e d . (3) S u b s t i t u t i o n i s den o t e d b y t,[s]. (4) B r a c k e t s w i l l be o m i t t e d whenever p o s s i b l e ; we w i l l w r i t e t s r for (ts)r. "By t h e l e n g t h o f a t e r m t we mean t h e number o f o c c u r r e n c e s o f v a r i a b l e s o r cons t a n t s i n t e x c e p t t h o s e i m m e d i a t e l y b e h i n d a A-symbol. 3)The t y p e l e v e l I T I o f a t y p e T i s d e f i n e d i n d u c t i v e l y by 101 :=O, (101 + 1, I T l ) .

IU+TI

= max

454

H. SCHWICHTENBERG

bound on t h e number o f r e d u c t i o n s t e p s , i s c e r t a i n l y n o t new t o any e x p e r t i n t h e f i e l d , However,

i t seems t h a t t h e s i m p l e e x p l i c i t d e s c r i p t i o n o f t h e bounding

f u n c t i o n o b t a i n e d below i s o f some i n t e r e s t . I t s h o u l d a l s o be n o t e d t h a t , by combining t h e r e s u l t s p r o v e d h e r e w i t h those o f Gandy [ G I , one can o b t a i n a u n i v e r s a l &

4

upper bound f o r t h e number o f reduc-

t i o n s t e p s w i t h r e s p e c t t o any n o r m a l i z a t i o n procedure. T h i s can be seen as f o l lows. F o r any t e r m t o f t y p e T , by Gandy's method one can d e f i n e a c l o s e d type-0term

I t 1 w i t h t h e p r o p e r t y t h a t i t s n u m e r i c a l v a l u e i s a bound on t h e number o f

r e d u c t i o n s t e p s , where i t does n o t m a t t e r i n w h i c h way t h e r e d u c t i o n s t e p s a r e choosen. Now t o o b t a i n a bound f o r t h e n u m e r i c a l v a l u e o f l t l , we f i r s t n o t e t h a t by t h e argument o f 52 we have an

g4

bound on t h e number o f r e d u c t i o n s t e p s t h e

s p e c i f i c n o r m a l i z a t i o n p r o c e d u r e g i v e n t h e r e w i l l c a r r y o u t t o produce t h e normal form o f

( t l ; t h i s bound i s i n terms o f m a x ( l h l t l ) , L ( l t l ) ) . S i n c e , by Gandy's conI t l , I h ( l t 1 ) depends o n l y l i n e a r l y on I h ( t ) , and L ( l t l ) = L ( t ) , we

struction of

a l s o have an &

4

upper bound on t h e number o f r e d u c t i o n s t e p s i n terms o f m a x ( l h ( t ) ,

L ( t ) ) . Next n o t e t h a t any r e d u c t i o n s t e p a t most squares t h e l e n g t h o f t h e o r i g i n a l term. So we have an

g4

upper bound on t h e l e n g t h (and hence on t h e numerical

v a l u e ) o f t h e normal form o f l t l , a g a i n i n terms o f m a x ( l h ( t ) , L ( t ) ) . T h i s g i v e s the desired result.

(The f a c t t h a t one can o b t a i n an g

work i n [ G I has been mentioned t o me b y G.E.

4

upper bound f r o m Gandy's

Minc and R . Statman).

§I.The p u r e t y p e s k a r e d e f i n e d i n d u c t i v e l y by 0 : = O , f i n e i t e r a t i o n functionals I I

k + l := k + k .

We de-

o f p u r e t y p e k + 2 by

:sAfXx f ( f (

w i t h n o c c u r r e n c e s o f f a f t e r XfXx;

. . . $( f x ) ...) ) , here f , x a r e v a r i a b l e s o f type k + l , k , res-

p e c t i v e l y . L e t f o g be an a b b r e v i a t i o n f o r Xx f ( g x ) , and l e t t = s mean t h a t t and s have t h e same normal form. W i t h t h i s n o t a t i o n we can w r i t e

I

= hf

f o f o

.... f

w i t h n occurrences o f f a f t e r i f . The main p o i n t o f o u r argument i s t h e f o l l o w i n g s i m p l e lemma, w h i c h can be t r a c e d back t o Rosser ( c f . Church [C, p. 301 1. LEMMA. -

lmln= I n

455

Normalization in typed lambdadalculus

As an immediate consequence we have

:=

tn

121*

... l 2

=

'2n 2 ( w i t h Z 0 := 1 , 2n+1 := 2 " ) . Now c o n s i d e r any sequence o f r e d u c t i o n s t e p s t r a n s forming t

i n t o i t s normal form,

and l e t S

d e n o t e t h e t o t a l number o f r e d u c t i o n

s t e p s i n t h i s sequence. THEOREM. Sn>2n-2-

n.

Ph006. The l e n g t h o f tn i s 3n.

... ( h x t ) s ...

-t

...t x [ s ] ...

Note t h a t any r e d u c t i o n s t e p

can a t most square t h e l e n g t h o f t h e o r i g i n a l term.

Hence we have Zn

< <

length o f l2

( t h e normal f o r m o f t n ) 2Sn

(length o f tn) 'n

=

n+Sn

< 22

(since 3nG2

2"

1,

and t h e theorem i s proved.

J 2 . Our a i m h e r e i s t o s e t up a s p e c i f i c n o r m a l i z a t i o n p r o c e d u r e f o r w h i c h

4

an .$

upper bound on t h e number o f r e d u c t i o n s t e p s can be o b t a i n e d e a s i l y .

So l e t

an a r b i t r a r y t e r m be g i v e n . Our n o r m a l i z a t i o n p r o c e d u r e i s an o b v i o u s one: we s e a r c h f o r redexes o f maximal t y p e l e v e l , and among t h o s e we t a k e t h e r i g h t m o s t one and c o n v e r t i t . Here by a redex we mean as u s u a l an o c c u r r e n c e o f a subterm ( A ~ ' t ~ ) ' ~ ~ s ' , and t o c o n v e r t i t means t o r e p l a c e i t by t [ s ]

.

I t s type level i s

the type level o f U + T . I n o r d e r t o g e t an e s t i m a t e f o r t h e number o f r e d u c t i o n s t e p s needed, we a s s o c i a t e a number w i t h any g i v e n t e r m and show t h a t t h i s number decreases w i t h any r e d u c t i o n s t e p . To o b t a i n such a number, we f i r s t a s s i g n t o any t e r m t a sequence a . . ( t ) o f numbers, as f o l l o w s : a . ( t )

i s t h e number o f redexes o r o f v a r i

a b l e s i n t w i t h t y p e l e v e l i + l . O b v i o u s l y o n l y f i n i t e l y many a i ( t )

w i l l be d i f -

f e r e n t from 0. Now l e t us c o n s i d e r a r e d u c t i o n s t e p and see how t h e a s s i g n e d sequence w i l l change:

456

H. SCHWICHTENBERG

. . . (Axu t T

U+T

s

u

... + . . . t;[s'l.

0

Here m + l

0

i s the level o f the type

o f s. ( N o t e t h a t ,

..

U+T

o f X x t , and n i s t h e l e v e l o f t h e t y p e

i f n = O , we'have a.(s) = O f o r a l l i ) .

ti

u

i s the contribution

t o o u r sequence o f t h a t p a r t o f t h e terms above t h a t i s denoted b y i f t h o s e terms a r e w r i t t e n as r [ ( X x t ) s l and r [ t [ s ] ] , Y Y X then c i = a i ( r ) - 1 .

precisely,

... .

More

respectively,

Now denote t h e sequence a s s o c i a t e d w i t h t h e o r i g i n a l t e r m by 9I and t h e sequence a s s o c i a t e d w i t h t h e reduced t e r m ( f o r w h i c h we o n l y gave an e s t i m a t e ) by 8 ' . We want t o have numbers 181, 18'1 a s s i g n e d t o %,8'such t h a t IN1 T h i s can be done as f o l l o w s .

e n t r i e s d i f f e r e n t f r o m 0 . L e t m be maximal w i t h d,>O.

1st

7 18' I .

L e t 9 = ( d i ) be a sequence w i t h o n l y f i n i t e l y many Then d e f i n e

w i t h d = max(do ,...,dm-l),

= g(m,dm,d)

where g(m,a+l ,b) = g(m,a,b 2 + 1 g(m+l ,O,b) g(O,O,b)

= g(m,b,b) = b.

Note t h a t g belongs t o t h e c l a s s E4 o f Grzegorczyk [ G r ] m t h e f u n c t i o n g(m,.,.)

I t i s easy t o check t h a t g(m,a,b)

Using t h i s ,

, and

t h a t f o r any f i x e d

3 b e l o n g s t o .$ , i . e . i s e l e m e n t a r y r e c u r s i v e .

l e t us show t h a t 1'111

>

i s monotone i n a and b f o r any f i x e d m.

IPT'I. Case 1 : c m + a m ( t ) > O .

Then we have

In1

= g(m,cm+am(t) +l,m) = g(m,cm+am(t),M 2 ) + 1

> g(m,cm+am(t), > 18' I + 1

with

M :=

max ( c i + a i ( t ) Ki<m

max (ci + a i ( t ) + a i ( s ) a n-1 ( t ) ) ) OQiG

+ai(s))

+I.

45 I

Normalization in typed lambda-calculus

Case 2: c + a ( t ) = 0 and M = O . Obvious. m m . t h a t c i + a i ( t ) + a i ( s ) = 0 . Then we have

Case 3: c m + a m ( t ) = 0 ,

i maximal such

1911 = g(m,l ,M) = g(m,O,M = g(m-1 ,M

> g ( i ,M' > IR'I.

2 2

) +1 2 ,M )

+1

,M')

Here we have made use o f t h e o b v i o u s f a c t t h a t g(m,b,b) c o n c l u d e s t h e p r o o f o f IRI

>

i s monotone i n m. T h i s

l a ' l . We can sumnarize o u r argument as f o l l o w s .

THEOREM. For any g i v e n t e r m t , t h e number o f r e d u c t i o n s t e p s f o r t h e p r o c e d u r e d e s c r i b e d above i s < g ( m , a m ( t ) , a ( t ) ) . a redex i n t , a ( t )

:=

z&

ai(t)

and a . ( t )

Here m + l

i s t h e maximal t y p e l e v e l o f

i s t h e number o f redexes o r o f

v a r i a b l e s i n t whose t y p e l e v e l i s i + l . COROLLARY. ( 1 ) There i s an g4 f u n c t i o n f such t h a t f o r a l l c l o s e d type-0term3 t t h e above n o r m a l i z a t i o n p r o c e d u r e t e r m i n a t e s i n < f ( m a x ( l h ( t ) , L ( t ) ) ) steps.

( 2 ) For a l l m t h e r e i s an e l e m e n t a r y r e c u r s i v e f u n c t i o n g

a l l c l o s e d type-0-terms m i n a t e s i n Qgm( I h ( t ) )

t with L(t)<m

such t h a t f o r

t h e above n o r m a l i z a t i o n p r o c e d u r e t e r -

steps.

REFERENCES

[ C]

Church, A.: The c a l c u l i o f lambda-conversion. Annals o f Math. S t u d i e s No. 6 . P r i n c e t o n 1941.

[GI

Gandy, R.O.: P r o o f s o f s t r o n g n o r m a l i z a t i o n . I n : To H.B. C u r r y : essays i n c o m b i n a t o r y l o g i c , lambda c a l c u l u s and f o r m a l i s m . London 1980.

[Gr]

Grzegorczyk, A . :

[S]

Statman, R.:The t y p e d A - c a l c u l u s Computer S c i e n c e 9 (1979).

[TI

T r o e l s t r a , A.S. ( e d i t o r ) : Metamathematical i n v e s t i g a t i o n o f i n t u i t i o n i s t i c a r i t h m e t i c and a n a l y s i s . S p r i n g e r L e c t u r e Notes i n Math. No. 344. B e r l i n

1973.

Some c l a s s e s o f r e c u r s i v e f u n c t i o n s .

Rozprawy Mat. 4(1953).

i s n o t e l e m e n t a r y r e c u r s i v e . Theor.

THE LEJ. BROUWER CENTENARY S Y M O S I m A S . Troeoelstra and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

459

NONSTANDARD MODELS AND CONSTRUCTIVITY C . Smoryfkki

429 S . Warwick Westmont , 111. 60559 USA

The Brouwer-Hilbert controversy, l i k e any good c l a s h of p e r s o n a l i t i e s , made f o r a f a s c i n a t i n g and f r u i t f u l episode i n t h e h i s t o r y o f mathematics. But, it i s history-- t h a t is t o say, past history- and I should not wish t o stir up t h e o l d a n i m o s i t i e s h e r e . Both s i d e s were r i g h t ; both were wrong; t h e r e was not t h a t much a c t u a l disagreement anyway; and nowadays only an occasional e c c e n t r i c yet pursues an a n a c h r o n i s t i c b a t t l e . Today, we recognise, not two competing views of mathematics, b u t two t y p e s of mathematics-- c o n s t r u c t i v e and nonconstructive-with c e r t a i n r e l a t i o n s o b t a i n i n g between them. Constructive mathematics i s t h e legacy of Brouwer's philosophy; and t h e r e l a t i o n between c o n s t r u c t i v e and nonc o n s t r u c t i v e mathematics i s t h e legacy of H i l b e r t ' s programme. H i l b e r t sought t o j u s t i f y t h e u s e of n o n c o n s t m c t i v e methods t o t h e i n t u i t i o n i s t by providing conservation r e s u l t s , which were, i n t u r n , t o be e s t a b l i s h e d by c o n s t r u c t i v e consistency p r o o f s . On Sunday, 7 September 1930, no l e s s a d i s c i p l e of Brouwer than Arend H e i j t i n g expressed f u l l agreement with H i l b e r t ' s p o s i t i o n , a s s e r t i n g t h a t such consistency proofs ( with t h e i r i m p l i c i t conservation r e s u l t s ) would indeed allow t h e i n t u i t i o n i s t t o accept nonconstructive arguments. I r o n i c a l l y , it w a s i n d i r e c t response t o t h i s p r o v i s i o n a l acceptance t h a t GBdel made h i s first ( as K r e i s e l would say: c h a r a c t e r i s t i c a l l y c a u t i o u s ) announcement o f h i s F i r s t Incompleteness Theorem: Consistency w a s not a source of g r e a t conservation and he had examples t o show t h i s . When GBdel destroyed H i l b e r t ' s programme, Brouwer should have won t h e b a t t l e by d e f a u l t . I n s t e a d , people j u s t f o r g o t about it. C o n s t r u c t i v i s t s developed c o n s t r u c t i v e mathematics alongside, r a t h e r than i n s t e a d , of c l a s s i c a l mathematics. The proof t h e o r i s t s of H i l b e r t ' s school-- w e l l , some simply q u i t proof theory. A f e w , f o r g e t t i n g H i l b e r t ' s reason f o r wanting a f i n i t i s t i c proof of consistency, expanded t h e s e a r c h t o allow c o n s t r u c t i v e i n f i n i t i s t i c proofs o f consistency. From o u t s i d e , as i t were, came a few who a c t u a l l y c a r r i e d on with H i l b e r t ' s r e a l programme. I n t h e 1930s, f o r example, both GBdel and Gentzen proved t h e conserv a t i o n of c l a s s i c a l a r i t h m e t i c PA over i n t u i t i o n i s t i c a r i t h m e t i c HA with respect t o negative sentences, i . e . sentences c o n s t r u c t e d f r o m atomic formulae with only t h e u s e of negation, conjunction, and u n i v e r s a l q u a n t i f i c a t i o n . Although f a r s u p e r i o r t o anything an i n t u i t i o n i s t i c consistency proof f o r c l a s s i c a l a r i t h m e t i c would have yielded, t h i s r e s u l t d i d not cause g r e a t excitement. And properly not-- H i l b e r t ' s emphasis on "real" statements w a s an underestimate of t h e i n t e r e s t s of t h e i n t u i t i o n i s t s . The i n t u i t i o n i s t s were not i n t e r e s t e d merely i n numerical i d e n t i t i e s , but mainly i n c o n s t m c t i o n s , and s o i n e x i s t e n t i a l s t a t e ments. An e x i s t e n t i a l a s s e r t i o n , s a y t h e i n f i n i t u d e of p a i r s of twin primes, ~ x 3 y y(> x & y , y + ~are prime),

(*I

given a nonconstructive proof would, under Geldel's and Gentzen's procedure, y i e l d a c o n s t r u c t i v e proof of a l e s s e r negative r e s u l t , i n t h i s case: V x l V y T ( y > x & y , y + 2 are prime). Of course, t h i s i s c o n s t r u c t i v e l y meaningful-- it t e l l s t h e i n t u i t i o n i s t not t o attempt t o f i n d an upper bound on t h e set of twin primes. But, it l a c k s t h e c o n s t r u c t i v e c o n t e n t of (*), which asserts t h e r e t o b e a method of generating

C.A. S M O R Y ~ S K I

460

l a r g e twin primes. [ I ignore h e r e t h e obvious o b j e c t i o n t o t h i s example t h a t w i l l be posed by any b e l i e v e r , l i k e myself, i n Markov's P r i n c i p l e . ] To d a t e , t h e b e s t g e n e r a l purpose conservation r e s u l t r e l a t i n g c l a s s i c a l and i n t u i t i o n i s t i c systems of mathematics i s t h e conservation of a c l a s s i c a l system over a corresponding i n t u i t i o n i s t i c one with r e s p e c t t o n2 sentences, i . e . t h e c l o s u r e of t h e i n t u i t i o n i s t i c system under Markov's Rule, MR. This has n o t t r a d i t i o n a l l y been an easy r e s u l t . And t h e metamathematics required f o r i t s proof, though c l e a r l y i n t u i t i o n i s t i c , h a s not always been e n t i r e l y obvious. A s r e c e n t l y as 1975, I was expounding ( [4] ) K r e i s e l ' s devious proof t h a t t h e c l o s u r e of HA under MR could be d e r i v e d i n HA i t s e l f . Today, however, we have t h e Friedman-Dragalin ( [2], [I] ) p r o o f , t h e metamathematics of which is simply P r i m i t i v e Recursive Arithmetic, and t h e a p p l i c a b i l i t y of which ( t o a given formal. system ) is almost immediately recognisable. The s i g n i f i c a n c e of t h i s new proof is t h a t it a c t u a l l y makes convincingly acceptable, as w e l l as p r a c t i c a l , t h e u s e of c l a s s i c a l t o o l s by t h e i n t u i t i o n i s t .

In [3], t h a t p a r t of my t h e s i s d e a l i n g with Heyting's a r i t h m e t i c , I emphasised j u s t t h i s p o i n t ( i n s e c t i o n 5.1.26 ):

My p r o o f s , though c a r r i e d o u t i n ordinary s e t theory, could r e a l l y be c a r r i e d o u t i n PA augmented by a few consistency statements. The conservation o f , s a y , PA -+ Con(PA) over HA -t Con(PA) = HA f Con(HA) with r e s p e c t t o IL2 sentences would t h e n r e a d i l y y i e l d t h e i n t u i t i o n i s t i c v a l i d i t y of t h o s e r e s u l t s e x p r e s s i b l e i n 112 form.

This simple

p o i n t seems n o t t o have-been w e l l understood. Indeed, i n [6] d e Swart made t h e exceptionable remark t h a t "C. Smoqfiski gave semantic proofs of t h e s e theorems with r e s p e c t t o Kripke models, however using c l a s s i c a l metamathematics. I n $1 we w i l l g i v e i n t u i t i o n i s t i c a l l y c o r r e c t , semantic p r o o f s . " What de Swart failed t o remark w a s t h a t h e had been unable t o prove more than a small f r a c t i o n of t h o s e theorems I could o b t a i n with t h e u s e of c l a s s i c a l l o g i c and Kripke models. For, h i s i n t u i t i o n i s t i c a l l y c o r r e c t p r o o f s could only u s e t h e equivalence: cp i s provable

iff

9 i s v a l i d i n all models,

while many p r o o f s r e q u i r e r a t h e r : cp

is unprovable

iff

cp

i s f a l s e i n some model.

(*)

Consequently, t o e s t a b l i s h a 112 sentence, d e S w a r t ' s i n t u i t i o n i s t i c a l l y c o r r e c t model t h e o r e t i c p r o o f s , when depending on (*), would only y i e l d d i r e c t l y t h e negative form of t h e d e s i r e d r e s u l t and he would have t o appeal t o c l o s u r e under Markov's Rule-- t h e same r e s u l t I require-- t o o b t a i n i n t u i t i o n i s t i c a l l y c o r r e c t proofs. Now, de Swart's proofs can be f o n a l i s e d i n a t h e o r y closed under Markov's Rule ( [a] ) ; b u t . . w e l l , l e t me quote D r a g d i n : "We note t h a t d e Swart a t t e m p t s an i n t u i t i o n i s t i c treatment of t h e Smoqfiski operation, b u t i n h i s work.. h e u s e s s p e c i f i c i n t u i t i o n i s t i c methods of argument ( Brouwer's c o n t i n u i t y p r i n c i p l e , t h e Fan Theorem ) which a r e unacceptable f r o m t h e c l a s s i c a l p o i n t of view. " O f course, conservation r e s u l t s render t h i s unproblematic-- t h e u s e of Brouwer's p r i n c i p l e and t h e Fan Theorem can b e bypassed i n e s t a b l i s h i n g a r i t h m e t i c theorems-- b u t t h e n e t r e s u l t i s t h a t d e Swart must appeal t o two conservation r e s u l t s and still u s e a s t r o n g e r metatheory t h a n I.

.

.

...

[ i n 5.1.26 ] g i v e s I n an unpublished n o t e , T r u e l s t r a says, "Smoqfiski a h i n t of how t o c o n s t m c t i v i s e many of h i s r e s u l t s . The h i n t i s very b r i e f and a l s o i n need of emendation and as we f e l t t h e t o p i c t o have ped ogic i n t e r e s t while a t t h e same time t h e d e t a i l s r a t h e r confusing t o many, wey t h i n k ] it worthwhile t o g i v e a l e i s u r e l y d i s c u s s i o n of t h e s e matters." While I , perhaps immodestly, do n o t r e a l l y s e e t h e need f o r an emendation, I must admit t o t h e f a i l u r e o f my message ( i n 5.1.26 ) t o have g o t through and t h a t a l e i s u r e l y discussion of t h e s e matters might indeed be worthwhile.

,..

The purpose of t h e p r e s e n t note i s t o e x h i b i t an i n s t a n c e of such a r o u t i n e

Nonstandard models

46 1

c o n s t r u c t i v i s a t i o n . Since it i s considered tacky among American l o g i c i a n s t o p u b l i s h a paper without a new r e s u l t , o r , at l e a s t , a new proof, I have chosen t o i l l u s t r a t e t h e technique by c o n s t r u c t i v i s i n g a "new" p r o o f . By "new" I simply mean unpublished: I knew t h e proof a l r e a d y i n 1972, but d i d not include it i n [Sl because i . it would have r e q u i r e d t h e i n c l u s i o n of an i n o r d i n a t e amount of background material, and ii. i n 3.2.32 of t h e same book ( [ ] ) T r o e l s t r a presented my a l t e r n a t e r e a l i s a b i l i t y proof of t h e r e s u l t . I did present t h e proof i n my Amsterdam Report 14.1, but t h a t t o o I never published.

f

1

The r e s u l t I wish t o c o n s t r u c t i v e l y prove i s t h e following: Theorem. PA i s an e s s e n t i a l l y unbounded extension of H A , i . e . no c o n s i s t e n t extension of HA by axioms of bounded q u a n t i f ier-complexity y i e l d s every i n s t a n c e of t h e Law of t h e Excluded Middle. THE NONCONSTRUCTIVE PROOF OF THE THEOREM I s h a l l be b r i e f i n sketching t h e proof. For one t h i n g , t h e d e t a i l s are similar t o t h o s e of my proof of Theorem 5.4.5 i n t h a t Markov's P r i n c i p l e i s not implied by any bounded set of i t s i n s t a n c e s . For another, w e need only a sketch of t h e main s t e p s s o t h a t we w i l l have something t o r e f e r t o i n d i s c u s s i n g the constructivisation.

[31

Let A be a set of sentences of bounded complexity, s a y A 5 C n and suppose HA

+

A y i e l d s PA.

W e wish t o show HA + A i s i n c o n s i s t e n t .

Assume HA + A is c o n s i s t e n t . Since it y i e l d s PA, w e see t h a t PA + A is c o n s i s t e n t . Let M b e a model of PA + A . By t h e Orey Compactness Theorem ( cf e . g . [51 ) , t h e r e i s a model M' of PA such t h a t i.

and

ii.

but:

iii.

t h e t r u t h d e f i n i t i o n f o r M', T r M , ( . ) , is d e f i n a b l e i n M under t h e canonical embedding of M i n t o M ' ( c f . [31 o r [51 ) , M and M ' agree on t h e t r u t h o r f a l s i t y of all Cn sentences with parameters f r o m M ( i . e . M ' is a xn-elementary extension of M ) ) ;

M' i s not an elementary extension of M .

Actually, we can b e more s p e c i f i c :

AFZn, whence T r M , ( * )can b e chosen

Now, an u r a l t e r e s u l t of nonstandard model theory affirms t h a t M ' is not An+l. an Zn+l-elementary extension of M: There a r e a xn+l-fomula cpvO . . . ~ m - l and elements ao, ...,a E I M I such t h a t m-1

Now, t h e r e a d e r with [31 before him can look a t 5.4.5, where t h e following observation is made: The Kripke model K obtained by p l a c i n g M ' above M, K: -

M'

I

a0 M , satisfies: i. ii.

Since

K

KI= HA a.

IC eZo..

iff

M

-

I= Go..,%-1,

0 E 2,

whence

iii. aoIV cpXo...%-l V l ( P X o where cp is as above. k HA + A, b u t K If PA, t h e proof i s complete.

K I=

A

C.A. SMORYNSKI

462 CONSTRUCTIVISING THE PROOF

The above proof i s c e r t a i n l y nonconstructive. To g e t a c o n s t r u c t i v e proof out of it, we need only s e e t h a t it can be formalised i n a c l a s s i c a l t h e o r y conservative over an acceptable i n t u i t i o n i s t i c t h e o r y with r e s p e c t t o a n i c e c l a s s of sentences, PA s e r v e s as a s u f f i c i e n t l y s t r o n g c l a s s i c a l t h e o r y and conservation with respect t o e i t h e r t h e negative sentences ( t h e r e s u l t of Gddel and Gentzen ) o r sentences ( c l o s u r e under Narkov's Rule ) y i e l d s t h e provab i l i t y of t h e r e s u l t i n H A . The first o b s t a c l e t o f o r m a l i s i n g t h e proof i n PA is t h e nonarithmetic n a t u r e of t h e r e s u l t t o be proved:

r i

v n V A ~ ~ n [ C o n ( P A + A ) ' 3 Y (PrpA( Y

[I The

1 1

) A 1

PrHA+a( F

proper premiss should, of course, be Con(HA+A)

.

))I.

C l a s s i c a l l y , however

( and I i n t e n d t o work c l a s s i c a l l y ) , t h e c a s e lCon(PA + A) i s immediate. ] While t h e formula

Fortunately, t h e proof y i e l d s a s t r o n g e r r e s u l t :

Q V ~ . . . V ~ - ~

constructed i n t h e above proof i s n o t uniform i n A , i t s complexity is. Thus we GI(v l v ) by t h e s t r o n g e r sentence can r e p l a c e t h e unprovable V vo.. . v ~ - ~

where T r ( ' ) i s a zn+l-truth d e f i n i t i o n f o r ~n+l-formulae. Thus, we have n' +I a c t u a l l y proven t h e s t r o n g e r r e s u l t : V n 3 Y VAzzn[Con(PA+A)-PrpA( rF i ) A 1 P r H A + n r ( Y i )]. T h i s i s c l e a r l y ( i . e . i n t u i t i o n i s t i c a l l y as w e l l as c l a s s i c a l l y ) equivalent t o t h e a r i t h m e t i c sentence: V n 3 ~Vcp Ezn[Con(PA+v )'PrpA(

r i Y

) A 1

r i

PrHA+cp(Y

11.

(2)

Thus o u r theorem h a s a s t r o n g a r i t h m e t i c formulation. Unfortunately, t h e formulation given i s n e i t h e r negative n o r X2. So we l o o k at (1). C l e a r l y t h i s sentence i s p r i m i t i v e r e c u r s i v e i n n , i . e . t h e r e i s a p r i m i t i v e r e c u r s i v e f u n c t i o n f such t h a t f(n)

=

I

Y I,

where Y is t h e sentence (1). PA

t

Also it i s c l e a r t h a t

v SrpA(fn).

Thus, (2) is equivalent t o ~

n cpvECn[Con(PA+ cp)+

l P r H A +CP (fn)

which i s negative [ and which i s c l a s s i c a l l y f r o m i t s c l a s s i c a l n2-equiva1ent

1.

3,

(3)

Il -- i n t u i t i o n i s t i c a l l y , it f o l l o w s 2

So now w e have a s s e r t i o n (3) of e x a c t l y t h e r i g h t form. To prove (3) i n HA, it s u f 3 c e s t o prove it within PA. And t h i s is p o s s i b l e by t h e u n i f o n n i t of t h e a r i t h m e t i s a t i o n within PA of t h e proof of t h e Completeness Theorem ( c f . f5] ) : Uniformly i n CP t h e r e is a c o n s t r u c t i o n within PA of a p r e d i c a t e Trn ( * ) which,

.

on assumption of Con(PA + CP ) , i s shown t o d e f i n e t m t h f o r a model M of PA + CP Now t h e f o r m a l i s a t i o n of O r e y ' s Compactness Theorem allows, f o r any n i n t h e "real world" ( b u t not for any nE I # ( ) , t h e c o n s t r u c t i o n of a Cn-elementary end extension M' of M as d e s c r i b e d i n t h e preceding s e c t i o n . Although M need n o t recognise t h a t M' )= PA, t h i s f a c t can b e proven i n PA+Con(PA+ cp). Thus, i n PA

463

Nonstandard models

we can p l a y with M and M ' as i n t h e preceding s e c t i o n and conclude t h e p r o v a b i l i t y of ( 3 ) . I fear t h e present sketch w i l l not convince anyone who i s not familiar with t h e a r i t h m e t i s a t i o n of semantic c o n s t r u c t i o n s ; b u t t h e a l t e r n a t i v e - - g i v i n g t h e f u l l d e t a i l s - - does n o t appeal t o me and probably wouldn't appeal t o most readers. My p o i n t i s not t h a t t h i s i s the way t o g i v e c o n s t r u c t i v e proofs, but simply t h a t it i s a. way and t h a t c r i t i c i s m of my nonconstructive proofs f o r t h e i r noncons t m c t i v i t y i s simply unfounded. A TINY ADDITIONAL OBSERVATION

The key t o t h e above c o n s t r u c t i v i s a t i o n is t h e a r i t h m e t i s a t i o n of model theory-- s p e c i f i c a l l y , t h e ( c l a s s i c a l ) c o n s t r u c t i o n of models with a r i t h m e t i c t m t h d e f i n i t i o n s v i a t h e a r i t h m e t i s a t i o n o f t h e Completeness Theorem. This i s unproblematic f o r c l a s s i c a l theories-- t h e u s u a l completeness proof goes by way of c o n s t r u c t i n g a s i n g l e completion of a c o n s i s t e n t r . e . t h e o r y and t h e r e s u l t F o r i n t u i t i o n i s t i c t h e o r i e s , i . e . t h e c o n s t r u c t i o n of Kripke models, i s A2. t h e r e i s no evident a r i t h m e t i c upper bound on t h e complexity of t h e model. can one do about t h i s ? My s o l u t i o n i s simple:

What

Work, not i n PA, b u t i n PA + 1-CON(PA). This theory with r e s p e c t t o negative and n2 sentences,

is conservative over HA + 1-CON(HA)

and i s s t i l l a somewhat weaker metatheory than i n t u i t i o n i s t i c a n a l y s i s . Here i s how one proceeds: By 1-CON(PA), w e can d e f i n e i n PA a model M of PA t h a t i s a cl-elementary extension of t h e u n i v e r s e of n a t u r a l numbers. The reason behind t h i s is two-fold: i.

c1-elementary, it recognises as c o n s i s t e n t only those r . e . t h e o r i e s t h a t a c t u a l l y a r e c o n s i s t e n t : F o r r . e . T, Since M i s

Con(T) *"M

PA ii.

Con(T)";

The f u l l t r u t h d e f i n i t i o n f o r M i s a v a i l a b l e t o u s .

One now s e m i - f o n a l i s e s t h e proof of t h e Completeness Theorem f o r Kripke models ( c f . 5.1.5-5.1.11 of [3] ) over M. Let me e l a b o r a t e t h i s last s t e p : The key s t e p of t h e proof of t h e Completeness Theorem f o r Kripke models c o n s i s t s i n c o n s t r u c t i n g a u n i v e r s a l model & of a given c o n s i s t e n t t h e o r y T by i. t a k i n g a h i e r a r c h y of languages C1,C2, and

...,

ii. l e t t i n g K c o n s i s t of all prime t h e o r i e s i n t h e s e languages. One then shows t h a t a t h e o r y Ti i n C i behaves p r o p e r l y because of t h e e x i s t e n c e of some prime

Thus, one could r e s t r i c t o n e ' s a t t e n t i o n t o T , prime

extensions Ti+l i n C i+l,

c2

t h e o r i e s i n C1,

c3

t h e o r i e s i n C2, e t c .

Where o r d i n a r i l y t h i s complexity

grows t o o l a r g e f o r PA t o handle, with M at o u r d i s p o s a l , i n PA + l-CON(PA) one all nonstandard nE 1M1 ) , of can d e f i n e K t o c o n s i s t of T and, f o r all n ( a l l prime t h e o r i e s T i n C n ( as C n looks i n M ) . One has s u f f i c i e n t

not

t o model T and, with t h e f u l l t r u t h c l o s u r e c o n d i t i o n s a v a i l a b l e t o prove d e f i n i t i o n f o r M , a s u f f i c i e n t hold on t h e c o n s t r u c t i o n t o c a r r y t h e proof out within PA + l-CON(PA). By way of example, l e t m e n o t e t h e following: Application. Proof:

HA + 1-CON(HA)

Work i n PA

+

proves t h e Existence Property f o r HA.

l-CON(PA).

Suppose cpv is such t h a t HA

cp fi f o r any n.

C.A. S M O R Y ~ S K I

464

By t h e Completeness Theorem, f o r each n , t h e r e is a Kripke model

Ir, v vfi.

Now: i. The c o n s t r u c t i o n i s uniform so we have a uniform t r u t h d e f i n i t i o n f o r t h e sequence En, i . e . a formula T r ( 9 , * { such t h a t , f o r each n, Tr(E, is a t r u t h d e f i n i t i o n f o r $; and ii. t h e model M over which t h e s e S ' s a r e 0 )

constructed c a n o n i c a l l y embedds i n t o t h e base s t r u c t u r e s o f t h e two f a c t s , t h e model recognised i n PA Remark.

+

(cEn)* c o n s t r u c t e d by

The metatheory, HA + 1-CON(HA),

For:

HA

f

By t h e s e

en's above M

l-CON(PA) t o b e a model of HA i n which 3 v w

HA + Con(HA)

[

collecting t h e

Knn's.

is

i s not true. QED

is j u s t r i g h t f o r t h i s a p p l i c a t i o n :

cl-EP.

Con(HA) + cl-Ep

1 I-CON(HA),

I b e l i e v e t h i s goes back t o K r e i s e l .

Cf.

b u t HA + Con(HA) 2 . 3 . 9 and 6 . 2 of [4].

l-CON(HA).

]

REFERENCES [ l ] A.G. Dragalin, N e w f o n s of r e a l i z a b i l i t y and Markov's rule, Soviet Math. Dokl. 21 (1980), pp. 461-464.

[2]

H. Friedman, C l a s s i c a l l y and i n t u i t i o n i s t i c a l l y provably r e c u r s i v e f u n c t i o n s , i n : G.H. Miiller and D.S. S c o t t , eds., Higher Set Theory, Springer-Verlag, Heidelberg, 1978.

[s]

C . Smoryiiski, Applications of Kripke models, i n :

[4]

__________ , Consistency and

[5]

__________ , Nonstandard

[7].

r e l a t e d metamathematical p r o p e r t i e s , Univ. of Amsterdam Tech. Report 75-02, 1975. Models of Arithmetic, Lecture Notes, Utrecht, 1980.

[ 6 ] H . de Swart, F i r s t s t e p s i n i n t u i t i o n i s t i c model theory, J. Symbolic Logic 4 3 (1978) t PP. 3-12.

[7]

A . S . T r o e l s t r a , e d . , Metamathematical I n v e s t i g a t i o n of I n t u i t i o n i s t i c Arithmetic and Analysis, Springer-Verlag, Heidelberg, 1973.

[8]

----------,

[9]

----------,

Markov's p r i n c i p l e and Markov's r u l e f o r t h e o r i e s of choice sequences, i n : J. D i l l e r and G.H. MLlller, e d s . , Proof Theory Symposion, Springer-Verlag, Heidelberg, 1975. C o n s t r u c t i v i s i n g r e s u l t s obtained f o r i n t u i t i o n i s t i c formal systems by means of classical semantics, Univ. of Amsterdam Tech. Report 77-08, 1977.

THE LBJ. BROWER CENTENARY SYMPOSIUM A S . Troelstra and D. van Dalen (editors) 0 North-HoNandPublishing Company,I982

465

ON THE ORIGIN AND DEVELOPMENT OF BROUWER'S CONCEPT OF CHOICE SEQUENCE A.S. Troelstra Mathematisch Instituut Roetersstraat 15, 1018 WB Amsterdam

I.

ZNTROQUCTZON

1.1. The incroduction of che concepc of a choice sequence wichin the concexc of incuicioniscic reasoning and ehe solipsistic exploicacion of che creacive subjecc are probably L.E.J. Brouwer's mosc original concribueions c o che foundacions and philosophy of machemacics. In chis paper we shall accempc co sketch che developmenc of che nocion of choice sequences in Brouwer's (published and unpub1ished)wOrk.

1.2. Oucline of ehe concencs. After a brief characcerizacion of che French semiincuicionist or realist school in Seccion 2, we discuss che views of one of che mosc our-spoken semi-incuicioniscs, E. Borel, in Section 3. In parcicular we describe Borel's view of che concinuum. In his writings choice sequences appear in conneccion wich discussiow around che axiom of choice and che "arichmecical" cheory of che concinuum. Borel is prepared co accepc countabZe secs (sequences) of choices ac least in probabilicy consideracions, chough co him che "choice" characcer is someehing primarily negative: choice sequences are nor given by a law. In Brouwer's early work, before 1 9 1 4 , we find essencially the same view of che concinuum, with one difference: Brouwer had become aware of che episcemological consequences of incuicionism also for logic (Section 4 ) . In 1914 Brouwer for che firsc cime seems co accepc choice sequences as incuicioniscically accepcable, chough his nocion of "secs" is still ehe one of his chesis. Bur by 1917 he had che spread concepc, and knew how co exploic choice sequences in che concexc of incuicioniscic episcemology: the firsc concinuicy argumencs appear. By chis step choice sequences take on a posicive aspecc, and it seems righc co say chae this for che firsc time made choice sequences inco something "sui generis" (Seccion 5 ) . . Wherecheviews of Borel, and Brouwer before 1 9 1 4 , may be described as "holistic", i.e. che concinuum had co be understood as a whole by direcc incuicion (and noc as the cocalicy of its elemencs), afcer 1914 Brouwer's views can be described as,"analycic" - che concinuum is decermined by choice sequences and how chey are Sven. We show by many quocacions chac Brouwer gave a good deal of choughc co che quescion as co how choice sequences are supposed co be given, and chae he changed his mind several cimes (Seccion 6 ) . In Seccion 7 , we cry t o reconscrucc some of che more incrigueing concinuicy arguments of Brouwer. From a hiscorical poinc of view, such reconscruccions are somewhat speculacive and che resulcs have co be accepced wich che necessary caucion. The need for reconscruccion, che filling ouc of che argumenes, is given by che face chac from che viewpoinc of recenc knowledge, chey seem co u s ambiguous (permiccing, ac first sight;, more chan one incerprecacion), presumably because coday we are aware of possible discinccions and picfalls nor: mencioned by Brouwer (he may indeed noe have been aware of chem). We have cried co presenc chose incerprecacions /reconscruccions,keeping as closely as possible co che spiric of Brouwer's proof. The very face chac che interprecacion in some cases is nor: unique is nevercheless of inceresc, ic seems t o u s . (For cechnical and concepcual informarion on recenc developmencs we refer the reader co Troelsera 1977, 1 9 8 1 . ) Seccions 8 and 9 are racher by way of appendix, and give some comments on che developmenc of che spread nocion and che proofs of che fan and bar cheorem respecc ively

.

A.S. TROELSTRA

466

1.3. Earlier work; acknowledgments. In his thesis (1971) van Stigt discusses Brouwer's theory of sets and the continuum in Chapter IV. In Troelstra 1977, Appendix A a brief account of the history of choice sequences is given; the history there, however, starts with Brouwer 1914. Heyting, in a lecture of I975 (published Heyting 1981) gave an account of the early history of choice sequences and pointed to the connection with Borel; we have incorporated many of his remarks in the present account. H. Jervell, in an unpublished note, remarked that the term "choice sequence" indeed has its origin in discussions concerning the axiom of choice, and we are grateful to him for his permission to make use of his note for this paper; W.P. van Stigt and D. van Dalen gave valuable help in locating relevant material from the Brouwer archives. We have quoted rather often, in translation; we think this makes it much easier to follow the line of reasoning and the basis for it. For quotations from unpublished sources we have given the original Dutch or German text in footnotes.

2.

E. BOREL AND SEMI-INTUITIONIST SCtlOOL

2.1. A. Heyting in his book (1934 and 1955, 11) calls any mathematician an intuitionist who subscribes to the following two tenets: (a) mathematics is not only formal, but also has content; (b) mathematical objects are grasped directly by the thinking mind; hence mathematical knowledge does not depend on experience. Then he observes that the second tenet may be interpreted in at least two different ways : (bl) one thinks of a mathematical object as having an existence independent of our thinking, but we can only conclude to its existence, and investigate it by means of a mental activity reconstructing the the mathematical object in our mind; and (b2) mathematical objects exist onZy as mental constructions, at least we cannot base a mathematical argument on their existence independent of o u r knowledge. Heyting calls the point of view represented by (a)+(bl) semi-intuitionism; (a) + (b2) corresponds to Brouwerian intuitionism. 2.2. Around the turn of the century a group of (mostly French) mathematicians held views which might be described as semi-intuitionistic in the above sense: E. Borel, H. Lebesgue, R. Baire, N. Lusinl. H. Poincari? may be regarded with some justification as a precursor both of (semi-)intuitionism and of Hilbert-style formalism. These so-called semi-intuitionists have not presented their views in a unified and coherent way, and there is also considerable variation of viewpoint within the group. It seems that Brouwer learnt of the views of the semi-intuitionists in the first place through the writings of Borel; and we may assume him t o have been familiar with Poincarg's writings, as appears from references in Brouwer's early work. As we shall see, there are some striking parallels between the way Borel expresses himself and some passages in the early writings of Brouwer. 3. 3.1.

BOREL'S VIEW OF THE CONTINUUM

Borel regards natural numbers as unproblematic and our intuition concerning them as sufficiently clear to exclude misunderstandings. Already in the first edition of his book (1898) Borel states that only countable sets can be given; the continuum he assumes to be given to u s directly by "geometric intuition", Areal number exists when it is computable, i.e. can be approximated with any required degree of accuracy, but the continuurn cannot be exhausted by any list of computable reals (Cantor's diagonal argument). Later, Borel (1908, 1909, 1912) returns to the matter and now gives a much more detailed exposition of his ideas. In his lecture before the International Congress of Mathematicians (Rome 1908, attended by Borel and Brouwer) he states

Brouwer's choice sequence

467

(Borel 1909)

I t i s necessary t o say something on the notion of the continuum, the only wellknown example of an uncountable s e t , t h a t i s t o say the only one of which the mathematicians have a clear idea i n comon (or believe t o have, which i n practice amounts t o the same t h i n g ) . I regard t h i s notion as obtained from geometrical i n t u i t i o n ; one knows t h a t the complete arithmetical concept o f the continuum requires t h a t one admits the legitimacy of a countable i n f i n i t y of successive choices. This legitimacy seems t o me t o be highly debatable, but nevertheless one should d i s t i n g u i s h between t h i s legitimacy and the legitimacy of an uncountable i n f i n i t y of (successive o r simultaneous) choices. The l a t t e r concept seems t o me, as I have remarked before, e n t i r e l y meaningless. Where a countable i n f i n i t y of choices i s involved, [...I one can a t l e a s t indicate a aourse which, being f i x e d i n advance, guarantees t h a t any choice w i l l be e f f e c t e d a f t e r a f i n i t e amount of time. [...I If the i n f i n i t y of choice i s uncountable, it i s impossible t o think of a method of defining it, C.. .I: it i s therefore impossible t o distinguish it as a mathematical object which can be used i n proofs. I' 3.2. In Borel 1908 the paradoxes are discussed; here Borel distinguishes between countable ("d6nombrable") and enumerable ("effectivement Gnum6rable"); the individually definable elements of the continuum are a countable collection, but not an enumerable one. We quote I t i s impossible t o indicate a method f o r determining a point on the real l i n e which does not belong t o t h i s s e t :of individually definable pointsl; the asser-

t i o n t h a t such points & i s true or f a l s e , depending on whether one admits or does not admit the p o s s i b i l i t y of an i n f i n i t e of successive choices; but t h a t i s a metaphysical question, i n the sense t h a t a p o s i t i v e o r negative answer w i l l never have any influence on the progress of science: a l l points one could ever need are f i n i t e l y defined; they form the practical continuum used by the mathematicians. In Borel 1912, these choice sequences are admitted at least for probability considerations:

it i s possible t o define a deci' I People w i l l also agree on the following p o i n t : mal number of bounded length by asking thousand people t o write down, a r b i t r a r i l y , some d i g i t ; thus one obtains a well-defined number, i f a l l the persons are arranged i n a row, and each one w r i t e s i n turn a new d i g i t a t the end of the sequence of of d i g i t s already w r i t t e n by t h e people i n the row preceding him. But look where the disagreement s e t s i n : i s it possible t o d e f i n e a decimal number of unbounded length by a similar process? I do not think anyone ever thought of asking an i n f i n i t y of people t o write down a d i g i t each, a r b i t r a r i l y ; but Zermelo and Hadamrd seem t o think t h a t one may regard such a choice as being carried out i n a p e r f e c t l y determinate way and thus t h a t it i s possible t o t a l k of a decimal number defined i n t h i s way, even i f t h e complete d e f i n i t i o n of such a number obviously requires i n f i n i t e l y many words. On my part, I regard it as possible t o ask questions of probability concerning decimal numbers obtained i n t h i s way, by choosing d i g i t s , e i t h e r e n t i r e l y arbitrary, o r imposing some r e s t r i c t i o n s which leave some arbitrariness, but t h a t it i s i m possible t o t a l k about a single individual o f such numbers, since i f one denotes such a number by a , d i f f e r e n t mathematicians, i n t a l k i n g about a , w i l l never be sure t o be t a t k i n g about the same number. In 1912A Borel remarks somewhere that in an earlier paper he has proved a theorem for "sBries choisis au hasard" (arbitrarily chosen sequences or sequences chosen at random). He describes (Borel 1912) his position as "empiriste" or "rgaliste" and repeats his view of the continuum understood as a whole, not as the set of its definable elements, in the following terms: O f course it i s possible t o reason about a c l a s s of mathematical objects, f o r example about a l l real nwnbers, o r about a l l continuous functions, i f t h i s class

i s f i n i t e l y defined, although the individual members cannot a l l be defined i n t h a t way. Thus one obtains the general properties of the class, but one never actually reasons about a f i x e d individual of t h e c l a s s , unless it concerns a very s p e c i f i c individual which can be distinguished from a l l others by a f i n i t e d e f i n i t i o n .

468

A S . TROELSTRA

A s pointed out by Jervell in his note, the terminology of "an infinity of choices" clearly derives from the debate concerning Zermelo's axiom of choice (cf. Borel 1904, Borel et al. 1904) which is rejected by Borel, at least in the uncountable case. It becomes quite clear that Borel felt so to speak "the same" with respect to the choice functions provided by the axiom of choice on the one hand, and the choice sequences required by the arithmetical interpretation of the continuum on the other hand: for logical reasons it seemed they had to be there, but one could not get hold of them, one could not exhibit or describe them completely. Thus a sequence given by a countable infinity of successive choices is something distinguished by a negative quality, a lack of something: there is no law describing it, no explicit definition singling it out. A s we shall see, Brouwer, from 1914 onwards, started to look at choice sequences in a different way. Thus for the comparison with Brouwer's early views, Borel's writings from before 1914 in particular are relevant. 3.3.

3.4.

In 1914A, Borel expresses certain doubts as to the reliability of the geometrical intuition of the continuum (see the footnotes). It is also interesting to observe that occasionally he comesclose to rejecting classical logic (as in Borel 1914, page 220) but soon discards the issue. We may sum up the sort of set-like objects regarded as "existing" in the sense of Borel as follows: 1 ) finite or enumerable sets, 2) countable, but non enumerable sets such as all definable reals, 3 ) certain classes directly given by our intuition, in particular the continuum. 4.

T H E CONTINUUM IN BROUWER'S EARLY WORK 1 1 9 0 7 - 1 9 1 4 )

4 . 1 . The essentials of Brouwer's philosophy of mathematics are already present in his thesis ( 1 9 0 7 ) , and there are considerable differences in outlook between Brouwer and the French semi-intuitionists: Brouwer stresses that mathematics exists exctusively as a free creation of the mind, that language is not essential for mathematics and plays a r6le only when we want to comunicate mathematics; logic is not independent of mathematics, only a part of mathematics, and may be described as the study of the regularities which may be observed in our linguistic description of our mathematical constructions. Nevertheless, since the semi-intuitionists wanted to restrict attention to mathematical objects which, even if having independent existence, can be grasped as mental constructions, there is much similarity in some respects, in particular in connection with the theory of the continuum and the theory of sets. 4.2.

For Brouwer, the continuum is given in intuition, and cannot be seen as just the collection of its elements. Some quotations make this clear (Brouwer 1907, p. 8-9, CW 17): Having recognised that the intuition of 'fluidity' is as primitive as that of several things conceived as forming a unit together, the latter being at the basis of every mathematical construction, we are able to state properties of the continuum as a 'matrix of points to be thought of as a whole'. ' I ; and on page 6 2 (CW 4 5 ) Brouwer writes 'I However, the continuum a whote was given to us by intuition; a construction f o r it, an action which would create from the mathematical intuition 'all' its points as individuals, is inconceivable and impossible. I t . 4.3.

Brouwer remarks that our mathematical intuition can only create denumerable sets of individuals; but a set of order type n (i.e. of the rationals) can be overlaid with a continuum. Obviously a set is meant here (just as for Borel) as something which can be specified via its elements. "Denumerable" corresponds with "6num6rable" of Borel 1908A. On the basis of this insight Brouwer describes three ways for generating "pointsets of the continuum":

Brouwer's choice sequence

469

Construction of discrete, individualized sets of points which are finite, of order type w , of order type ri , and combinations of these (by taking ordered sums etc.). Given a set of type (A), we may consider whether it is dense in an interval or not; if it is, we can represent the approximations by an infinite binary tree representing a set' of order type q ; this may then be "overlaid with" a continuum. We may obtain further pointsets by deleting a dense set of points from a continuum in an interval. Apparently Brouwer regards this as an exhaustive' description of the intuitionistically possible pointsets; as a result, he regards the continuumhypothesis estab1ished Accordingly, on page 149 (CW 83) of his thesis he lists as the possible cardinalities for sets (now in the wider sense, including the continuum) lo. the d i f f e r e n t f i n i t e nwnbers 20. the denwnerably i n f i n i t e 30. the denumerably unfinished 4 O . the continuous. l r . Here "denumerably infinitel' is Borel ' s "6numdrable" (1 908A) ; "denumerably unfinished" are sets to which Cantor's diagonal procedure applies. Brouwer's first example of a denumerably unfinished "set" is the totality of all ordinals. He then remarks (1907, p.150; CW 83) But i f we introduce the logical e n t i t y : t o t a l i t y of the points of the continuum, abandoning the i n t u i t i o n of a continuum, we shall be forced t o define the points of the continuum, and t h i s i s only possible by definable laws ro r G i o n for a q p y o x i y t i n g dua! fractions. Now i n t h i s sense the c o n t G m isPde:umerabZy unfinzshe , and so zs the second nwnber c l a s s . So Brouwer's conclusions are quite similar to those of Borel 1908, 1909 (cf. end of 3.3, 3 . 2 ) . Brouwer 1908A summarizes the theory of cardinalities of his thesis; he distinguishes between "Menge" as something which can be specified by its elements, "Methode" (a principle of generation, e.g. for denumerablyunfinished sets the diagonal method for finding new elements) and "Matrix" (something completed, not a set, such as the continuum). Already in footnote 3 on page 142 of his thesis (1907, CW 7 9 ) Brouwer expresses his doubts3 as to the solvability of every mathematical problem, and in 1908C he has become fully aware of the fact that his intuitionistic epistemology conflicts with classical logic - a step not taken by the semi-intuitionists (except Borel in some of his later writings, long after Brouwer; e.g. Borel 1947, 19478, in a halfhearted way). In Brouwer 1914, the description of the intuitionistically possible sets is essentially still the same list as in his thesis. In 1917, Brouwer notes that one can also permit a set of order type n to be represented by the nodes of an inf i n i t e l y branching tree, making construction principle (C) above redundant. In the same paper he announces his new theory of sets (spreads), which then appeared in 1918B; it does not seem far-fetched to assume that the idea of a branching tree representing an order type ri plus a part consisting of isolated points, which Brouwer had been considering before, now suggested the more general spread concept. But the spread concept itself is not enough to obtain a new theory of the continuum; for that one has also to accept choice sequences as legitimate objects in intuitionistic mathematics.

.

4.4.

Special interest attaches to Brouwer's inaugural address (1912A, CW 123-138).

Cn reading it becomes clear that Brouwer tends to identify his views with those of the French semi-intuitionists. In form, it is an exposition of two opposed views: and two points of view may here be distinguished, intuitionism (largely French) and formulism (largely German) As in his thesis, Brouwer does not distinguish here between formalists 1 la Hilbert or logicists as Frege, or Platonists such as Cantor. Brouwer discusses among other things set existence and in particular the theory of the continuum:

...

A S . TROELSTRA

470

" Finally t h i s basal i n t u i t i o n o f mathematics, i n which the connected and the separate, the continuous and the d i s c r e t e are united, gives r i s e immediately t o the i n t u i t i o n of the linear continuum, i . e . of the 'between', . . . The intuitionistic theory of sets Brouwer then sketches is essentially still the one of his thesis. In connection with choice sequences, the following passages are illuminating (Brouwer 1912, CW 133-134) I' Let us consider the concept: ' r e a l number between 0 and 1 '. For the formalist t h i s concept i s equivalent t o 'elementary s e r i e s of d i g i t s a f t e r the decimal p o i n t ' , f o r the i n t u i t i o n i s t it means 'law f o r the construction of an elementary s e r i e s of d i g i t s a f t e r the decimal p o i n t , b u i l t by means of a f i n i t e number of operations'. And when the f o r m a l i s t creates the ' s e t of aLl r e a l nwnbers between 0 and 1 ', these words are without meaning f o r the i n t u i t i o n i s t , even whether one t h i n k s of the r e a l numbers of the f o r m a l i s t , determined by elementary s e r i e s of freeZy seZected d i g i t s , o r 'I and somewhat later " f o r the i n t u i t i o n i s t can only construct denwnerable s e t s of mathematical objects, and i f , on the b a s i s of the i n t u i t i o n of the linear continuum, he admits elementary s e r i e s of f r e e s e l e c t i o n s as elements of construction ... ' I . These quotations sound like an echo of the discussion in Borel 1912: at the outset choice sequences are a formalist concept, but the intuitionist might just conceivably accept them as objects; compare Borel's greater willingness to consider countable sets of choices as compared to uncountable sets of choices, and Borel's remark that sequences generated by choices may at least be used in probability considerations. Brouwer's identification of his own views with the semi-intuitionist views becomes quite apparent at the end, where he presents a "Brouwerian counterexample" to the proof of the Cantor-Bernstein theorem (in the spirit of Brouwer 1908C), as an example of intuitionist criticism of "formalist" reasoning.

".

...

...

4.5.

Later, when the differences between his own views and those of the semiintuitionists have become more apparent, Brouwer referred to them as "Alt-Intuitionisten" (Brouwer 1930; he lumps Borel and Poincar6 together), and still later "Pre-intuitionist school" (1952B, 1954A); in his Berlin lectures, he still refers to his own viewpoint as "Neu-Intuitionismus" (cf. 6.6 below). 4.6.

Sununing it up, we can say that in the treatment of the continuum, both the semi-intuitionists and Brouwer's views till 1914 might be described as "holistic", the continuum must be regarded as a whole, and cannot be understood as the totality of its (individually definable) elements. 5.

CHOICE SEQ-UENCES COMING INTO TffElR OWN; T H E FTRST CONTlNUlTY ARGUMENTS (1914-1919)

5.1.

Brouwer's review of the book by Schoenflies and Hahn (1914) is perhaps the first place where choice sequences are accepted without reserve from an intuitionistic point of view. In this review Brouwer states that each well-constructed set in intuitionism is composed of a countable part and an uncountable, (continuumlike) part, a consequence of the analysis of intuitionistically possible sets as given in his thesis (cf. 4 . 3 ) ; the elements of the second part are determined "by a sequence of choices from a finite set or a fundamental sequence". In 1917 (CW 145-146) Brouwer gives some corrections to his thesis. There he remarks that his earlier analysis was based on two assumptions, which we shall not describe here, but that it had become recently clear to him that these two assumptions could be dispensed with "as I hope to explain in a paper that will shortly appear", obviously referring to Brouwer 1918B, so in 1917 Brouwer must have had the concept of a spread ("Menge") with choice sequences ("Wahlfolgen") as its elements.

Brouwer's choice sequence

471

5.2. This is confirmed by the notes for his course on theory of pointsets (preserved in the Brouwer archives under BMS 91). Apparently Brouwer gave the course based o n these notes twice, for the first time in the academic year 1915-1916, and, so it appears, a second time in 1916-1917; on this occasion he added notes in pencil to the manuscript. In the original notes (written in ink) Brouwer proves the non-denumerability of the continuum by means of a diagonal argument, but in pencil we find a new proof (hence presumably not later than 1917) based on a continuity argument, where he writes4 I' The i m p o s s i b i l i t y of mapping a l l elements of f l t o d i s t i n c t elements of p follows from the f a c t t h a t the choice of the elements of p would have t o take place a t a c e r t a i n point of the (forever unfinished) choice sequence, and i n t h i s way a l l continuations of such a f i n i t e choice-branch determining the element of p w i l t obtain the image i n p . l'. Clearly, at this time Brouwer knew how to exploit the fact that he was dealing with choice sequences, i.e. unfinished objects. Continuity arguments are incompatible with classical logic, as Brouwer realized. The continuity argument above reappears,essentially unchanged,in Brouwer 1918B, at the end of Section 1 (CW 100). We also find the spread concept in the pencilled notes, in the following form5: " A mathematical object i s e i t h e r an element of a fundamentat sequence constructed i n advance (governed by induction, such as the sequence p ) F , o r a fundamental sequence f (forever unfinished, not governed by induction) of a r b i t r a r i l y chosen elements of F . (With such a sequence one can work q u i t e well, i f f o r a thing d o r fundamental sequence r t o be obtained from t h i s sequence one atways, a t each stage, needs t o work with a suitable i n i t i a l segment f o n l y ) , ( r i s then, generally speaking, also forever unfinished). Now a s e t i s a law, by means of which a d o r an r can be obtained from an f ; t h i s r may c o n z n , f o r example, also r e l a t i o n symbols as eZements ( e ; g . ord e r i n p ) , and thus t h e taw may lead t o e.g. well-ordered s e t s o r other ordered s e t s o r functions (however, i n t h i s manner one cannot obtain the s e t ordered s e t s or the s e t d w e l l - o r d e r e d s e t s ) .

".

5.3. Brouwer 1918B contains the first published definition of a spread. Basically, a spread is given by a) a method for determining a subtree (without terminating branches) of the universal tree consisting of all finite sequences of natural numbers, and b) a method for assigning to the nodes of the subtree other mathematical objects. (German: . "Zeichenreihe". In particular Brouwer permits the assignment of "nothing" to a sequence in the subtree, but in the discussion below we shall not take this into account.) (a) corresponds to Brouwer's decisions w.r.t. sterilized and non-sterilized nodes (German: "gehemmte" and "ungehenimte Folgen"), (b) to the assignment of a "Zeichenreihe". In 1918B, both the methods under (a) and (b) are rolled into one and assumed to be given by a taw ("Gesetz"); in 1920B and later Brouwer relaxes this: the method may involve a choice sequence. Accordingly Brouwer henceforth distinguishes between spread laws and spread d i r e c t i o n s (1981, 19548). We return briefly to this matter in Section 8. The term "Wahlfolgen" in 1918B refers t o the finite and infinite sequences of choices in the subtree. In the (incomplete) Dutch text of Brouwer's Berlin lectures (cf. 6.4 below) we find a more precise version of (b) above6: '' spread taw, according t o which, if an arbitrary natural number i s chosen as 'index' again and again, each of these choices e i t h e r pvoduces a ' s i g n ' i . e . mathematical thought-object ( a l s o 'nothing' may be regarded as such), or r e s u l t s i n the halting o f the process

...

... ".

5.4. In 1919A, 85 we find a statement classically equivalent to the BolzanoWeierstrasz theorem for the Euclidean plane; its proof is remarkable, inasmuch here without further explanation use is made of (what afterwards became known as the) fan theorem:

AS. TROELSTRA

412

...

'I hence i n generating the points of p one must be certain a f t e r a f i n i t e choice sequence M of fi3ced cardinal number m that a f t e r a f u r t h e r f i n i t e sequence of choices, of cardinality m l dependent on M , a square q i s generated

...

I'

(our underlining: the word "finite" implies a continuity argument, "fixed" amounts to uniformity on a finitely branching tree, i.e. the fan theorem). In 3952C Brouwer returns to the Bolzano-Weierstrasz theorem and now proves a form similar to the one of 1919A, but this time explicitly appealing to the fan theorem.

5.5. Summing up, we can say that after 1914, Brouwer's view of the continuum becomes "analytic": a spread somehow "generates" all its elements, and one can prove facts about the continuum by reflection on the way an element (choice sequence) is supposed to be given. This aspect will become much clearer in the next section. Notealso that, incontrasttoBore1 (cf. 3.3)Brouwer'sviewof choice sequences now has a positive turn: it is the (positive) result of the free activity of the mathematician; this positive view is combined with the insight how to exploit the intuitionistic epistomologicalposition, in the formof continuity arguments; and it is by this step that choice sequences come into their own. 6. RESTRICTIONS ON CHOICE SEQUENCES ( 1 9 2 4 - 1 9 5 2 ) 6.1. A s noted above, adopting choice sequences as elements permitted Brouwer to think of a spread as the totality of objects generated by sequences of free choices in a tree - thus liberalizing the semi-intuitionist position of Borel. More important, he realized one could turn the peculiar epistomological status of choice sequences as unfinished objects to advantage: the continuity arguments. 6.2. Initially, Brouwer does not say anything about possible restrictions on choice sequences; later, however, he becomes more specific and keeps on revising his ideas up till 1952. The historically earliest example of choice sequences subject to a particular sort of restriction we found in a letter of Brouwer to A. Heyting, dated 26-VI1924, where he writes7 I n the present context l e t me a l s o i l l u s t r a t e by an example the aforementioned assertion t h a t the doubling of the r e l a t i o n of support between a point and a point-species produces a new r e l a t i o n ( i n general): l e t R be the species of rationals, V the species of ' f r e e ' numbers of the continuum ( i . e . those numbers such t h a t i n t h e i r approximation by a convergent sequence of A-intervals the

choice o f t h e n e x t A-intewal within the previous one remains completely f r e e ) ; C the f u l l continuum. Then V i s contained i n the support R' of R , and the support R" of R' coincides with C . However, R' and C certainzy do not coincide. ' I . Here " X supports x " or " x belongs to the support X' of X " is defined by x where f

E

X'

=

-iVyeX(x

y)

f

denotes apartness; therefore we have

R' R"

=

{x : iVyeR(x

=

{x : iVzcR'(x 6 z)}

f

y)}

.

Below (7.3) we shall see how indeed Brouwer's argument can be completed in the context of the modern theory of lawless sequences. Here we only want to note that Brouwer in this case explicitly considers a collection of choice sequences subject to a specific second order restriction: no (first order), restriction on future choices of values.

Brouwer's choice sequence

413

6 . 3 . In 39258, Brouwer essentially repeats his spread definition of 1918B: he appends a footnote to the phrase "Each sequence of signs3, generated in this manner by an unlimited choice sequence . . . ' I , and in footnote 3 we read: '' Including the feature of t h e i r freedom of continuation, which a f t e r each choice

can be limited a r b i t r a r i l y (possibly t o being f u l l y determined, but i n any case according t o a spread law).

".

6 . 4 . In 1927, Brouwer lectured in Berlin; at one time he planned to rework his notes into a book on intuitionism. Nothing came of it, however; but in the Brouwer archives under no. BMS 3 2 the manuscripts of his lectures in several variants are preserved. From numerous cross-hatchings and additions and corrections scribbled in the margin it is obvious Brouwer must have worked on these notes after 1927. One of the revisions in the Dutch version consists of a very elaborate footnote on restrictions': " The freedom t o proceed with the choice sequence can a f t e r every choice be arbidependence on events i n the world of mathematt r a r i l y restricted (possibly i n i c a l thought of the choosing person, imposed on the choosing person) (resulting e.g. i n complete determination, or determination by a spread l a w ) . The arbitrary nature of t h i s r e s t r i c t i o n , permitted a t each n e w choice as long as the p o s s i b i l i t y t o proceed i s retained, i s an essential element of the f r e e becoming of the element of the spread, as i s the p o s s i b i l i t y t o link t o every choice a r e s t r i c t i o n of the freedom t o make further r e s t r i c t i o n s of freedom e t c . 'I.

...

Three points are worth noting here: (a) Brouwer refers here explicitly for the first time to the possibility of making choice sequences dependent on the acts and ezperiences of the (idealized) ma thematcian; (b) (first-order) restrictions may consist of something other than restrictions to spread laws only; (c) higher order restrictions are explicitly mentioned. 6 . 5 . In his edition of Brouwer's Collected Works Heyting mentions (CW 5 9 0 ) that Brouwer in 1929 revised his paper 1925A by inserting corrections in all his copies. The footnote on the possibility of continuation reads in this version: '' Including the feature of t h e i r freedom of continuation, which a f t e r each choice

can be limited a r b i t r a r i l y (possibly t o being f u l l y determined, but anyway according t o a spread l a w ) . The arbitrariness of t h i s possible r e s t r i c t i o n , which does not violate the p o s s i b i l i t y of continuation, adds a new element of arbitrariness t d t h i s choice sequence and i t s continuations. I t i s also possible t o j o i n a well-ordered species of r e s t r i c t i o n s t o the spread (e.g. a r e s t r i c t i o n of the existing freedom of adding r e s t r i c t i o n s on future choices).

".

Here first order restrictions are supposed to be given by spread laws; there is no explicit reference to the activity of the idealized mathematician; the restrictions may be presented in a well-ordered hierarchy. This time, Brouwer does not state that the restrictions should always be given by spreads; on the other hand, he now permits higher order restrictions. Another passage' in the original Geman text of BIG 3 2 reminds us of the "completely free sequences" in the letter to Heyting (cf. 6 . 2 ) . " I n f a c t , by admission of i n f i n i t e choice sequences Neo-intuitionism moves s t i l l further away from the principle of the excluded third, since it not only makes it 6.6.

inadmissible (which was already the case), but even completely nonsensical; the arbitrary & f i n i t e binary expansion, which may also be constructed .in forever remaining freedom can not possibly be regarded as e i t h e r d e f i n i t e l y rational, or d e f i n i t e l y (negatively) irrational; here it i s even meaningtess respectively incorrect t o assert that it i s either rational or irrational - nevertheless t h i s . meaningZess assertion is non-contradictory

. .

Here we have rendered the German "widersinnig" by "meaningless". We do dot see how to reconstruct this argument in terms of the modern theory of lawless sequences however: a lawless binary expansion would be nonrational.

A.S. TROELSTRA

414

6 . 7 . A shortened version of the footnotes cited in print in Brouwer 39428 " arbitrarily restricted mination, or according to a spread law) to each second order restriction limiting the arbitrariness

...

...

6.5 and 6 . 4 above appears in

(for example to complete deterrestriction can be added a of further restrictions etc.".

6.8. In the text of his Cambridge lectures, dating from around 1950 (see van Dalen's editorial preface to Brouwer 1981) Brouwer has second thoughts on higher order restrictions. On the continuation of a choice sequence (there called arrow) he notes (page 13 of Brouwer 1981) Finally the freedom of proceeding, without being completely abolished, may a t some p v undergo some r e s t r i c t i o n , and l a t e r on f u r t h e r r e s t r i c t i o n s . to which he appends a footnote: I n some former publications of the author r e s t r i c t i o n s of freedom of future res t r i c t i o n s of freedom, r e s t r i c t i o n s of freedom of future r e s t r i c t i o n s of freedom of future r e s t r i c t i o n s of freedom, and so on, were also admitted. But a t present

the author i s inclined t o think t h i s admission superfluous and perhaps leading t o needless complications. ' I . In 1952B, Brouwer strengthened this (page 142, footnote) to However, t h i s admission i s not j u s t i f i e d by close introspection, and moreover would endanger the s i m p l i c i t y and rigour of f u r t h e r developments. If.

... .

6.9. Creative subject arguments. As we have seen, the actions of the "idealized mathematician" ( IM for short) or "creative subject" are already explicitly mentioned in connection with choice sequences in Brouwer's Berlin lectures (implicitly of course the idea of the IM appears already much earlier - perhaps as soon as weak counterexamples are presented). But the type of arguments involving what one might call the "solipsistic exploitation" of the IM first appear in print in Brouwer 1948A (though Brouwer there remarks he had used such examples in lectures since 1927). The first example af solipsistic exploitation was a weak counterexample (of the form "we have no grounds to assert that") but already in 19498 Brouwer gave a strong counterexample ("it is false that") based on solipsistic exploitation. van Dalen observes that ( I ) the Cambridge Lectures, (Brouwer 1981), even while in their final version dating from after 1949, use the solipsistic exploitation only in weak counterexamples, and ( 2 ) that only in 3954F Brouwer published a strong counterexample with "solipsistic exploitation" in English. The principal new element in the solipsistic exploitation is not so much in thinking of the activity of the IM going on in a countable sequence of discrete stages, and the construction of mathematical objects by reference to those stages, but in using at the same time as a mathematical t o o l the insight that a l l mathematical activity is carried out in those stages and hence that something, in order to be found true by the I M , must be established at some stage. 6.10.

Summing up, the foregoing display of footnote-scholarship shows beyond doubt that Brouwer spent a good deal of attention on the question as to how an individual choice sequence was supposed to be given, and that he changed his mind several times . A s will become clear in Section 7, for some of Brouwer's arguments it was certainly a relevant question. It is quite understandable that Brouwer rejected the idea of higher order restrictions because of the (mind-boggling) complexities which might arise. This in itself is no argument against very special and simple second order restrictions (such as imposing no higher order restriction, or restricting first order restrictions to spread laws). What Brouwer meant to say by "moreover not j u s t i f i e d by close introspection" remains conjectural; e.g. he may have felt higher order restrictions did not agree with the freedom of the idealized mathematician in creating choice sequences.

415

Brouwer's choice sequence 7.

ANALYSIS OF CERTAIN CUNTINUITY ARGUMENTS

7.1. In this section we shall indulge in a, strictly speaking, ahistorical activity: the reconstruction or completion of certain of Brouwer's proofs in the light of present knowledge. Nevertheless we think this may also help u s to understand Brouwer's views. These reconstructions may be anachronistic where they use distinctions and refinements for which we can find no evidence in Brouwer's writings. Such reconstructions may nevertheless be historically relevant, for example in showing that an argument may be reconstructed in several different ways (supposing each of them in itself compatible with Brouwer's views): from this one learns to interpret the available data with a good deal of caution. 7 . 2 . We first make a general remark concerning continuity assumptions for choice sequences. Ifweare thinkingof a choice sequence as a process which is completely free, that is to say at any stage we only know a finite piece, plus the possibility of continuing this (according to a spread law say), it is clear that any assertion concerning such a sequence must be made on the basis of an initial segment (finite piece) alone. This might be termed a "local" argument: it applies to an individual sequence. At first sight, this reasoning is behind Brouwer's first application of continuity (see the quotation above (5.2) from his course notes), though we cannot be certain. However, where we are considering a continuum containing not only completely free elements but also more or less restricted or completely determined elements, we cannot conclude that a mapping from such a continuum to e.g. the natural numbers must be continuous by the local argument; for a completely determined sequence we have much more information than just an initial segment. Here we need a global consideration: an operation which should be applicable to aZZ choice sequences and which always yields a definite object (e.g. a natural number) should be determined from initial segments, since it ought to apply everywhere, also to completely free sequences. That is to say, the operation should be continuous because in general we cannot assume that we know more than an initial segment of any given choice sequence. Brouwer's19240 and 19278 papers postulate continuity without any elaborate argumentation, in the words (CW 393): It Then there e x i s t s i n l~ a detachabte countabZe subspecies P I of continuable f i n i t e choice sequences, such t h a t alZe elements of M belonging t o the same element of u 1 .get the same naturaZ number 5 assigned t o them, C.. .I. ( A continuable

element of l~ shouZd be counted as belonging t o p l if, and onty if, f o r t h i s element - but not f o r any of i t s proper p a r t s - according t o the algorithm of the assignment the decision concerning 5 i s not postponed t o f u t u r e choices; / Since Brouwer at this time was regarding determinate choice sequences as part of the continuum it is plausible to assume a global consideration like the one given above is the basis for this continuity statement.

... ".

7 . 3 . However, we shall now return to the argument involving completely free sequences in Brouwer's letter to Heyting from 1924 (see 6 . 2 above). We shall identify Brouwer's "completely free" with "lawless" as used in the recent literature. Brouwer's argument can be completed if we adopt the following principles (a) open data: what can be asserted for a lawless sequence, can be asserted on the basis of an initial segment and should hold for all lawless sequences starting with the same initial segment. (b) If a natural number is assigned to each lawless sequence this is done on the the basis of an initial segment. Let 0 be-such an assignment; Take any sequence 5 ; consider <, , gn = etc. and think at any stage of these finite segments as if they were initial pieces of a lawless sequence_, and keep trying to apply 0 on the basis of the information given by gn alone (that is to say, we keep trying to apply 0 to

...

...

A.S. TROELSTRA

416

-in for ever increasing n , "forgetting", or "abstracting from" any extra information concerning 5 beyond En). Then Ip will ultimately yield a number. This we call the "extension principze" since it permits us to extend Q to all sequences. Now we briefly sketch a formal proof of Brouwer's assertion in the theory of lawless sequences, as expounded e.g. in Troelstra 1977. Readers not familiar with the recent theory may skip this excursion. To keep technical complexities down, we replace R by

Q

{cpn

=

: n

Ri}

E

with cpn

=

n * Xx.0

I

We have to show Proposition. (i)

LS c Q'

(ii)

Q'

(iii)

mRi c

Proof.

For

#

IP (LS)'

,

LS c Q'

#.

Q" =

i.e.

we have to show VaWtl(a+cpn),

i.e.

VaWn3x(ax # Qnx)

,

. .

then also Vy3xry(ax # cp- x) , so Vy3x>y(ax # 0 ) aY To refute this, apply first open data, and let m be such that Assume Vdx(ax # qnx)

choose y

VyVB E m3x 2 y(6x # 0) ;

m,

a

E

=

lth(m)

,

then

VBEm3x21th(m)(Bx x

the extension principle to

0) .

=

B

is found continuously in

5

from a sufficiently long initial segment; apply =

m * Ax. 1

to find m' z m

and an x

2

lth(m)

such

that m

*

Ax. 1

consider a lawless

E

m', YE

m"

Vy

,

m

*

E

m'(yx

Ax. 1

0) ;

= E

m"

,

lth(m")

>

x ; then a contradiction

follows.

Q' # ININ

is obvious, since Ax. I

Finally, assume Vu3x(ux # Ex) assume a

cp

y:a

for all

n

element of

,

hence

1".We

Xx.1

4 Q'

.

shall tacitly

to range only over lawless sequences not entering into the construction

of

6.

gy

and an x

-

, 5

With continuity and the extension principle we can find an initial segment such that

.

VB E SY(BX # 5x) We may assume y > x ; let now a

E

C y , then we have a contradiction.

Remark. NH might in this argument be interpreted as consisting of all sequences obtained by applying lawlike continuous operations to n-tuples of lawless sequences for all n . 7.4. Brouwer's argument for weak continuity in 1927B. Some preliminary definitions: A fuzz function on C0,ll is a real-valued function which is everywhere defined on C 0 , l l . f is said to be negativeZy continuous in x if for each sequence <%>n converging to x , n i s said to converge negativety to a if 13pVn3m(Ian+m - al > 2-P) (p,n,m E a) It has puzzled certain authors (e.g. Parsons 1967, p.448 footnote g) that Brouwer later establishes a much stronger theorem in the same paper. I don't

.

411

Brouwer's choice sequence

think there is much of a mystery here; Brouwer may well have regarded this as a simple warming-up exercise. (We should also point out that Parsons' remark that the stronger statement to the effect that a continuous (in x) modulus of continuity for f at x requires the fan theorem for its proof, is incorrect, as observed by Veldman (1981, Stellingen no. 3 ) . Heyting (1981) mentions at least three possible interpretations - but these need to be supplemented before they can be compared. Below we shall first outline one possibility, then proceed to sketch briefly another, less likely reconstruction. But first we shall sketch Brouwer's proof of the Theorem. A full function is everywhere negatively continuous. Proof. -

Let

f be a full function, x

reals converging to Assume for some p If(xn,) - f(x)l

> 2 -'

a real number, and

;

there is an n r > n

we write x'

such that

f o r this

x n' ' as follows. Start from a sequence of rationals


converging to x , then construct x for the time being, for each n 2

a sequence of

X.

that for each n

Now we define a real x

i

<x>

already considered we let a! = ai for all

...

n , but retain the freedom to decide, once a; a' m have been chosen, that are determined in such a way that either i converges to

all further x

or to some x'

Then for x hence

.

lima' f(x,) w n n 13pVnVm(\f(xAm)

is undefined, contradicting the fullness of

=

-

f(x)l

>

2-'

f.,

.

Brouwer claims to have had this theorem (and presumably with this proof) since 1918. Our first reconstruction of the reasoning rests on the assumption that the proof is based on an "abstraction argument", as indicated above under (b) of 7 . 3 . To be more specific, let can,$m be a sequence of rationals such that IX;

I < - a n o

z-"-~,

We consider the following tree-arrangement

478

A S . TROELSTRA

Each branch in this tree obviously converges to a real number in

C0,ll ;

f

is therefore defined for the limit of each branch, and a rational approximation of the value of

f up to

think of

can be given.

2-'-'

Now consider n, and for any n

as the initial segment of a sequence in the tree T which is

as yet completely free, but afterwards may be narrowed down so as to coincide with

or become of the form aO'. - .y a n+m,a,i+m,l,a,i+m,2,... , and try to approximate n n f up to 2-p-l on t h i s assumption ( s o we "forget", or "abstract from" the information that all the

we are considering, are initial segments of the given

) ; ultimately we shall find an answer on the basis of or x as limit, both with the But then the possible continuations may have x ' "0

same approximation up to

I f(x' ) "0

-

f (x) I < 2-'

,

2-p-l

for

f(x' "0

) , f(x) ; and this would imply

which is contradicting our assumption

If (x:)

- f (x) I

>

2-'

.

Our reconstruction above may be regarded as an elaborate version of Heyting's first interpretation (1981, page 131). It is e s s e n t i a l , however, to bring in something like the abstraction argument - otherwise one runs into the sort of difficulties described by Posy (1976, p.110-112). (In Posy 1976, e.g. on page 97, in the description of " B " , the use of the abstraction process seems to be misunderstood; this is indeed corrected in Posy 1980, footnote 8.) Actually, the abstraction argument is certainly compatible with (limited) "proto-Christian charity" (cf. footnote 18 in Posy 1976). Instead of this type of reconstruction, Posy (1976, p.113-114 and 1980, p.148) proposes a creative subject argument, to which we shall briefly return below.

419

Brouwer's choice sequence

7.5. Second reconstruction. Heyting's second suggestion (Heyting 1981) for the interpretation of the proof is, that for all y represented by branches of the spread T described above f(y) should be defined, and that the continuity argument applied to the construction of a rational 2-P-I-approximation of f , for all y , yields a contradiction. On the one hand this is fairly close to the preceding reconstruction, but on the other hand it comes so close to a global continuity argument,which also seems to be underlying Brouwer's proof of the uniform continuity theorem in the same paper, that the presentation of a separate proof becomes more puzzling; in addition, the explicit reference to "retaining the freedom to continue" does not seem to play a r81e in this reconstruction, which we regard therefore as less likely. Heyting suggests still a third interpretation of Brouwer's proof: a mathematician constructs < a : > , and keeps on choosing :a = an without committing himself to this course of action for the future. Heyting states this brings us close to the theory of the ideal mathematician. Not very essentially s o , however; at least this is still far removed from the "solipsistic exploitation" of the idealized mathematician found in Brouwer's later papers. Also the interpretation is not too different from the first interpretation offered by Heyting (which corresponds roughly to the much more detailed reconstruction in 7.4 above). Another possibility for expanding Heyting's "third interpretation" is making it into a weak counterexample argument, as follows. Let Vn(Pn v 1Pn) but

3nPn unknown at present. We construct n n as long as 13kSn Pk a* = a n n a* = a'k,n+l-k when 3k
defining x

be known,

as follows

.

Then, at presentwedo not know whether will converge to x (and thus n n x = x ) or to one of the x' But since we know that f(x,) is defined,

.

such that and a subsequence <XI> of <x > n n n n However, this yields a weaker conclusion than the one

this means we cannot indicate a If(xi)

-

f(x)l

>

ZPp now.

p

stated by Brouwer: we only get that x

f cannot now be shown to be discontinuous at

(f is discontinuous at x when

<XI> and p as above can be found). n n Posy's reconstruction (1976, p.113-114) resembles this argument, though he does seem to mean something slightly different" (and upholds the strong conclusion of Brouwer's theorem) but we cannot completely follow his reasoning, which in any case seems to us somewhat anachronistic as an interpretation of Brouwer's thought. A reconstruction which would be based on the "solipsistic exploitation" of the creative subject seems to us to be undoubtedly anachronistic; though Brouwer (19488) states he had been using such arguments since 1927, he certainly did not use them in his publications before 1948, and in addition he states that theorem 1 of 1927B was known to him since 1918, as already mentioned. (It perhaps deserves mention here that Brouwer already in 1921 (CW 236) remarks, between parentheses, that a discontinuous function cannot be everywhere defined. Allin all, we think the first reconstruction is still the best we can do at present. 7.6. In I9498 (and later) Brouwer combines the "solipsistic exploitation" of the IM (cf. 6.9) with the fan theorem. We note here that the argument in the second part of Brouwer I9498 (reproduced in Heyting 1956, 8.1.2) makes essential use in the freedom of imposing or not imposing f i r s t order r e s t r i c t i o n s on our choice sequences.

480 ti.

A S . TROELSTRA

THE NOTION OF A SPREAQ

8.1. Over the years Brouwer's description of spreads varies somewhat, becoming generalized in one direction, restricted in another. Heyting (1981) gives a concise account in the first half of Section 3. Brouwer treats spreads and all the notions and terminology in extenso in his Cambridge Lectures (Brouwer 1981); here van Dalen's comments are quite helpful. Most of the variations in the formulation of the spread notion are of a technical nature and therefore not directly relevant to our understanding of Brouwer's ideas concerning the continuum and choice sequences. Here we shall briefly consider such aspects as touch on the concept of a choice sequence. 8.2. We recall (5.3) that a spread required two methods (a) and (b). In Brouwer 1918B, (a) and (b) are assumed to be given together in a law ("Gesetz"); this definition is repeated in 1919D, 1925B. In 1926B, Brouwer generalizes the spread notion by permitting the method (a) to be given by a choice sequence: I' such t h a t by means of an i n f i n i t e sequence of decisions f o r each non-sterili z e d f i n i t e sequence of choices it i s decided whether it w i l l be s t e r i l i z e d or

...

.

remain non-sterilized This feature is retained in later publications. In 1942A,C, Brouwer considers a further generalization, in which also may be specified by a choice sequence a taken from a spread M the method ( b ) (in the original sense), and then proves in 1942C that such a second order spread M, is a subset ("subspecies" in the orthodox intuitionistic terminology) of a spread M I constructed from M ; M I is the union of the M, if a ranges over M . Similarly one may define higher order spreads; they can be reduced in the same way. For a modernized version of Brouwer's argument, see Troelstra 1969, 11.6. The relaxing of (b) to a non-lawlike method is not considered anymore in Brouwer's later papers. The generalization of the "method" under (a) to a choice sequence is retained, however. Heyting (1981) observes that to avoid circularity in the explanations, one may assume the choice sequence determining the subtree to belong to a lawlike spread. This is in keeping with Brouwer 1942C.

9.

BAR THEOREM ANQ FAN THEOREM

9.1. Brouwer obviously felt dissatisfaction with the semi-intuitionist treatment of the continuum as a primitive notion; neither did he regard the reduced continuum as a good basis for mathematics. A s he observes in 1930A, the "reduced continuum" has measure zero (in the sense that at any moment we have only countably many points, which can'be covered by a sequence of intervals of arbitrarily small measure). This does not mean he does not consider the reduced continuum as a meaningful species - at least he discusses in 1930A a number of its properties. A l s o in Brouwer 1981 a lot of attention is given to the reduced continuum. 9.2. However, for a satisfactory theory of functions and a good measure theory Brouwer wanted t o guarantee the uniform continuity of full functions. For this he needed the fan theorem, which he simply accepted without proof in 1919A. In 1923, he attempts to prove the uniform continuity theorem by considering the possible forms an operator defined on all choice sequences could have; then in 1924D, 1924G and again in 1927B he gives the "proof" based on a fundamental assumption concerning intuitionistic proofs of well-foundedness. With variations this proof appears in many of Brouwer's later papers; it has been frequently discussed and analyzed (e.g. Kleene-Vesley 1965, Dummett 1977, Martino & Giaretta 1981). The analysis of this proof, however, does not seem to teach us anything more about the concept of a choice sequence as such, beyond what is already used in justifying continuity arguments.

Brouwer's choice sequence

48 1

The justifications of bar theorem and fan theorem do not seem to profit very much from the analytical approach to choice sequences. Therefore we shall not enter into these discussions here. 9.3. Concluding remarks. It seems there have been two basic motives for the introduction of choice sequences by Brouwer. The first one is, to do justice to the intuitive content of our idea of the continuum, and the second, to provide a basis for the theorem that a function on a bounded closed interval has to be uniformly continuous. How are we to judge of the success in attaining these two aims? Whether the intuitive content of the continuum has been caught by the notion of a choice sequence will remain to some extent a matter of personal insight and feeling. On the one hand, it seems to us, Brouwer's later theory of the continuum is far more coherent and successful in catching the underlying intuition than the approach of the semi-intuitionists. On the other hand, further development of the "analytic point of view" has taught us that the intuition of the continuum is a thorny affair, much more complicated than is suggested by Brouwer's writings. O f lasting interest, however, is the discovery of choice sequences as objects forcing non-classical logic upon us. As to the second motive: ironically, nowadays the development of constructive recursive analysis (as expounded e.g. in Kulner 1973) and Bishop's constructive mathematics (Bishop 1967) has taught us that a good measure theory for, what in Brouwer's view would be the reduced continuum, is possible. Constructive recursive analysis can even e q l a i n the apparently contradictory outcome that C0,ll has measure zero: C 0 , l l can be covered by intervals <[an,bn1>,, an,bn rational, with Xn,o Ibn-anl < E for any E > 0 , but such a sum does not converge positively when E < I As pointed out above, the struggle with the question "how are choice sequences given" (the analytic approach) has left no visible traces in Brouwer's proofs of the bar theorem and the fan theorem; and, as we noted, bar theorem and fan theorem are not essential for a constructive development of mathematical analysis. In addition, Kleene (in Kleene & Vesley 1965) argued more or less successfully that some of the assumptions underlying the proof of the bar theorem are certainly not simpler than the bar theorem itself. Whether these assumptions are more evident than the bar theorem itself, remains perhaps debatable, and it would carry us too far to discuss this point here. On the other hand, Brouwer's proof of the bar theorem suggested the philosophically interesting notion of "canonical proof" (Dummett 1 9 7 7 ) . Thus perhaps choice sequences have not yielded the harvest Brouwer expected but their study has been rewarding in ways not foreseen by Brouwer, and without doubt they will remain of interest to the philosophy of mathematics.

.

NOTES I)

Hadamard i s only marginally connected with this group. Of the semi-intuitionists, E. Bore1 and N. Lusin have given the fullest explanation of their ideas. An excellent brief account is in Heyting 1934, 1 9 5 5 (12); a lengthy survey with an extensive bibliography is tb be found in Bockstaele 1949. As to Poincar6, see Mooij 1966.

2)

Apparently, Brouwer regarded it as perfectly clear that the intuitively graspable construction principles could yield only such pointsets and no others, when afterwards (Brouwer 1917) he realizes that some strongly limiting assumptions are introduced, which are not really needed, he remarks: " Now it is true that from the intuitionist point of view the unrestricted comprehension axiom cannot be used (see below under 7 ) ; therefore it is impossible to avoid special hypotheses about the way in which the pointsets under consideration are constructed, and This implies the right to consider as contained in the construction principles such hypotheses as are desirable for the viability of the theory. However, "

... .

...

A S . TROELSTRA

482

and then follows the announcement of the spread concept. As a defence of Brouwer's earlier position this is none too convincing. Among the notes which Brouwer made in preparation of his thesis, we find a remark " Is logic a reliable means for discovering natural laws? No, since nature i s not a mathematical system. Are the principles of contradiction and the excluded third true? (It will appear that they are not true: the issue of the unsolvable mathematical problems ) In Dutch the quotation reads " Is de logica een betrouwbaar middel tot het opsporen van natuurwaarheden? Neen, want de natuur is geen wiskundig systeem. Zijn de principes contradictionis en tertii exclusi waar? (Het zal blijken dat ze zelf & waar zijn: kwestie der onoplosbare wiskundige problemen ) (I owe this quotation to D. van Dalen.) On the other hand, the misinterpretation of the principle of the excluded third on page 131 of Brouwer 1907 (CW 75) seems to indicate that Brouwer had not yet completely realized the consequences of his own views.

... ".

... ".

The Dutch text reads: " dat het omgekeerd onmogelijk is alle elementen van f l af te beelden op verschillende elementen van p , v o l g t z r u i t , dat de keuze van het element van p dan zou moeten plaatsvinden op zeker punt van de (iuimers nooit aflopende) keuzenreeks, en op deze manier krijgen alle verlengingen van zulk een eindige keuzentak, die het element van p bepaalt, hetzelfde beeld in p 'I.-

...

.

The Dutch text reads: " Een wiskundig ding is 6f een element uit een te voren geconstrueerde fundamentaalreeks (door inductie beheerst,zoals de rij p ) F , , bf een fundamentaalreeks f (die nooit af is en niet door inductie beheerst wordt) van willekeu(Met zulk een reeks kan men zeer goed werken rig gekozen elementen uit F als men voor later uit af te leiden ding d of functiereeks r altijd maar in elke fase met een passend beginsegment van f heeft te werken), (r is dan _-i.h.a. 66k nooit af). Een verzameling is nu een wet, waarmee uit een f een d of een r wordt afgeleid; deze r kan danb.v. als elementen ook relatie-symbolen (b.v. ordenende) bevatten, zodat de wet b.v. tot welgeordende verz. of andere geordende verz. of tot functies kan voeren (overigens kan men zo niet komen tot de verz. geordende verz. of welgeordende verz.).

.

".

The Dutch text reads: " spreidingswet, volgens welke als telkens een willekeurig natuurlijk getal als "index" wordt gekozen, elk dezer keuzen hetzij een "figuur" d.w.z. een wiskundige denkbaarheid (als hoedanig ook het "niets" kan fungeren) voortbrengt, hetzij de sluiting van het proces . From handwritten lists with the names of students attending Brouwer's courses, and from the official list of courses given at the University, it appeared Brouwer lectured twice on the theory of pointsets: in 1915-1916, and again i.n 1916-1917. We must assume the pencilled notes in BMS 91 date from 1916-1917.

...

.. ".

The Dutch text reads: " Zij R de soort der rationale getallen, V de soort der 'vrije' getallen van het continuum (d.w.2. der getallen bij welker benadering doormiddel van een convergente intervalreeks van A-intervallen de keuze van ieder volgend hinterval binnen het vorige volkomen vrij blijft); C het volledig getallencontinuum. Dan behoort V geheel tot de aanleuning R' van R , en de aan1eunin.g R" van R' valt samen met C. Doch er is Eeen sprake van dat R ' met C samenvalt. "

483

Brouwer's choice sequence

8) The Dutch text of the footnote reads: I' De vrijheid van voortzetting der betrokken keuzenrij (eventueel in den kiezer opgelegde afhankelijkheid van gebeurtenissen in de wiskundige gedachtenwereld van den kiezer) kan na iedere keuze willekeurig (b.v. tot volledige bepaaldheid, of ook volgens een spreidingswet) worden beperkt. Het willekeurig karakter dezer bij iedere nieuwe keuze onder behoud der voortzettingsmogelijkheid geoorloofde beperking der voortzettingsvrijheid is een essentieel onderdeel der vrije wording van het spreidingselement, evenals de mogelijkheid om aan iedere keuze een beperking der vrijheid van verdere vrijheidsbeperkingen te koppelen, enzovoort. Een element waarvoor de nummerkeuzea van den aanvang af door een wet eenduidig zijn vastgelegd, wordt scherp genoemd. It is not absolutely certain that the pages in Dutch of BMS 32 belong to an incomplete draft in Dutch of the Berlin lectures, through they were found together with these; it is possible they should be dated somewhat later (though pre-war)

".

.

9)

The German text reads: " Indessen entfernt sich der Neu-intuitionismus durch diese Zulassung der unbegrenzten Folge freier Wahlen noch weiter vom Satz V. a. D., indem er ihn nicht nur unzulzssig (was er schon war), sondern sogar ganz widersinnig macht; der beliebige unendliche Dualbruch, der ja eventuell in unbegrenzt bestehen bleibender Freiheit konstruiert werden kann, kann ja unm6glich als sei es sicher rational, s e i e s s k h r (negativ) irrational gelten; es ist also hier sogar widersinnig bezw. falsch zu behaupten, dass er entweder rational oder irrational sei; trotzdem ist diese widersinnige Behauptung widerspruchsfrei.

".

10) There is an obvious sense in which epistemic notions appear in Brouwer's

writings before 1948, for example in the distinction between "absurdity" and "not (proven)"; Brouwer certainly distinguishes between the conclusions drawn from weak counterexamples and contradictions. So far we agree with Posy (1976) It is also clear that weak counterexamples fit readily into the picture of the idealized mathematician (the novel element in Brouwer (19488) is the use of Vn(kN

+-&).

On the other hand, for Posy's "second reading of -P" (page 115) we see no evidence in Brouwer's writings. Posy's formula (32) (page 114) ought to hold equally well for a non-lawlike f , provided f is given to u s right from the beginning as an everywhere defined function: the successive choicesentering in the construction of f are only permitted as long as there is a guarantee that f will turn out to be everywhere defined, from which we conclude that this part of Posy's argument cannot be an essential ingredient.

REFERENCES The numbering of Brouwer's publications corresponds with the numbering of the Collected Works (Brouwer 1975). Page numbers of the collected works are preceded "CW" e.g. "CW 73-75" refers to pages 73-75 in the Collected Works. Similarly, 2refe;s to Borel 1972. Bishop, E. 1967 Foundations O J ~constructive analysis (HcGraw Hill, New York) Bockstaele, P. 1949 Het intuitionisme bij de Franse wiskundigen, Ve'erh. Koninkl. Acad. Wetensch. B e l g i c I I , no. 32. Borel, E. 1898 Lefons sur Za thSorie des fonctions (Gauthier-Villars, Paris). The later editions (Borel 1 9 1 4 , 1928, 1950) contain the same text with additional notes. I904 Quelques remarques sur les principes de la thzorie des ensembles. Math. Ann. 5 9 , 514-516. A l s o (E 3 , 1251-1252.

.

484

A S . TROELSTRA

Les "Paradoxes" de la thGorie des ensembles, AnnaZes de Z'&cole Normaze (3) 25, 443-448. Also in Borel 1914, 162-166, and (E 3, 1271-1276. 1909 Sur les principes de la th6orie des ensembles, in: A t t i deZ N Congresso InternazionaZe dei Matematici, Roma, 6-11 ApriZe 1908. Academia dei Lincei, Rome; also in Borel 1914, 160-162, and 03 3, 1267-1269. 1912 La philosophie mathgmatique et l'infini, Revue du Mois 14, 218-227. Also in Borel 1914, 166-174, and (E 4, 2127-2136. 1912A Notice sur Zes travaux s c i e n t i f i q u e s (Gauthier-Villars, Paris). Also in IE I , 119-190. 1914 Legons sup Za the'orie des fonctions (Gauthier-Villars, Paris, second edition) Third edition, 1928. Contains the text of the second edition, with the same pagination; Note VII has been added. Fourth edition, 1950. Addition of Note VIII. 1914A L'infini mathgmatique et la rgalitl, Revue du Mois 18, 71-74. Also in Borel 1914, 174-181, and (E 4, 2137-2150. 1947 Sur l'illusion des d6finitions numgriques, C.R. Acad. S c i . Paris 224, 765767. Also (E 2, 979-981. 1947A Sur les difficult& des dsfinitions asymptotiques, C.R. Acad. S c i . Paris 224, 1597~1599. Also (E 3, 1379-1381. 1972 Euvres (Editions du Centre National de la Recherche Scientifique, Paris, 4 volumes) (Abbreviation 0 3 ) . Borel, E. et al. 1904 Cinq lettres sur la th6orie des ensembles, BUZZ. Soc. Math. France 32, 261-273. Reprinted in Borel 1914, 150-160, also 03 3, 1253-1265. Brouwer. L.E.J. I907 Over de grondsZagen der wiskunde, Ph.D. Thesis, Amsterdam (Maas en van Suchtelen, Amsterdam). Reprinted with additional material, and D. van Dalen as editor, in the series MC Varia 1 (Mathematisch Centrum, Amsterdam, 1981). English translation in CW. 1908A D i e m E g l i c h e n M s c h t i c h k e i t e n , A t t i deZ N Congresso IntemazionaZe d e i Matematici, Roma, 6-11 ApriZe 1908. Academia dei Lincei, Rome, 569-571. I908C De onbetrouwbaarheid der logische principes, T i j d s c h r i f t voor Wijsbegeerte 2, 152-158. English translation in CW. 1912A Intuitionisme en formalisme, Inaugural address, University of Amsterdam. English translation: Intuitionism and formalism, BUZZ. A.M.S. 20 (1913), 81-96. Also in CW. 1914 Review of: A. Schoenflies und H. Hahn, Die Entwickelung der Mengenlehre und ihrer Anwendungen, Jahresber. Deutsch. Math.-Verein. 23, 78-83 (kursiv). 1917 Addenda en corrigenda over de grondslagen der wiskunde, VersZagen der KoninkZijke NederZandse Akademie van Wetenschappen 25, 1418-1423. English translation in CW. 1918B Begrkdung der Mengenlehre unabhangig vorn logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Allgemeine Mengenlehre, VerhandeZingen der KoninkZijke NederZandse Akademie van Wetenschappen, lste sectie 12, no. 5. 1919A Begriindung der Mengenlehre unabhzngig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil, Theorie der Punktmengen, VerhandeZingen der KoninkZijke Nederlandse Akademie van Wetenschappen, lste sectie 12, no. 7. 1919D Intuitionistische Mengenlehre, Jahresber. Deutsch. Math.-Verein. 28, 203-208. 1921 Besitzt jede reelle Zahle eine Dezimalbruch-Entwickelung?, Math. Ann. 83, 201-210. 19238 Begrindung der Funktionenlehre unabhangig vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil, Stetigkeit, Messbarkeit, Derivierbarkeit, 1908

Verhande lingen der Konink Zijke Neder Zandse Aakdemie van Wetenschappen, lste sectie 13. no. 2. 1924D, G Beweis, dass jede volle Funktion gleichmzssig stetig ist, Proceedings of t h e KoninkZijke Nederlandse Akademie v a n Wetenschappen 27, 180-193. Bemerkungen zum Beweise der gleichmzssigen Stetigkeit voller Funktionen, Ibidem, 644-646.

Brouwer’s choice sequence

485

I9258 Zur Begrcndung der intuitionistischen Mathematik, Erster Teil, Math. Ann. 93, 244-257. 1926B Intuitionistische Einfihrung des Dimensionsbegriffes, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 29, 855-863. 3927B ijber Definitionsbereiche von Funktionen, Math. Ann. 97, 60-75. 1930 Die S t r u k t u r des Kontinuwns, Wien, 14 pages. 19428 Zum freien Werden von Mengen und Funktionen, Indag. Math. 4, 107-108. 1942C Beweis dass der Begriff der Menge h’dherer Ordnung nicht als Grundbegriff der intuitionistischen Mathematik in Betracht k o m t , Indag. Math. 4, 274-276. I9488 Essentieel negatieve eigenschappen, Indag. Math. 10, 322-323. English translation in CW. 19498 De non-aequivalentie van de constructieve en de negatieve orde-relatie in het continuum, Indag. Math. I I , 37-39. English translation in CW. 1952B Historical background, principles and methods of intuitionism, South African J. Science 49, 139-147. I952C Over accumulatiekernen van oneindige kernsoorten, Indag. Math. 14, 439-441. 19548 Points and spaces, Canadian J . Math. 6, 1-17. 1954F An example of contradictority in the classical theory of functions, Indag. Math. 16, 204-205. I975 Collected Works, Vol. I: Philosophy and Foundations of Mathematics, edited by A. Heyting (North-Holland, Amsterdam), (Abbreviation CW). Brouwer’s Cambridge lectures on i n t u i t i o n i s m , edited by D . van Dalen 1981 (Cambridge University Press, Cambridge). From a manuscript dating from 1951, with later revisions. Dumett, M.A.E. 1977 Elements of i n t u i t i o n i s m (Clarendon Press, Oxford). Heyting, A. 1934 Mathematische Grundlagenforschung. Intuitionismus, Beweistheorie (SpringerVerlag, Berlin). Photographic reprint 1974. 1955 Les fondements des mathimatiques. Intuitionisme, Thiorie de Za ddmonstrat i o n (Gauthier-Villars, Paris; Nauwelaerts, Louvain). French translation of Heyting 1934, with additions. 1956 Intuitionism, an introduction (North-Holland, Amsterdam). Second edition 1966; third edition 1971. 1981 Continuum en keuzenrij bij Brouwer, Nieuw Arch. Wisk. (3) 29, 125-139. Jervell, H.R. 1978 From the axiom of choice to choice sequences - a historical note. Manuscript. KuEner, B.A. 1973 Lectures on constructive mathematical analysis (Nauka, Moscow), (Russian). Martino, E . and Giaretta, P. 1981 Brouwer, Dummett and the bar theorem, in: S. Bernini (editor), A t t i d e l Congress0 Nazionale d i Logica, Montecatini Terme, 1-5 Ottobre 1979 (Bibliopolis, Napoli), 541-558. Mooij, J.J.A. 1966 La philosophie des math6matiques de Herd Poincard (Gauthier-Villars, Paris; Nauwelaerts, Louvain). Parsons, C. 1967 Introduction to: L.E.J. Brouwer, On the domains of definition of functions, in: J. van Heijenoort (editor), From Frege t o Gb’del. A source book in mathematicaZ logic 1879-1931 (Harvard University Press, Cambridge, Mass.), 446-453. Posy, c. 1976 Varieties of indeterminacy in the theory of general choice sequences, J. PhiZos. Logic 5, 91-132. 1980 On Brouwer’s definition of unextendable order, History and PhiZosophy of Logic 1 , 139-149. Stigt, W.P. van 1971 Brouwer’s Intuitionism, a re-appraisal of Brouwer’s contribution to the study of the foundations of mathematics, Ph.D. Thesis, Univ. of London.

486

A S . TROELSTRA

Troelstra, A . S .

PrincipZes of intuitionism (Springer-Verlag, Berlin). Choice sequences (Clarendon Press, Oxford). A supplement to "Choice sequences", Report 79-04, Dept. of Mathematics, Univ. of Amsterdam. Contains corrections to Troelstra 1977. 1981 Analysing choice sequences, Report 81-05, Dept. of Mathematics, Univ. of Amsterdam. To appear in J . PhiZos. Logic. Veldman, W.H.M. 1981 Investigations in intuitionistic hierarchy theory, Ph.D. Thesis, Katholieke Universiteit Nijmegen. With a leaflet "Stellingen". Vesley, R . E . 1980 Intuitionistic analysis: the search for axiomatization and understanding, in: J. Barwise, H . J . Keisler, K. Kunen (editors), The KZeene Symposiwn (North-Holland, Amsterdam), 317-331. 1969 1977 1979

THE L.EJ. BROUWER CENTENARY SYMPOSIUM A S . Troelshu and D. van Dalen (editors) 0 North-Holland Publishing Company, 1982

487

AN APPLICATION OF PROJECTIONS OF LAWLESS SEQUENCES

Gerric F . van der Hoeven Deparcmenc of Machemacics Universicy of Amscerdam Amscerdam THE NETHERLANDS

We give a survey of che resulcs obcained in che 'reduccionisc program' of building projeccion models for nocions of choice sequence. The resulcs are used co show chac chere is a correspondence becween validicy in copological models over l0,ll and validicy uniformly in a paramecer ranging over che 'free reals' in 10,Il. This paper consises of cwo pares.

In pare one we describe che progress made

in building incuicioniscic models for cercain nocions of choice sequence (subdomains of incuicioniscic Baire-space) by means of projeccions of lawless sequences. We only scace che resulcs, for proofs che reader is referred co CHT801 and CH821. In pare cwo we consider che correspondence becween validicy in copological models and (ineuicive) incuicioniscic validicy uniformly in a suicable paramecer. CK58l we find che firsc example o f chis:

In

ic is shown chere chac incuicioniscic va-

lidicy uniformly in a paramecer ranging over che fan of lawless 0-1 sequences corresponds co validicy over Cancor-space, for proposicional formulae. We give our own exposicion of chis resulc, which deviaces slighcly from che one given in rT771.

In lT8lAl a parcial correspondence is escablished becween validicy over che closed incerval C0,Il

c

IR and validicy in a paramecer ranging over 'free reals'.

Free reals are Cauchy-reals indexed by elemencs of a special domain of choice sequences, che free sequences, which have been modelled by means of projeccions of lawless sequences in IHT801. We improve on che resulcs of CT81AI by choosing a slighcly differenc concepr: of free real, based on anocher Cree represencacion of che incerval. PART

ONE

1.1.

INTRODUCTION. We consider incuicioniscic Baire-space N

che sense of CT811. Thac is co say, we

enciey, nor do we regard quancificacion over chink of

from che 'analycic' viewpoinc in

N as a single primicive N as incuicively clear. Racher we

do noe chink of

N as consisring of individual objeccs, generaced by mencal conscruccion

processes which may be non-predecerminace.

Several nocions of sequence can be dis-

G.F. VAN DER HOEVEN

488

cinguished according co che sore of daca on cheir graph chae may become available during che conscruccion process. lawless sequences LS;

N

One of che subdomains of

is che universe of

che daca char we can have on a lawless sequence ar: any

scage o f ics conscruccion consisc of merely an inicial segmenc of che sequence. We assume che reader r o be familiar wich che concepcs of lawlike and lawless sequence. Lawless sequences are creaced e.g. in CT771 wich furcher refinemencs in CT811.

The simplicicy of che 'possibly available daca' on a lawless sequence, permics an intluirively convincing axiomaeisatlion.

(For the

&,-axioms see 1.3 below.)

Moreover, ic suggescs co use che lawless sequences as a basis for conscruccing ocher domains of sequences wich bencer manhemacical closure propertlies via 'projeccions'.

The idea of and che mocivacion behind projecrions of lawless sequences are

discussed excensively in CHT80, 1.1-1.41

and rDT701.

1 . 2 . NOTATION.

N is che see of nacural numbers, N

(a)

verse of lawless sequences, K

is Baire-space, LS

is che uni-

is che see of neighbourhoodfunccions, induccively

defined by K1

Vm(Xn.m+l E K ) ,

K2

aO=O

K3

Va(A(a,Q)

where

A(a,Q)

A

Vn(Xv.a(*v)cK) +

acQ)

+

+

Va(aeK

:h(a=An.m+l)

V

acK,

+

acQ),

(aO=O

A

Vn(hv.a(*v)

E

Q)),

(See also (d) below.) (b)

k, m, n, u , v, w

range over

numbers as finice sequence codes. cp like elemencs of 5

N , a, 6,

over subdomains U of (c) ky (d)

6

N

ocher chan LS

and

w

range over

over LS, e, f

y,

and

g

especially over nacural

N,

a

over K

and

over law-

b

and

E,

n

and

and che universe of lawlike sequences.

j , j,, j,,

Vp, (Lp)n,

u*v, u d v , Ich(u),

( u ) ~ , G(n), cp

E

u

see CT771 seceions 1.8, 1 . 9 . As in CT771 we wrice

continuous operacions Ye: N

.. .

+

e(cp)

N and

and

re:

el($ N

+

for che image of cp N

E

N under the

wich neighbourhoodfunccion e

..

E

K

,cp ) abbreviaces e\up(cpl,. ,cp ). P P We adopc che conveneion char all choice paramecers in a formula are shown

respeccively. el ( c p , , (e)

Ji

Pairing, p-cuple coding and finice sequences are creaced as in CT771,

for che explanacion of and

N, u, v

and

in nocation.

We puc

489

An application of projections

LL.

1.3. THE SYSTEM

has che axioms of density

The cheory of lawless sequences, LS,

cidable equality (LS4)

(LSZ),

open data

and che lawlike.choice axiom from numbers t o tawlike sequences

LSI

vv3a(a€v),

LS3

!a[A(a,8,,

LS4

a! ]...!a

..., B p ) P

3a A(a

-t

3v(crcv

],..., ap,a)

A

yycv A(y,Bl,.

Vn3a A(n,a)

1.4. THE SYSTEM

+

de-

(AC-NF):

.. .BP))l,

+

3eVv'vlevfO+ 3a!alckyv AC-NF

(LSI),

and c o n t i n u i t y f o r lawlike sequences

(LS3)

...?a ckpv A(al,. ..,ap,a)l,

P

P

3bVn A(n,(b),).

E.

The universe of lawless sequences is noc closed under non-crivial concinuous operations, hence ic is noc suiced for che foundacion of incuicioniscic analysis. Formal syscems which are adequace for che foundarion of inmicioniscic analysis have been proposed by Kleene and Vesley CKV651 and by Kreisel and Troelscra CKT701. Concepcually, che second syscem is che mosc incerescing one, ic is called and in consiscs of che axioms of closure

tinuity

cS3

(CS3),

VE3q-continuity

Vs3a A(s,a)

+

3eVvCev#O

(CS4)

+

(CSI), and

a n a l y t i c data

(CSZ),

G,

Vs3a-con-

AC-NF:

3aVscv A(~,a)l,

(In [KT701 closure is noc an axiom buc a cheorem of

G.

We have puc i c among che

axioms here co scress ics imporcance.)

-

CS has all che feacures we would like a formal syscem for incuicioniscicanalysis co have, bue unforcunacely ic is noe easy co specify a subdomain of N for which che validicy of che axioms can be rigorously juscified. There are (cf. CHT801):

KWO

ways co approach che problem of finding such a subdomain

informally, i.e. by scudying primicive nocions of choice sequence,

or via projeccions of lawless sequences, i.e. by scudying universes which are conscrucced from L S .

In praccice ic is advancageous eo use boch approaches simulca-

neously, since chey supplemenc each ocher.

490

G.F. VAN DER HOEVEN

1.5. GC-SEQUENCES. I n CT691, CT69Al T r o e l s c r a h a s proposed a p r i m i c i v e n o t i o n o f c h o i c e s e q u e n c e

as a model of c h e operacions).

che GC-sequence

g-axioms:

(GC f o r Generaced by C o n t i n u o u s

The a n a l y s i s of c h i s n o c i o n i n CT771 and che e l a b o r a c i o n of t h a c

a n a l y s i s by Dummetc i n CDu771 show ( i n f o r m a l l y ) ellac c h e u n i v e r s e of GC-sequences s a c i s f i e s c l o s u r e , a n a l y t i c d a e a and

VtZ3a-concinuicy.

I n o u r accempcs t o c o n s c r u c c a p r o j e c c e d u n i v e r s e which i m i c a c e s c h e b e h a v i o u r

of che u n i v e r s e of GC-sequences, changed.

c h e d e s c r i p c i o n of che n o c i o n h a s been s l i g h c l y

We ‘ d e f i n e ’ che u n i v e r s e GC of GC-sequences a s f o l l o w s .

GC : { e l ( E n

,...,E ; ~ )

-I v e r s e of GC-carriers,

: e

GCC z

{

E

K, c n ,

E

: ~ n

-

-1 E

GCC},

E

N I.

where

i s che c o u n c a b l e u n i -

GCC

One may c h i n k of

2

in

E~

name of a n unbounded (mencal) r e g i s c e r f o r s c o r a g e of n a c u r a l numbers. quences

€2

E

are g e n e r a c e d s i m u l c a n e o u s l y , c h e g e n e r a c i o n p r o c e s s i s a mencal

GCC

;, n

p r o c e s s of f i l l i n g c h e r e g i s c e r s

register

as che The se-

E

~- ~ ( mi s) che m-ch v a l u e p l a c e d i n

N.

; during c h i s process.

The g e n e r a c i o n p r o c e s s o f

GCC

i s d i v i d e d i n c o c o u n c a b l y many s c a g e s .

Ac

e a c h s c a g e w e c a n puc a f i n i r e number of c o m p l e c e l y a r b i c r a r y v a l u e s in a f i n i c e number o f r e g i s c e r s ( a s i f we were g e n e r a c i n g a c o u n c a b l e u n i v e r s e o f l a w l e s s seq u e n c e s ) bue a e che same rime w e may d e c i d e co make c h e f u r c h e r v a l u e s of o n e car-

rier

t~~

dependenc on che v a l u e s of one o r cwo o c h e r s

continuous f u n c c i o n wich n e i g h b o u r h o o d f u n c c i o n we c a n choose a c r i p l e

(~,,k,g) o r a pair

c i d e chac f u r c h e r v a l u e s of respeccively; values f o r from i n i c i a l segmencs o f E”

E~

(E,$

(cm,

-

or

E L , ~ Z ~v )i a a

Thac i s co s a y , a c e a c h s c a g e

e.

c o g e c h e r wich a n

w i l l b e che v a l u e s of

e

E

e l ( E ~ , E ~o )r of

- -

K

and deeltZm

-

a r e no l o n g e r f r e e l y g e n e r a c e d b u t chey a r e compuced cE c ~ srn , o r o f srn a l o n e , v i a e. I f w e d e c i d e co make

dependenc on one o r cwo o c h e r carriers a e some momenc of che g e n e r a c i o n p r o c e s s ,

c h e s e o c h e r s muse be f r e s h c a r r i e r s a t chac momenc, i . e . chey muse b e independenc of o c h e r s .

Bue of c o u r s e chey c a n b e made dependenc on o c h e r c a r r i e r s a c a l a c e r

scage. The f o r e g o i n g g i v e s a s i m p l i f i e d p i c c u r e of c h e GC-sequences,

e.g.

che g e n e r -

a c i o n p r o c e s s musc be adapced co g u a r a n c e e c h a r a l l c a r r i e r s w i l l become c o c a 1 sequences e v e n c u a l l y .

F u l l d e r a i l s a r e i n rH821.

Ac e a c h s c a g e of c h e g e n e r a r i o n p r o c e s s o f E

= el

,**-,En

EE 1 9 1

a n e q u a c i o n of c h e form

are f r e s h a c s c a g e

-p,q The a v a i l a b l e d a e a on

regiscers E

..,ESP )

E

E

GCC

w e h a v e f o r e a c h GC-sequence

= e, 1 (cn

-I,]

, . . . ,cn

-P,9

),

where

Z.

a c scage

z

c o n s i s c of

e,

and c h e names of c h e

~ l , l , . . . , ~ p , q ; c h e l a c c e r p l a y a rBle i n e x c e n s i o n a l r e l a c i o n s beeween

and o c h e r GC-sequences.

An application of projections

AND PROJECTION MODELS.

GC(C)-SEQUENCES

1.6.

49 1

The nocion of GC-sequences can be relacivised. In creacing dependencies becween GC-carriers we can rescricr ourselves co dependencies via some a subsec of

Relacive co che GC-carriers w.r.r. C, defined as

GC(C)

E

{el(En -1

If C i.e.

,...,E GP )

E

C,

C

che universe of GC-sequences w.r.c. C : E ~ .E GCC(C),

eeCcc3nVa(JnIa=eIa)),

u6(C) = i f 1 6 : f

E

MI

chen GC(C)

for some M

c

e

is

C].

E

-1

J: N

can be enumeraced by a mapping

enumerable C

e

We call che resulcing nocion of sequence GC-carrier w.r.t. C.

K.

K

+

K (ac lease modulo equivalence,

can be imicaced by a projecced universe

(dependenc on

C ).

The rescriccion co

arises from che face chac in creacing dependencies becween projecced

sequences we use che numerical values of muse be possible co selecc any

e

E

C

6

co selecc an element: of

C,

and ie

in chis way.

Using an eliminacion cranslacion incroduced by Dragalin (ID741) one can show chac

U,(C)

CS(C)

fulfills che syscem CS(C),

ac lease if

is che following relacivised varianc of

CS(C)3

VE3a A(E,a)

plus

AC-NF.

-f

3eVv[ev#O

+

C

is suicably chosen.

CS:

3atlcEv A(~,a)l,

Nore chac W(K) = CS,

in parcicular

CS(K)4

Ic is co be expecred chac a projeccionmodelforfull under che assumpcion of excended Church's chesis ECT,,.

t

CS(K)3

+

AC-NF k CS4.

CS can be conscrucced (Bue noc all derails have

yec been verified.) 1.7.

FREE SEQUENCES. In IHT801 more special, simpler versions of

projeccions.

GC(C)

are imicaced by means of

The simplificacions are che following:

(a) a sequence can be made dependenr on only one ocher sequence ac che cime, (b) a sequence can depend on anocher only via a poincwise acting concinuous funccion, i.e. via an

e

E

K

such chae elE(n)

can be decermined from n

and

Eh).

The simplesc nocion of chis cype is che f r e e sequence:

if a free sequence is made

dependenc on anocher chen che cwo coincide from chae point: onwards; che firsc one depends on che second one via che idencicy mapping. The projecrionmodelfor free sequences (a special case of che projecced universes of CHT801) sacisfies che axioms of closure under replacement o f i n i t i a l seg-

ments

(FSI),

open data i n one parameter

(FSZ),

c o n t i n u i t y f o r lawlike sequences

492

G.F. VAN DER HOEVEN

(FS3),

and

VE3q-continuity

FSI

VEVV~~(Q=V/E),

FS2

VELA(€)

FS3

V E ~ . . . V EA(E~ ~~~

+ ~V(EEV A

(FS4):

Vqev A(q))I,

,...,~ ~ , a ) +

3eVv[ev#O FS4

V ~ 3 qA(E,~)

VIE

where

-f

3aVElck;v

-f

...VEPckpv P A(E~,...,€,,a)],

V E ~ VA(E,vIE),

is defined by

1.7.1. DEFINITION.

Lee v

n < lch(v)

chen

(v),

N, chen vlV

be a finice sequence, cp an element: of

is che sequence obtained by subscicucing v

for G(lch(v)),

i.e.

vlQ(n)

= if

else cp(n).

We use che free sequences for che incroduccion of che free reals in pare Kwo below.

FS

is che universe of free sequences,

is che axiom syscem FSI-FS4

plus AC-NF. PART TWO

2.1. INTRODUCTION. In chis pare we will show char validicy in copological models over che closed incerval [ O , I I corresponds co validicy uniformly in a paramecer ranging over CO,llFs.

The elemencs of

[O,IlFs

by free sequences (see 1.7 above).

are Cauchy-reals, indexed in some scandard way Before we curn co che free reals however, we

give a shore accounc of che well-known correspondence becween validicy in Bechmodels 0ve.r ehe cree of 0-1 sequences (Cancor-space) and validicy uniformly in a paramecer ranging over che lawless elemencs of

ZN

.

2.2. VALIDITY AND VALIDITY UNIFORMLY IN A LAWLESS PARAMETER. We rescricc ourselves co pure firsc order predicace logic, wichouc equalicy and funccionsymbols. The language has a councable lisc of predicacesymbols

... .

Po, P I ,

P. has a(i)

argumencs, a

a mapping from N

co

N.

2.2.1. DEFINITION. (a) An interpretation of predicatesymbols in the domain D, a map

ID which assigns co (b)

and

If A

Pi a predicace

is a formula, VD

ID(Pi)

a map from che free variables of

A

ID an incerprecarion of che predicacesymbols in D, chen AD(ID,VD)

cained from A

by subscicucing ID(Pi)

for Pi, V (v. ) D =P

variable in A ) and relacivising all quancifiers co (c)

A

D

is valid iff eyIDbJD A (ID,VD).

is

c

D.

for v iP

inco D is ob-

(v. a free IP

An application of projections

In chis definieion che predicaces

493

may depend on choice paramecers. Valid-

ID(Pi)

icy uniformly in a lawless paramecer ranging over

ZN

is a rescricced form of che

above validicy nocion. DEFINITION. An incerprecacion of predicace symbols

2.2.2.

l e s s parameter

iff for all

a

Pi

chere is a P?CLSXD'(~) o(1) . )

sacisfies Vd l...Vdu(i)CID(Pi)(dl,...,d

VD

D,

PP(a,dl,...,du(i))l.

is v a l i d u n i f o n z y in a lawless parameter iff for all

A formula A for all

c-t

is lawlike in a lawID which is lawlike and

for all incerprecacions ID

lawlike in a

a

E

LS,

and for all valuacions

V ).

of che free variables, we have AD(I

D' D

Validicy and validicy uniformly in a lawless paramecer are discussed ac lengch in CT771, CT77A1 and CT81AI. BETH-MODELS OVER THE BINARY TREE.

2.3.

In chis section cp ranges over

ZN,

v

and w

2
range over

2 . 3 . 1 . DEFINITION. (a) A Bech-model over che binary cree of finice 0-1 sequences

is a pair

*

<*,D>, D

: Pi w P :

(b) usual.

che domain for che incerprecacion of variables,

Pr c 2
*

a mapping,

where

is, for any given Bech-model, defined as

In parcicular we have for prime formulae:

v Pi(dl and for A v Ik A

,...,do ( 1.) ) Z

iff V W v 3 n PT(&n),d

,*.

.,do(i)))

3xB(x)

iff VWv3n3d~D(G(n) 1 1 B(d))).

For any formula A, fies v

[A(dl

E

,...,dP)I

,...,

d )I P iff VlKv3n(G(n)~[A(d~

che sec [A(d,

E

,,...,

d ) I sacisP Sets satisfy-

{v : v 11 A(d

,...,dp)])).

ing chis condition are in 1-1 correspondence wich che opens of

2H.

If we assume rhe principle V W v 3 n A(G(n))

FAN

cp

+

3mVw(lth(w)=m

+

3w'dv.w

A(w')),

chen che forcing clauses for primeformulae, disjunccion and exiscencial quancificacion can be formulaced equivalencly as v I 1 Pi(dl v 1 1A

V

,...,do(i))

B

iff

v 1 1 3xB(x)

iff

3mVw(lth(w)=m

3mVw(lch(w)=m

iff

hVw(lch(w)=m

-+

v*w 1 1A

+

+

3w'dv*w Pr(w',dl

V

v*w 1 1 B),

,...,do(i))),

3d(v*w 1 1 B(d))).

Of course we can also define che forcing relacion chis way (wichouc assuming FANv ) .

v

E

[A(dl

In ehac case che see [A(d l,...,dp)]

,...,dp)]

iff

hb(lch(w)=m

-+

3w'

will satisfy v*w(w'c8A(dl

,...,d,)])).

In the nexc seccion we work wich che second definition of is incorporaced.

Il-

in which FANw

494

G.F. VAN DER HOEVEN

2 . 4 . CORRESPONDENCE BETWEEN BETH MODELS AND INTERPRETATIONS UNIFORMLY IN A LAWLESS

PAFMBTER. ID be an incerprecacion of predicace symbols uniformly in a, lee P p

Lee

be such chac Py(a,dl

,...,do(i))

tf

,..., do ( 1.) ),

ID(Pi)(dl

and ler Aa(dl

,. . . ,v.

,..., dp)

VD maps O* il IP . respectively. Puc P. (v,dl,.. .,d ) E Vatv Py(a,d ., P dP) Lee <*:Pi-$, D> be a Bech-model,'puc Pro(a,dl d ) S 3n PT(a(n),dl ,...,dp). P m, Assume che following principles for lawless sequences ranging over 2": be AD(ID,VD)

where A

concains che free variables v

which

,,..

..

onco dl,. ,d

,...,

OD

VcrlA(a,dl

,...,dp)

Va3n A(a(n))

FAN,

+

-t

3nVfka(n) A(B,dl ,

3mVw(lch(w)=m

+

. . . ,dp)1 ,

3 w ' ~ v * wA(w'))

Then che following hold: , (a) < * o * : p . ~ P f ~ * D> 1

<*:P.wP;, D > .

is a Bech-model, ic is equivalenc co

1

(b) The predicaces

and

P* :O

Po

.

i

are equivalenc for all

(c) v IF A(dl,. ..,dp) ++ Vatv Aa(dl ,...,dp), where on che lefc-hand side we have forcing over che model righc-hand side we have che incerprecacion which maps

P.

<*,D> on

and on che

*O

P.

.

The proofs of chese faces are scraighcforward. is a domain of lawlike objeccs like e.g. che nacural numbers,

Noce chac if D

chen v IF A(d

che lawlike sequences or che elements of K,

=El.

formula in che language of (some excension) of is provable in

is provable in

ml

is a lawlike

Hence if V m v Aa(d

or some suicably adapced varianc of M,

v IF A(d l,...,dp) nation cheorem.) 2.5.

] , . . . , dp)

( o r some varianc).

dP)

chen (Proof via che elimi-

FREE REALS. There are several ways co associare wich each

U, E

N

a Cauchy-sequence of

racionals. We choose che following one (rescriccing our accencion t o 1 0 , l l ) . Firsc we fix an enumeracion n n 2.5.1. DEFINITION. p

of che rarionals in [ O , l ] .

is rhe mapping from

N inco

Qn[O,l]

Then we puc

which sacisfies the

recursion equacions:

'p (v) ocherwise. We use 2.5.2. (pc

p

co conscrucc Cauchy-sequences of racionals as follows.

DEFINITION. Lee

U, E

N,

chen cp*: N

-t

9

is che sequence Xn.p(&n+l)).

is a Cauchy-sequence, xcp is che real generaced by

2.5.3.

FACT.

Icp*(k+l)-cp*(k)(

<

2-(k+1)

for a l l

cp

E

cp*.

N, k

E

H.

An application of projections

495

Equalicy and ordering on reals and rationals are defined in che usual manner, in cerms of che generaEors Q*. 2.5.4. DEFINITION. "(P = x* XV < 9

iff

tlk3m~n(1cp*(m+n)-,,,*(m+n) I < 2-k),

iff

3k3mVn((~*(rn+n)+2-~ < 4).

q < xu xv < x,,,

iff

3k3mVr1(q+2-~< cP*(n+m)),

iff

3q(xQ
x

iff

3q(xQ
< x +q' V u i x -q' < x,,,

v

iff x

<

(

P

q <x,,,),

A

qcx,,,)

A

x +q'. *

2.5.5. DEFINITION. U(n,xV)

is che see

{x : x -2-" < x < x +Z * c p * c p

2.5.6. DEFINITION. ~O,llLs is che subsec of lawless Peals in [O,IlLS : t xa : a [O,l~,,

(FS

-n

1.

rO,ll, defined by

Lsl;

E

are che free reaZs in

LO,]],

defined by

[O,lIFs

{xE :

5

6

E

FSI.

is che universe of free sequences, cf. 1.7.)

2.5.7. REMARK.

is noe closed under non-rrivial concinuous operaeions, FS

LS

closed only under a small see of concinuous operaeions. Lies of

[O,llLs

is

ConsequenEly, che proper-

[O,1IFs depend on che mechod one chooses co associace a

and

Cauchy-sequence of rationals wich an elemenc of che represencacion of

or

LS

FS

respeccively. E.g.

in rT81Al is differenc from che one we use here,

[O,lIFs

and leads co differenc propercies. 2.5.8. From che propercies of

one can show char: [O,lILs

LS

is noe closed under

non-crivial concinuous operaeions, ir: is nor: a field of reals. Yer che copology of

[O,llLs,

We ask ourselves whecher

[O,llLs

VxrCO, ]ILs [A(x)

+

3kVycU(k,x)nCO,

There is a problem.

If A(x

cia1 segmene v

a

mine a

k

of

sible chae chere is a FSI

B

which

(1.7.),

E

v

A(y)l. E

(Y

for all

sacisfy xB

E

LS,

chen we can find an iniFrom v we can decer-

f? E V.

U(k,xa),

bur ic is very well pos-

wich a differenc inicial segmene such chac y*

y

In order LS

]ILs

holds for some

)

such chae A(xa)

such chac all

a limir in U(k,x,).

induced by che usual meEric, is incerescing. saciesfies r e a l open d a t a :

KO

escablish A(x

does nor: have.

)

also has

we need a closure properey like

Y Ic curns our chae che scruccure of [0,llFs

is more incerescing in chis respecr. 2.6. PROPERTIES OF

[O,llFs.

2.6.1. From here on we use che free reaZs. isfy A(X)

A

x

X, y

All predicaces A =

y

+

and

z as variables for elements of [O,llFs,

of free reals below are excensional (chey sar-

A'(y)).

We show in chis subseccion, chae

[O,llFs

satisfies

G.F.VAN DER HOEVEN

496 FRI

Vq~LO,ll~lk3X(X~U(k,q))

(density),

FR2

VX[A(x)

( r e a l open data),

FR3

VXEU3n A(X,n)

U and V

3kVcjEU(k,x) A(y)l

-+

-+

Vq1q2EU3V([q1,q21cUV

A

in FR3 are basic opens of r 0 , l I

WEV3nVXEV A(X,n)).

V

(of che form U(k,q)),

chere is an n finice cover

E

v

N

such char A(X,n),

chen each closed incerval i n

of basis elemencs such chac for each V

isfying A(X,n)

for all

X

E

E

V

is

x

f i n i t e colleccion of such basis elernencs. FR3 says chae if wich each

E

a

U

U has a

chere is an n

sac-

V.

is a dense subset of N then CO,1Iu E {xcp : cp [O,llFs s a t i s f i e s FRI.

2.6.2. LEMMA. I f

U

density property

FRI, i n particular

E

Ul

has the

PROOF. Trivial. 0 xcp E U(n,cp*(n)).

2.6.3. LEMMA. PROOF. By 2.5.3

cp*(r1)-2-("+')+2-~

cp*(n+l) <

E

U(n,cp*(n)),

k such chac By induceion w.r.c. m

i.e. we have a

cp*(n+l) < cp*(11)+2-("+')-e-~.

one shows

(using 2.5.3)

J

2,-k

2-k

t p (v)-2-"

c

4 -

P(V)

@(no)

Fig. I

P(v)+~-"

An application of projections

i.e. co find

qi+l we bisecc che incerval

che locacion of

491

Iqi,p(v)+2-n- 2-(k+l)l.

Fig.2 shows

ql.

Noce chae

2- (i+l) (2-n-2-(k+ I ) (a) che discance becween q; and 9i+] is ) and hence - < 2-(n+i+l) qi+l qi hence there is an m such char (b) , converges co p(~)+2-"-2-(~+'), A similar sequence n

can be conscrucced co che lefc of in ( I ) is larger than n

W.1.o.g. we may assume chae no in follows chae @(no+m),

m

p(v).

(=lch(v)-I).

as in (b), lies in che incerval

p(~)+2-~-2-~). We assume chae @(no+m)

p(v),

2

From ( I )

(p(~)-2~+2-~,

che ocher case is symmecric.

From (h) ic follows chae chere is an i < m such chae q . 5 @(no+m) < qi+]. Now consider che finice sequence of racionals u = wich lengch I+ng+m-n. By (a), che discance becween che k-ch and k+l-ch elemencs of

u

Lee w

lr:h(w) = lch(u)'l

=

v*wlW

of

I ( u ) ~ - ( u ) ~ +I ]= 0 .

u

we can associace an index in che enumeracion n n' be che finice sequence of nacural numbers such chac

Then v*w IJJ

in face for k > i

is smaller char: 2-(n+k+1),

Wich each element ( u )

ng+m-n

=

and

sacisfies p(v*w) we have

=

V k < lch(w)

@(n+m)

and

Vkho+rn(@k=$*k),

((~)~+~=r(~)~). lch(v*w) = no+m+l, hence for

i.e.

xQ = xJI.

n

As an immediace corollary of 2 . 6 . 3 and 2 . 6 . 4 we find 2.6.5.

LEMMA. I f Vnrv A(x,)

where

ci

then

lch(v) = n+l VxrU(n,p(v))

f-f

ranges over

FS and

A(x),

x over [O,llFs.

PROOF. From lefc t o righr: by 2 . 6 . 4 1 closure of segmencs and excensionalicy of A, From che definicions of 2.6.6.

FACT. If x

E

U(n,q)

Real open daca for 2.6.5 2.6.7.

x < q, y < x+2

then

CO,llFs

+

3kV+U(k,X)

-k

under replacemene of inicial

ecc. ( 2 . 5 . 4 . )

U ( k , x ) c U(n ,q)

0

f o r some

one easily derives k.

follows immediacely from open daca for LS,

and 2 . 6 . 6 .

LEMMA. VXrA(X)

FS

from righc co lefc by 2 . 6 . 3 .

A(y)l.

G.F. VAN DER HOEVEN

498 PROOF. Assume

A(X),

be such chac

chis yields a

lch(v) = n+l.

x

holds by 2.6.5, and chac U(k,X)

c

C-N

we have e

KFS

E

Lee Q

(We may assume

U(n,p(v))

E

v

.

x

=

Apply open daca

and VBEV A(xg).

Lee n

By 2.6.6 we find

a

k

A(Y)

such

0

we have VyaU(k,x) A(y).

FR3. Noce chac as a special inseance of che concinuicy

+

3eVv[ev#O

+

FS3 and hence

iff Vvw(e(v)#O

3nVacv A(a,n)l.

-f

C-N

wich

e

e(v)=e(v*w))

Fan(Q) E < >EQ

A

Vv(v€Q

Q

tf

In a weaker variane

KFS,

ranging over che see

defined

and Va~FS3n(e(a(n))#O).

be a see of finice sequences and let

ing char che elemenes of

Fan(Q)

be che formula express-

are nodes of a finicely branching decidable cree:

3n(v*)~Q))

A

3mVv(card({n FAN

a

lch(v) > 0.) Then VyEU(n,p(v))

by 2.6.3.

for chis k

x

a E FS,

such char

ranges over che induccively defined see K.

In chis schema e by

v

for free sequences, we have

Va3n A(a,n)

of

E

U(n,p(v)),

Finally we prove axiom FS3

x be generaced by

lee

a,

for FS w.r.c.

: v*EQ}) < m)

A

Vv(vrQ

V

vbQ).

is che following principle

FAN

Fan(Q)

FAN

+

Ve3nVv~Q(lch(v)=n

is provable if

e

Thac is co say, FR3

U

e(v)#O).

ranges over K.

The proof of

FR3 below depends on

can be derived from che scrong varianc of

cively from che weak variane and 2.6.8. LEMMA. Let

+

be

U(kO,q).

FAN.

or alrerna-

C-N,

FAN.

I f f o r a22

x e U there i s an n such t h a t

then

A(X,n),

Vq q ~U3V([q~,q~lcUVA W E V ~ ~ V XA(X,n)), ~ V 1 2

where

V ranges over f i n i t e s e t s o f opens U(k',q')

PROOF. Lee

q1,q2 be rationals in U,

q 1 < q2, and lec

(m,k) , m

ping which assigns co each pair

5

Q iff Vk3m

5

2k+1(U,(k)=$(m,k))

Vk(Irw(k)-r~(k+l ) I i.e. if

U, E

Q chen

5

(p* : N + Q

[O,llFs. 6 : NL

-t

IN be a map-

Zk+l, an index for rhe rarional

ql+m*(q -q )*2-(k+1) in che enumeracion 2 1 n n' -(k+l) r6(m,k) - q1+m-(q2-q1)-2 Define che sec Q c N by U, E

of

i.e. if m < 2-(k+1)

chen

and (q2-q,).2-(k+2)), is an infinice parh in che parcially ordered

syscem of fig.3, and conversely, wich each infinice pach

x

in chis syscem chere

499

An application of projections

is a cp

E

Q

such char: (P* = X.

We define r:he fan Q

{G(n) : n

as

(a) if v E Q, lch(v) = k+l k+ 1 some m 5 2 ,

(b) if

E

N, cp

E

0,).

Noce char:

chen p(v) = ql+m.(q2-ql).2 -(k+l)

for

m 2 Zk+l chen chere is a v E Q such chat lrh(v) = k+l and 2- (k+ 1 ) P(V) = q,+m-(q2-q1) The incerval [q1,q21 is conrained in U(ku,q). Lec k l be such char:

.

ql-2-k1 E U(kO,q)

and

@(kl)

E

whence u(kl,(P*(kl))

hence

xW

[q,,q,l, E

U(kg,q).

q2+2-k1

whence by

C-N

ev#O By

FAN

A

lr:h(v)

+

lch(v)=kl+l

2

k2

k2+1

-t

c

2

+

ev # 0,

-f

By (a) and (b) chis yields

E

3nVa~vA(xa,n).

k l such char: for all v

3nVa'aev A(xa,n)).

E

Q

BY 2.6.3,

such chac for all v

hence VveQ(lch(v)=k2+1

U(k0,q).

A(xa,n)),

chere is an e

chere i s a

If G(kl+l)

U(k0.q).

by assumpcion we have

So

Va3n(a(kl+l)EQ

E

E

Q

Q

r:hen xcp

f

U(kl,(P*(kl)),

G.F. VAN DER HOEVEN

500

One easily verifies char rhe colleccion - (k2+1 k2+ 1 V ! {U(k2,m*(q -q )-2 ) : m 5 2 } 2 1 REMARK. The proof of open data and

C-N

covers

0

Cql,q23.

in che projeccionmodel for free sequences

as given in CHT801, deals only wich formulae in che language of analysis, che principles do noe hold (in che model) for arbicrary predicaces.

However, one can easi-

ly describe che class of "lawlike" predicaces for which chese principles do hold: chey are che predicates which sacisfy a permucabilicy condieion (see [HTBOI). Below we use 2.7.

FR2 and

FR3

for arbierary lawlike predicaces.

THE CORRESPONDENCE BETWEEN TOPOLOGICAL MODELS OVER [O,II AND INTERPRETATIONS LAWLIKE IN A FREE REAL PARAMETER. Incerpreeacions lawlike in a free real parameeer are defined exaccly as incer-

precacions lawlike in a lawless paramecer in 2.2.2. We give a slighcly non-scandard definicion of a copological model over [ O , l l , analogous co che definicion of a Bech-model over che binary cree in 2 . 3 . A copological model over [0,11 is a pair for che incerprecacion of variables and dicace symbol P. a predicace

11-

define che forcing relacion

P:

c

*

<*,

D>, where

D

is che domain

is a mapping which assigns co each pre-

NxQxD'(i)

.

Wich each eopological model we

becween basic opens U(k,q)

and formulae A

in-

duccively as follows: U(k,q) U(k,q)

It Pi(d ,,...,da(1). ) iff VxeU(k,q)3k'q'(xeU(k',q') It A A B iff U(k,q) It A A U(k,q)lkB,

U(k,q)

I!- A 11- A

U(k,q)

11- Vv B(v)

U(k,q)

3v B(v)

U(k,q)

V

B

iff Vx€U(k,q)3k1q'Cx€U(k',q')

+

B

iff VU(k',q')cU(k,q)CU(k',q')ltA

A

A

P:(k',q',d

(U(k',q')ll-A +

V

],..., d

. I),

u(1)

U(k',q')ll-B)],

U(k',q')l~-Bl,

iff VdED(U(k,q)kB(d)), iff VxEU(k,q)3k' ,q'3dcD[xEU(k',q')

U(k',q')ll-B(d)l.

A

...,

The see %A(d l,...,d ) ] {U(k,q) : U(k,q) 11- A(dl, dp)} sacisfies P E [A(d l,...,dp)] iff Vx€U(k,q)3k'q'[x€U(k',q1) A U(k',q')E[A(d, dp)]l.

,...,

U(k,q)

Such sees are in 1 - 1

correspondence wich che opens of CO,ll, i.e.

under che U(k,q) I!- A(d,, ...,d ) corresponds co U(k,q)nCO,Il c [A(d l,...,dp)B P If we assume the local compactness more usual definicion of che valuacion U - I . principle Lcx

Vx€U(k,q)3U(k',q')Cx€U(k1.q')

A

A(k',q')l

-f

Vqlq2~U(k,q)3V(Cql,q21cUV

where

V

A

W(k' ,q')EV

ranges over finice sees of basic opens of che form U(k,q),

A(k' ,q')), chen che

forcing clauses for primeformulae, disjunccion and exiscencial quancificacion are equivalenc co che following: U(k,q)

It Pi(dl,..

iff Vq,q2'U(k,q)~VW(k',q')EV

l'~(kt,q',dl,. ..,do(i)),

An application of projections U(k,q)

I!- A v B

U(k,q)

\I- 3v B(v)

where

V

501

iff Vqlq2EU(k,q)3VW(k',q')EV(U(k',q')ll-A

V

U(k',q')lkB),

iff VqlqZEU(k,q)3VW(k',q')EV3d(U(k',q')l!-B(d)),

ranges over finicecoversof

[q1,q21 consisring of basic opens U(k',q').

Of course we can also define che forcing relacion chis way, not: assuming LCx.

E

'iA(dl

...,

d ) I sacisfies: P iff V q l q 2 E U ( k , q ) ~ ~ ( ~ q l , q 2 1 cAU ~VCKA(d l,...,dp)l).

Then che see [A(dl,

U(k,q)

,...,dp)l

I!-,

We use che second definicion of

in which

LCx

is incorporaced.

is a finice see of opens of che form U(k',q'),

V

Noce chae since

iff Vq3E[ql,qZ13U(k',q')EV(q3EU(k',q')),

[ql,q21 c UV

i.e.

V

covers

[ql,q21

It

A(d ]...., d ) can be expressed in cerms P of racionals and nacural numbers, ic involves no quancificacion over che reals in iff ir: covers

[ql,q21nQ. Hence U(k,q)

co.11. : Pi

Lee

Pp

c

* Pp

assign to each predicace symbol Pi ]...., dp)

lee A"(d

[O,llFs~Da(i),

che predicace

be che formula obcained from A

by sub-

scicucing Xvl,...,V~(~).P~(X,V~,...,V ) for Pi, d l ,...,d for che free vario(i) P vip of A and relacivising all quantifiers in A LO D. ables vil,

...,

o* (k,q,dl,...,du(i)) Puc Pi

Lee

D>

<*,

VXcU(k,q)

P?(X,dl,...,du(i)).

be a copological model over CO,l],

puc

,dl,....dp)).

(OD follows from FR2: real open data, LCx Here coo, Vq3E[q

Cq,,q,I

c

UV

from FR3 and

FRI:

density.

can be expressed in cerms of racionals:

q313U(k,q) cV(q3cU(k, 4)) * ) Then che following hold: 3

2.7.1. LEMMA. (a) The topological models (b) For a l l

i

D>

<*,

the predicates

and

Pp and

<

*o*

o*O

Pi

, D> are equivalent. are equivalent.

k A(dl ,...,d ) iff VxeU(k,q) Ax(dl 9 * * ,dp) 3 where II- on t h e l e f t - . P hand sic& is over <*, D> and the i n t e r p r e t a t i o n of p r e d i c a t e l e t t e r s on the *O right-hand side is P i & Pi

(c) U(k,q)

.

PROOF. (a) and (b) scraighcforward, (c) by formula induccion. We consider cwo cases. (i)

A E B + C.

From OD

one proves chae VXEU(k,q)

W (k',q 1) cU (k,q) CVXeU (k ' ,q ' ) Bx

-f

Now apply induccion hypochesis.

VXcU(k'

,q1) Cxl.

x x

(B +C )

is equivalent: co

502

G.F. VAN DER HOEVEN

(ii) A :3v B(v),

then VxcU(k,q) Ax

VXcU(k,q)3k'q'[X~U(k',q')

f

VXEU(k,q)3d

Bx(d).

A

3dVgcU(k',q')

BX(d)l,

A

3d(U(k',q')Il-

B(d))]

By

OD

this yields

which is equivalent to VXEU(k,q)3k'q'CXEU(k',q')

by inductionhypothesis. Now apply

LCx, this yields

Vq,qz~U(k,q)3~(Cq,,qzlcU~A IflT(k1,q')cV3d(U(k'

,q') Il-B(d)))

as desired. 0 Noce finally that if

D

is a lawlike domain, chen U(k,q)kA(dl,

...,dp)

is

a lawlike formula. Free reals are defined from free sequences which have a model in the lawless sequences.

This model leaves che lawlike parn of the theory of

free sequences unchanged.

Moreover

part.

Hence, if D

is lawlike chen

inself i s conservative over its lawlike kVxcU(k,q) AX(d,, ...,d ) P

...,

d ) is provable in rhe lawlike part of U(k,q) 11- A(dl, P system of classical analysis).

iff

(which is a sub-

REFERENCES CDr741

DRAGALIN, A.G.,

Constructive models f o r i n t i u t i o n i s t i c theories o f choice

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Elements o f Intuitionism (Clarendon Press, Oxford, 1977). Projections of tawless sequences, Ph.D.thesis, Uni-

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van der HOEVEN, G.F.

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Holland, Amsterdam, 1980) 265-298. CKV651 KLEENE, S . C . & R.E. VESLEY, The Foundations of I n t u i t i o n i s t i c Mathematics,

especially i n r e l a t i o n t o recursive functions

(Norch-Holland, Amscerdam,

1965). CK58l

KREISEL, G . ,

A remark on f r e e choice sequences and t h e topological com-

pleteness proof, [KT701

J. Symbolic Logic z(1958) 369-388.

KREISEL, G . & A . S . TROELSTRA, Formal systems f o r some branches o f i n t u i t -

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TROELSTRA, A . S . ,

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Principles of I n t u i t i o n i s m (Springer, Berlin, 1969).

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TROELSTRA, A . S . ,

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TROELSTRA, A . S . ,

Anazysing choice sequences, Reporc 81-05 Depc.MaCh.Univ.

Amscerdam (1981). [TBlAl

TROELSTRA, A . S . ,

On a second order propositional operator

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in i n t u i t i o n i s t i c

THE L.EJ. BROUWER CENTENARY SYMPOSILM AS. Troelstraand D. van Dalen (editors) 0North-HolkmdPublishing Company, 1982

505

L.E.J. BROUWER, THE SIGNIFIC INTERLUDE Walter P. van Stigt The Roehampton Institute London University "Language, the slave of the illusion of reality, cannot be an instrument of truth.. ." (1) "Mathematics is languageless activity of the human mind ..."(2 ) These are just two of many quotations which seem to sum up Brouwer's most fundamental views on language and the role of language in mathematics. Such expressions can be found in his early writings as well as in his lectures and publications of later years. They lend support to the conclusion of a remarkable consistency of Brouwer's views. Any occasional expression diverging from this line of thought can easily be contributed to his mischievous love of controversy and be dismissed. There is also a tendency in the study of the views of a great man and innovator such as Brouwer to analyse such views statically. Brouwer was a great genius , but a human being, a complex character, and he lived a long life. Complete understanding of his personality and his views at every moment of hislife is impossible; the best we can hope for is to follow the broad waves in the curves of his development with continuity almost everywhere.The graph of Brouwer's views on language is not completely constant, there is an important wave, almost a double U-turn. In the next hour I wish to draw your attention to an important shift in Brouwer's thinking on language which is not only remarkable for its magnitude but which has had important repercussions on the development of his philosophy of mathematics at a critical time and enabled him to embark on the re-construction of mathematics on Intuitionist lines. I am referring to the period 1914 - 1926. Underlying Brouwer's early works Life, A r t and Mysticism,and The Foundations of Mathematics are two almost obsessive convictions : one concerning human language and the other the application of mathematics by man. On both issues his vievts are negative, they are inspired by a dislike of what he finds, an unwillingness to accept these aspects of the human reality, and a moral condemnation. They originate in the person of Brouwer himself, his background, his character, perhaps most o f all in his inability, especially during his adolescence, to accept and love his fellow human beings and come to terms with human society. In Foundations of Mathematics the emotional rejection of the use of mathematics in man's conquest of his environment is even stronger than that of human language. One only has to read the original version of his thesis as rejected by his promotor Korteweg (3). I shall not attempt to analyse his attitude towards applied mathematics nor concern myself here further with the development of his views on it. It is, however, interesting to note a parallel shift in his views on this issue around 1913. Following his appointment to the chair of mechanics, generously vacated by Korteweg, there is a softening in his attitude towards applied mathematics. In 1918 in his efforts with Lorenz 0 2 to be appointed to the Royal Commission on Defence, Brouwer even claimed " that the setting up of a laboratory for photogrammetry was his life's ambition ".(4) There is also in 1927 a return to the depressed, sombre mood of his early years and the use of mathematics is branded as "causal acting", as the evil ,"the cunning act". The change is one o f attitude , there is a temporary lull in the offensive, perhaps a pragmatic acceptance, but no sign that Brouwer's conception of the nature of applied mathematics had changed. There is a similar softening o f (moral) attitude towards language in that same period , but there is a more fundamental change in his views on the role of language in the re-construction of mathematics.

506

W P.VAN STIGT

We have already remarked t h a t Brouwer's stand on both t h e s e i s s u e s during t h e e a r l y years o r i g i n a t e s i n h i s person and c h a r a c t e r . In p a r t i c u l a r h i s c r i t i c a l a n a l y s i s of t h e r o l e of language i n t h e l o g i c i s t and f o r m a l i s t philosophies of mathematics was inspired by a d i s l i k e of human communication and s o c i a l i n t e r c o u r s e . His two and only books, w r i t t e n in t h e s e e a r l y y e a r s , a r e c l e a r evidence of t h i s obsession with language and of h i s f i e r c e independence; a b r i l l i a n t mind, b u t a t t h e depth of depression, lonely and i s o l a t e d , searching f o r j u s t i f i c a t i o n of a personal b i a s , brooding over t h e i s s u e s of t h e individual i n human s o c i e t y , human thinking, and His outspoken r e b e l l i o n a g a i n s t s o c i e t y i s more than an adolescent mathematics, unwillingness t o accept t h e conventions of h i s e l d e r s . He sees human s o c i e t y a s a dark force enslaving t h e i n d i v i d u a l , f o r c i n g on h i m not only a p a t t e r n o f behaviour, And language i s t h e instrument of a moral code, b u t even a p a t t e r n of thinking. s o c i e t y ' s domination of t h e i n d i v i d u a l . In many well-known passages Brouwer describes the purpose of language as " t h e imposition of man's w i l l on o t h e r s " . We would be wrong t o i n t e r p r e t t h i s as a somewhat naive and biased h i s t o r i c a l i n t e r p r e t a t i o n and a n a l y s i s of t h e o r i g i r and essence o f human language: man Brouwer sees language a s t h e c a r r i e r of p a s t giving commands t o o t h e r s . experience and c u l t u r e . Through interpersonal signs, through language, s o c i e t y transmits t o i t s new members i t s ideology, r e l i g i o n , i t s moral view of t h e world To Brouwer in h i s and e x e r c i s e s i t s c o l l e c t i v e control over t h e i n d i v i d u a l . depressed s t a t e t h e accumulated wisdom of man, passed through language t o new generations, i s t h e e v i l hoard of man's t r i c k e r y ; through language s o c i e t y imposes on t h e individual - and perpetuates - i t s m a t e r i a l i s t i c vision of the world. In t h e Augustinian theological t r a d i t i o n o r i g i n a l s i n was passed on t o Brouwer seems t o a l l f u t u r e generations through t h e a c t of sexual i n t e r c o u r s e . blame language a s t h e source of a l l e v i l , t h e instrument by which t h e s i n s of t h e past a r e transmitted t o f u t u r e generations. In L i f e , A r t a n d M y s t i c i s m t h e s o l u t i o n f o r t h e individual l i e s i n what Brouwer describes a s "turning i n t o oneself", "looking i n t o oneself" ( c f . I n t u i t i o n ) , l i b e r a t i o n from t h e shackles i n which language has p u t our thinking and turning away from s o c i e t y ( 5 ) . Brouwer did a c t on h i s own advice; from 1904 onwards he withdrew i n t o t h e s o l i t u d e of his h u t on t h e Laren heath, working i n complete i s o l a t i o n on h i s t h e s i s on t h e foundations of mathematics, t r y i n g t o look a f r e s h a t t h e fundamental i s s u e s , analysing t h e nature of mathematical thought, t h e nature of language and c r i t i c a l l y looking a t t h e i r confusion i n p a s t and c u r r e n t philosophies of mathematics. The main conclusion of his t h e s i s i s t h e independence of t h e mathematical thought process from language in a l l i t s a s p e c t s . I t i s a c l e a r statement of his simple conviction of t h e e s s e n t i a l d i f f e r e n c e between t h e s e two aspects of human a c t i v i t y , each operating i n i t s own sphere w i t h i t s own r u l e s , a conviction which d i r e c t e d t h e a t t e n t i o n of his b r i l l i a n t mind t o t h e flaws i n c u r r e n t foundational p r a c t i c e In h i s l a t e r " h i s t o r i c a l surveys" he and inspired his revolutionary diagnosis. w i l l describe " h i s i n t e r f e r e n c e " a s the " f i r s t a c t of Intuitionism", the "separation o f mathematics and language", a programne of s e p a r a t i n g o u t t h e entangled confusion of a long t r a d i t i o n . Brouwer was well aware of t h e complex i n t e r a c t i o n between language and human thought; he recognized t h e r o l e of language in t h e genesis of mathematical thought i n each human i n d i v i d u a l , a s d i s t i n c t from what he c a l l s " t h e primordial happening". He saw t h a t i n the educative process of t r a n s f e r of ideas language has a r e s t r i c t i n g and d i s t o r t i n g e f f e c t , i t burdens them w i t h a host of a l i e n a t t r i b u t e s and prevents t h e b i r t h of notions, ideas which do not f i n d a place i n o r s u i t t h e c o l l e c t i v e wisdom of s o c i e t y , Furthermore, language has c o n s t r i c t e d the human mind t o a s t r a i g h t - j a c k e t completely a l i e n t o t h e a c t i v i t y o f l i n k i n g those i d e a s , human thinking, i n p a r t i c u l a r mathematical c o n s t r u c t i v e thinking. Brouwer's search was f o r t h e pure mathematical, fundamental concepts born i n t h e human mind f r e e from t h e h e r e d i t a r y l i n g u i s t i c deadweight, f o r t h e pure process of linking mathematical c o n s t r u c t s , f r e e d from t h e a l i e n f e t t e r s of t r a d i t i o n a l western deductive logic. In his d i s s e r t a t i o n he holds out t h e p o s s i b i l i t y t h a t

The signific interlude

507

" w i t h the same organization of t h e human i n t e l l e c t and t h e r e f o r e with t h e same mathematics a d i f f e r e n t language of understanding had emerged i n which t h e r e was no room f o r our language o f l o g i c a l reasoning. Probably t h e r e a r e people l i v i n g o u t s i d e our c u l t u r a l sphere where t h i s would be t h e case. Neither can we preclude t h e p o s s i b i l i t y t h a t a t a f u r t h e r s t a g e o f development l o g i c a l reasoning w i l l l o s e i t s p o s i t i o n in t h e language of our c u l t u r e ".(6) Brouwer's promotor , Professor D.3.Korteweg ,did not s e e t h e point and remarked: be a r a t h e r s t r a n g e race o r people t h a t does not reason l o g i c a l l y " ; " i t must t o which Brouwer r e t o r t e d : "Mathematical reasoning i s not l o g i c a l reasoning; because of t h e poverty of our language i t has t o use t h e connectives of logical reasoning The misconceptions e.g. of set theory have a r i s e n not from inadequate mathematical i n s i g h t . Mathematics , f o r lack of a pure language , had t o make do with t h e language o f l o g i c a l reasoning; i t s thoughts a r e not l o g i c a l but mathematical reasoning, something q u i t e d i f f e r e n t " . ( 7 ) In his d i s s e r t a t i o n Brouwer does not embark on the c r e a t i o n of a new, pure language with vocabulary and syntax s u i t e d t o mathematical thinking. In h i s depressed and p e s s i m i s t i c mood of t h e time he seems resigned t o accepting t h e e s s e n t i a l inadequacy of any human language a s a means of communicating s p i r i t u a l concepts,despairing of t h e p o s s i b i l i t y of exact and adequate r e p r e s e n t a t i o n of mathematical thought and communication between mathematicians. A pessimism and scepticism which i s again found i n Brouwer's l a t e r work and which has made commentators c l a s s i f y Brouwer as a diehard s o l i p s i s t . During t h e f i v e years following his doctoral promotion Brouwer continued t o work i n t h e i s o l a t i o n of his home i n Laren. His ambition was a professorship with i t s f i n a n c i a l s e c u r i t y . Under some g e n t l e pressure of Korteweg he d i v e r t e d his a t t e n t i o n t o more d i r e c t l y mathematical problems and so prove his f i n a n c i a l worth. I need not e l a b o r a t e here on t h e t r u l y b r i l l i a n t work t h a t Brouwer produced i n topology during t h e s e y e a r s . His achievements were immediately recognized and these achievements alone have placed h i m among t h e g r e a t i n t h e world of mathematics. They a r e f i r s t o f a l l due t o his mathematical genius, b u t undoubtedly they a r e the r e s u l t of Brouwer's c h a r a c t e r and of circumstances. In p a r t i c u l a r t h e i s o l a t i o n , both physical and psychological, i n which Brouwer had placed himself enabled him t o look a t problem; a f r e s h and unbiased , approach them and s o l v e them with a simple o r i g i n a l i t y which i s t h e hallmark of a l l Brouwer's topological work. On t h e o t h e r hand h i s concentrated involvement i n t h e c r e a t i o n of new mathematics, t h e excitement of achievement helped t o l i f t h i m out of h i s i s o l a t i o n ? gradually d i s s o l v e some o f his pessimism and dispel h i s m i s t r u s t in human communication. The tool of language had served h i m well , he was understood and was welcomed i n t h e i n t e r n a t i o n a l company o f g r e a t mathematicians. He enjoyed t h e i r recognition, their respect and t h e hand of f r i e n d s h i p of H i l b e r t , Klein , Hadamard and e s p e c i a l l y Poincare', who s h o r t l y before his death wrote t o Brouwer: " I am happy t o have had t h e opportunity t o come i n t o contact w i t h a man of your eminence". (8) A t home t h e mathematics department of t h e University of Amsterdam , Korteweg and de Vries , were working t i r e l e s s l y t o secure him a place i n t h e i r team. They persuaded him i n 1909 t o accept t h e part-time unpaid p o s i t i o n of "Privaat-docent". In 1912 Brouwer was appointed "extraordinarius". A y e a r l a t e r Korteweg generously stood down and vacated his own c h a i r i n favour o f his great pupil. Brouwer had emerged from his i s o l a t i o n . He discovered t h a t he enjoyed the g l a r e of p u b l i c i t y and s o c i a l l i f e . He could now speak out openly on t h e i s s u e c l o s e s t t o h i s h e a r t , In h i s Inaugural Address Intuitionism and FormaZism he r e t u r n s t o t h e i s s u e of t h e foundations of mathematics, c l e a r l y taking over t h e mantle of leadership from Poincar; and p u t t i n g forward his I n t u i t i o n i s m a s an a l t e r n a t i v e philosophy t o r e s t a t e s Brouwer's p o s i t i o n o f Formalism. In many ways Intuitionism and FormaZism 1907, but t h e r e i s a remarkable s h i f t of emphasis: his own a n a l y s i s of t h e Primordial I n t u i t i o n of time has become t h e prominent base f o r an a l t e r n a t i v e philosophy. There i s no long moral d i a t r i b e a g a i n s t the e v i l and inadequacy of language and logic. The offensive i s now d i r e c t e d a g a i n s t those w h o claim language t o be t h e exclusive

...

WP.VAN STIGT

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s e a t of mathematical e x i s t e n c e and deny mathematics a higher e x i s t e n c e i n t h e Brouwer's more o p t i m i s t i c mood in these y e a r s i s p a r t i c u l a r l y human mind, r e f l e c t e d i n h i s a t t i t u d e towards language; h i s e a r l i e r s o l i p s i s t i c defeatism i s replaced by a new hope, a new t r u s t i n t h e p o s s i b i l i t y of human communication, a trust t h a t higher, s p i r i t u a l concepts including mathematical concepts can be expressed i n language, but t h a t language must be improved, a new language be created t o r e f l e c t more a c c u r a t e l y t h e processes of human thinking and t h e concepts of t h e human mind. Brouwer remains t r u e t o h i s o r i g i n a l t h e s i s t h a t t h e primary o b j e c t s of mathematics a r e t h e mental c o n s t r u c t s . His search i s now f o r a language which f a i t h f u l l y r e f l e c t s t h i s inner r e a l i t y . Brouwer's d i s s e r t a t i o n i s evidence of h i s e a r l y and o r i g i n a l a n a l y s i s of human thinking and s i g n i f i c a t i o n by means of language ( c f . h i s d i s t i n c t i o n between mathematics and metamathematics, o r t o use h i s o r i g i n a l terminology between mathematics of f i r s t and second o r d e r ) . Yet Brouwer was very much i n tune with t h e thinking of h i s time. I n t e r e s t i n language i n a l l i t s a s p e c t s - e s p e c i a l l y the e l u s i v e l i n k between meaning and s i g n - i s shared by philosophers, s o c i o l Ryle speaks of the theory of meaning a s t h e o g i s t s and mathematicians. occupational disease of t h e twentieth century Anglo-saxon and Austrian philosophy. I t would be an o v e r s i m p l i f i c a t i o n t o confine t h i s obsession with language t o t h e Anglo-saxon world: i t follows a long t r a d i t i o n i n Germany in t h e work of von Brentano, m u t h n e r , Husserl and Meinong, and i n France where t h e philosophical consideration of language was named " l i n g u i s t i q u e " ( c f . de Saussure and A. M e i l l e t ) . P a r t i c u l a r l y r e l e v a n t t o our t o p i c i s t h e concern of some "non professional" philosophers with t h e s o c i a l and behavioural aspects of language, language as t h e means of i n s t i l l i n g and perpetuating a c e r t a i n c u l t u r e , e x e r c i s i n g s o c i a l control And i f I over t h e i n d i v i d u a l ; a concern shared by Brouwer from his e a r l y days. mention t h e name of Lady Victoria Welby ( a l s o known a s t h e Honourable Victoria S t u a r t Gregory o r Victoria Wortley) i t i s not only because she coined the name S i g n i f i c s : her aims and a t t i t u d e s were very much p a r a l l e l t o those of Brouwer and she i n d i r e c t l y contributed t o t h e b i r t h of t h e S i g n i f i c Uovement i n t h e Netherlands Through her various f o r which Brouwer can r i g h t l y claim f u l l parentage. publi.cations and books such a s Sense, Meaning and Interpretation (1896), what i s mecmzng (1903) and S i g n i f i e s and Language (1911) and her correspondence with philosophers such a s C.S. Peirce she succeeded i n concentrating t h e a t t e n t i o n of many on t h e s o c i a l aspects of language and t h e problems of s i g n i f i c a t i o n and meaning. She was a s o c i a l campaigner, an i d e a l i s t who believed i n t h e c r e a t i o n She c a l l e d on t h e i n t e r n a t i o n a l community of philosophers of a new language. and l i n g u i s t s t o co-operate i n preparing a d i c t i o n a r y of t o o l s of l i n g u i s t i c Response t o this i n v i t a t i o n i n study, new words, new analogies and metaphors, Holland came from Frederik van Eeden, one of the g r e a t Dutch poets of t h e turn of t h e century, a medical doctor and a s o c i a l i d e a l i s t . He v i s i t e d Lady Welby a s He was i n s p i r e d by her conviction and t h e l i n g u i s t i c reformation e a r l y a s 1892. became one of t h e a s p e c t s of his programme of s o c i a l reform. The personal f r i e n d s h i p between van Eeden and Brouwer dates back t o 1915. Van Eeden recognized i n Brouwer a man of genius " t h e g r e a t e s t genius I have ever met" who could provide t h e i n t e l l e c t u a l power needed i n t h e r e a l i z a t i o n of his They shared a romantic idealism and a naive s o c i a l and s i g n i f i c i d e a l s . confidence t h a t s o c i e t y and t h e l o t of the poor masses can be changed by t h e benign Brouwer was f l a t t e r e d by t h e i n t e r f e r e n c e of a few wise men l i k e themselves. admiration of t h e famous poet and he shared van Eeden's ambition f o r a leading I n i t i a l l y they sought a platform f o r the r o l e i n a campaign of s o c i a l reform. launching of t h e i r programme i n t h e Amersfoort Academy, which Brouwer and van Eeden themselves had helped t o form. Under t h e chairmanship of Brouwer t h e aim of the Amersfoort Academy was defined t o be "the deepening of the philosophy of Life" (verdieping van levens en wereldbeschouwing). When, however, Brouwer demanded g r e a t e r power f o r the e l i t e executive, and an i n t e r n a t i o n a l involvement of o t h e r g r e a t minds he was forced by " t h e majority of mediocrities" t o hand over

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t h e chairmanship t o Reiman. He formed a break-away group o r c i r c l e . In a c i r c u l a r l e t t e r t o the members and the n a t i o n a l press Brouwer, together w i t h van Eeden and. two o t h e r friends (H. Bore1 and H.P.J. Bloemers) announced t h e s e t t i n g up of a comnittee w i t h the task t o form "An I n t e r n a t i o n a l Academy o f p r a c t i c a l I t s aim was described as "the Philosophy and Sociology a t Amsterdam". r e v a l u a t i o n o f the fundamentals o f l i f e f o r t h e i n d i v i d u a l and f o r society". The foundation o f t h e Amsterdam Academy can r i g h t l y be c a l l e d the b i r t h o f the The S i g n i f i c Flovement i n t h e Netherlands w i t h Brouwer as i t s p r i n c i p a l author. "Preparat0r.y Hanifesto" which was included i n t h e c i r c u l a r l e t t e r and l a t e r published i n t h e f i r s t o f t h e o n l y two issues o f C o m n i e a t i o n s of the Internutiom2 I n s t i t u t e f o r PhiZosophy a t Amsterdam i s c l e a r l y o f Brouwer's hand and sets out t h e immediate o b j e c t i v e s o f t h e Academy: 1. " t o crea4e words o f s p i r i t u a l value f o r t h e languages o f western nations and thus make those s p i r i t u a l values e n t e r i n t o t h e i r mutual understanding";

2. " t o p o i n t o u t and brand those words o f t h e p r i n c i p a l languages which f a l s e l y suggest s p i i i t u a l values f o r ideas u l t i m a t e l y o r i g i n a t i n g i n the d e s i r e f o r m a t e r i a l satiety and comfort, and i n so doing t o p u r i f y and c o r r e c t t h e aims o f democracy towards a u n i v e r s a l comnonwealth".(9) They are the two aspects o f Brouwer's S i g n i f i c programme, the two aspects a l s o o f h i s I n t u i t i o n i s t programme, a d i a g n o s t i c one and a c r e a t i v e one: f i r s t , t o diagnose the misuse o f language and t h e tyranny o f language i n d i s t o r t i n g t h e underlying deeper r e a l i t y , causing t h e b i r t h o f deformed ideas and repressing t h e b i r t h o f pure ones; secondly, a c r e a t i v e one, i n generating a new language, new words, t r u t h f u l l y r e f l e c t i n g t h e pure human thought and thought processes, There seems t o be a confusion o f terminology, i n t u i t i o n i s m and s i g n i f i c s are Brouwer introduced a l e c t u r e by van Eeden, e n t i t l e d almost i d e n t i f i e d . Intuitive Signifies. I n t h i s i n t r o d u c t i o n Brouwer explains t h e aims o f the "The S i g n i f i c Movement, and h i m s e l f r e f e r s t o i t as " I n t u i t i v e S i g n i f i c s " : words o f our present languages are nothing b u t commandosigns o f s o c i a l - l a b o u r Those i n search o f happiness and h i g h e r s p i r i t u a l l i f e w i l l n o t regulations. f i n d i n these languages any i n c e n t i v e t o a c t i v e and powerful thought Attempts by reformers could n o t succeed, s i n c e they had t o r e l y on ordinary language as a they had t o compose t h e signs f o r t h e i r new s o c i e t y from means o f communication those o f a c o r r u p t s o c i e t y and they remained subjected t o the dominating i n f l u e n c e o f o l d associations. The I n t u i t i v e S i g n i f i e s movement concerns i t s e l f w i t h t h e c r e a t i o n o f new words forming a new code o f elementary means o f communication"(l0).

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I n Brouwer's own mind t h e r e was no confusion o f terminology, h i s mathematical i n t u i t i o n i s m f i t t e d i n n a t u r a l l y w i t h h i s f a r wider v i s i o n o f s i g n i f i c reform, and h i s personal ambitions a t t h a t time went f a r beyond mathematics, extended t o academic leadership over a wide range, and even included p o l i t i c a l influence, Closer t o home, h i s ambition was t o t u r n h i s mathematics department f n t o an i n t e r n a t i o n a l c e n t r e o f research, a second Gllttingen, where under h i s leadership the task o f r e c o n s t r u c t i o n o f mathematics on I n t u i t i o n i s t l i n e s would be undertaken. He took steps w i t h the Amstergam City Council and t h e M i n i s t r y o f Education - and p a r t l y succeeded i n o b t a i n i n g t h e i r approval o f the appointment o f f o r e i g n scholars such as Hermann Weyl.

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One aspect o f h i s new optimism i n the p o s s i b i l i t y o f human communication was h i s new f a i t h i n the a b i l i t y o f scholars t o co-operate i n t h e generation o f a new I n the place o f h i s e a r l i e r language and the r e c o n s t r u c t i o n o f t h e i r d i s c i p l i n e . s o l i p s i s t i c despair t h e r e i s a new t r u s t i n t h e c o l l e c t i v e a b i l i t y o f what he c a l l s "a group o f p u r e - f e e l i n g and independent t h i n k e r s " . I n the Preparatory Mmcifesto he describes the f a i l u r e o f e a r l i e r attempts a t p u r i f i c a t i o n o f language as " i n e v i t a b l e and due t o the i n d i v i d u a l character o f t h e work o f such philosophers; t h e i r words were o n l y u s e f u l f o r memorizing t h e expressed thoughts i n the mind o f

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t h e w r i t e r and i s o l a t e d r e a d e r s , they never found a place in t h e mutual understanding of t h e masses.., B u t i f now this task i s undertaken c o l l e c t i v e l y , by a c i r c l e of pure-feeling and independent t h i n k e r s , then t h e i r i n s i g h t s , formed i n mutual understanding, w i l l acquire a l i n g u i s t i c accompaniment which can f i n d a place i n t h e mutual understanding of the masses. As t o t h e r e a l i z a t i o n of these i n s i g h t s , one should bear i n mind t h a t t h e thought a s t h e embryo of a c t i o n has f a r g r e a t e r chance of development and r e a l i z a t i o n i f i t i s t h e c o l l e c t i v e intimate conviction of a group of human beings than i f i t belongs t o a s i n g l e i n d i v i d u a l , however courageous he may be and however l a r g e t h e crowd of half-understanding followers surrounding h i m " . ( l l ) Within t h e c i r c l e of t h e Amsterdam Academy members s e t about t h e dual s i g n i f i c I t was well accepted by members t h a t i n h i s d i s s e r t a t i o n Brouwer had task. completed t h e d i a g n o s t i c p a r t of t h e programme f o r mathematics, a model t o be used One of t h e members, Jacob I s r a e l de Haen, a poet, gay in o t h e r d i s c i p l i n e s . author and lawyer, attempted a S i g n i f i c reappraisal of l e g a l terms in h i s d i s s e r t a t i o n LegaZ Signifies AFebruary 1916) i n which he f r e q u e n t l y quotes from Brouwer's Foundations of Mat ematics , t o which he r e f e r s a s "Brouwer's mathematical s i g n i f i c s " . After lengthy discussions new members were e l e c t e d and i n v i t a t i o n s t o j o i n s e n t together w i t h a copy of t h e Manifesto t o Norlind, B j e r r e , Romain Roland, Landauer, Buber, Gutkind, Carus, Ehrlich, H.G. Wells, Tagore and Peano. On t h e retirement of Korteweg in September 1917 Mannoury was appointed as extraordinary professor a t the University of Amsterdam, an appointment mainly due G e r r i t Mannoury, a self-made s c h o l a r , who durinq t o Brouwer's i n s i s t e n c e , Brouwer's s t u d e n t days combined t h e posts of primary school t e a c h e r and p r i v a a t docent a t t h e University of Amsterdam, had i n s p i r e d in Brouwer an i n t e r e s t i n t h e foundations of mathematics; a b r i l l i a n t mind and t h e most g e n t l e of c h a r a c t e r s , In s p i t e of t h e i r d i f f e r i n g views on language, l o g i c and mathematics they remained l i f e - l o n g f r i e n d s . Vannoury shared Brouwer's i n t e r e s t i n the s o c i a l aspects of language, y e t r i g h t from t h e s t a r t his a t t i t u d e t o language, l o g i c and axiomatics was p o s i t i v e and we may even say p o s i t i v i s t . A t t h e time of t h e s e t t i n g u p of t h e Amsterdam Academy, Mannoury worked f a r away i n Vlissingen as a Soon a f t e r h i s move t o Amsterdam he was welcomed by Brouwer secondary teacher. Van Eeden was impressed and Brouwer f e l t he could leave t o the inner circle. the co-ordinating t a s k i n t h e running of t h e Academy t o his t r u s t e d and Mannoury was e l e c t e d chairman, allowing Brouwer t o conunambitious f r i e n d . c e n t r a t e on t h e re-construction of mathematics on I n t u i t i o n i s t l i n e s and t h e s i g n i f i c t a s k of c r e a t i n g a new language f o r i t , Warmhearted and g e n t l e , Mannoury was a l s o a strong p e r s o n a l i t y with a mind of his His chairmanship of t h e c i r c l e was not a f o r m a l i t y , nor t h e impartial own. adjudication of an o u t s i d e r e f e r e e . He became t h e a c t i v e l e a d e r and changed t h e course of t h e S i g n i f i c C i r c l e , so much so t h a t he i s widely but mistakenly considered t o be t h e founder of t h e S i g n i f i c Movement i n t h e Netherlands. The change of aim and approach i s immediately evident from h i s Opening Address ; the tone and language a r e markedly d i f f e r e n t from t h e somewhat aggressive and Brouwer's g r e a t mathematical patronizing Manifesto of Brouwer's hand. reputation on t h e one hand and Mannoury's humane c h a r a c t e r and his s o c i a l i s t and communist leanings on the o t h e r have led comnentators t o c r e d i t Mannoury f o r t h e s o c i a l e t h i c trend i n t h e S i g n i f i c Movement, which would have caused Brouwer t o Nothing i s f u r t h e r from t h e truth: l o s e i n t e r e s t i n t h e S i g n i f i c Movement. Brouwer's S i g n i f i c programme was a revolutionary moral campaign a g a i n s t t h e corruption and tyranny of s o c i e t y and language; i t s immediate and only o b j e c t i v e Mannoury, with a l l h i s love f o r h i s fellow was t h e c r e a t i o n of a new language. men and h i s concern about the misuse of language, did not s e e S i g n i f i c s a s a movement of s o c i a l reform, but saw i t and made i t i n t o a science of language, a A t the end of his opening address on t h e s c i e n t i f i c and methodical system.

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o c c a s i o n o f t h e o f f i c i a l f o u n d i n g o f t h e Academy, Mannoury prompts h i s audience, u s i n g t h e words o f van Eeden i n de KZeine Johannes :"Hake a s c i e n c e o f t h e word He as y o u have made a s c i e n c e o f t h e s t a r s , e q u a l l y e x a c t and t h o u g h t f u l " . d e f i n e d S i g n i f i c s as " t h e t h e o r y o f means o f communication" v e r y much a k i n t o t h e science o f l i n g u i s t i c s . Under Flannoury's l e a d e r s h i p and l a t e r o f h i s son-in-law P r o f e s s o r D. Vuysje, S i g n i f i c s became a science, " t h e s y s t e m a t i c s t u d y o f t h e mental and l i n g u i s t i c phenomena o f c o m m u n i c a t i o n ' ~ ( l 2 ) , on t h e l i n e s o f Charles W. M o r r i s ' s o r C.S. P e i r c e ' s " s e m i o t i c " , t h e t h e o r y o f s i g n s . Using M o r r i s ' s t e r m i n o l o g y , we can d i s t i n g u i s h t h r e e main branches i n t h i s scignce, 1 s y n t a t i c s , t h e s t u d y o f t h e f o r m a l r e l a t i o n s o f s i g n s t o one another ; 2 semantics, t h g s t u d y o f t h e r e l a t i o n o f s i g n s t o t h e o b j e c t s t o which s i g n s a r e a p p l i c a b l e ; 3 pragmatics, t h e s t u d y o f t h e r e l a t i o n s between s i g n s and t h e interpreter. Mannoury was i n t e r e s t e d i n l o g i c , which would f a l l under 1 s y n t a t i c s ; a l s o , h i s r e s e a r c h and t h a t o f h i s f o l l o w e r s i s pragmatic, t e n d i n g t o Brouwer's concern i s w i t h language as i t c o n c e n t r a t e on language as i t e x i s t s . ought t o be. He s t a r t s f r o m t h e mental o b j e c t , i n p a r t i c u l a r t h e mathematical mental c o n s t r u c t , The o n l y r e l a t i o n s t h a t m a t t e r t o him a r e t h e immediate r e l a t i o n s between these o b j e c t s , not t h e f o r m a l s y n t a c t i c a l and l o g i c a l r e l a t i o n s H i s programme i s t o f i n d a b e t t e r l i n g u i s t i c between words t h a t r e p r e s e n t them. representation. I t i s i n t e r e s t i n g t o n o t e t h a t t h e o r i g i n a l aim o f t h e Academy was c o n f i n e d t o t h e p r o d u c t i o n o f words, a d i c t i o n a r y ; words t o r e p r e s e n t concepts a d q u a t e l y as w e l l as t h e r e l a t i o n s between concepts, Even a f t e r h i s "conversion", Brouwer s t i l l saw language as an a r b i t r a r y , a r t i f i c i a l means o f The M a n i f e s t o r a t h e r communication; he does n o t l o v e n o r r e s p e c t language, a r r o g a n t l y assumes t h a t language can be manufactured c l i n i c a l l y by some s u p e r i o r b r a i n s and g i v e n t o t h e masses. The d i c t i o n a r y never m a t e r i a l i z e d , Buber accepted t h e i n v i t a t i o n t o j o i n b u t had h i s r e s e r v a t i o n s about t h e M a n i f e s t o : "However h i g h I v a l u e t h e p o s s i b i l i t y o f a community o f l i k e - m i n d e d men and t h e i r c r e a t i v e power, t h i s t o me seems i n p r i n c i p l e t o go beyond t h e f u n c t i o n o f such a committee.., The c r e a t i o n o f words t o me i s one o f t h e most m y s t e r i o u s processes o f s p i r i t u a l l i f e . . , a mystery enacted i n t h e e n l i g h t e n e d u n i v e r s a l s o u l o f man" (13). S i m i l a r doubts were expressed by G u t k i n d and o t h e r s , Peano was a l i t t l e more encouraging b u t he saw " d i f f i c u l t i e s c r e a t e d by t h e m u l t i t u d e o f languages". He o f f e r e d h i s c o - o p e r a t i o n i f Brouwer "wanted t o adopt h i s i n t e r n a t i o n a l language"(14). B u t Brouwer s e t about t h i s s i g n i f i c t a s k i n h i s own d i s c i p l i n e singlehandedly. I n accordance w i t h h i s p h i l o s o p h y he s t a r t e d f r o m and c o n c e n t r a t e d h i s mind on t h e mathematical o b j e c t s themselves as c r e a t e d i n t h e human mind, such as t h e continuum, t h e r e a l numbers and t h e i r s t r u c t u r e and found he c o u l d n o t c a p t u r e them i n n o t i o n s and words as were c u r r e n t i n mathematical p r a c t i c e , He a r r i v e d a t new n o t i o n s , more n a t u r a l t o these concepts and t r i e d t o c r e a t e new words f o r them, sometimes u s i n g known words such as "Menge" and g i v i n g them a new meaning, I t i s i n c r e d i b l e how Brouwer managed t o combine w i t h h i s busy s o c i a l , p o l i t i c a l and academic l i f e h i s programme o f r e c o n s t r u c t i n g mathematics a l o n g I n t u i t i o n i s t l i n e s d u r i n g these y e a r s . I n t h e e a r l y t w e n t i e s t h e achievement o f h i s i d e a l s and a m b i t i o n s seemed i n s i g h t . However, l i k e a mirage, i t was s h o r t l i v e d and A f t e r t h e d i s a p p o i n t i n g r e a c t i o n t o h i s S i g n i f i c Manifesto, t h e illusory. Movement developed a l o n g t h e l i n e s s e t by Mannoury; Brouwer remained an i n t e r e s t e d bystander.

I n s p i t e o f t h e e u p h o r i c f a i t h i n t h e c o - o p e r a t i o n o f w i s e men, Brouwer had n o t changed temperamentally, he h i m s e l f remained t h e main o b s t a c l e t o c o l l a b o r a t i o n , a l i e n a t i n g f r i e n d s and c o l l e a g u e s and h i s own s t u d e n t s , s u s p i c i o u s and ambitious. The One by one he f e l l o u t w i t h them and l o s t t h e i r f r i e n d s h i p and support. programme o f r e p l a c i n g c l a s s i c a l mathematics b y an i n t u i t i v e , c l e a r c o n s t r u c t i v e system g r a d u a l l y began t o r u n aground and became l e s s a t t r a c t i v e as t i m e went by.

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D i s i l l u s i o n m e n t reached a climax i n 1927; h i s removal from the e d i t o r i a l board o f the Mathematische Annalen , t h e b e t r a y a l by h i s f r i e n d s , l e f t him embittered and u t t e r l y d i s i l l u s i o n e d . I n a l e t t e r t o a f r i e n d he w r i t e s : " A l l my l i f e ' s work has been wrested from me and I am l e f t i n f e a r , shame , o p p o s i t i o n and m i s t r u s t by my cruel torturers". The I n t e r l u d e was over. He f e l t r e j e c t e d and abandoned and t h e r e were doubts about h i s own a b i l i t y . The clouds o f pessimism and gloom o f h i s e a r l y years returned. While Wittgenstein i n 1928 l e f t Vienna a f t e r a p e r i o d o f i s o l a t i o n , f u l l o f renewed hope, Brouwer spoke i n t h a t same c i t y o f h i s despair o f t h e p o s s i b i l i t y o f human communication and the goodwill o f h i s f e l l o w men, t h e l a s t time before a r e t u r n t o i s o l a t i o n and t h i s time t o s i l e n c e , I t was t h e end o f Brouwer's c r e a t i v e l i f e , t h e end o f a s h o r t p e r i o d o f hope and optimism, o f a temporary t r u s t i n t h e human a b i l i t y t o communicate which had allowed him t o make a p o s i t i v e and unique c o n t r i b u t i o n t o our treasure o f mathematics.

NOTES AND REFERENCES

, Leven , Kunst en Mystiek (J.Waltman, D e l f t 1905 ) p.40. Brouwer, L.E.J. Brouwer, L.E.J., Guidelines o f I n t u i t i o n i s t i c Mathematics, KNAW Proceedings 50 (1947), p.339. van S t i g t , W.P. , The r e j e c t e d p a r t s o f Brouwer's d i s s e r t a t i o n on t h e Foundations o f Mathematics, H i s t o r i a Mathematica 6 ( 1979) , pp. 385 -404. Brouwer t o Loren<, 16 .02.18 C f . a l s o Brouwer's c o n t r i b u t i o n "Over Moraal' i n t h e students' j o u r n a l P r o p r i a Cures ( j g 16, no 10, p.110 ) I' S e l f - l i b e r a t i o n f o r the woman w i l l lead her i n sweeping moves o f f l i g h t t o acts a g a i n s t the laws o f t h e land, against c u r r e n t m o r a l i t y ; f o r these are d i r e c t e d against God, serve t o perpet u a t e l i f e as i t i s , t h e realm o f the d e v i l . I t w i l l move her t o a c t s against r e l i g i o u s m o r a l i t y , f o r t h i s does not stop greed, i t o n l y f o s t e r s . r e s i g n a t i o n t o imprisonment and k i l l s thoughts o f s e l f - l i b e r a t i o n . She w i l l s c a t t e r these acts on her path unabashed , making her way w i t h wonderful b r i l l i a n c e between t h e laws and p u b l i c opinion', p r o t e c t e d 6y no6le dtsgust from h u r t i n g h e r s e l f , Her way w i l l lead her from murder, t h e f t , l i e s and a d u l t e r y w h i l e t h e world has no hold over her u l t i m a t e l y t o sujcide. S e l f - l i b e r a t i o n o f man w i l l g r a d u a l l y move him away from s o c i e t y ; i t w i l l teach him t o seek the n i g h t , the heath and s o l i t u d e . I t w i l l f i n a l l y b r i n g him t o an uninhabited i s l a n d where he w i l l l o o k towards t h e horizon, surrounded by b i r d s l o o k i n g a t him i n s u r p r i s e b u t whomhe does n o t see. He does n o t eat , b u t he does n o t die. O r he becomes a lawyer and smokes a pipe 'I. Brouwer L.E.J.. Over de Grondslaaen der Wiskunde l Maas & van Suchtelen. Amsterdam , 1907-) p. 129. Brouwer t o Korteweq , 23.01.07.See a l s o r e f . (3). .. Poincar; t o Brouwer , 1911. Comnunications o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r Philosophy a t Amsterdam, no 1 March 1918. Brouwer, L.E.J. , I n t r o d u c t i o n t o van Eeden's l e c t u r e on I n t u i t i v e S i g n i f i c s , 13.03.18., BMS 228. See ref.(9). Vuysje , D. , S i g n i f i c s , i t s Tendency, Methodology and Application,Proceedings o f the American Academy o f A r t s and Science, V01.80 no 3 (1953). Buber t o t h e Secretary (Brouwer) o f t h e C i r c l e , 18.03.17, published i n t h e Communications o f t h e Academy. Peano t o Brouwer, 25.10.19.

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TH6 LXJ. BROUWER CENTENARY SYMPOSIUU A.S. Troekim and D. van Dalen (editom) 0Nonh-Holland Publishing Company, 1982

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THE CONSTRUCTIVE CONTRAPOSITIONS O F TWO AXIOMS OF COUNTABLE CHOICE

W I M VELDMAN

Mathematisch I n s t i t u u t Katholieke U n i v e r s i t e i t Nijmegen, t h e Netherlands

INTRODUCTION

The following two p r i n c i p l e s of reasoning set one t h i n k i n g , i f one i s an intuitionist:

(We u s e m , n , . . .

t o range over w, t h e s e t of n a t u r a l numbers and a , B ,

... t o

range

o v e r t h e s e t of a l l i n f i n i t e sequences of n a t u r a l numbers, conceived i n t u i t i o n i s t i c a l l y , F.e.:

t h e u n i v e r s a l spread: ww.

W e w r i t e A(n,m) f o r :
E A.

an i s t h e n-th subsequence o f a , when t h e i n f i n i t e sequence a i s s p l i t up i n t o a

sequence of i n f i n i t e sequences, by means of some p a i r i n g f u n c t i o n < > on w: f o r a l l m,n E w:

an(m)

:=

a()1.

When r e a d c o n s t r u c t i v e l y , t h e s e two p r i n c i p l e s r a i s e s u s p i c i o n s which soon prove t o be well-founded. The c l a s s i c a l mathematician a c c e p t s them, t h o u g h t l e s s l y looking i n de m i r r o r (61'&nrpou

zv

aiv;ypcrri

,

Morgan's

1 Cor. 13,121 and i d e n t i f y i n g them with t w o

i n t u i t i v e l y sound axioms o f c o u n t a b l e c h o i c e , v i z . :

W. VELDMAN

514

ACO1:

For a l l s u b s e t s A of

wXww:

I f Vn3a[A(n,u) 1 , then 3aVn[A(n,an)1

( c f . Kleene and Vesley 1965, x2.2

and x2.1,

Veldman 1981, 1.1 and 1 . 3 ) .

The p r i n c i p l e s ( i )and (ii) s i m p l i f y t h e t a s k of c l a s s i f y i n g d e f i n a b l e s u b s e t s of w and ww,

i n c l a s s i c a l d e s c r i p t i v e s e t t h e o r y and i n r e c u r s i o n t h e o r y , s u b j e c t s

which a r e r u l e d by t h e s p i r i t of c l a s s i c a l d u a l i t y . ( c f . Kleene 1955, s e c t i o n 3 . 4 ,

and Moschovakis 1980, s e c t i o n 1C).

We f i r s t show t h a t t h e p r i n c i p l e ( i ) ,t o g e t h e r w i t h commonly a c c e p t e d p r i n c i p l e s of i n t u i t i o n i s t i c a n a l y s i s , l e a d s t o a c o n t r a d i c t i o n . T h i s may be seen a l s o i n d i r e c t l y , from t h e f a c t t h a t some of t h e s i m p l i f i c a t i o n s which a r e made p o s s i b l e by ( i ) ,do n o t hold i n i n t u i t i o n i s t i c h i e r a r c h y t h e o r y . The union of two c l o s e d s e t s , f o r example, i s n o t , i n g e n e r a l , c o - a n a l y t i c a l ,

whereas, thanks t o ACOO, a l l a r i t h m e t i c a l s e t s a r e a n a l y t i c a l ,

as expected.

( c f . Veldman 1981).

The p r i n c i p l e ( i i ) ,which i m p l i e s ( i ) ,is a l s o f a l s e .

Looking f o r a n o t h e r q u e s t i o n , we r e s t r i c t t h e range of t h e f u n c t i o n q u a n t i f i e r s i n ( i )and ( i i ) t o t h e s e t of a l l i n f i n i t e sequences of 0's and l ' s , i . e . :

the

b i n a r y spread, o r fan: O 2 . Using t h e f a n theorem,

( c f . Kleene and Vesley 1965, * 2 7 . 8 ) we a r e a b l e t o prove

t h e r e s u l t i n g p r i n c i p l e s ( i ) ,and ( i i ) 2' More g e n e r a l l y , we study, f o r a l l d e c i d a b l e , i n h a b i t e d s u b s e t s K of w t h e p r i n c i p l e s ( i ) ,and ( i i ) K :

(i), :

For a l l s u b s e t s A of oXK: I f VaEWK3n[A(n,a(n) )I

,

t h e n 3nVrnEK[A(n,m)1 .

( i i ) K : For a l l s u b s e t s A of wXwK: I f VaEWK3n[A(n,an)1, t h e n 3nVaEWK[A(n,a) 1.

( A s u b s e t K of w i s c a l l e d

decidabZe i f Vn[nEK v n@K], and inhabited i f 3n[nEK]).

W e prove, f o r a l l d e c i d a b l e , i n h a b i t e d subsets K of w: ( i ) , i f and o n l y i f

( i i ) , i f and o n l y i f K i s f i n i t e .

515

Contrapositions of countable choice

The c o n t e n t s of t h i s a r t i c l e c o i n c i d e roughly with c h a p t e r 15 of t h e a u t h o r ' s d o c t o r a l d i s s e r t a t i o n , Veldman 1981.

1.

DOUBTS ABOUT THE PRINCIPLE

Let us d e f i n e a sequence y i n w '

( i ).

such t h a t , f o r a l l nEw: y ( n ) # O i f and only i f n

i s t h e number o f t h e p o s i t i o n i n t h e decimal development of

71

a t which t h e f i r s t

block of ninety-nine 9 ' s i s completed.

Let u s d e f i n e a l s o k:=pn[y(n)#O] ( i . e . t h e s m a l l e s t number n such t h a t y ( n ) # O ) .

This does n o t p r o p e r l y d e f i n e a n a t u r a l number, we do n o t know i f k e x i s t s . k i s sometimes c a l l e d a

v o l a t i l e number.

Despite t h e i r u n c e r t a i n e x i s t e n c e , such numbers have t h e i r u s e s . We d e f i n e a d e c i d a b l e s u b s e t A of wxw by:

For a l l new, mEo: A(n,m) := n t k v m S k

Observe t h a t t h i s s e t A f u l f i l s t h e c o n d i t i o n of t h e p r i n c i p l e ( i ) : f o r a l l aEWw: 3 n [ A ( n , a ( n ) ) ] , a s : e i t h e r A ( O , a ( O ) ) , o r A ( a ( O ) , a ( a ( O ) ) ) .

Observe t h a t , i f k e x i s t s , then Vm[A(k,m)] and, i f n o t , then Vm[A(O,m)]. We a r e n o t very f a r from t h e conclusion of t h e p r i n c i p l e ( i ) . But f a r enough, indeed.

Suppose: 3nVm[A(n,m)1 C a l c u l a t i n g a number n

n

such t h a t Vm[A(n ,m)], w e may decide whether

2 k o r no < k .

I f no 2 k , t h e n , a p p a r e n t l y , t h e r e i s a block of ninety-nine 9 ' s i n t h e decimal development of IT. I f no < k , then:

Vm[m 5 k ] , and w e a r e c e r t a i n t h a t such a block of 9 ' s w i l l n o t

appear.

C l e a r l y , t h e p r i n c i p l e ( i )i s l e a d i n g u s t o i r r e s p o n s i b l e , reckless d e c i s i o n s

2.

THE PRINCIPLE ( i ) IS CONTRADICTORY

Let u s d e f i n e , f o r each yEWw, a s u b s e t A

Y

of w x w by:

516

W.VELDMAN For a l l nEw, mEw: A (n,m) :=

31 5 n [ y ( l ) # 01 v Vl 5 m [ y ( l ) = O ]

Observe t h a t , l i k e t h e s e t A from s e c t i o n 1 , each s e t A

Y

f u l f i l s t h e c o n d i t i o n of

the principle ( i ) . Applying t h e p r i n c i p l e ( i )and reasoning a s i n s e c t i o n 1 , we conclude:

W e appeal t o t h e following weak c o n t i n u i t y p r i n c i p l e :

CP

For a l l s u b s e t s A of 'wxw: If Va3n[A(a,n) 1 , then Va3m3nVB[im=&

-t

A(a,n)1 .

( c f . Kleene and Vesley 1965, e 2 5 . 1 5 , Veldman 1981, 1 . 8 ) .

(8m

i s ( t h e code number o f ) t h e f i n i t e sequence of l e n g t h m which is an i n i t i a l

p a r t of t h e i n f i n i t e sequence 8 : Bm=
..., B ( m - l ) > ) .

W e find:

S p e c i a l i z i n g t o y :=

3.

0

:=

An0 ( t h e zero-sequence), we f a c e a c o n t r a d i c t i o n .

SOME NOTATIONS; SPREADS AND FANS

U k w w be a f i x e d one-to-one mapping from t h e set of f i n i t e sequences k Fw of n a t u r a l numbers o n t o t h e s e t of n a t u r a l numbers.

Let < >:

> i s a coding of t h e f i n i t e sequences. Every n a t u r a l number now s t a n d s f o r a f i n i t e sequence of n a t u r a l numbers. We d e f i n e a f u n c t i o n lg from w t o w such t h a t , f o r a l l aEw: l g ( a ) := t h e l e n g t h of t h e f i n i t e sequence coded by a .

For a l l aEWo,aEwwe d e f i n e :

aEa := Elg ( a ) =a

i .e . t h e f i n i t e sequence

coded by a i s an i n i t i a l p a r t of t h e i n f i n i t e sequence

[Y

Contrapositions of countable choice *:2w

-f

w i s t h e b i n a r y f u n c t i o n on w which corresponds t o c o n c a t e n a t i o n , i.e.

517 for

a l l mEw,nEw:

m*n

:=

t h e code number of t h e f i n i t e sequence t h a t one g e t s by concatenating t h e f i n i t e sequence coded by m and t h e f i n i t e sequence coded by n.

For a l l aEWW,BEWw we d e f i n e ~ E B := v ~ [ B ( & ) = o ] . uE'w

i s c a l l e d a spread i f and o n l y i f :

u ( < >)=O and Va[u(a)=O -+ 3 n [ o ( a * < n > ) = O ] ] . (< > is

t h e (code number o f ) t h e empty sequence, of l e n g t h 0 ) .

A s p r e a d uEWw i s c a l l e d a

finitary spread, o r a f a n , i f and only i f

If K i s d e c i d a b l e i n h a b i t e d s u b s e t of w,


.. , an-1 >Ew: U

K (
we may d e f i n e U Eww such t h a t , f o r a l l K

,...,an-l>)=O = ( a0EKA. ..ha n- 1EK).

uK i s a s p r e a d and t h e set {alaEu 1 c o i n c i d e s with WK. K

I f K i s a f i n i t e s e t , then uK i s a f a n . The f a n theorem i s a famous r e s u l t i n i n t u i t i o n i s t i c a n a l y s i s :

For a l l d e c i d a b l e s u b s e t s A of w and f a n s uEWw: I f VaEu3n[A(&) I , then 3mVaEu3n[n 5 m

A A(&)

I.

Combiningthis theorem with a s t r o n g c o n t i n u i t y p r i n c i p l e (ACl0 i n Veldman 1981, *27.2

i n Kleene and Vesley 1965) w e f i n d a more g e n e r a l v e r s i o n of it:

For a l l s u b s e t s A of

'WXw

and f a n s uEWw:

I f VaEdn[A(a,n) I , t h e n 3mVaEdn[n 5 m

A

A(a,n) 1

( c f . Kleene and Vesley 1965, *27.8). W e u s e t h i s (more g e n e r a l ) theorem i n t h e s e q u e l .

518

4.

W. VELDMAN PROOF OF THE P R I N C I P L E ( i ) 2 .

Let A be a s u b s e t of 0 x 2 ,

such t h a t V a E W 2 3 n [ A ( n , a ( n ) ) l .

m 0Ew such t h a t : VaEW23n[n < mo

Using t h e f a n theorem, we c a l c u l a t e

A

A ( n , a ( n )1 1 .

For any aEw2, we may e s t a b l i s h t h e t r u t h o f : g n [ A ( n , a ( n ) ) ]by i n s p e c t i n g only

...v

f o r : A(O,a(O)) v A ( l , a ( l ) ) V

amo,

A(mo-l,a(mo-l)).

Carrying o u t t h i s i n s p e c t i o n f o r a l l members of {im (a.EW21, which, as a whole, 0

i s only a f i n i t e t a s k , we w i l l d e t e c t a number of t r u e f a c t s among t h e s t a t e m e n t s

A ( n , i ) , where n<mo and i < 2 .

W e now claim t h a t e i t h e r A(0,O) and A ( 0 , l ) a r e both among t h e s e f a c t s , or A(1,O) and A ( 1 , l ) a r e b o t h among them, or...or

A(m

0

-1,O) and A ( m u - 1 , l ) a r e b o t h among

them.

For, i f n o t , we d e f i n e a s p e c i a l sequence aEW2 such t h a t t h e s t a t e m e n t A(O,a(O)) and t h e s t a t e m e n t A ( l , a ( l ) ) and ... and t h e s t a t e m e n t A ( m - l , a ( m - 1 ) ) do, a l l of 0 0 them, n o t belong t o t h e evidence we used d u r i n g our g e n e r a l i n s p e c t i o n . But, we a l s o i n s p e c t e d : a m o ,

d i d n o t we?

We conclude, t h a t 3n < m [A(n,O) 0

A

A(n,l)l

This completes t h e proof of t h e p r i n c i p l e ( i ) 2' The c a r e f u l , metamathematically f l a v o u r e d reasoning w e used, a l s o goes through

i n c a s e w e do n o t know t h a t A i s a d e c i d a b l e s u b s e t of w X 2 . I f A i s , indeed, a d e c i d a b l e s u b s e t of wx2, we may argue with l e s s circumstance.

We l e a v e it t o t h e r e a d e r t o g e n e r a l i z e t h e r e s u l t of t h i s s e c t i o n . Using t h e same method, w e may prove, f o r each f i n i t e , i n h a b i t e d s e t K of n a t u r a l numbers, the principle (i)K.

5.

PROOF OF THE PRINCIPLE ( i i ) 2 .

We u s e t h e r e s u l t of t h e p r e v i o u s s e c t i o n , t h a t f o r each f i n i t e , i n h a b i t e d set K of n a t u r a l numbers:

(i)K and , apply t h e f a n theorem one more t i m e .

519

Contrapositions of countable choice

Let A be a s u b s e t of w X w Z such t h a t VaEW23n[A(n,an)].

Applying t h e c o n t i n u i t y p r i n c i p l e CP ( c f . s e c t i o n 2 ) which may be r e l a t i v i z e d t o any spread a , i n t h i s case: 0 = ~ 2( c f . Kleene and Vesley 1965, *27.5, o r Veldman 1981, s e c t i o n 1.91, we f i n d :

k

(We u s e t h e f a c t t h a t : VkVl[8 (l)=B()

A

2

11.

T h i s i s a reasonable assumption on t h e p a i r i n g f u n c t i o n < > ) .

We d e f i n e a s u b s e t A* of w x w by:

Observe t h a t : VaEW23n3m[A* (n,anm)1 .

Using t h e f a n theorem, we determine MEW such t h a t : -

VaEW23n3m5 M[A*(n,anm)1 , and, t h e r e f o r e : -

VaEW23n[A*(n,anM) 1 .

W e d e f i n e : K := {&M(aEw2} and observe t h a t K i s a f i n i t e , i n h a b i t e d s e t of n a t u r a l numbers.

We c l a i m t h a t : VyEWK3n[A*(n,y(n))

For, given some sequence

-

1.

yEWK, we may c o n s i d e r a sequence aEW2 which f u l f i l s :

-

vn[y(n)=anM], and c a l c u l a t e a number n such t h a t : A*(n,anM), and t h e r e f o r e : A* (n,y(n))

.

Using t h e p r i n c i p l e ( i ) ,we f i n d a number n such t h a t : VmEK[A*(n,m)] and, t h e r e -

-

f o r e : VaEW2[A*(n,anM)1, and: VaEW2[A(n,a) 1.

T h i s completes t h e proof of t h e p r i n c i p l e (ii) 2’

W. VELDMAN

520

We leave it to the reader to generalize the result of this section. Using the same method, we may prove, for each finite, inhabited set K of natural numbers, the principle (ii) K' 6.

REMARK:

FOR ALL DECIDABLE, INHABITED SUBSETS K OF w :

(ii)K + (i)K.

This is an easy remark.

Suppose: K is a decidable, inhabited subset of w such that (ii) K' Let A c - w X K be such that VaEWK3n[A(n,a(n))l We define a subset A* of wxWK by:

FOS all nEw,aEWK: A* (n,a) := A(n,a(O)). We claim that: VaEWK3n[A* (n,an)1.

For, given some sequence aEWK, we consider a sequence a*EWK such that vn[a* (n)=an(0)I. We calculate a number n such that A(n,cr*(n)) and observe that: A*(n,an) Applying the principle (ii), we find nEw such that VaEWK[A*(n,a)1, and, therefore: VmEK[A(n,m)].

This establishes the remark.

We have seen, in section 2, that the principle (i)=(i)

leads to a contradiction.

The principle (ii)=(ii) apparently does the same.

7. AN INSTRUCTIVE EXAMPLE.

Let k be some volatile number (cf. section 2 ) . then K := I0.k) is a decidable, inhabited subset of w, and the set WK "is" a spread (cf. section 3 ) . We make three observations:

(a)To assert that this set K is a finite set, is reckless.

(b) To assert that wK satisfies the fan theorem, is reckless.

521

Contrapositions of countable choice ( c ) To a s s e r t t h e p r i n c i p l e ( i ) Ki s , reckless.

Ad ( a ) : Is K i s a f i n i t e s e t , then K h a s one element o r K has two elements, so: k does n o t e x i s t o r k does e x i s t , and w e a r e n o t a b l e t o decide t h i s .

Ad ( b ) : Remark: VaEwK3m[a(0) 5 m]. Applying t h e f a n theorem, we c a l c u l a t e mEw such t h a t : V ~ E ~ K L ~ (5 Om) l .

Now, e i t h e r k 5 m and k e x i s t s , o r k does n o t e x i s t . We are n o t a b l e t o decide t h i s .

Ad ( c ) : We d e f i n e a s u b s e t A of wXK by: f o r a l l nEw,mEK: A(n,m) := n 2 k v m=O. We observe t h a t t h i s s e t A f u l f i l s t h e c o n d i t i o n of t h e p r i n c i p l e ( i )

K'

a s , f o r any aEwK, e i t h e r a ( 0 ) = 0 o r a ( 0 ) = k, t h e r e f o r e : e i t h e r A(O,a(O))

or A(a (0),a( a ( 0 ) )

)

.

Applying t h e p r i n c i p l e ( i ) we c a l c u l a t e nOEw such t h a t Vm€KIA(no,m)]. We may decide whether no L k o r n

< k. 0 I f no E k , t h e n k e x i s t s , and i f no < k , t h e n VmEK[m=O], and k does not

exist. We a r e n o t a b l e t o d e c i d e t h i s .

The argument given "Ad ( c ) " i s very s i m i l a r t o t h e argument given i n s e c t i o n 1 .

Generalizing t h e remark "Ad ( b ) " , one r e a l i z e s , t h a t , f o r a l l d e c i d a b l e , inhabit e d , s u b s e t s K of w:

i f WK s a t i s f i e s t h e f a n theorem, t h e n K i s f i n i t e .

We now prove:

8.

FOR ALL DECIDABLE, INHABITED SUBSETS K OF w:

IF

(l)K THEN ,

K I S FINITE.

W e c a l l a d e c i d a b l e s u b s e t K of w transparent i f :

Observe t h a t t h e s e t K t h a t we mentioned i n s e c t i o n 7 cannot s a f e l y be s a i d t o be transparent.

Let K be a d e c i d a b l e , i n h a b i t e d subset of w such t h a t ( i ) . We want t o prove, f i r s t , t h a t K i s t r a n s p a r e n t .

W. VELDMAN

522 Let PEW. We define a subset A of wXK by:

for all n€w,mEK: A(n,m) := (n>pAnEK) V m 5 p.

We observe that, for any aEWK, either a(0) Ip or a ( 0 ) > p, therefore either A(O,a(O)) or A(a(O),a(a(O)),

and: 3n[A(n,a(n))l.

Applying the principle (i)K, we calculate n Ew such that Vm€K[A(n ,m)], and dis0 0 tinguish two cases: Either: no>p or

:

A n EK, therefore: 3n[n>p A nEK1, 0 no 5 p v nofK, therefore: VmEK[m 5 p].

This completes the proof that K is transparent.

Now we prove that K is bounded and, therefore, a finite subset of w.

We define a subset A of w X K by:

for all mEK: A(0.m)

:=

31 l>m

for all nEo,mEK: A(n+l,m)

:=

We claim that VaEwK3ntA(n,a(n)

1.

A

lEK]

VlEK[l
Let aEwK and consider a ( 0 ) . We know that K is transparent and may distinguish two cases: Either: 31[1>a(O) or

:

A

lEK], and: A(O,a(O))

VlEK[l 5 a(0)], and: A(a(O),a(a(O)).

This establishes our claim. Using (i) once more, we calculate n Ew such that: VmEKIA(no,m)] 0 Suppose, for a moment, that n 0=O. Then: Vm€K31[l>m A lEK]. K, being inhabited, is an infinite decidable subset of w, and, therefore, @K is, naturally, isomorphic to the universal spread, 'w. Therefore (i), is equivalent to (i) =(i), and not true (cf. section 2 ) .

Apparently, n >O and: Vm€Ktm<no]. 0

Contrapositions of countable choice

This completes our proof: we have seen that every inhabited decidable subset K of w which satisfies (i)K, must be finite.

We did everything we promised in the introduction.

REFERENCES: S.C. Kleene (1955) Arithmetical predicates and function quantifiers. Trans. Am. Math. SOC. 79, pp. 312-340. S.C. Kleene and R.E.Vesley (1965) The foundations of intuitionistic mathematics, especially in relation to recursive functions. Studies in Logic and the Foundations of Mathematics. Amsterdam (North Holland Publ. Co.). Y.N. Moschovakis (1980)

Descriptive set theory. Studies inLogic and the Foundations of Mathematics. Amsterdam (North Holland Publ. Co.). W. Veldman (1981) Investigations in intuitionistic hierarchy theory. Thesis. Katholieke Universiteit Nijmegen.

523