Methods in Mathematical Logic

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1130 Methods in Mathematical Logic Proceedings of the 6th Latin American Symposium on Mathematical Logic held in Caracas, Venezuela August 1-6, 1983

Edited by C.A. Di Prisco

Springer-Verlag Berlin Heidelberg New York Tokyo

Editor Carlos Augusto Di Prisco Instituto Venezolano de Investigaciones Cientfficas Departamento de Matem&ticas Apartado 1827, Caracas 1010-A, Venezuela

ISBN 3-540-15236-9 Springer-Vertag Berlin Heidelberg New York Tokyo ISBN 0-38?-15236-9 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting, re-use of illustrations,broadcasting, reproduction by photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

PREFACE The Vl Latin American Symposium on Mathematical Logic was held in Caracas, Venezuela from the i st to the 6 th of August 1983. The meeting was sponsored by

Asociacidn Venezolana para el Avance de la Ciencia

(AsoVAC), Consejo Nacional de Investigaciones Cientificas y Tecnoldgicas (CONICIT), Fundacidn Polar, IBM de Venezuela, Instituto Venezolano de Investigaciones Cientificas (IVIC), Universidad Central de Venezuela, The National Science Foundation of the United States of America, The British Council, The French Government, Division of Logic, Methodology and Philosophy of Science of the International Union for History and Philosophy of Science, Organization of American States. The Program Committee was formed by Xavier Caicedo (Universidad de los Andes, Bogota, Colombia), Rolando Chuaqui (Pontificia Universidad Catdlica de Chile, Santiago, Chile), Newton C. A. da Costa (Universidade de Sao Paulo, Brasil) and Carlos Augusto Di Prisco (Instituto Venezolano de Investigaciones Cientlficas y Universidad Central de Venezuela). IVlC's Centro de Estudios Avanzados sponsored a seminar consisting in five short courses of ten hours each. The courses were: Non-Clsssical Logics by Newton C. A. da Costa, Some Aspects of the Theory of Large Cardinals by Wiktor Marek (Warsaw University), Mathematical Practice and Subsystems of Second Order Arithmetic by Stephen Simpson (Pennsylvania State University), The w-rule by E. G. K. Lopez-Escobar (University of Maryland), and Logic, Real Algebra and Real Geometry by M~ximo Dickmann (Universit~ de Paris VlI-CNRS). The courses were attended by participants fron Argentina, Chile, Mexico, Italy and Venezuela. The Ateneo de Caracas and the Consejo Nacional para el DesarroIio de la Energfa Nuclear hosted a panel discusssion on the Philosophy of Mathematics with the participation of Newton C. A. da Costa, Rolando Chuaqui, George Wilmers (Manchester University) and Vincenzo P. Lo Monaco, Pedro Luberes and Juan NuNo of the Institue of Philosophy of the Universidad Central de Venezuela. The local Organizing Committee included Arturo Rodriguez Lemoine, (Universidad Central de Venezuela), Pedro Lluberes who organized and chaired the panel discussion on Philosophy of Mathematics and Jorge Baralt (Universidad Simdn Bolivar. Professor Wiktor Marek provided invaluable help organizing the symposium.

IV The final version of these proceedings was prepared by Mrs. Magally Arvelo-0sorio.

My sincere appreciation for her skilled work.

Carlos Uzc~tegui proofread large portions of the book. This volume was edited with the collaboration of Xavier Caicedo, Rolando Chuaqui and Newton C. A. da Costa. Carlos Augusto Di Prisco

LIST OF PARTICIPANTS

Jos~ A. Amor Ayda Arruda Jorge Baralt

Wiktor Marek Maria V. Marshall Adrian R.D. Mathias

Susana Berestovoy

Gisela M~ndez

Lenore Blum

Irene Mikenberg

Everett Bull

Julia ~ l l e r Kenneth McAloon

Xavier Caicedo Ana Cavalli

Anil Nerode

Rolando Chuaqui Roberto Cignoli Peter Clote

Juan NuNo Jeffrey Paris

Manuel Corrada Luis Jaime Corredor

Ruben Preiss Alexander Prestel

Newton C.A. Da Costa

Cecylia Rauszer

MAximo Dickmann Carlos A. Di Prisco

William Reinhardt Arturo Rodriguez Lemoine

Itala M. D'Ottaviano

Rafael Rojas Barbachano

Sergio Fajardo

Gerald Sacks

James Henle Jorge Herrera

Maria G. Schwartze Antonio M. Sette

Jaime lhoda

Stephen Simpson

Ramdn Pino

Thomas Jech

Roger Soler

E.G.K. L6pez Escobar

Jacques Stern

Alain Louveau

Elias Tahhan

Maria Jimena Llopis Pedro Lluberes

Carlos Uzc~tegui Carlos Vasco

Menachem

George W i l m e r s

Magidor

Jerome I. Malitz Professor Michael Makkai was unable to attend the meeting. is, nevertheless, included in the proceedings.

His paper

TABLE OF CONTENTS Xavier Caicedo Failure of interpolation for quantifiers of monadic type. Rolando Chuaqui & Leopoldo Bertossi

13

Approximation to truth and theory of errors. 32

Peter Clote Partition Relations in Arithmetic.

69

Manuel Corrada On the axiomatizability of sets in a class theory.

76

Maximo A. Dickmann Applications of model theory to real algebraic geometry.

A survey.

Carlos A. Di Prisco

& Wiktor Marek

151

On the space (i) < Itala M.L. D'Ottaviano The model extension theorems for ~ 3 - theories Sergio Fajardo

157 174

Completeness theorems for the general theory of stochastic processes. James M. Henle, Adrian R.D. Mathias & W. Hugh Woodin

195

A barren extension. E.G.K. Lopez-Escobar

208

Proof functional connectives. Michael Makkai

222

Ultraproducts and categorical logic. Jerome I. Malitz

310

Problems in taxonomy, a floating log. Jeffrey Paris & A. Wilkie

317

Counting problems in bounded arithmetic. Ramon Pino & Jean-Pierre Ressayre Definable ultrafilters and elementary end extensions.

341

VII

Alexander

Prestel

351

On the axiomatization

of PRC-fields. 360

Cecylia P~uszer Formalizations

of certain intermediate

logics.

Part I. William N. Reinhardt

& Rolando

Chuaqui

385

Types in class set theories. Jacques

395

Stern

Generic

extensions which do not add random reals.

FAILURE OF INTERPOLATION FOR ~UANTIFIERS OF MONADIC TYPE Xavier Caicedo Universidad de los Andes - Departamento de Matem~ticas Apartado A~reo 4976 Botogt~ D.E.,

Colombia

ABSTRACT It is shown that no proper extension of first order logic by LindstrDmMostowski quantifiers of monadic type, that is quantifiers of the form QXl...Xn($1(x I) ..... ~n(Xn)), satisfies the many sorted Craig's interpolation lenmm or even the one sorted, if closed under relativizations. For example L or any of its admissible fragments can not be generated by any number of these quantifiers. This generalizes previous results of the same type shown under stronger hypothesis. In contrast, all monadic logics generated by cardinal quantifiers satisfy interpolation. §0.

INTRODUCTION.

In the context of abstract model theory few

"natural" logics seem to satisfy Craig's interpolation ienmm. No proper compact extension of first order logic satisfying this property is known; and besides L and its admissible fragments there are not too many uncompact examples. On the other hand, there are several general non-interpolation results, starting with Lindstr~m [9], who shows that if a logic extends L ~ w in infinite models, is generated by finitely nmny quantifiers, and satisfies the downward LDwenheim-Skolem Theorem, then it does not satisfy even Beth's definability In [7]

'

theorem.

Friedman shows that no logic between

L~

(Ch)

or L

(Q)

with

e I and the large logic L w(Ch,Q~l~ c Ord) satisfies Beth's theorem, where Ch is Chang's quantifier and Q~ is the quantifier "there are at least m ...". In [3], we observe that this generalizes to sublogics of L ( Q I Q E Mon) where Mon denotes the class of all quantifiers of monadic type. Makowsky and Shelah prove in [ii] that no logic of the form Lwm(Qi i e I) with Qi ~ Mon satisfies many sorted interpolation,

provid-

ed it satisfies Robinson's joint consistency lemma, the Feferman-Vaught

property for sums of structures, and also to be a strong limit cardinal•

II I ~ ~

where ~

is assumed

See also Makowsky [I0] and

Mundici [13]

for related results. In this paper we show that for quantifiers of monadic type, the above hypothesis,

as well as the assumption that the logic contains one of the

quantifiers L(Qili

Ch

or

Q~, ~ ~ i, are unnecessary.

No logic of the form

~I), Qi cMon, satisfies many sorted interpolation, and it does

not satisfy even single sorted interpolation in case it admits relativizations of sentences. After considering some simple applications we show an analogous result for extensions of

L .

Then, we observe that in contrast, the result

does not hold for monadic logics, since all extensions of first order monadic logic by cardinal quantifiers satisfy interpolation. §I•

PRELIMINARIES.

We assume that the reader is acquainted with the

basic notions of abstract model theory as presented, for example, in [i], [8] or by

[ii].

Universes of structures

A,B,C ..... respectively,

For a formula

~(x~)

IAI

~,~,

~,... will be denoted

denotes the cardinal number of

in any logic, ~

=

A•

{(a I ..... an) cAnl@~

~[a I ..... an]}. Elementary equivalence between two structures with respect to a logic L will be denoted "~mod L", elementary equivalence in

L

will be simply denoted

"~

~"

We will consider quantifiers in the sense of Mostowski [12] as generalized by Lindstrom [8].

A ~uantifier

Q

is a class of structures of some

finite type closed under isomorphism, to which it is associated a syntactical rule which allows to form, from formulae ~i~i ) . ~n(~n), . . the . new . formula~ . . ~° = Q~I' ~Xn(~l(Xl)~.... '~n(Xn ))'with the meaning: ~ ~ o ~> ( A , ~ ..... ~ ) e Q o Note that we use the same symbol for the quantifier itself and for its syntactical expression. Given a family of quantifiers Q l i~I, the smallest logic closed under the first order logical operations and the quantifiers Qi will be denoti ed L ( Q li c I) and we will say that it is ~enerated by the quantifiers Ql For cardinals K ~, LK~(Qili ~ I) is constructed allowing °

>

conjunctions of size less than

K;

the subscript

K = ~ will denote

closure under conjuctions of arbitrary sets of sentences.

Finally,

L~w(QIIi ~ I) represents the sublogic obtained by restricting to sentences of quantifier rank less than the ordinal y (cf.[3]). Note that the infinitary character of a logic is a relative matter, and definitively it is not a model theoretical property,

since under very

weak conditions any l ~ s

the form

L(Qili

lies (could be proper classes) of quantifiers.

c I) for adequate famiThis includes the in-

finitary logics construed above (for Y limit). This is also the ease of any logic satisfying the interpolation lemma. This paper studies the case of logics generated by quantifiers of monadic type or . Well known examples of these are the cardinal quantifiers Qe = {(A,B) I IBle~ } as well as Chang's Ch = {(A,B) I IBI = IAI} and Hartig's H = {(A,B,C) I IBI = ICi}. One could consider many other possibilities b where

such as

Qx I .... Xn(%l(X I) ..... ~n(Xn )) ~> P ( I ~ I ..... I~n~l) = 0 P(x I ..... Xn) is a given diofantine polynomial,

Qxyz(~(x),~(y),o(z))

or

~=~ I~~] ÷ (1~I) 2

in the sense of Erd6s-Rado partition calculus. Obviously these do not include the quantifiers of Henkin, Magidor and Malitz, and all sort of order quantifiers (cf.[ll]). Without misgivings about classes of classes, let Mot be the class of all quantifiers of monadic type (cf. [8]). We will abbreviate L~w(Mon) for L

(QIQ ¢ Mot) .

Given formulae ~l(X) .... ,~n(X), which may allow extra free variables, we introduce for each function 6:{0,1 .... ,n - I} + {0,I} the formula ~: where

%i 0

is

~i

and

%i I

=

A ~i+l(X) 6(i) i
{PI ..... Pn }, a corresponding monadic structure ~ = (C, P ~ ..... Pn~) is completely determined up to isomorphism, by the cardinals K~ =

IP 6(x)~I

6 c 2n"

Hence, the associated quantifier Q ~ = {~'I~'i~} is expressible, for ICi~ m , in terms of the cardinal quantifiers QB' B ~ ~+i, and @=m (there are exactly m), m < w: Q ~ x I ..... Xn(P I .... Pn ) <~ / ~ ( Q ~ x P ~ ( x ) A -, QB+IXP~(x)) ^ m (@=mx p~(x)). = ~B K~ = m<~ We have taken the quantifiers ~=m as primitives in order that the two equivalent formulae above have the same quantifier rank. by a simple induction, for every ordinal (even finite)

Therefore,

L Y~ (Q~IICI <-we) ~ L~w(Q B, ~= ml B ~e+l, n<~) where ~ r u n s

(i)

through all monadic structures of size at most

we have the following lemma, shown in [3] for ~ = I LEMMA I.i. Let ~ a n d ~ b e each ordinal y, ~ m o d L Y (Mon).

w e.

Then

and y = ~.

structures of power at most w e , then for LY~ (QB' ~= mlB ~ ~' m<~) implies ~ m o d

PROOF. We can dispense the quantifier Q~+I since it acts trivially in and ~ . Hence, if we have the first elementary equivalence, by (i) it is enough to show that ~ F m o d L ¥ (Mon).

L¥~ (Q~I

ICl ~ w e)

implies ~ m 0 d

This is done by a back-and-forth argument as in [3],

using

the characterization of elementary equivalence for arbitrary quantifiers given by the author there 4.2). Note that if

(cf. theorems

3.1, 3.3

and lemmas

cof(y) ~ m, we may ignore the quantifiers

hypothesis of the lemma, because they are definable

~= m

(from

4.1,

in the

~)without

raising the quantifier rank up to y. Now, we exploit a well known counterexample for interpolation in Let

K eM

(D,E) ing

be arbitrary cardinals,

such that M

IDl = K

and

E

then define

[2]

for

K = M = ml'

M' = m,

K.

L(QI).

to be a structure

is an equivalence relation in

equivalence classes, each one of power

appears in

N(K,M)

D

hav-

The following lemma

and quantifiers of type .

As Makowsky observes in [7] , its generalization to all cardinals and all quantifiers of monadic type is straightforward: LEMMA

1.2.

Obviously,

If

K e M e M'~ w

then

D(K,M) z D(K,M') mod L ( M o n )

this lemma does not hold for

M

or

M' finite°

However, we

have the following finite version. LEMMA 1.3. If mod L ~ ( M o n ) .

K e M e M' e n

with

n~

then

N(K,M) E N(K,M')

PROOF. By lemma I.i, it suffices to show elementary equivalence in the n logic L~w(Q B, ~=mI~ ~Ord, m ~ ) . For this purpose we introduce the following back-and-forth relation between the two structures [3])

(cf. definitions

with parameters in

then for

(n,<).

Let

(x I ..... Xr) , (Yl ..... yr ) E AruB r

(Xl ..... Xr) ~ P (Yl .... Yr ) a partial isomorphism, and

2.1, 2.2 and theorem 3.3 in

N(K,M) = (A,E), and

~(K,M') = (B,E');

p = 0, I ..... n-l:

if and only if the assignment r ~ n-p.

xi ~ Yi is

Let us recall that given [a]~

=

{xcA I ~x , p~

here, wea write and b e B: [b]~

:

To c o n c l u d e

(i) (ii)

~ = (a I .... ,ar)eA r

~x

~a}

{xcB I ~x ~ p that

we h a v e

,~P 9, Given

~b}

~

~

~ ~

for any

a~X

Analogously,

~ ~ Br and

for

equivalence

we m u s t

verify:

is the empty sequence. with

~A

such that (A) ~a,~P ~f(a) for any (B)

then

.

elementary

where

acA,

,

(a I ..... dr,X) .

for

and

r

and

~ c B r,

there exists

f:A+B

a~A,

XcA

and any quantifier

[a]~ ~ Q a

implies

Q

generating the logic,

~ [f(a)]~cQ. aEX p

(iii)

As (ii), interchanging the role of

(iv)

If (x I 'Xr)'~P (YI' ,yr ) then the assignment xi~Y. is a partial isomorphism between the structures to which x and belong, respectively.

We verify only the extension property ~ ~, ~ E A r, b~B r equivalence classes in

B.

(ii), the rest is trivial.

a i or

If and

or b i, and so we can

f establishes a bijection between the equivalence class of and that of bi, with

(ii)

and

then r ~n-(p+l)
E', respectively) not occupied by any define f:A ÷ B such that (i)

A

ai

f(a i) = bi, for i = I ..... r.

f establishes a bijection between the union of the equivalence classes of A not occupied by any a i and the union of the equivalence classes of

B

not occupied by any b i.

~p~

It is easy to see that a a b f(a), because we still have a partial isomorphism and length of (~ a) ~ n-(p+l)+l=n-p. Also, for any aEA, [[a] ~ I = [If(a) ]~ I and disjoint sets of the form a b to disjoint sets [f(a)]~ by f.

[a]~ a

are sent

b Hence, for any

X ! A

and any quantifier

Q = Q~ or

~=m

we have the

the extension property: ~J [a]~ cQ acX a

~ [f(a)]P EQ. aeX a

implies

Therefore,

in the notation of [3]: =m (n,<) (D (K,M), ~ ' ] )e,m (D(K,M') , which shows elementary equivalence.

Q~' ~

=m

)~,m

§2. FAILURE OF INTERPOLATION. For simplicity, we consider first the case of single sorted logics closed under relativizations. The many sorted case will follow readily. A logic is closed under relativizations if for every sentence of type T and monadic predicate V, there is ~V of type Tu{V} such that for all ~ +V

~> ~ r V ~'~+.

Most known logics have this property. Recall that a class K of structures of type T is PC (projective) logic L if there is a sentence 0 in L of type T'~T such that

in a

K = {~PT[ ~ 0 } . The logic has the interp01ati0 p property if any two disjoint PC classes may be separated by an elementary class. In the next lemma we only need closure under negation and conjunctions. 2.1 LEMMA. Assume that L ~ L (Mon) and L is closed under relativizations. If L has the interpolation property, then for any pair of monadic structures (of finite type) ~ z ~ i m p l i e s ~z ~ mod L. PROOF. tures

Suppose that the conclusion does not hold then there are struc~

= (A,PigI) i

(i) ~ ~ ~

and

~

= (B,Pi~)i

mod L

(2)

~

such that

~ ~

mod L.

By the downward LSwenheim-Skolem Theorem in L ~ we may assume without loss of generality that IBId,. Then by (I) we have the following possibilities for each ~E2n:

IP~I By (2), ~ Choose

e IP~I + ~,

and so

+EL such that

=

IBI

=

~.

IF~q/I > Ip~51=w ~ ~

and

~

4.

for some 6.

Hence

IAI>IBI = ~.

For each monadic predicate

Pi

introduce a new predicate Pi' let V be another new monadic predicate. With the help of additional (binary) predicates to express the cardinality relations indicated symbolically below, it is possible to say in the logic L the following sentences:

i) ii)

Vx [Pi(x) ÷ V(x))

iii)

~--m x P~(x)^~ =m x ~ ( x ) A V ( x ) )

for

iv)

(IP6nV 1 : IV1) ^ [IP6nVI ~ IPI)

for

v)

+A~ ~V

Then

( ~ f , ~ .... ), with V interpreted as B and Pi interpreted as Moreover, any ( ~ ' , ~ ' , . . . ) satisfying satisfies the sentences. l ' the sentences must have IA'I > IB'I e ~, because if IA'I = IB'] we would have by (iii), (iv), IP6nV I = I P 6 1 for all 6e2 n, and so by (ii) ~ ' = ~', contradicting (v). Now, consider sentences with an additional predicate additional predicates as necessary to say :

vi) vii) viii)

"E

E

and as many

is an equivalence relation of the universe"

"Each equivalence class is equipotent to the universe" "The number of equivalence classes is equipotent to V".

Then, if @ is the conjunction of

(i)

to

(viii), the class

K I = {(A',E') 1 ( ~ ' , ~ ' , .... E'...) ~0 for some ~ ' , ~ ' , . . . } is PC in L and contains ~(IAI, IBI) with ~ a n d ~ a s above. Moreover, it is disjoint of the class K 2 = {(A',E')I(A',E')

m D(K,K), K ~Card}

which is trivially PC in L. Since we have D(IAI, mod L ( M o n ) by lemma 1.2, interpolation fails in L.

IBI) ~ D(IAI,IAI) D

REMARK. Clearly the conclusion of the lemma follows from the weaker hypothesis: Interpolation ( L , L ( M o n ) ) . 2.2

THEOREM.

If

L = L(Qlli

relativization and extends properly

e I)

with

Qi ~Mon is closed under

Lw~ then it does not satisfy inter-

polation. PROOF. If the logic has the form L = L ( Q i l i c I) with each Qi of monadic type, then for any monadic structures ~ f , ~ such that ~ ~, we have by the l e m a : ~ Qi if and only if ~ Qi This means that Qi does not distinguish infinite cardinals in the sense of Mostowski

[12]

and LindstrSm [8]. Applying Corollary 3.2 in LindstrSm's paper, we have that L must satisfy the downward LSwenheim-Skolem Theorem. By LindstrSm's corollary I in [9], there is a PC class K in L with the following properties:

i)

(A,B)~ K

implies

IBI < ~ .

ii) For all n c ~ there is (A,B) ~ K such that IBI = n + I. Let ~(P) be the sentence which characterizes K (with the help of additional predicates) when P is interpreted as B. Add new predicate symbols

<, (binary), and

V (monadic).

Then the conjunction of the

following sentences "<

is a linear order without last element"

"E

is an equivalence relation"

Vx(P(x) ÷ V(x)) ^ ~(p)V ^ V x ~ y ( P ( y )

^ yEx)

shows that the class of equivalence relations: K = {(A,E) I

IAi e ~,

E has finitely many equivalence classes}

is PC in L, because ~ ( P ) V class is disjoint of the PC K'

=

{(A,E)

if and only if class

IP ~ I< ~.

Since this

J (A,E) m D(K,K), K c Card},

and by le=~a 1.3

we have

N(~,n) ~ D(~,m)

mod

Ln

(Mon),

these two classes are inseparable by any finite quantifier rank formulae and so by

L.

Up to now we have considered single sorted types only.

If we consider

many sorted types then any sentence may be stated about each one of the sorts (or universes) of the type. sentences

~V,

Hence, wherever we used relativized

we may use instead the sentence

ing about a second sort.

~,

construed as speak-

Thus, we have:

2.3 THEOREM° Let L = L ~ ( Q i l i ~I), Qi ~Mon, be a logic satisfying many (two) sorted interpolation then L E L This improves considerably theorem

6.15

in Makowsky and Shelah [Ii] ,

shown under stronger additional hypothesis

(see introduction) which imply

full compactness. §3.

AN APPLICATION.

Note that theorems

extended to arbitrary sublogics of

L(Mon)

2.2

and

2.3

logic satisfying interpolation and having relativizations. can be said of any admissible fragment of est one,

AL(Qo)

,

can not be

since L~I ~ is a such subL~m,

The same

for example the small-

where A denotes closure under

A-interpolation

(cf. [Ii])• However, these logics are generated by adequately chosen families of quantifiers. But ALw~(Qo) may not be generated by finitelymany

quantifiers because it satisfies the downward L S w e n h e i m ~ k o l e m

Theorem (Lindstr@m [9]). For quantifiers of monadic type we have more generally: 3.1

COROLLARY.

Lm~ m and its admissible fragments are not generated

by any number of quantifiers of monadic type. As an interesting example, introduce for each set of natural numbers the quantifier Q S where

S

~ QSx~(x) ~> I~ ~I e S. Obviously proper. Lm~m.

L~m(QSISJ m) < Lml ~ and by the corollarythe extension must be However,

this logic is strong enough to have

In the same way, if

L~(QSIscA)

is a proper

A

AL(QSIs!~)

is a countable admissible set,

sublogic of

LA

whose

Now, we turn our attention to the extensions of analogue of theorems 2.2 and 2.3.

A-closure is

L A-

L~I~, showing an

3.2 THEOREM. A proper extension of L iW of the form L I~( Q i li E I) with QI e Mon does not satisfy the many sorted interpolation lemma. If it is closed under relativization interpolation either. PROOF.

If the logic

(call it L)

it does not satisfy single sorted

satisfies interpolation

then lemma 2.1

applies and each class of monadic structures which is elementary in is characterized by its countable models. Hence, for each i e I, QiXl ..... Xn(Pl(Xl ) ..... en(Xn) ) ~ / { ~ i ~ where

~

is a

"Scott sentence" for ~ in

E Qi IA I ~ ~} L i~.

L

(i)

Since there are only

countably many non-isomorphism structures of a given (finite) monadic type, then the disjunction in the right hand side of (i) may be taken in L~I~. By an obvious inductive argument, L ~L~I ~. D Actually, by the remark after lemma 2.1, we have under the hypothesis of theorem 3.2 that interpolation ( L , L ( M o n ) ) fails also. §4.

MONADIC LOGICS.

The non-interpolation

result does not hold in

monadic logics. In [4] it is shown, using compactness, that M ( Q I ) satisfies interpolation (we use M to denote monadic logics in general). Since the logic Mmm(QI,Q 2 ..... Qm ) is also countably compact, as shown in [5], the same method yields that this logic satisfies (many sorted) interpolation (cf. [6] theorem 1.3.4, for generalizations). However, we may give a more general result. 4.1 THEOREM. For any class of ordinals J, the logic has the interpolation property.

M(Q~I~

c J)

10

Since interpolation is a validity matter and any sentence uses finitely many quantifiers, lemma. 4.2

LEMMA.

by the remarks above it is enough to show the following

Let % be a sentence of

Mwm(Q~ I ..... Q~ ), ~l<...<~m , m and let %' be the sentence of M (QI ..... Qm ) which results of substituting the occurrences of Q~. in ¢ for Qi' then: ~¢<=> ~¢'. PROOF. Associate to each monadic structure ~ = (A,P~ ,[..,PJ) ]~ ~ K , 6c2 n, another structure ~' = (A',P~', .... P £ ) by

ip'~' I

=

K~

if

K6 < ~ I

~i

if

~

if

ei ~ ~ K~. em

wm

A straightforward back-and-forth (~"QI

..... Qm )'

< K6

<

ei+l

argument shows that

in the notation of [3]

.

with

(~,Q~i,...,Q~m)

Hence, ~' ~ '

<=>~

and

so

The converse implication is more delicate and we give the details. N >qr(¢),

N c w. Given ~

as before, associate

K~ N

6O

~

Let

as follows:

if if

K6 < N N ~ K~ < w I

if

w i ~ K~ < ~i+l ' I _
if

~m ~ K~

(2)

~m

Now, define a back-and-forth relation between ~f and -~, with parameters in (N,<), by : a p ~ ~=> a i ÷ b i is a partial isomorphism and length(a)

=

length(~)

P+w~l ~, then P

K = length(a) = length(b)
This permits to define acP~

-{aili
Since [a] p a and

<- N - p, p ~ 0,i ..... N-I.

and

{b i} , or

#

f:A ÷ B

f(a) ~ P ~ [f(a)] p b

~ <=> P6

and so by (2), for each 6:

-{bi I i < k}

such that -{bil i
and

Pf

#

f(a i) = b i

~. and for

Obviously

have simultaneously

~ ~0 -{aili
Assume now that

~a ,~P

~f(a).

the possible forms

-{bili
,

{a i}

we have by (2)

11

acX

IUEf(a)]~ I > ~ acX a i

a

the extension property in one direction. Hence

(cf. theorem

(l~i~m),

The other direction is similar

3.3, [3] ),

( ~ 'QI ..... Qm ) and so ~ ~ ~'

,

(N,,_._.<) ( ~ ' Q ~ I ..... Q~m )

<=> ~

= ~. ~

The claim follows from

Thus,

=> (I)

~,

(3)

and (3).

As an application,

M~(Qo)

satisfies interpolation.

This logic is

also equivalent to

M w(Q I) in validities and so it is recursively axiom-

atizable~ REFERENCES [ I] Barwise,K.J.

Axioms for abstract model theory.

Ann.Math.Logic

7(1974),pp.221-265. E 2] Caicedo,X.

Maximality and interpolation in abstract logics. Thesis, University of Maryland (1978)

[ 33

Back-and-forth

systems for arbitrary quantifiers,

IN: "Mathematical Logic in Latin America", Proc. IV Latin American Symposium on Math.Logic, [ 4]

Holland (1980). On extensions of

L ~ ( Q I ) , Notre Dame J. of Formal

Logic 22 (1981),pp. [ 5] Fajardo,S.

North

85-93.

Compacidad y decidibilidad en idgicas mon~dicas cuantificadores

cardinales,

Rev.Colombiana

de Mat.

14(1980), pp. 173-196. [ 6] Flum, J.

Characterizing

[ 7] Friedman, H.

Beth's theorem in cardinality logics. Math. 14(1973),pp.205-212.

[ 8] Lindstrom,P.

First order predicate calculus with generalized quantifiers.

E 92

logics.

Preprint

(1982). Israel J.

Theoria 32(1966), pp. 187-195.

On extensions of elementary logic, (1969), pp. I-ii.

con

Theoria 35

12

[i0] Makowsky, J.A.

Characterizing monadic and equivalence quantifiers, Preprint (1978)

[ii] Makowsky,J.A. and Shelah,S.

The theorems of Beth and Craig in abstract model theory I, Trans.AMS 256(1979) pp°215-239~

[12] Mostowski,A.

On a generalization of quantifiers0 44(1957),pp.12-36.

[13] Mundici, D.

Quantifiers, an

owerview.

Fund.Math

Preprint (1981).

APPROXIMATION TO TRUTH AND THEORY OF ERRORS "I~(~

Rolando Chuaqui (2)

Leopoldo Bertossi (3)

Universidad

Cat61ica de Chile

Casilla II4-D Santiago,

i.

Chile

Introduction. The purpose of this paper is twofold°

On the one hand, we shall give

in Section 2, a model for the theory of errors based on the probability structures

of Chuaqui

1983 and 1984.

recent result in non-standard

The construction

analysis°

tion of the notion of "approximation

of this model uses

On the other hand, a formaliza-

to truth", which includes

bility of random errors in measurement,

will be presented

the possi-

in Section 3.

This last section is mainly the work of the first author, who takes full responsability

for it.

A formalization

of the notion of approximation

to truth was present-

ed in a paper inspired by ideas of N.da Costa(Mikenberg,

da Costa,

Chuaqui

of random errors

198+,MDC,

for short), but there,

the possibility

in measurement was not taken into account.

As an introduction

ideas in the present paper, we shall briefly sunmmrize

and

to our

the formalization

of MDC. A relational physical

structure ~ = < A , R i > icl is thought of as a theoretical

structure about the objects

in A.

We think of this total struc-

ture (ioe. for each n-tuples of elements of A, it is determined whether it belongs

to the n-ary

for the objects

in A,

relation R i or not) as what the theory gives us A natural way to formalize

to consider scientific

theories

Suppes 1957, Chapter 12.

tuples

this point of view, predicates,

is

as in

What is actually known about the objects in

A,

= where each R i' is only a 1 icI' defined relation over elements of A. That is, only for some

is in the Rartial partially

as set-theoretical

structure

~'

it is determined whether they belong R i or not.

the tuples,

it is undetermined.

for (or compatible with) ~',

A theoretical

if ~ i s

with

an extension

R' where the latter is defined. i' in ~' if it is true in all extensions

theoretical

structure

adequate

i.e. what is true in ~'

structure of ~ ' ,

For the rest of ~is

adequate

i.e.

We define a sentence

Ricoincides

to be true

with the same universe A.

A

for ~' is one of these e x t e n s i o n s , ~ ,

is also true in ~,

and nothing that is false

14

in ~' is true in ~.

Thus, there may be several theoretical structures

which are adequate for ~', theories.

each one determined

(possibly) by different

When our knowledge about the objects in

obtain another partial structure

~",

A

increases we

which is an extension of

were compatible with ~2~ , but not with~j",

~'.

If

then it must be changed and

the corresponding theory also. Another possibility for the rejection of the theory is the following. Suppose that

~f'

is what we know for the objects

theory T, say classical mechanics, universe

~ r domain ) A, that is compatible with

we take a d i f f e r e ~ d o m a i n

rejected,

and that a

~f'

Assume that if

T

is incompatible w i t h ~ ' .

is what we

determines, Then

T

For instance,

at slow velocities and ly true for

T

if we take

A

for B,

should be

but we would say that it is still approximately true

objects in A.

for the

as the medium sized objects

as classical mechanics,

then

T

is approximate-

A.

This picture, however, structure

A

B (which may include A), then ~ '

know about the objects in B, and that the theory a total s t r u c t u r e ~ t h a t

in

determines a total structure ~f, with

~,

~f' exactly.

given by T,

is not completely accurate.

The theoretical

in general does not extend the partial structure

We usually say that ~ coincides with ~', except for possible

errors in measurement.

It is this last factor that we want to formalize

in the present paper. In the measurement of a certain quantity, we usually assume that there is a theoretical value, actual value,

given in the total structure ~f,

obtained,i.e,

of errors in the procedure of measurement. for this error.

Sometimes,

procedure itself.

Second,

but that the

occurring in ~', may differ from it because There are three main sources

there is a systematic error derived from the there may be an error produced by the limit

of precision of the measurement method.

Lastly there may be random errors.

We shall disregard the first two types of error and consider just random errors, which,

in general are the most important.

The account presented

in Section 2 could be modified so as to take into consideration the other types of errors. Thus, if we include random errors in measurement, structure ~ m i g h t

anyway, be considered compatible with it. sentence true

That is, there might be a

in ~', but false, strictly speaking,

fact being enough ground to reject

~(and

this situation, we do the following. with~another

then the theoretical

not be an extension of the partial structure 4', but,

hence T).

in'f, without this In order to formalize

In the first place, we associate

structure that we call an error-structure ~

.

Now, a

15

sentence

~,

instead of being true or false in ~ ,

P ~ e (~)' which depends on ~ e. has low probability ~e'

where~is

compatible

If there is no sentence

according

theoretical

~'

that

according to

structure of ~f,

then @~is

with~'. is thought of as "degree of partial

for a justification

for this view).

error, we cannot get to truth; as possible

to low probability, try to approximate

Because of the possibility

~ true in

198 +, there is a discussion need sequences

falsehood.

~' but with to zero~

~e' have to be complicated somewhat

~fe~ that formalizes

of the experiments

and define probabilities

of Chuaqui

P~e"

falsehood with a Thus, our structures we construct

from

an unlimited number of repetitions for sentences

These probabilities

according to

are defined using the

198+'

This presentation

is offered,

not as a program for practical

but only as a way to illuminate

and evidence,

In Chuaqui

In order to do this, we

for this purpose:

~e' the structure

mentation,

i.e. given any e >0,

of how to approximate

decreasing

but just

In fact, we should

P~e(~ ) <~.

of trials of the same experiment.

~ e ~, derived from

of

In a similar way, a false

falsehood as much as possible,

sequence of probabilities

(see Chuaqui

but we cannot get to falsehood,

which is approximate

to try to get a sentence

truth"

thus, we should strive to get as close

to it, i.e. to high probability.

consequence means rejection,

methods

true in

to ~e' and high probability

an alternative

Probability 1977,

has a probability

and between

the relations between

truth and probability,

in science.

impletheory

It is clear,

that the models presented here are a preliminary version that is oversimplified.

In particular,

we just consider deterministic

hope to improve these models

in the future,

theories.

We

and include nondeterministic

theories. 2.

A theoretical

model for the theory of errors°

In this section, we present a model for the theory of random errors. In this theory, the procedure combination

a quantity

is supposed to have a theoretical

of measurement

introduces

of a large number of independent

causes,

a very small error in the positive or negative for a particular measurement by the different

value, but

a random error that comes from a each one producing

direction.

The total error

is obtained by adding up the errors produced

causes.

In the compound probability

structures

of Chuaqui 1983, the causes are

represented by a causal tree T, with a partial ordering relation. here the causes are independent, related,

two different

elements

i.e. one has no influence upon the other.

Since

of T are never

The fact that there

16

is a large number of causes, terms,

is a non-standard is an infinite, associate error

~k

or

Chuaqui

-ek

=

{ <{-~k,~k}

at

tk,

of

the probability {-~k,ek}

,

consists measure

GKk.

this set and the models

number

,

of all subsets

structure

structures,

if

~,

namely,

K k.

obtains,

The algebra

the GKk - invariant

contains

measure,

if

of

In order to obtain

we need a group of permutations it clearly

the

as in

, {~k } > }

E k is.

of

T

the

In order to describe

The first structure

and the second, ~k'

That is,

With each cause t k we

<{-ek,~ k}

{-ek,~ k}

In this case,

~k'

n~* ~ ~~°

analytic

where

ek and t k may cause

of two relational

is

-ek is produced,

events naturally

Joe.

set.

a simple probability

, { -~k } > Kk

in non standard

T = { to,t I ...... ~ }

with equal probability.

K k consists

The universe

internal

infinitesimal

tk, we introduce

1984o

set

infinite natural number,

but hyperfinite

a positive

action of

Kk

can be represented,

by taking T to be the internal

of

all permutations

assigns

of

1/2 to each of

of Kko

The probability

structure

structure

with

with bounds,

we must assume

causal

for the action of all causes

tree



that

.

~ ~ k=0

In order = ~

is a compound

to keep the total error

where e is a finite posi-

rive number. The set of compounds H i.e.,

H

=

~

consists

~(tk)CK k for each (we assume

The result as follows.

~k

of the functions

$ with domain represents

Xk(~) (ioeo,

Then.

EeH,

measure

~k

value to be measured

~ defined on subsets represented

k~n, we first define

reals or hyperreal

if or

T

and such that

a possible

measurement

is 0).

of

H

As in Chuaqui

is the product

k
of the measurement

= ~k'

~k =

Each

for

For each

(the non-standard

,

k~n.

that the theoretical

of the

is

< K k : tkET >

1983 the probability measure

outcomes

~(t k) = ~k =

by an outcome

numbers)

<{~k,-~k}

,

$cH is given

a random variable by,

{~k } >

-~k )°

the result of the measurement

in outcome

~eH

is

Xk÷*~

17

f(~) =

~

k=0

k(~)

In order to study the distribution of f,

we need the central limit

theorem with the Lindeberg condition in non-standard form as given in Stoll 1982: If n~* I~ - lq and < Yk : k~n > is an internal sequence of *independent random variables in an internal probability space (g,A,~) such that ~(Yk ) = 0 and mal weights

~(¥k 2) = I, and < ak: k- ~k c* ]R such that k=~o~k2 =

~([ ~ ~kYk<-~]) k=0

i* ~ Q ~- ) for all

is a sequence of infinitesi-

2

with

0 <°~e~ +,

then

Xe*~.

Here *~ is the non-standard normal distribution with mean

0

and

standard deviation I. Taking standard parts, one can show that if ~ is Loeb's measure generated by o , then ~([o ~ ~kYk_<~] ) k=0

= ~(o -~) , for all ~c~, a

where ~ is the standard

normal distribution. This work of Stoll is based on Loeb 1975 Thus, we have for our random variable tribution :

and Anderson

1976.

f:H ÷ * ~, the following dis-

~([f-<~]) = ~([ ~k=O ~k [~]-<~])

*+(oA) for each

,

~c* ~.

Then, its standard distribution is, ~([of<~]) for each

= ~(o~),

~e~R

Thus, the result of the measurement is normally distributed with mean

0

and standard deviation °E.

It is easy to modify the construc-

tion so that the mean (i.e. the theoretical measurement) r cA.

is any number

Thus, in order to obtain a model for the measurement of a certain

quantity, we must be given two parameters:

the theoretical measurement

18 r

and the standard error

compound probability probability

°Co With these two numbers, we construct

structure as above

structures)°

Then the random variable

actual value of the measurement, mean

r

and standard deviation The mean

r

For instance,

will be normally

f,

the theoretical measurement

The standard deviation ioeo

and may be

from the data.

of length is usually obtained directly

the data, but other quantities may be consequences

smaller,

the

distributed w i t h

from the theory or be estimated

a measurement

method of measurement.

which gives

o~°

represents

obtained by calculations

a

(that will be called an error

from

or the theory°

°E,on the other hand,

depends

If the method is more accurate

on the

E ° will be

there may be less causes of error of the error produced by

each cause may be smaller.

There may be several procedures

ment for the same quantity have different

standard deviations.

mated from the distribution is roughly estimated

of measure-

all should give the same mean, but possibly This standard error is usually esti-

of actual measurements,

from theoretical

but occasionally

considerations

concerning

it

the

supposed precision of the procedure. These values of the mean and standard error are,then, with a series of actual values using the usual statistical the distribution the theoretical

of the actual values is very improbable distribution

compared

techniquesoif

according to

then this last one is rejected as a model

of the real state of affairs° A l t h o u g h the mean has a theoretical error has not, Thus,

since we are just considering

significance,

the standard

deterministic

phenomena°

it is usually important just to test the appropriateness

theoretical

value of the measurement

obtained.Which

is the theoretical

viation of the error probability

as compared with

standard error structure)

of the

the actual values

(ioeothe standard de-

is not important°

A statis-

tical test for testing the mean with unknown standard deviation Student's

Test.

For using this test, we calculate t

when

r

M - r

-

actually obtained,

values. We can then compute for any error p r o b a b i l i t y of

a

the quantity

is the theoretical va~ue of the measurement,

of the values

M

the mean value

and S M the standard deviation of these

the probability

of

structure with mean r.

for which these probabilities

structure)

is

are less than a certain

deviation of the error probability

~o

Itl
a oR,

We can obtain a value

(one for each error probability Thus, whatever may the standard

structure be,

the probability

of

19

ItI
If we take ~ sufficiently small, then the pro;tl
will be small for

cal value of the measurement° value, is to be rejected. 3.

r

being the theoreti-

Here, the hypothesis that

r

is this

Structures with errors in measurement° We are now ready to introduce a theoretical structure that includes

measurement with random errors, and the corresponding partial structure representing what we actually know. types of theoretical structures

In fact, we shall introduce two new

: a pure measurement structure or

structure, and the error structures associated with it.

]R-

We shall discuss

later the structures that represent our knowledge and which correspond to the partial structures of the old setting sketched in Section Io An

]R-structure is a system of the form ~=

where each Rj, for and each fi~, for

i~l,j~J'

j~J, is an nj-ary relation between elements of icl, is an ni-ary operation from A into

also have operations from

A

into

]Ro

A,

We could

A, or distinguished elements of A,

but, for simplicity, we shall not include them, since they can be replaced by relations° We think of ~ as what we accept theoretically to be true of the elements of A. The R.~'s represent possible relations between these elements, and the f.~ J l 's , measurements performed on them. Thus, f~.~_(ao .... ,ani_l) is a real number that measures some property of the system~r (ao ..... an_ l) of elements of A. There could be two measurements f ~ and f~, with i jk, of the same quantity. In the JR-structure they could coincide. However, in the error structures and the partial structures to be introduced below, they may differ. As in Section I , ~ what the scientific theory prescribes for the elements of Ao

is

In this

paper, we only consider deterministic theories. The language for

]R-structures is a one-sorted language, with variables

x,y,z, .... that constains the following types of atomic formulas: x=

y

Rj x O ..... Xn _i

for each

j~J,

and [fi(Xo ..... Xni_l) er]

, for each each

rc~ (the rational numbers), and

i~lo

These formulas are combined in an

L~lu-language with negation,

countable conjunctions and disjunctions, and finitely many quantifiers°

20

The variables are assigned elements of Ao

For each formula ~ and each

assigment s of the variables in A, we define when s satisfies ~ in ~ , in symbols ~ I =~[s]o Most of the clauses are the usual ones, plus I=[fi(xO ..... Xni_l)~r][s]

iff

f~

(S(Xo) ..... X(Xni_l))~ro

We could also have a two-sorted language with variables and operations for the real numbers, but we shall not need this in this paper. Most of the mathematical results obtained in MDC could easily be extended. to m-structures, with the natural notion of partial ~-structure. We shall not pursue this line here. Instead, we shall introduce another type of theoretical structure: the error-structures (or briefly, E-structures) associated with the m-structure ~ . While ~ d e t e r m i n e s whether a formula is satisfied by an assignment or not, an E-structure ~e,determines only the probability that is assigned to the formula. An E-structure associated with ~ i s ~ e =
~e

for each

~e>icl,jcj

= ~,~

for

ao, .... ani ~A,

a system of the form

,

j~J, and,

if

f~

is an

then

ni-ary operation,

f ~ e (a ° .... ,ani_l) is a random variables whose

distribution is given by an error probability structure

(see Section 2):

fi e(ao .... ,ani_l) is a random variable having a normal distribution with mean f ~ for each i (ao ..... ani - I). . = fi(ao ..... ani_l)

+

variable with mean

0,

icl .

More precisely, . . .

~i(ao ..... ani_l),where

fi~e(ao ,

ei(ao ..... ani_l)

,an _i ) i is random

which represents the error in the measurement.

The distribution of structure

ei(ao, .... an._ I) is determined by an error p r ~ i l i t y l (of Section 2) whose universe H is the domain of

~i(ao ......an _i) o I Each fi~, for iel, represents one method of measurement for a quantity. There may be several methods for the same quantity indexed by different elements of Io Thus f ~ and fk~C may be measurements of length, say by a ruler and by wavelengths° fk~ (a O ..... an_ l) and

fk~ e

for every

(ao ..... an_ l)

In this case, fi~f(ao..... an_ 1 )

a ° ......an_l,~A.

Hence,

=

fi~e(ao ..... an_ l)

are random variables with the same mean.

But,

their standard deviations may be different. This deviation depends on the procedure of measurement, and may be determined by the theory of the method or the empirical data. as was explained in Section 2.

21

It may be more reasonable to assume that for each

icl, f.~e i

is

defined only for a subset of A, namely, for those objects that are possible to measure with the procedure involved° This would introduce additional inessential complications assume that for each tuple fi%le(ao ..... ,an_ l) is defined in

to our models, so that we shall

a ° .... ,an_ I of elements of

A,

(ioeo it is a random variable with values

~).

The language for the E-structures ~e associated with ~ , is the same as that for ~. However, instead of satisfaction, ~ e determines a probability° For each formula ~ and each assignments of the variables in A, we define the probability that s assigns to ~ in ~, in symbols P q ( ~ , s ) . The definition that will be given below is based in that of Scott and Krauss 1966. The main differences with Scott and Krauss are that we use assignments instead of constants, and that we give the definition jointly for all relations and operations instead of doing it separately, and then joining them by their method of independent unions. and Rj, for

In any case ' this means that we assume the different f i~ iel and j~J, to be stochastically independent.

We now proceed to state the definition of

P4lin several stages. Let

Re B i (do,-.... ,an_ l)

be the measure algebra of the error probability structure where f ~ e (a O ..... an_ 1 ) is defined, and ~i@/e(ao ......an i ) , its measure. We shall always consider, now and in what follows, strictly positive measures, i.e. measures that vanish only on the zero of the algebra, and their corresponding measure algebras° This is needed because in our definitions we must have complete algebras, ioe. algebras where the suprema and infima are always defined.

If necessary,

to achieve a strict-

ly positive measure, we take the algebra (and the measure) modulo its null sets. In what follows, we shall suppose that this is done, without mentioning it. @ f For each icl, Bi ne is defined to be the product algebra. B'~el and

,,.an_l)

ui~e its corresponding For

and

= ~
each j~J,

: ao,..o,an_ I c A > .

(strictly positive) measure.

Bj ~

is the two element measure algebra {0j ,lj} , uj~e the measure that assigns i to lj and 0 to 0j°

We consider, now, the product B~e

of all these algebras:

22

8~e = ~ < 8 #

: kcluJ>,

and its corresponding product measure, An element in

B ~e is a system

= <~k on its turn, if

: k~luJ > ;

icl, then

~i = <~i

ui is a system

(ao' ....

an.-1) 1

We call the unit of ~fge, will

be the

corresponding

i i = < li(a ° .... ,an._l): 1

0i =


of

Bk~ ,

0

Similarly,

for

k~IUJ,

a ° ......a n . _ l e A > , 1

: a o .....

where,

Ik,0 k if

and

a n . _ l CA >

For each formula ~ and assignment a valuation

a n . - 1 eA > 1

I , and its zero

elements

i~l,

, a n ~ _ 1)

: ao .....

s

of the variables in A, we define

h ( ~ ; s ) e B ~/'e, b y r e c u r s i o n :

(i)

h(x=y;s)

= {~', otherwiseif s(x) = s(y)

(ii)

h(Rjx o ..... Xn_l ,s)

lj, Pj

=

~ , where its components ~j are given by

if

< s ( x o) ..... S(Xn_10 >eRj~

= 0j, otherwise,

and (iii)

~k = Ik ' for all

keluJ

h([fi(x O ......Xn_l)er]

with

; s)

=

~ , where

~i(S(Xo) ..... S(Xn_l)) = (i.e. the corresponding element of ~i(ao ..... an_l)

=

[fi~e(s(Xo ) ......S(Xn_l)) er] S i ~ ( s ( x o) ..... S(Xn_l)),

li(a ° ..... an_ l) for

(S(Xo) ..... S(Xn_l)) , and ~k = Ik' for (iv)

h(-~ ¢;s) =

(v)

h(

v nc

(vi)

h(

~n;S)

k ~io

~ - h(~,s) =

1.I

A Cn;S) n c 11

kcluJ,

v nc

=

j ~k.

h(On,S ) 1~I

^ h(~n,S ) n e lq

(a o ......an_ l)

23

X

(vii)

h(@x~;s)

=

a~A h(~'Sa)

(viii)

h(V~;s)

=

^ h(~,s~) aeA

Here,

sxa

possibly,

is the assigment that coincides with

on

x

where it assigns

Now we are ready ~o

s

everywhere,

except,

a.

to define P~ (~,s), the probability that

s

assigns

This is simply given by P~e($, s)

=

~e(h(,; s)) .

We shall now proceed to the discussion of the structures that represent what we actually know and their relation to the theoretical structures~ In order to study this relationship, it is not enough to consider one ~-structure ~f, but need to consider all of its alternatives, as well. An alternative to the l~-structure ~/is an l~-structure~, with the same universe A and the same similarity type. That is, if ~=

< A,fi~ ,

Rj#$>icl,jej,

then 7

=

i~l,j~ J

where fi~

and R.~j are of the same arity as f.~l and Rj~ respectively (The similarity type T determines for each keluJ, where the symbol indexed by k is an operation or a relation, and its arity) o The set of alternatives with universe A and similarity type r, we call the A,T-alternatives, For each a l t e r n a t i v e ~ t o ture.

If

by ~ e '

@re is an

~f

we construct the corresponding

E-structure associated with~/,

the E-structure associated to ~ i n

E-struc-

we shall designate

which the distribution of

i~, e(ao ...,an_ 1 ) has the same standard deviation as that of fi~e(ao ..... an_l) O Now, we define an A z-partial verse (i.e. a nonempty set) and of the form: ~f' where and

fi Rj

, for , for

structures, functions,

=

IR-structure ~' , where A is a uniT a similarity type. ~f' is a system

< A,f i~f' , Rj ~f~>ie I, jcJ

iel, is an

ni-ary partial function from A into

jeJ, is an nj partial relation.

~,

For describing partial

it is better to replace relations by their characteristic i.e. we write

24

Rj

(ao ..... an _i) 3

=

I,

if < a ° ......an._l>ERj ~' 3

=

0, otherwise

Then, a partial relation is a partial function from A complete extension ~ o f A,T-alternative type

T)

(ioe, an

the

A,T-partial

A

~-structure

R-structure with universe

A

such that the operations and relation of ~ a r e

those in ~'o

We already have defined satisfaction for

We can now define satisfaction for partial MDC, namely, for any formula

into {0,I}o

R-structures

~ , and assignment

s

9~f' I=T~[S] iff for every complete extension ~ o f

in ~f',

~f'

is an

and similarity extension of R-structures. ~/'

as in

A: we have

~l= ~esl ~'

~F~[S]

iff

@/' I=T ~ [ s

]

~[' ~ U ~[S] , otherwise Thus, a formula may be satisfied, not satisfied, or left undetermined by an assignment

s

in

~'.

Notice that for atomic formulas, the definition of satisfaction given above ~'

can be translated to ~T

:

_ [fi (Xo ......Xn_ I) >r][s] iff

defined and ~TRj

A partial

~-structure

structures ~ c o m p a t i b l e

i~' (S(Xo) ......S(Xn_l) , is

~ r;

x O ......Xn_ 1 Is] iff and equal to I.

least, accept and are not

f

~'

5

(S(Xo) ......S(Xn_l))

defined

represents what we actually know, or, at

willing to change°

with

is

In

MDC,

the theoretical

~/' (i.e. that are possible g i v e n S ' )

the complete extensions of ~f'.

are

Here, the situation will be different.

There may be compatible theoretical structures which are not extensions of @/'. Now we are ready to relate ~ t o

~'o

We say that ~ '

is incompatible

with the total ~-structure ~/(given ~e ) , iff there is a formula ~ and an assignment s in ~ s u c h that, (i)

~/' I=T #Is]

,

2S (ii)

P~/e(~,s ) is low,

(iii)

P~ (#,s) is high, for some alternative to ~/,~.

and

In the account without considering random errors of M D C , @ / w a s to be rejected, if it was not an extension of ~o That is, if a sentence true in ~/, was false in @/' (or, more precisely,

if there is a formula ~ and

an assignment s such that ~ [ s ] , but ~f' I=T ~ ~[s] ). In our present account, ~fmight not be an extension of ~', but anyway compatible with it, if

P~(~,s)

is high for all ~ and

is, everything that is approximately

s

with

true in ~(i.e.

~'

l=T#[S].

That

has high probability

in ~fe) is true in ~', and there is nothing true in ~' that is approximately false in ~(i.e. has low probability in ~e ) . How low the probabilities should be to reject ~ , depends fundamentally on the alternatives available. If there is a "reasonable alternative that assigns high probabilities to all sentences true in ~', then we might reject ~ , even though the probabilities in ~e might not be very low. With no reasonable alternative, we would need very low probabilities, in order to reject ~. The following is a possible explanation of what a reasonable alternative is. First, a definition. We say that the theory • (in the similarity type T ) is confirmed by the B-T-partial structure (given ~[e) if the total ~-structure ~ d e t e r m i n e d by T for the objects in B, has the property that for all formulas ~ and assignments s

in

B, if

~

l=T~[S] then

P~(~,s)

is high.

Suppose that if

B is

a set of objects that has been studied in a science, then ~ is the B-T-partial structure that is accepted as true, and assume that T is confirmed by all such ~ . Then, if ~ is the total ~-structure determined by such a

T

for the objects in

A, it is a reasonable alternative

to ~. The account given up to now is unrealistic in that it assumes that we measure each object just once. We could solve this problem by having several measurements, but assign one value to f i ~ ( a o , .... an_l) , namely, their average. However, by using this procedure we lose some of the statistical power that may be available° In particular, with just one value assigned, we have no real hope of getting rid of ~e in the definition of incompatibility. As given, we d e f i n e d ~ i n c o m p a t i b l e with 91[' (given ~ ) ° The standard deviations included in ~fe usually, important for scientific theories.

are not,

26

In order to include repetitions

of measurements,

we introduce,

for

w

each ~ a n d ~e' the structure ~e' called an w-E-structure, with a language for this structure and a definition of probability for its formulas. To all operations over

and relations

w, the natural numbers;

for each

ao, .....an_leA and

fi~e(ao ......an_l); Rj~(a o .... an_ 1 )

in ~,

~w

fie

we add one more place to range

(ao,...,an_l,t)

is a random variable

tow, with the same distribution as

similarly, with relations,

for all a o ..... an_l,eA,

Rj~ ( a o ,

.... an_l,t) =

tom.

The language is now a two-sorted language with variables for elements of x

=

y

m

=

n

A, and

Rj x ° ..... Xn _l,m J

,

[fi(Xo .... Xn._l,m)->r]

m,n

for elements of ~.

for each

x,y,z...

The atomic formulas are:

jcJ

, for each

rc~, iEl.

]_

This language will be a two-sorted quantifiers elements

for both types of variables. of

A

the other sort. Let

L

for the variables Just as for

~fe' ~e

-language with finitely many IThe assignments

assigns probabilities

B i e(a o ..... ,an_ 1 ) be the product algebra of

w-times,

and

~

e (ao,..o,an_l)

s,ncw, adscribe

x,y,z .... and elements of ~ for the

B

to formulas.

(a ° .... an_ 1 )

its product measure.

Then,

~w Bie

= ~
(a o, ... ,an_ 1 ) : a o ..... ,an_leA >

~w and

~i e is its corresponding product measure. Bj~e 8~

and let ~

~j~ e are defined analogously.

and

= ~ <8k ~e

= kcJ,

Finally,

jeJ, let

k~IuJ >,

be its corresponding product measure.

An element ~e

If

:

For

then

8%

is a system

<~k

: kcluJ >.

~k =

< ~ k (t):tew >

where

~k(t)c Bk~fe.

If

icl, then

27

~i = < ~i(ao ......an-l't) : ao .....an-~A'tcm ~ where

~i(ao .....an_l,t)~Bi (aO .....an_l)

for each t ~ o h, now, assigns to each formula ~ and assignment language an element of B~ , as follows:

= i I, if (i)

s

of the new

s(x) = s(y)

h(x=y;s) ~, otherwise~ ~, if

s(n) = s(m)

h(n = m;s) = ~, otherwise° (ii)

h(Rj x o ..... Xn_l,m;s )

= ~

lj, if ~j(s(m)) and

~k(t) =

(iii)

=

where

R ~ ( S ( X o ) ..... S(Xn_l))

~j, otherwise,

ik, for all

keluJ,

t~m, with

h([fi(x o ...... Xn_l,m) er];s)

= ~

~i(S(Xo) ...... S(Xn_l),s(m)) and

~k(ao ..... an_it )

=

(v), ~vii), and

(viii)

k~j

or

t ~ s(m).

where =

ik(a ° ..... an_l) ,

(a o ...... an_l,t) (iv),

= I

[fi(S(Xo) ..... S(Xn_l)) e r] for all

k ~ i

or

~ (s(x O) ......S(Xn_l),s(m)) are the same as before.

We need two more clauses: (ix)

h(~n~;s)

=

v

h(~,s~ )o

tom

(x)

h(¥n~;s)

=

^

n). h(~,s t

t~

Just as before,

the probability P~

Now, the repetition),

(~,s)

~e

= ~

in ~

is given by:

(h(#;S))o

A,t- partial m-structures Bare of the form

(or partial

structures

with

28

~

=
R~>

'

J

i~l, j~J'

--

where

f~

n.

is a partial operation defined on

1Axm into

~

and

R.~is ]

n.

a partial relation on

]Axe.

(Here,

nA is the set of

These functions may be partially defined on

A,m

n-tuples of A).

or both;

eogo

f~(a o .....,an_l,t) may be defined only for some ao, .... an_l ~A In general, if we assume that ~ for each

ao, ....,an_leA

and

t~m.

represents our actual knowledge, then,

there will only be finitely many

tc~ with

f ~ ( a o ..... an_i t ) defined. A complete extension ~ of ~ will have these functions defined everywhere in A and ~ , and extend those of ~ o Observe that in ~,_ or in any of its extensions fi~(ao,...,an_l,V), for

~

t,vem with

, we may have

f~ (ao,o.o,an_l,t)~

t ~v°

Satisfaction for ~ is defined just as for the partial structures without repetitions ~'. In the language that we have introduced there is a formula ~ and an assignment s such that i=T¢[s]

iff

Iti(a° ..... an_l) I < a

where ti(ao,ooo,an_l) is Student's t for the measurement f~(a o .... ,an_l,V) with

vE~

that are defined in

~

p

and

a e~.

That is M-

fi~f(ao ..... an_l)

ti(a o ..... an_ l) = SM where

M

is the average of the sequence


fi~(ao ..... an_l,V) is defined in ~ > and

SMiS its sample standard deviation° As we mentioned in Section 2, there is an

a ~,

such that P ~ ( ¢ , s )

29

is low for all

u-E-structures

~e

, associated with ~o

Thus, the

following definition makes sense° is incompatible with ~ iff

We say that the partial ~-structure there is a formula ~ and an assignment (i)

~

(ii)

P~(~,s)

~e

~T~[S]

s

such that,

, is low, for every

~-E-structure

associated w i t h ~ .

(iii)

P ~ ( ~ , s ) is high, for a certain alternative to ~/, ~ , ~e and a certain m-E- structure ~ associated w i t h ~ o e If a certain

~-E-structure

reasons, over all other

~e

is preferred, because of theoretical

u-E-structures associated with@/,

might relativize the definition of compatibility to this changing

(ii)

to

(ii)'

: P~

then we

~,

by

(~,s) is low.

However, the definition given (with (ii) instead of (ii)') is preferable, because it is independent of inessential theoretical features, such as standard deviations. It can be shown, by arguments similar to those presented in Chuaqui 198+, that the statistical tests for hypothesis are a special case of these definitions for the situation of this paper.

In particular, we

can explain, in this fashion the approximation to falsehood by a sequence of probabilities decreasing to zero° Two possible extensions of the models discussed here may be mentioned. In the first place, than the normal one.

fi R

(ao, .... an_ 1 ) may have a different distribution

This may happen with some methods of measurement.

The second possible extension is to non-deterministic theories.

In

this case, the theoretical structure Sfitself may have random variables, i.e. fi@/(ao ..... an_ I) may itself be a random variable. This is a possible line of inquiring that we have not yet pursued.

30

REFERENCES Anderson, RoMo

[1976]

A non-standard representation for Brownian Motion and Ito integration, Israel J.Matho volo 25, pp~15-46o

Chuaqui, Ro

[1977]

A semantical definition of probability,

in

Non-Classical Logics, Model Theory, and Computability, Arruda,da Costa, and Chuaqui (editors), North Holland PublicoCoo Amsterdam pp~ 135-167o [1983]

Factual and cognitive probability,

to appear in

the Proceedings of the V Latin American Logic Symposium, Caicedo (editor), Marcel Dekker Inc., New York° [1984]

Models for probability,

to appear in the

Proceedings of the First Chilean Symposium on Analysis, Geometry, and Probability, Chuaqui (editor), Marcel Dekker, Inco, New York°

Loeb, P°Ao

[1985]

How to decide between statistical methods. To appear in Mathematical Logic and Formal Systems. (volume in honor of N.C.A.da Costa), de Alcantara (editor), Marcel Dekker Inc., New York.

[1975]

Conversion from non-standard to standard measure spaces and applications in probability theory, Trans~AmoMathoSoc~volo 113-12 2

211, ppo

Mikenberg, Io, NoCoAoda Costa and RoChuaqui

[198+]

Pragmatic To appear

Scott, Do and P. Krauss

[1966]

Assigning probabilities to logical formulas, in Aspects of Inductive Logic, Hintikka and

truth and approximation to truth.

Suppes (editors), North-Holland Pubblic. Co., Amsterdam, ppo 219-264o

31

Stoll, Ao

Suppes,

A non-standard

[1982]

Po

[1957]

construction

of L~vy Brownian

motion with applications

to invariance

principles,

(Mathematik),

Diplomarbeit

Universitat,

Freiburg,

BRDo

Introduction

to Logic,

D. Van Nostrand Coo,

Inco, Princeton°

(i)

This paper was partially and Technological American

Development

Universidad

de Investigaci6n

of

(DIUC) of the

Cat61ica de Chile°

The paper was partially written when the first author was at the Institute

for Mathematical

Stanford University, Memorial (3)

Program of the Organization

States and the Direcci6n

Pontificia (2)

supported by a grant of the Scientific

Studies

in the Social Sciences

Foundation Fellowship°

The authors would like to thank NoCoAoda Costa for many useful comments o

at

financed in part by a John Simon Guggenheim

PARTITION RELATIONS IN ARITH~L~flC P. Clote I U.E.R.

UNIVERSITE PARIS Vll de Math~mmtiques et Informatique

Tour 45-55

5~me ~tage

75230 Paris

§0.

Cedex 05, France

Introduction. Recall

ble scheme, we give Peano

- 2 Place Jussieu

the folklore

is provable

(infinite)

equivalents

Part of the original

a certain arguments

in certain

The results

presented

herein

, as a defina-

arithmetic.

In this paper,

for certain

subsystems

of

for this work was

to allow one to formalize

subsyqtems

contribute

of Peano arithmetic

~neorem

motivation

amount of machinery

binatorial

of fragments

that Ramsey's

in first order Peano

combinatorial

arithmetic.

to produce

result

com-

of arithmetic.

to the proof theoretic

- specifically

that of

study

Z n induction

(I E n) and E n collection or bounding principle (BEn). To see combinatorial significance of these subsystems, recall the well-known result

of J. Paris

[20]

recursive (or even Kalmar positive integer m gn,m(X)

that a recursive elementary)

= least

y

such that

function

in the function

Ix,y]

.... >

f

is primitive gn,m for some

(n+2)~ +I

if and only if

IE n

~

V x ~y "f(x)

= y"

if and only if BEn+ I ~

i

Vx~y "f(v) = y"

These results were obtained and presented while giving a course Models of Arithmetic in the fall semester of 1982-83 at the Universit~ Paric VII. AutHor's present address: Department of Computer Science, Boston College, Chesnut Hill, MA 02167 USA.

in

33 This result ]inks essentially

the growth rate of a fast grewing recur-

sive function with the proof-theoretic vably recursive

this paper was to find infinite of subsystems

the combinatorial of the recursion

wizardry

sets of

Ramsey Theorem-type

of [20].

The main result,

to be essentially

[6].

into

(possibly non-standard)

M

set.

~ B~n+ I

This characterization

M

satisfies

initial segment

I

BEn+ I.

iff

M

>

of the collection

of

~n induction.

of

M

is

M satisfies

if and only if

satisfies

a certain infinitary An

definable

non-standard)

En formulas,

subset of

(Theore~ 4).

En induction

~n

M

I is a model

is partitioned

and J. Paris

[9]

formulation :

a model

: when any

into (possibly

corollary

go to

(Corollary

for Boolean combinations

even prefixed by bounded quantifiers.

give an alternative

equivalent

then at least two elements

induction

an

induction if and only if

This yield the useful

implies

[9] :

A n pigeon hole principle

bou~dedly many pieces,

the same piece

sub-

there

scheme then answers a question

Mills and J. Paris in

n-Ramsey

M

~-element

(M) n . <M

Along the way, we give an easy combinatorial A model

En+ I

Schematically

about initial segments posed by G of

formalization

boundedly many pieces,

AI

6) that

Theorem ii, turns

a proof-theoretic

Our result is that a model

is an unbounded homogeneous

unbounded

characterizations

if and only if for any recursive partition of

M

in

theoretic analysis of Ramsey's Theorem done by C.G.

Jr. in

collection

Our original motivation

of Peano arithmetic with the intent to simplify some of

out (in one direction) Jockuscb,

notion of a function being pro-

in a system of arithmetic.

In Theorem

of 8

we

and proof of a result due to G. Mills M

satisfies

E2n

collection

if and

only if the filter product of the Fr~chet filter with itself n-times F x

F x

F x

.

.

.

x

F

n- times is Ao-Complete.

34

The techniques

used are model

ultraproducts)

and more especially

formalized theorem,

versions

Lemma

answer

!.

(involving

recursion

end extensions

theoretic

Forthcoming

some of the questions

low-basis

work by the author has further

theorem and other recursion

and

(involving

of the limit lemma and the Jockusch-Soare

14).

ed the low-basis

theoretic

theoretic

exploit-

techniques

to

posed in this article.

Preliminaries. A formula

in the language

said to be bounded

of Peano arithmetic

if it is built up from atomic

Boolean operations

and by bounded

Ao = F~o = ~o

=

{~:

quantification

is a bounded

{+,.,0,I,<}

formulas : 3 x
formula

is

by the and V x < y.

in the language

of

arithmetic}.

Let

In+ I be the class of formulas

m cN

and 0 is in

form

V x I .... ,V Xm~

in the language

~n;

also

it requires

a certain

a theory

T

Nn+l

be the class of formulas

of the

where

mc N

and 0 is in

is a theory

An(T)

In(T)

equivalent

= In(T)

amount of induction

quantifiers

thus obtaining

The theory P- is the usual semi-ring

IE n (in-induction

finitely

(see [ii],

I n,

and in

n ~n(T).

to allow one to pass bounded

of unbounded

ly ordered

then

are provably

~n formulas;

where

let

of arithmetic,

of formulas which

on the form @ x I .... ~ Xm0

Hn(T) T

to

~n respectively

or collection quantifiers T-equivalent

axiomatized

is the scheme

T

are the classes

As shown in

p. 200).

w i t h parameters)

If

[ll,p.201] (see below)

in

to the right formulas.

theory of a discrete-

35

v ~E ( ¢ ( 0 ,u) ÷

&

÷ v x(¢(x,u)

÷ ¢(x+l ,u))) +

÷ ] + v x¢ ( x , u)

w h e r e ¢ is a Zn formula. BEn(In-Collection

or b o u n d i n g principle w i t h parameters)

~ [V x < a

is the scheme

~ y¢(x,y,~) ÷ ~ b V x < a q y < b¢(x,y,~)]

where ¢ is a Zn formula. We remark that

A° Wilkie has shown that

1E o is equivalent

to the scheme of Zo-induction without parameters, n e i

the scheme with parameters

Proposition

3).

Peano arithmetic,

with all of the induction convention

is stronger° Throughout

is denoted as DEFINITION.of M

X ~Zn(M), M

> An

a model

both a Zn and a ~n ) formula w i t h

is definable

if its graph is definable).

means

that for any

This

A n definable unbounded

A n partition F:X ÷ a (for partitions,

numbers with the set of their predecessors a c M, there exists

so that

we identify

F : X ÷ {0,i, .....a-l})

i < a such that F -I (i) is unbounded

This is the infinitary A n pigeonhole unbounded

A~in

+ 1Z o .

etc...

(M)~M

and any

from

this paper, we adopt the

f will be said to be Zn(respectively

in M (a function

P"

denoted P, is the theory P- together

schemes.

M if it is definable by a Zn(resp° parameters

where

(This latter follows

that M is a coun:tab!e model of the w e a k base theory P

A set X or function

set X

over

although of course for

scheme

in M.

: any A n partition of a

An

subset into b o u n d e d l y many pieces has one piece with unbounded-

ly many elements. Recall the w e l l - k n o w n FACT I.-

If M I=P- + IZ n for n e i, then one can define functions by Zn

recursion

in M.

More precisely,

then so is f, where ÷

÷

->

f(O,x) = g(x) PROOF. total in

Since

if g,h are total Z n functions

in M,

f is defined by

n e 1

f(a + l,x)

and all primitive

->

=

->

h(a,x,f(a,x))° recursive

functions

are provably

IZl, we have

f(a,x)=y <--> @ s ~ S e q [ l h ( s ) = a + l df &

(s) a = y]

& (S)o=g(x)

& V i
(s)i+l=h(i,x,(s)i)

36

<---> Ys~Seq Since

[lh(s)=a+l

& (S)o=g(x) the above

IE n implies BEn,

& V i
formulas

(s)i+l=h(i,x,(s)i) are equivalent

+(S)a=y]

to a EnreS p.

~n formula. Recall

D

that M' is an n-elementary

M ~ M') iff

M' is an extension

end extension

of M (denoted by

of M satisfying

~,e

(i)

for

(ii)

~ c M and ~ a

E n or

M' is an end extension ment of M' - denoted by ÷ @ b eM

(M'

M I=0(~)

M c M'): e

~ mcM

iff

M'l=e(~)

M is an initial V a ~M'

(M'I=

and

seg-

a ~m

l=a = b))o

Our point of departure THEOREM 2 o

Nn formula,

of M (equivalently,

is from

(Kirby-Paris

[ii] Theorems

A and B) Modulo

P- + I E o ,

for

ne0 IEn+ I ----->BEn+l ~--> and these implications of P- + IE o and for

BKn ==~

are strict°

II n Furthermore,

model

n>-l,

M ~ BEn+ I

iff

M M

iff

An (M)I M admits

a proper n + 1-elementary

extension which REMARK°

for M a countable

By induction

ne I

on

is a model

end

of P- + IEoo

using Fact i and Theorem

2, it is easy

to show that M

> (M) I M A

where

iff

• (M) I M A

n

the latter means

(i.e. the partition subset

i < a for which

relation

a partition

defined

the partition

F-l(i)

many pieces,

of the entire universe)° relations

partition

on an unbounded

partition

relations

partition

F : M ÷ a

is unbounded

on the left concerns

into boundedly

concerns

n

: given any A n definable

where a ~ M, there is an unbounded

M

a partition

whereas

of an

that on the right

The reason

given before Fact

in M

that we have

i in terms of a

subset will be clear when considering

(Theorems

5 and 9)°

other

37

We remark here that the equivalence is more or less implicit

in [II];

however,

were first introduced by E. Kranakis, in the context of set theory For instance,

suppose that

formula.

Let

Suppose

i

definable p a r t i t i o n relations

who proved the above equivalence

(Theorem 4°3 of [12]). M

M ~ IE 1 by proving that every non-empty a least element°

of BEn+ I with M ~ " ~ ( M ) < M n

that

M

i , (M)<MO We show first that A1 I 1 definable subset of M has

~ @ yO(x,y) where

8 is a bounded

F : M ÷ a + 2 be defined by least x ~ a such that ; y <s@(x,y)

if such exists

f(s) = a + i otherwise° If i ~ a + I is such that ly

i

is the least element

definable u n b o u n d e d

F-l(i)

in the given

is u n b o u n d e d Z I set.

subset of M then by Fact I,

in M, then clear-

Now if

X-is

a AI

X is order isomorphic

with M via a A I function and so i , (M)<M implies AI

M

For hypothesis

n >i

i > (M)<M.

M AI

argue in the same fashion,

and T h e o r e m 2 to have that M

a formula of the form ¥ y < s @ w h e r e is

Hn)iS equivalent

~ BE n and hence

@ is E n ( r e s p e c t i v e l y

in M to a En(respectively

Before going on, several that in the proof that for M ~ BEn+ I implies

but using the induction

First is

n e I of a proper n + 1-elementary

end extension K of M w h i c h is itself a model of P

definable

3)

~n)formula.

things are w o r t h noting.

the existence

Kirby and Paris constructed

([ll]Lemma

~ y < s ~where

a

"E

n

ultrafilter"

U on

subsets of M by defining an u-sequence

+ IZ o, Zn(M)

the En

X o ~ X I ~ X 2 ~ ooo such

that

(1)

each X i is an u n b o u n d e d

(2)

given the i th En definable partial

A n definable

subset of M,

function fi with bounded

38

range in an u - l i s t i n g w i t h infinite repetitions of all such functions

(recall that E

parameters

definable means definable w i t h n - it is at this point that c o u n t a b i l i t y of the

model M is used),

either Xi+ I is disjoint from the domain

of fi or fi is constant on Xi+l. The " u l t r a f i l t e r " U is then defined to be i e ~(X = Xi)} -

U is a "E °

U = {X~Zn(M)

u l t r a f i l t e r " in the sense that for a In n

d e f i n a b l e subset X of M, either X c U or there is a disjoint Zn set Y w i t h K

=

Y c Uo

K

definable

D e f i n i n g the r e s t r i c t e d u l t r a p r o d u c t

{f : f is a Zn partial function w i t h dom(f)~ U

then by u s i n g the fact that holds

:

: if ~ is a

M

~

and rng(f) c M } / U ,

IZn, the following form of Bog' T h e o r e m

Zn formula w i t h m free v a r i a b l e s then

~ ~([fl ] ..... [fm ])

iff {a E M : a is in domain of fl, .... fm and

One then easily verifies

M

~ ~(fl(a) ..... fm(a))}~U°

that K is a proper n + 1-elementary

end e x t e n s i o n of M w h i c h satisfies P- + IE . Each instance o

~(0,~) of a

En or

the form

&

v x(~(x,~)

~ ~(x + l,~)) ÷ v x ~(x,~)

~n i n d u c t i o n a x i o m w i t h parameters r e s t r i c t e d to M is of

En v Z n + I V ~ n + l

(resp. En v En+ I v ~n) and so true in

S i m i l a r l y every instance of a En_ I or

K.

N n - i i n d u c t i o n a x i o m w i t h para-

meters in K is true in K° Thus it is a natural q u e s t i o n w h e t h e r given a m o d e l of for n ~I,

BEn+ I

one can construct a proper n + 1 - e l e m e n t a r y end e x t e n s i o n

K satisfying

P

P R O P O S I T I O N 3.

+ IEn°

A n e g a t i v e answer follows from

If K is a proper n + 1-elementary end e x t e n s i o n of M

s a t i s f y i n g IE n w i t h n e 0, then M is a m o d e l of B E n + 2. PROOF. for n ~ i

(The case for n = 0 is part of T h e o r e m B in [ii]°

The p r o o f

arose in a c o n v e r s a t i o n w i t h Z° A d a m o w i c z and A~ Wilkie) o We

show that M

~ B~n+lO

39

Suppose

that M

~ V i
q t V x 9, where

~ is En. Then for every

d c K - M w e have Vi < a @ t < d Fix

e 0 ~ K - M.

M ~ Vx ~(i,t,x).

Then for every V i~ a @ t < d

d ~ K - M, K > V X<eo~(i,t,x).

K > V i < a ~ t < d V x < e o ~ ( i , t , x ).

Since

K

~ IEn,

let d o be the least d satisfying

every d ~ K - M satisfies M QUESTIONS. Kaufmann

this formula,

~

q d V i
i)

The v e r s i o n

(see [8]

t
Vx

d o ~ M.

in arithmetic

2)

fail,

n e I and any countable model

a proper n + 1-elementary P

+ BE

n

attempts

completeness

theorem

since the arguments

We mention

to solve this p r o b l e m affirmatively

a tree argument

(as in Lemma

([14], po 252)°

end

o

Does T h e o r e m 2 hold for uncountable models?

by trying to formalize metized

of a question posed by Mo

for

extension of K of M w h i c h is a model of

that the natural

As

$.

po 102) is whether,

M for P- + BEn+ I there is n e c e s s a r i l y

the above formula°

It follows that

proceed

14) or by the arith-

However,

these attempts

appear to require more than the desired amount

of collection. 3)

To our knowledge,

it is an open p r o b l e m to characterize

those models M with a proper n - e l e m e n t a r y K

~ P- + Em or

J-P Ressayre

even

K

~ P.

: characterize

tary end extensions,

i.e.

those models M having a k-tower of n-elemenM

~ M I ~ M2 ~ n,e n,e n,e

connection we also mention an open p r o b l e m results)

due to J. Paris and A. Wilkie

P- + IE ° + BE1,

P- + IE

to the Barwise compactness

"'" n~

~

In this

e

(with u n p u b l i s h e d partial

: if M is a countable model of

then does there exist a proper

w h i c h is a model of

2.

end extension K for which

This is related to a p r o b l e m due to

end extension

(i.e° an arithmetic o theorem)?

analogue

K

of

M

of a corollary

INDUCTION SCHEMES° In [5] Ho Friedman defined a weak subsystem of second order arith-

metic

"arithmetical

comprehension

axiom w i t h r e s t r i c t e d

induction" ACA o

40

w i t h language

{+,.,0,1,<,e}

induction axiom

and as axioms

: P-, extensionality,

the

(and not the second order scheme)

V x[(0 c X & V x ( x ~ X and arithmetical

÷ x + i cX))

+ V x (x~X)]

comprehension X V x (x ~ X + - + ~ ( x ) )

where

~ is an arithmetical

viduals

formula

(i.e. quantification

only over indi-

and not sets) possiblywithfree second order variables other thanX.

If M is a model of P then (M, Def M) is a model of ACA o, where Def M is the class of subsets of M definable w i t h parameters.

Thus ACA o is

a conservative

to Ei-CAo(where

arithmetical

extension of P.

comprehension

Since ACA o is equivalent

is replaced by E 1 comprehension),

show that ACA ° is finitely axiomatizable is equivalent Lermna, m ÷ produces ACA o

to certain second order combinatorial

(~)~)o

theorems

By the Paris-Kirby m i n i a t u r i z a t i o n

indicators

for stron~

one can

and that its essential

initial segments

axiom

(KSnig's

technique,

I of M (i.e.

where RMI = {A n I : A c Def M}) and hence independence

this

(I,KMI~>

results

for

Peano arithmetic. The motivation

for the following easily established result was to

solve the equation

P ACA

O

IEn and to possibly ?

simplify

independence 1

results

for the subsystems

T H E O R E M 4.

For

ne i

IE n.

See also the discussion at the end of [2].

and M a model of P- + IE o, the following are

equivalent. i) M ~---> (2) 1 n <M 2) M = V a ~ b V x-< a(~y~ ~

>~ y_
# is any Hn-i formula.

3) M > V a ~ b I B b2...B b n V x_< a (QlXl ...Qnxn ~ < QnXn < - b n ~) 4)

(M,DefoM) ~ w -

5) M > IE i

where

> Q l X l _ < b I...

~ is any bounded formula and the

Qi quantifiers °

EnCA o.

n"

The referee has pointed out that seems to be the natural

IE~ (second order

Eno induction)

theory for solving this equation.

41

1

Here M

- ~ n (2)~M means

that for any

of M and any A n definable p a r t i t i o n i < a w i t h card F-l(i) e 2o pigeonhole

principle

A n definable unbounded F : X ÷ a, where

This is a seemingly

asserting

a~M,

trivial

subset X

there is an

infinitary

that any A n p a r t i t i o n of a A n u n b o u n d e d

set into boundedly many pieces has a piece with at least two elements. DefoM designates sets of Mo

the collection of

The subsystem "weak

theory with language

h ° definable

In comprehension"

{+,.,0,1,<,c}

where

~ is an

(x c X

sub-

w - EnCA ° is the

and as axioms P-, extensionality,

induction axiom and the "localization" V m 4 X V x-<m

(with parameters)

of En comprehension ,

~-->~(x))

En formula possibly with free second order parameters

other than X° PROOF°

I) ÷ 2)°

Let X = {<x,b>

By the induction hypothesis, X is An definable° unbounded, violates

: M

> x < a & ~(x,b)

M is a model of IEn_ I hence of BEn_l,

If X is bounded,

we obtain our conclusion°

then let F : X ÷ a be defined by F(<x,b>)

the hypothesis

By induction on the number of quantifierso

3) ÷ 4)°

Since M is a model of lEo,

n

4) ÷ 5)° M

comprehension

follows

(M,DefoM)

then M

If X is

But then F

~ axiom of induction°

from the hypothesis

One shows the least element principle

I= @ x~(x,~)

= Xo

so

I).

2) + 3).

Weak E

& V b'
~ ~(a,u)

for some a cM;

3)

for In formulas:

if

obtain X ~ D e f o M with

÷

(M,DefoM)

~ V x ~ a (x c X

and apply the induction 5) + i)° (*)

~--> ~(x,u))

axiom°

First show that P- + IE I

~-- "for all a and all finite functions

f : a ÷ a - I

there exist two elements w i t h the same image"° Now since M is a model of IEn, (*)

for F : X ÷ a a given An partition,

We remark that the eauivalence

of

2) and

ently and m u c h earlier by Ho Friedman

5) was noticed independ-

in E43o

42

M

~ V x ~ s ~ Seq(lh(s) = x & ¥ i <x((s) i is the (i + i) st element

of x)). Let g be a finite function in M w i t h d o m a i n {0, .... a} is the (i + I) st element of K°

The f = Fog satisfies

such that g(i)

(*~,so that i)

holds. i)

REMARKS °

Notice that

M

> (2) M

iff

M

> (m)<M

An This holds as w e l l

2)

for any

m~M°

An for

n = 0o

In this notation,

for an initial segment i of M, I is semi-

> (2) I and I is regular iff I

regular iff coded 3)

>(I)
BE n asserts that "the range of a E n function w i t h b o u n d e d

domain is b o u n d e d " and

IE n asserts that "the range of a ~n partial func-

tion w i t h b o u n d e d domain is bounded". C O R O L L A R Y 5o

The E n d e f i n a b l e elements of a model of P- + IZ n are not

cofinal in the model° PROOF.

Recall that an element a c M is E n - d e f i n a b l e if there is a

En

formula ~ such that M

~ @

~ x~(x)

&

0(a) o

There is a E n s a t i s f a c t i o n p r e d i c a t e Sat n for and apply M

E n formulas°

Let a ~ M ~ N

2) to o b t a i n

~ ~ b ~ x
(" q y s a t i s f y i n g the x

C O R O L L A R Y 6o

~

> @ y
E n formula coded by

satisfying the E n formula coded by x").

IZ n is equivalent over P- to IB~ , w h e r e the latter is the

induction scheme for the closure under b o u n d e d q u a n t i f i c a t i o n of the c o l l e c t i o n of B o o l e a n combinations of En formulas. SKETCH PROOF.

To o b t a i n a least element for a n o n - e m p t y set defined by

y ~ v ~ Y0 w h e r e ~(respo ~ is Hn_l(respo M Apply

~ @ y~(a,y)

v V y@(a,y) o

2) to o b t a i n b such that

En_~,suppose

that

48

M

~ V x~a((;y¢(x,y)

Now apply

Ln

<

)

~ y
<

> V y
to obtain a least element

X = {x-
: M

i= ~ y < b ~ ( x , y )

vV

&

(V y~(x,y)

for the set y
We remark that the above corollary has been independently R. Kossak,

observed by

J° Paris and doubtless many others°

If M ~ K then

K ~ M - BE n means

ac M and any En formula

that for any

~(x, y,m) ÷ where ÷m e K K

I= ;f x < a @ y~(x,y,m) + ] b ¥ x < a ~ y < b ~ ( x , y , m ) .

A I n - u l t r a f i l t e r U on M is said to be En-COmplete set X c M 2 and a cM, V i< a (X)ieU where

~

if for any Z n - d e f i n a b l e

n (X) ie U. i< a

(X) i = {z : (i,z) e X}.

If U is a

En-Complete

En-ultrafilter

K = {f:f is an ~n partial

on M, then letting

function with dom(f) ~ U and

rng(f) c M}/U, it is easy to see that K is an n + 1-elementary that K

~ M - BE

end extension of M and

n

The following yields

some insight about question

i) after proposi-

tion 3. PROPOSITION

7.

If

n e 2 and M is a countable model of P- + IEn+ !, then

M has a proper n - e l e m e n t a r y PROOF. Using U on M. Let

end extension ~ w h i c h is a model of

IEn+l, we first construct

{X m : m ~ }

a

En-COmplete

be an infinitely repetitive

P- + BE

n"

ln-ultrafilter

u-list of all En-definable

(with parameters) repetitive

subsets of M 2 and let {a : m ~ w} be an infinitely m w-list of all elements of Mo Let <,> be a b i j e c t i v e pair-

ing function and ( )oa%d( )I be the first

and

second projection

function.

44

We construct

an u-sequence

Yo = YI of

Z -definable n

=°''~ Yk

~°°"

sets such that

(i)

each Yk is unbounded

(ii)

Y

o given Yk'

(iii)

in M

= M

is unbounded

if Yk

n

~ i<(k)o

(X(k) ) . i i

in M then set Yk+l to be this set;

can find the largest Yk

n i<j

is unbounded

by IZn+ I we

j < (k) o such that (X(k)l)i

in M and we can find

Yk

otherwise

£ ~ M such that

r-h n i<j+l

is bounded by Zo

(X(k) l)i

In this case set

Yk+l = { x ~ M

: x>£

and

x c Yk n

~ i<j

(X(k)

) }. I i

Now let U = Clearly

{Zc Z n (M)

U is a

: there exists k ~ ~ such that

Zn-complete

Zn-ultrafilter

ed, let K be an n + 1-elementary Now form a parameters)

"Zn_ I ultrafilter"

2) if

f

for some

U' on

of K which additionally

I) every element is a

of

partial

As previously

of M satisfying

En_ I definable

discuss-

M - BE n .

subsets

(with

satisfies

U' is an unbounded

Zn_I(K)

on Mo

end extension

Z ~ Yk}.

Zn_ I subset of K

function with dom(f) c U' and rng(f)

m c M, then there exists

B ~ U' with B c dom(f)

! m

and f constant

on B. Now let L = {f:f

is Zn_l(K)

partial

Then !~o~' Theorem holds

function with d o m ( f ) E fl' and rng(f)

in the sense that

i K}/U'°

45

L

~ 0([f I] ..... [fr])iff

for all En_ I

formulas

a proper non-cofinal

{ieK:K

e.

~ e(fl(i) ..... fr(i))} cU',

It is now straightforward

n-elementary

to check that L is

extension of K, which furthermore

is

an end extension of Mo (L is said to be a cofinal extension of M if ¥ a c L ~ m ~M (L ~ a ~ m ) o

K ~ L to mean that L is an n,M extension of K where both K and L are end extensions of M)°

n-elementary

Also we write

Let K be the cofinal closure of K in L: = {a~L

:

Now since M

~ K~L, n+l n

K

and

~ IEn_ I

CLAIM.

L

induction on Lo

~

a-<m)}.

by the discussion before Proposition

3, we have

~ IEn_ 2.

K n~_l,e L~

PROOF OF CLAIM°

Kec

~ mcK(L

Clearly L is an end extension of

k_
Consider the case

is ~k and [g] e K and & @ x#(x,g(i))}

cU'o

K k~e L.

mcK

and

We show by

This is clear for

k + I : suppose that

Since

Ko

K ~ [g]-<mo

k = 0 since

L ~ ~ x~(x,[g])

where

Then A = {ic K:K ~g(i) -<m

K ~ IEn_ 2, by Theorem 4, there is a b c K

such that K ~ V z _<m ( @ x~(x,z) < ....> ~ x_< b~(x,z)). Now let dom(h)

=

A and

h : i :

>least x e K such that

Then h is a In_l(K) ~ ~([h],Eg])o CLAIM.

M

function and

L ~[k] _
Thus K ~ ~ x~(x,[g])o

~ K. n,e

PROOF OF CLAIM.

Clearly K is an end extension of M and so

show that M n~Ko formula.

partial

K ~ ~(x,z)°

Suppose that K ~ @ x#(x,m) where m e M

Then K ~ ~([f],m)

L ~ ~([f],m).

We

and ~ is a Hn_ I

for some [f] c K and so by the previous

claim,

Thus

A = {i~K and B c { i ~ K

M ~
: K ~

: K ~-~(f(i),m)} ~(f(i),m}°

JU'

so there is a B ~ U '

with BnA=~

46

(We write this in terms of a double negation only for En_ 1 formulas)°

since Los' Theorem holds

Let i ° be such that K = ~(f(io),m)o

Then

{ic M : M ~ ~(f(io)(i),m )} ~ U, so that M ~ ~ x ~(x,m) o Now since M has a proper n - e l e m e n t a r y has a proper n - l - e l e m e n t a r y Proposition REMARKS°

end extension K w h i c h itself

end extension L ~ P- + IZn_2,

so by

3, K ~ P- + BE n o

I)

Can one show that M ~ P- + BEn+ 1 iff there exists

tary end extension K which right to left follows

is a model of P- + BE ? (The direction from n

from Proposition

2)

Notice

M

A > (2) 1<M o

an n-elemen-

that inthe

3)°

proof of Theorem 4, we actually

iff

M ~ I~lo

iff

M ~ BE2o

show that

One can also show that M A Recall

1 > (M)<M o

that the primitive

are provably

above schemes° Wrathall's Bennett's

recursive

functions

Also recall

Theorem Theorem

[18]o [I]o

the results

The linear time hierarchy Rudimentary

or not,

Go Wilmers

the linear time hierarchy

raised the question w h e t h e r 1 ~ (2)<M

than M

En

En+ 1

fiers b e g i n n i n g w i t h bounded existential

M

of bounded quanti-

quantifierso

showed that the answer is "no",

> (2)IM implies E

theory:

= Rudimentary°

w h e r e an E n formula is one with n alternating blocks

Ao Wilkie

in the

= DefoNo

1 > (2)<M is strictly weaker

M

those which

and hence

in complexity

Since it is still an open question whether collapses

are exactly

total in P- + IZ 1 (theorem due to Minc)

the exact level where

that M is a model of IE 1 depending on the logical

n

complexity function

of the

A ° formula expressing

(in the proof of Bennett's

be E 1 is an open question.

the graph of the exponential

Theorem)°

Whether

We present Ao Wilkie's

the graph can

proof.

47

PROOF°

Let n o be such that ~(x,y,z)

is an E n

formula r e p r e s e n t i n g O

x y = Zo If M ~ Va,b~c(a b =c) for some a,b ~ M then let X = {

: M ~ ay = z

&

y~b}o

If X is bounded then by I~o, X has a m a x i m u m element - say o then e X t h u s is u n b o u n d e d contradicts

contradicting

in M, then let our assumption

maximality

F : X÷{0,ooo,b} that M E

of o

by F()

> (2) 1 <M °

= y.

If

But

X

But this

(This part of the argu-

n o

ment is due to Go Wilmers)o M

>

Hence

I

(2)< M implies

that

M N Exp,

En o

w h e r e Exp i s ([3],po31),

the

statement

¥ x ¥ y ~] z d 0 ( x , y , z ) .

P - + 1E O + Exp p- N a t i j a s e v i c '

1'I s a t i s f i e s

the

least

÷

M ~ ~ x0(x,u)

M~

element

principle

By D i m i t r a e o p o u l o s

Theorem. for

We now show t h a t

Z1 f o r m u l a s .

Suppose that

÷

& 0(a,u) where

V ~ (0(x,~)

<

9 is EIO

By Matijasevic'

, ~ w(p(x,u,w)

q(x,u

=

where p and q are polynomials with non-negative

Theorem,

,w))

integer

coefficients°

Coding ~ by a single element w, let X = {<x,w>

: M ~ x < a & p(x,u,w)

= q(x,u,w)

& V w' < w p(x,u,w')

#

q(x,u,w')}o If X is bounded above by d then M ~ V x -] ~ <÷d ( p_( x , u,w) +

= q(x,u,w))÷ + )

and as M is a model of leo, we are finished° define F : X ÷ { 0 .... ,a} by F(<x,w>) M

If X is b o u n d e d in M, then

= x and we have a v i o l a t i o n of

> (2) IMO

D

En o Notice also that the infinitary w i t h the finitary A ° pigeonhole

A ° pigeonhole

scheme

scheme has little to do

(due to Ao Macintyre) o Ao Macintyre's

question P- + IZo

"finitary A o pigeonhole

scheme"

48

is still an open question, V ~ (V x(V i ~ x

where

~ o' y < x @ ( l ,"y , u +)

the latter scheme is ÷ ~ i # i' ~ x ~ y < x(0 (i,y ,u) ÷ & +

e (i' ,y,u) ) ) where @ is a Ao formula. A. Woods

by an ingenious

argument,

[17] has shown that ~-- "finitary A ° p i g e o n h o l e " + V

P- + I E o 3.

In this connection,

x ~ y(x -< y <

2x & y is prime)

C O L L E C T I O N SCHEMES° Let us first introduce

DEFINITION.

the

For M a model of P- + leo,

the Fr~chet

the subscript M if clear from the context) co-bounded

FM (we drop

is the collection of all

subsets of M (sets A whose complement

A _c M k+l and a c M

filter

is bounded

in M).If

then the a th section of A is A a = {(b I ......b k)

(a,bl,..°,bk)~A}o

Now let

:

Fk be the usual k - f o l d filter product of F;

that is, FI

=

F;

= Fx

F k+l

Fk = {A c M k+l -

: {a~ M:A a c Fk} c F}.

Similarly a set A -~ M k+l is unbounded

if {a c M:A a unbounded

in M k} is

unbounded. A filter F on M k is An-Complete and

a ~ M, if

~ i < a X i ~ F then

In this terminology,

if for all A n definable

N X i ~ ~o i
we can rephrase

a theorem due to Go Mills and J. Paris The following the Mills-Paris LIMIT LEMMA.

lemma simplifies

sets X _c ~ + i

and give a different proof of [9]

°

a small step in the proof we give of

result and will be used later in the proof of Theorem ii.

If

M ~ P- + BEn+ I where n e 0

and if f c En+I(M)

function from M into M, then there exists an "approximating" tion g ~ En(M ) from M 2 into M such that for all a E M f(a)

=

lim g(a,s) s~M

Here,

in the obvious

sense,

o

is a total

total func-

49

lim g(a,s) = b s~M PROOF°

iff

M ~ ~ s V t e s g(a,t) = b

This is trivial if n = 0 - one just bounds the existential

q u a n t i f i e r to o b t a i n an approximation;

the general result is just an

a r i t h m e t i c a l v e r s i o n of the w e l l - k n o w n limit lemma in r e c u r s i o n theory that a A 2 f u n c t i o n can be a p p r o x i m a t e d by a A 1 function° p r o o f due to P u t n a m ([16],p° 51)~ of P- + IE n and if tion

h:M2÷{0,1}

X E An+l(M),

We adapt the

We show first that if M is a model

then there exists a An d e f i n a b l e func-

w h i c h a p p r o x i m a t e s the c h a r a c t e r i s t i c f u n c t i o n of X.

Suppose that

Define

acE

~

> M ~ ~ y V z~(a,y,z)

a ~ X

<

> M ~ q y ¥ z~(a,y,z),¢

0(a,y,s)

<

(V z ~ s~(a,y,z)

and $ b o t h En_ I formulas

y is the least element in M satisfying &

7 V z_< s~(a,y,z))

&

-~ V z _< s~(a,y,z)) .

or

(V z_< s~(a,y,z)

Now define 0

if

~ y ~ s(e(a,y,s)

i

otherwise.

& V z ~ s~(a,y,z)

& -~

z ~ s~(a,y,z))

h(a,s)=

Since M ~ BE n the above is a A n d e f i n i t i o n in M. lim h(a,s) s

= 0

lim h(a,s) = I s IDEA O F PROOF. z¢(a,y,z)

ac X

<---> a ~ X .

Suppose that

a e X.

Let Yo = least y such that

- this exists by

l~n..

Thus

V y~yo and by BE n

~

One than shows that

@ z

~ ~(a,y,z)

and

V y < y o @ z ~ ~(a,y,z)

we can b o u n d all these z by an s larger than yo o Then

any t larger than s, we have h(a,t) = 0° Now if f : M ÷ M

is In+ I definable,

X = {

: f(a) = b}

for

One argues similarly if a ~ X.

the graph of f

50 is An+l definable°

By the above discussion,

ximmtion of the characteristic I least y ~ s(h() g(a,s) [ 0

let

function of Xo

h e En(M) be an appro-

Define

= 0) if such exists

otherwise

If f(a) = b then M~

V y
@ s ¥ t>_s(h()

So by BEn+l, T H E O R E M 8°

g eventually (Mills-Paris).

M ~ BE2k PROOF.

iff

the hypothesis

direction

&

@ s V t>_s h()

= Oo

takes on the value bo

D

For M a model of P- + I~

Fk is &l-complete

We first recall

The difficult

= I

o

and

k e I

(or even Ao-Complete) o

the underlying

idea in the proof of Mills-Pariso

is from right to lefto

Suppose that M satisfies

but not the conclusion of an instance of BH2k_I

N2k-i and so is of the form ¥ x 2 ooo ¥ X2k@ where @ is bounded°

: say ~ is Suppose

that M ~ ~ i < a @ x I ~ x 2 o.. @ X2k_l ¥ X2kO~ Mills-Paris

then show that modulo BE2k_2 any N2k formula is equivalent

to a U k formula, where,

as in recursion

theory,

preted as "there exist unboundedly many"~

the quantifier

U is inter-

Hence by the induction hypo-

thesis, M ~ V i < a ~ Uy I oooUYk ~ where ~ is bounded° But this is equivalent V i < a X i ~ Fk By the hypothesis

to saying that

where X = {(Y2 ..... Yk ):M = ~ ~(Yl ..... Yk )}° that Fk is &o-COmplete,

M ~ ~ U y I °oo Uy k V i < a~ M ~ V i
~-~ X i ~ Fk and so (essentially) i
~ x
The upshot of the argument

is that since V @ is equivalent

order to obtain the &l-completeness formulas.

of

to U, in

Fk, we need collection

for E2k

51

Using what has gone before, that for k ~ I

and

M ~ BZ2k+n

iff

First we introduce Me

FACT.

F : X ÷ a

Mk

differently

to prove

F k is An+l-completeo

the following

that for any A n definable

partition

slightly

definition

due to J-P Ressayre:

> (M k) i <M

An means

we proceed

ne0

unbounded

set X _c M k and any A n definable

where a ~ M, there is an i < a with F-l(i)

> (~)~M

iff

Fk

is

unbounded.

An-COmplete.

~n This fact follows reader.

easily

from the definitions

Now it is easy to verify M ~ BE2k+n

For the converse,

implies

Mk

and is left to the

that --~ An+l

we proceed by double

(Mk)~M

°

induction

: for k fixed, we

induct on n. If k = i, then this is just Theorem n = 0

2.

Now suppose

that k = 2 and

and that M2 --Al> (M2)~M" The idea is to "step down" the exponent

complexity

of the partition

(removing

a factor of 2 to the complexity "step down" of Theorem

is originally ii)°

one occurence

of the partition).

due to E. Kranakis

Thus we propose

2 to I and add 2 to the

to prove

[13]

of "unbounded"

adds

Such an idea of a (see also the proof

that

I >(M)<M

M A3

which by Theorem 3 is equivalent LEMMA A°

For n ~_ 0

~n' there exists

with BE4o

and every En+ I formula

a formula

~ ~ Hn (P- + IEn)

of the form g x ~ where such that

~ is

52

P- + PROOF°

IZ n

~- ~ x¢ <---->~ 'o x#°

This fact was observed

in r e c u r s i o n

theory by [ 19] - notice

the formula ¢ is Hn and not simply of the form #(x)

& V i<x

that

-~¢(i)which

w o u l d be equivalent

to a "~ & Z " formula° This is proved by induction n n on n: for n = 0, let ¢(x) be ~(x) & ~ i < x -~ ¢(i) - this uses IEoo For

n = I, P- + IZ I Let

~

~ ~ x¢ <.........> .......~ x(¢(x)

¢(x) be equivalent

induction hypothesis,

to

& V i<x

-~¢(i)).

@ ye(x,y) with e bounded.

¥ i <x ~

~(i)

is equivalent

By the

to ¥ i < x @

~ ye*(i,y)

and so BE I + IE I to ~ x ¥ i < xe*(i,(z)i) , using the usual p r o j e c t i o n functions th for the i element of a sequence coded by z. Thus

~ x~ is equivalent

to

Proceed in a similar manner We leave it to the reader

~

~ u(~((U)o)

for

& ¥ i < (U)oe*(i,(u)l)i)). B

n ~ 2°

to define ~ : x I V x 2 @

: X3ooo

• ..¥ X2m @ : X2m+le° For instance,

@ : x V

y

@ ~ ze is an abbreviation

x(~ y @ ~ ze'(x,y,z) LEMMA Bo

M2

> (~)~M

implies

& ~ i ~x ~

for

(~ y ~ ze(i,y,z)))0

that M is a model of IZ2°

41 PROOF°

The hypothesis

implies that M ~ i

of BE2°

To see that M ~ A 2

so IE2,

suppose

> (M)~M

(M)~M

and so M is a model

and hence that M satisfies

that F : M ÷{0, .... a - I} is a A 2 p a r t i t i o n and a •M;

by the limit lemma let g be a h I total approximating lim" g(x,s) s Then g : M 2 ÷

= i"

function w i t h

<......> F(x) = io

{0 ..... a - i} and if (by the hypothesis)

it is clear that F-l(i) works

B E 3 and

is unbounded

in Mo

g-l(i) • F 2, then

(A variant of this argument

if we had begun w i t h F : X ÷ {0 ..... a - i} where X is a A 2 definable

unbounded suffices).

subset of M; but by the remark after Theorem 2, this argument D

53

LEMMA C

M2

> (M 2)
t h a t M ,A,3> (M)I M.

AI PROOF.

Suppose

that F : X x cX

÷a

<

F(x) = i

that X is a

E 3 definable

is a E3 partition

unbounded

and a ~M.

subset of M and

Then by IE2,

> M ~ ~ ' y ¥ z @ '. w4~(x,y,z,w) <

> M ~ @ ' y V z 9 'o w~(x,i,y,z,w)

where $ and ~ are bounded. Define Z = {(<x,i,Yl,Y2>,)

: M ~ i< a &

$(x,Yl,Z,U z) & ~(x,i,Y2,Z,Vz)}. Consider

the picture:



1 | ! I

,

I

I

<x,i,Yl,Y2>

Then it is easy to see that x E X and F(x) = i iff the section Z<x,i,Yl,Y2>

is unbounded

for some unique yl,Y2.

G(<x,i,Yl,Y2>,) = i.

Then Z is a A o definable

set of M 2 and G is a A o partition Clearly

if G-l(i)

Let

of Z into

is unbounded

{0,...,a

then F-l(i)

"unbounded"

- i}.

is an unbounded

X. In a similar manner, M2

we have

.----~(M2)IMAn implies

that

M

A-~+2(M)~M

sub-

.

subset of

To obtain ~ + I

>

(Mk+l.l )<M implies ~

a E 3 unbounded

(Mk) ~M , suppose

> A3

AI

that X is

subset of M k and F : M ÷ {0,.°.,a - I} is a E 3 p a r t i t i o n

of X into a-many pieces where a e M By IE 2 let (x I .... x k) c X

<

> M I= @

F(x I ...... x k) = i <

'oy¥ z ~

'. w

> M ~ ~ ~ y V z @ ~ w

where ~ and ~ are b o u n d e d Define

Z = {(Xl,X 2 ......Xk_l,<Xk,i,Yl,Y2>, M ~ i
:

& ~(x I ..... Xk~l,Z,U z) & ~(x I ......xk,i,Y2,Z,Vz)}

and G : Z ÷ { 0 ..... a-l}

by G((x I ..... Xk_l,<Xk,i,Yl,Y2>

Then Z and G are A o definable if G-l(i)

)

is u n b o u n d e d

and Z is unbounded

, ) = i. in ~ + i o

in M k+l then F -1 (i) is u n b o u n d e d

Clearly

in Mko

The result now follows by induction° We state some of the trivial by-products vious

of the proof

: (with the ob-

intended meanings) ~

(Mk) I <M

iff

M ~ BZ2k_I

(~)I

iff

M~

for

ne i

n Mk

-~ O

<M

BE2kO

We define a An-based u l t r a f i l t e r sets of M k (not n e c e s s a r i l y (i)

A c U ÷there

(ii)

A,B E U ÷ A n B ~ U

(iii)

A~ U

(iv)

A c An(M) ÷ e i t h e r

and

in Mko

A c B c M÷BcU A~U or

A = M k - A E U.

we m e n t i o n

(Pino [15])o

For n,k e i, if M is a countable model of

BE2k+n_l , then there is a An-COmplete T H E O R E M i0o

such that

exists B e An(M) with BoA and B~U and B unbounded

With this notation, P R O P O S I T I O N 9o

U on M k to be a collection of sub-

An-definable)

(Ressayre-Pino

An-based ultrafilter

U on Mko

[15] )o For n,k e i, if there is a An-COmplete

55

An-based ultrafilter elementary

U on M k, then there is a k-tower

of proper n+l-

end extensions

Mn~l,e

MI n+l,e ~ M 2 n+l,e ~

In view of Theorem 9, [15]) whether

~ °°° n+l,e Mk ~ P - + leo"

it is natural

the converse

question

or Proposition

(due to Ressayre-Pino

9 and Theorem

I0 holds°

(By

the limit len~na, it is easy to show that if there exists a An-Complete An-based ultrafilter

U on M k then M ~ BEn+k) O

We come now to a final combinatorial tion schemes.

The result

C.G Jockusch's

recursion

proof requires motivation retical

angle°

formalization

analysis

much of the machinery

of Ramsey's

collection

of combinatorial

version of The

Our original schemes

so as to facilitate

type arguments

[6]°

[i0] from a proof theo-

was to find combinatorial

schemes

collec-

Theorem

thus far introduced°

the "2-3 question"

Another motivation

to certain

of certain

turns out to be an arithmetical theoretic

was to consider

equivalent

equivalence

in fragments

the

of Peano

arithmetic° DEFINITION°

M--~-~ (M)~M means that given any A m unbounded subset X m of M and any A m partition F : IX] n ÷a, where a ~ M, of the n-element subsets of X into a-many pieces, Y c X (ioe. there exists A word of warning:

there

by the remark after Theorem F : [M] n ÷a. Iio

(÷)

the exponent partition°

set

the second order variable

2, it suffices

to consider

Yo

in M, but Also,

partitions

For n,m-> i

M --~--> (M)nM_ m PROOF°

homogeneous

it turns out that Y will be definable

the above scheme is written w i t h

THEOREM

is an unbounded

an i < a with F"[Y] n = {i})o

iff

M

BZn+m~

This part is easy and uses of a partition By induction

that n = k + i and . . k+l M --~--> (•) <M

m

while

the limit lemma to step down

stepping up the complexity

on n : if n = i, this is Theorem

2°

of the Suppose

56

We show that M

> (M)kM

.

km+l Using

BZm+I,

apply the limit lemma to obtain a

Am

approximation

G : [M] k+l ÷ a w i t h lim G(Xlm ..... Xk,S) = F(x I ..... Xk). S

If

Y

is homogeneous

homogeneous

for

(÷ )

for

G

and u n b o u n d e d

in M, then

Y

is also

F. This direction

is more difficult and proceeds by a series

of lemmas. First, v ~(v

Ik

let

÷ x(¢ (x,u)

<

>

be the scheme

n

¢(x,u)) ÷

where ¢ is LEMMA 12. PROOF.

En For

and ~ is

+

((¢(0

Suppose that

and trivial

a e X e An+I(M).

the set

for

n = 0.

for non-empty

For

for the characteristic

function of

~ g(x,s)

n ~ i, we show

An+ I definable

By the limit lemma,

M ~ V x ~ a ~ s V t ~ s g(x,s)

sufficiently

+ 1,u)))+

~n"

X s = {x~ a : M Then

+ ¢(x

V x¢(x ,u))) +

that the least element principle holds

A n approximation

& v x(¢(x,u)

n a 0, P- + IE ° + BEn+ 1 D IAn+ 1.

This is w e l l - k n o w n

sets.

, u÷ )

X;

let

g

be a

let

= 0}.

= g(x,t),

so by BEn+ 1 , let

large to bound all these values of s.

Now apply

s o be IZ n to

X s to obtain a least element - this is also the least element

of X. REMARK. In fact,

Independently

and much earlier,

in [4] it is stated that

over the base theory

BEn+ I

H. Friedman noticed and

IAn+ I

this lemma.

are equivalent

P- + 1E o + V x @ y(2 x = y).

Before giving details of the proof of T h e o r e m ii, we give a sketch of the idea behind the proof that M ~ BE 3

implies

2 M -----> (M)<M AI

57

First, tree

in the usual manner,

associate

T ! M < M with the p a r t i t i o n

a

Al

definable

F :[M] 2 ÷ a

branch B of the tree is "pseudo-homogeneous" bi,bj,b k ~ B

and

b i
& b i
then

in order to obtain a contradiction, matic

subset of B.

"Erdos-Rado"

such that an u n b o u n d e d in the sense that if

F(bi,bj)

= F(bi,bk).

that there is no unbounded monochro-

Then

M > V i
~ x V y,z e x ( y , z E B & y < z ÷ F(y,z) # i).

A trivial basis t h e o r e m produces thus if M is a model of M ~ ~ t V i
BE 4

an u n b o u n d e d branch B w h i c h is A 3-

then

@ x ~ t V y,z e x ( y , z ~ B & y < z ÷ F(y,z)

which is absurd since B is u n b o u n d e d

branch B whose jump B' is the

"low basis"

infinite finite b r a n c h i n g

S I satisfaction

A3

of the jump operator)

for

tree, E1

To formalize

and obtain an unbounded

in the extended

a A 3 formula

then there exists an infinite

formulas

branch B, so that via the satisfaction mulas

language

BZ3, we for-

theorem [7]: if T c w <~ is a

definable.

predicate

# i)

and F is defined on [M] 2.

To push through this argument w h e n M is only a model of malize the J o c k u s c h - S o a r e recursive

Suppose,

A3

predicate

{+,.,0,1,<,B}

this, we modify

(to obtain an analogue definable all

"l-generic"

S I and

are equivalent

91 forin

M

to

: then V y,z ~ x ( y , z ~ B

is equivalent

to a

A3

& y < z ÷ F(y,z)

formula and so

BE 3

suffices

# i to obtain a con-

tradiction. For the following discussion, of P- + 1E o . We say that parameters)

Recall T

is a

that

fix a (not n e c e s s a r i l y

Seq is a

Sn-tree in

subset of M,

M

definition of sequence numbers.

if

T

M ~ V s,t("t c T" & lh(s) ~ lh(t)

retic conventions

is a

~n definable

: M ~ Seq(a)}

(with

and

& V i < lh(s)((s) i = (t)i÷ "seT").

we work in

: "t ~ T" abbreviates

with parameters

model M

AI

T c Seq M = {a ~ M

To make things more readable,

countable)

M

and adopt usual set theo-

~(t,u) where

finition of

T

lh(s)
& V i
u, s ~ t

T is said to be finite b r a n c h i n g

means

~ is the

Seq(s)

(in M) if

En

& Seq(t)

de&

58

M ~- V s e T A definable for any

function

ac M

~ a V b->a

(s

< b> CT).

f :M ÷ M is said to be piecewise

there is an

coded in M if

s ~ Seq M such that

V i
f(i) = (s) i & lh(s) = a.

(This n o t i o n is due to R. Kossak). A branch

f

of

T

is a definable piecewise

such that for all a ~ M where

f

~ T.

coded total function f : M ÷ M Let L(f) = {+,.,0,1,<,f}

is a new function symbol of one argument.

In an obvious way,

there is an effective manner to associate

to every bounded formula

~(~)

in

in

L(f)

a bounded

formula

that if the function symbol

f

fs = {(i'(s)i) then for any

~*(Zo,~)

: i < lh(s)}

(One replaces is then

occurrance Seq(v)

Furthermore, of bounded

such

where

s ~ Seq M,

~ e M, (M,f s) ~ ~(~)

~*

L = {+,.,0,I,<}

is interpreted by the function

if

iff

of f(x) = y

M ~ ~*(s,~).

in ~ by

(V)x = y to obtain ~'.

& ~').

M ~ P- + I~ ° + Exp,

satisfaction

then the usual

AI

formalization

allows ->

iff

(M,f s) ~ ~(a) where

M ~ Sat o ( r- ~, -l

<s,a>)

~ is bounded.

FACT.

If

M ~ P- + 1E o + BE n for

function

f

is piecewise

PROOF.

By hypothesis,

given.

By

Bln let

b

n e 2,

then any total

En

definable

coded in M.

the graph of

f

is

be sufficiently

M > V i
~ y
An

definable.

Let

a cM

large so that f(i) = y.

Now M ~ @ s ~p~+l by induction on w o r k in

BE n

(s e Seq & lh(s) = a + I & V i ~a((s) i = f(i)))

i ~a

for

using

n e2

A formula in the language formulas

(including

IA n.

Here

Pb = b + i - st prime;

to be able to use the function L(f)

is b o u n d e d

Pb by

we Ill).

if it is built up from atomic

f(x) = y) by Boolean ooerations

and b o u n d e d quan-

sg

~ifiers LEMMA

] x
13.

function formula

and

Suppose

V x
in a model in

L(f)

f : M ÷ M is a definable

M

of

P- + IE °

with those

more that for each

a ~M,

and that

free variables

s a e Seq M

piecewise

~(x,~)

coded total

is a bounded

indicated.

Suppose

further-

and

s a = Then for (i)

b ~ M the following

are equivalent

(M,f) ~ ~ x ~(x,~)

(2)

] ao~MV

(3)

V ao ~M

a->a ° M ~

] x~(x,y), M ~ Exp,

(2')

M ~ ] zVuez(u~Seq

(3)

M ~Vz ~ u e z ( u c

PROOF.

is

let $ be the formula

then

(i) is equivalent

~ x < lh(u)

with

& lh(u) ->z&Vi)

that ~ is in disjunctive @x I < r I ...Qx n < r n W

~

÷Sato(

normal

First, we show by induction that for

on

(atomic or negation

n = number of bounded

quantifiers

formula or negation

cases where

is anything

there

to prove

of atomic are when

formula,

t I ~ {c~b I .... ,bk,0,1,2 .... }

sufficiently n = j + i. fiers by

and

large

(so that

conjunction

large, we obtain (It is important

@ x
and

is a term in L(f)).

V x
tI

pose

(here

b

is

then the only

~ is of the form

f(tl) = t2, f(t I) # t2,f(t I)
Disjunction

in

÷ @ a ° ~ M V a ~ a o M ~ ~ (Sa,C,b).

If ~ is an atomic

a o sufficiently

of

formula).

c,b c M

(~ (M,f) ~ ( c , b )

with

~ ,u <x,b>).

form:

atomic

~(x,~)

~*(u,x,y).

the following

Seq&lh(u) >-z&Vi < lh(u)((u)i=f(i))

We can suppose ~(x,~)

+ I ~ (Sa,X,b)

] a>-a o M ~ ~ x < a + l~*(Sa,X,b).

Given the formula If additionally

~ x
t 2 ~ f(tl),

b I ..... bk)~

is in the domain of

Sao) (+)holds.

no problem and by choosing

the induction at this point rather

than

Then for

a

o step for showing ~ ) w h e n

that we define bounded @ x
and

quanti-

¥ x
t

60

It now follows that

for

¢(x,~)

(M,f) = ~(c,b)

a bounded

formula in

L(f),

iff

~ a ° ~ M ¥ a ~ a o M ~¢(Sa,C,b)

iff

V a°

÷

Notice

that

for ~(x,y)

(++)if

a bounded

M > ~ (Ss,C,b)

relation

formula

in

L(f),

then for all

From this fact and the previous If the exponential

M ~ a e a ° M >~(Sa,C,b).

discussion,

a' e a Mi = ¢ (Sa,C,b).

(I) and (2) easily follow.

function is total in M, then the usual satisfaction

for b o u n d e d

formulas

is

AI

definable and yields

(2') and

(3') A

Zn

D definable

total function

formula of the form formula in L(f), in L;

f in M is said to be l-~eneric

~ x#(x,b)

or

is equivalent

i.e. there are

if every

V x~(x,b), where ~ is a bounded

in

M

Zn formulas

to both a

0, ~ and

~n

Zn

and a

formulas

Hn

formula

0',~'

in L

such that

and LEMMA

(M,f) ~ } x~(x,b)

iff

M ~ 0(b)

iff

M De'(b)

(M,f) = V x¢(x,b)

iff

M D @(b)

iff

M D~'(6)

14.

If

te b r a n c h i n g

M ~ P- + BEn+ 2 En

branch f of T.

tree in

where

n-> i

and T is an unbounded

M, then there is a An+ 2 definable

(This is the arithmetized

fini-

1-generic

version of the "low basis"

theorem) . PROOF.

We w o r k in M.

numbers

of the levels:

k(0)

First define a function k bounding

= 0 = sequence number

k(a+l)

= least x such that

for <> V y>-x Vs-

Here k(a + i) = . M ~ ¥ a ~ y k(a) Now define in

the sequence

Using

Ign+l,

it follows

~ T).

that

= y.

M

g(0) = 0 = sequence number

for < > f%

0

if

gm ~ teT(lh(t) ->m & -~ ~y
I

otherwise

g(1) =

Sato((0)o,t,

(i)l))

61 and g(2a + 2) = least

x(¥ m ~ tcT(lh(t) e m & < g ( 0 ) , g ( 2 )

V i ~a(g(2i+l)=0

+ ~ @ y
..... g(2a)>'~<x> ~ t &

Sato((i)o,t,~(i)l))))

0 if V m ~tcT(lh(t) e m & ~ t Vi ~ a(g(2i + I)=0 + ~

&

@y < lh(t) Sato((i)o,t,y(i)l)))

g(2a+3) =J @y
Sato((a+l)o,t,~(a+l)l )

1 otherwise. Recall that

~ @y < lh(t) Sato((i)o,t,~(i)l ) is the formalization of

"there is no y < length of

t

bounded formula ¢, t ~ Seq

and

ce coded by (i)l".

such that (i) o is the GSdel number of a ¢(y,~)

Thus when

i

is true, where

The function

g

PROOF.

Notice that for

is the sequen-

runs through M, we consider all

possible bounded formulas with parameters CLAIM.

~

in

M.

is total in M. m ~ M, if

g(m) is defined,

then

g(m)~ ko(m),

where ko(2m) = k(m)

and

ko(2m + i) = I.

We wish to show that M ~ V m @ y-
= y"

e (m) where the defining clause for

g(m) can be written in the form

k(m+l)+l s -<~k(m+l)+l

(s ~ Seq &

Hn+l

&

~n+l )

v

P

Then M ~ 0(m) <

>@ z(k o(m) = z & @ y sz~(m)) An+ 1

M ~ e(m) <

An+ 2 (BEn+ 2)

> V Z(ko(m) = z + @ y-
An+ 2 (B Zn+2)

62

and h e n c e

0 is

An+2(P-

CLAIM.

M ~ 0(0)

PROOF.

Suppose

are defined.

+ BEn+2).

& V x(0(x) that

÷ 0(x+l)).

M ~0(a)

where

M ~ V m @ t c T(lh(t) e m V i ~b(g(2i+l) Let

a = 2b + 2 , so g(0) ..... g(2b + i)

Then & ~ t

= 0 ÷ ~ @ y < lh(t)Sato((i)o,t

X = {t c T : 4 t V i ~ b(g(2i+l)

Then

X

is an u n b o u n d e d

be d e f i n e d

~ ( i ) l))).

&

= 0 ÷ ~ ~ y < lh(t)Sato((i)o,t definable

AI

set.

Let

(i) l)))}.

F:X ÷ {0 ..... k o ( 2 b + 2 ) }

by F(t)

Since

&

= x

4 t .

if

B i n + 2 implies > (M) i <M

M

'

An+l there

is an X

x_
+ 2) such that

= {t ~ T: ~ t

X

V i sb(g(2i+l) is u n b o u n d e d . X

X

Define

&

= 0 ÷ ~ ~ y < l h ( t ) S a t o ( ( i ) o , '~'(i)l))}

g(2b + 2) to be the least

XSko(2b

+ 2) such that

is u n b o u n d e d .

To show that Now

M

g(2b + 3) is d e f i n e d

is a m o d e l M ~ 0(0)

of

is s i m i l a r

B E n + 2 and h e n c e

& V x(0(x)

of

and even easier.

IAn+ 2

QED claim.

and 6 E A n + 2 ( P - + B E n + 2)

÷ 0(x + i))

hence M ~ V m 9 y "g(m) By lemma

13 it is clear

= y"

that g is a 1 - g e n e r i c

An+ 2

definable

branch

in

the tree T. LEMMA

15

If

M

is a m o d e l

of P- + BEo

•

PROOF.

Since

order-isomorphic

~

M ~ IEl, with

any u n b o u n d e d M

via a

AI

then M '

AI

> (M) 2 A1

definable

isomorphism,

<M"

subset

of

M

so w e can suppose

is that

63

w e are g i v e n

a

gl

partition

F : [M] 2 ÷ a, Define

the E r d o s - R a d o

where tree

T = {t ~ S e q

: V i <j < l h ( t ) ( ( t ) i <

V i < lh(t) of where

a ~ M.

- l[(t)i+l

(t) j & (lh(t) > 0 ~ ( t ) o

is a m i n i m a l

compatible

= 0 & extension

< (t) ° ..... (t) i > ]

the e x p r e s s i o n

in square b r a c k e t s

+ "compatibility"

is

V j < k ~
and ++ "minimality

of c o m p o r t m e n t "

V y ( ( t ) i < y -< ( t ) i + l & V F (t)j ,y))

& F((t)j,y)

Clearly,

T is a

CLAIM.

T is finite b r a n c h i n g

PROOF.

Al

We m u s t

ments

partition,

possible,

then there

so at most

X = {s ~ T: there

T

in M.

X is u n b o u n d e d

PROOF.

If not,

t.

= i

that

is u n b o u n d e d many

if

~ T)). F : [M]

2

a-many n e w m i n i m a l successors

in M,

÷ a

was

the

comport-

of each n o d e

s ~ T.

let

x > (S)lh(s)_ 1 such that

- l(F((s)i,x ) = F((s)i,(s)i+l))}-

in M. ~2 d e f i n i t i o n ,

by

IE 2 the set

Then

Y = {x:x > (t) lh(t)_l in M.

the fact immediate

then since X has a

element

x V y>z(s'~

are at most

are u n b o u n d e d l y

CLAIM.

H(=O

= m÷~

a-many

V i < lh(s)

is u n b o u n d e d

and u n b o u n d e d

II 2 n o t i n g

To see that

a maximum

= (t)i+l).

show

can be done u s i n g

original

= F ( ( t ) i , ( t ) i + I) ÷ y

= F((t)j,(t) k , ) =

tree.

M ~ V m V s ~T(lh(s) This

j
Let iff

& V i < lh(t)-l(F((t)i,x ) = F((t)i,(t)i+l))} H

: Y ÷ a be a F((t)lh(t)_l,X)

Al

partition = i.

defined by

X

has

Then by

M

(M)<M,I there is an

io < a

such that

H-l(io ) is un-

AI bounded

in M.

Let

x ° be the least element

of H -l(io) .

Then

t' = <(t) ° ..... (t)lh(t)_l,Xo> is in

X

and is larger

X is unbounded By Lemma

14, let

construction

f

be a

of T, for

by

PROOF,

application

of

: x~M}

an

= df

Thus

branch

of T. By

c i < a.

a

+ I)). such that

~ x

G-l(i)

IV yex(F(f(y),f(y

the expression

is unbounded

in M.

+ I)) # i)].

in square brackets

a contradiction.

is A3, so a final

This final claim completes

15.

If M is a model of P- + Bln+2,

PROOF. Suppose this has been shown for a partition F : [M] n + l ÷ a , let T ={s Seq

1-generic

= {m : ~ x-<m(m = f(x))}+

i
BS 3 insures

the proof of Lemma LEMMA 16,

of t.

then

M ~ V i
maximality

in M.

definable

= F(f(x),f(x

There exists If not,

A3

= F(f(i),f(k))

G : {f(x) G(f(x))

CLAIM.

t, contradicting

i <J
F(f(i),f(j))

Now define

than

and hence T is unbounded

then n

M "AI

(M)TM"

and consider

n + i : given

: V i <j < lh(s)((s)i<(s) j). & Vj <-n(lh(s) >-j÷Vi<j((s)i=i))) V i-< lh(s)

- l[(i >-n ÷(s)i+l

is a minimal

compatible

&

exten-

sion of < (S)o ..... (s)i>)]}, where

the expression

in square brackets

is

compatlbility" ¥ i < ... < i n < k < k ' - < i ( F ( ( s ) i I ..... (s)i

,(S)k ) = F((s)i I ..... (s)i n

F((S)il ..... (s)i+l)) 4-+ "minimality

of comportment"

,(S)k,)= n

65

V y((s)i < y ~ (s)i+! & V if<...<

i n < k
,(s) k) = n

F((S)il ..... (s) i ,(S)k,) n

= F((S)il ..... (s) i ,Y)) n

V if<" ...
IE2,

-> y = (S)i+l). T is a finite branching unbounded tree tlh(s) has at most ~ n-I )- many immediate successors). By

s cT

14, let

construction

f

be a 1-generic

A3

: ~ x~m

is pseudo-homogeneous

for

f(x) = m} F;

i.e.

F(f(x I) ..... f(Xn),f(Xn+l)) An examination BE 3 then

low

zation,

is meant

that there exists

partitions

one obtains

whose

that if

low This observation M > BEn+ 2 (*)

Then by

for

: x ~ M}

xI < x2

<..~

Xn+ 2

= F(f(xl) ..... f(Xn),f(Xn+2)). that if M is a model of

2 (M)<M

an unbounded

graph is low M

A 2.

is a model of

homogeneous Similarly

set for

by relativi-

BEn+ 2 then

(M) 2 <M

An+ I yields

that

then M ............ > A3

(M) n-I <M

let G : If" M] n-I

by

= {f(x)

together with the induction hypothesis

low As usual,

T.

A2

M ............ >

if

branch of

of the proof of Lemma 15 reveals M ............ >

appropriate

definable

of T, {mcM

by which

= F((S)il ..... (S)in-l'

one can show that

in M (a node Lemma

&

G(f(x I) ..... f(Xn_l))

.... >

a be the induced partition

defined

= F(f(x I) ..... f(Xn_l) , f(Xn_ I + I)).

66

By the property

(*) there is an unbounded

by p s e u d o - h o m o g e n e i t y This completes

set homogeneous

thus also homogeneous

the proof of Lemma

for

collection

If

RM(I)

I

16 and hence of T h e o r e m

is a proper initial

and

Ii.

D

We first introduce

segment of

M, then the

of reals of I or coded subsets of I is the collection

of traces on I of

M-definable

sets:

ZM(I) = {A n I : A definable with parameters Similarly,

G

F.

The following corollary was pointed out by J. Paris. some notation.

for

we can speak of functions being coded,

in M}.

etc... An initial

segment I of M is n-Ramsey_ if for any a c I and any coded

F :[I] n ÷ a

there exists an

i
In .

segments

M

write

I

of

I ~ BE* n

such that

w h i c h are

to mean that

I

scheme, w h e r e coded parameters

is unbounded

in

Initial

are also called regular.

satisfies are allowed

the usual

We

E -collection n

(see [14], p. 251).

In [20]

it is shown that for

I

is

l-Ramsey

iff I b BE E and in Propo-

sition

I

is

n-Ramsey

then

3

I c M, e [20] that if

F-l(i)

l-Ramsey

of

In a remark on notions

of

p. ii of

n-Ramsey

[9],

and

corollary of Theorem

I ~ BEn+ I.

it is suggested that for

BZn+ 1

do not necessarily

ii, we have the surprising

n > 2, the

coincide.

As a

fact that these notions

do in fact coincide. COROLLARY iff

17.

If

I c M e

with

I ~ P- + IE

then

I

is

n-Ramsey

o

I > BEn+ I.

PROOF.

in M.

(÷)

By Proposition

(÷)

Suppose that

3 of [20]. F : [I] n

Repeat the proof of T h e o r e m

Thus there is an So then

F-l(i)

i
is unbounded

ACKNOWLEDGEMENTS. especially Corollary

17.

X c I in

II

where

ae I

Special

for several

and

F is coded

but with a single coded parameter.

unbounded

in

I

with

F"[X] n = {i}.

I n,

0

I would like to thank J.P. Ressayre,

J. Paris

this paper.

and

÷a

conversations

A. Wilkie and

on the subject matter of

thanks to J. Paris for kindly pointing

out

67

REFERENCES [ I]

BENNETT,J.H.

On spectra, Ph.D.dissertation, sity (1962).

[ 2]

CLOTE, P.

Anti-basis

Princeton Univer-

theorems and their relation to indepen-

dence results in Peano Arithmetic, in Model Theory and Arithmetic, Springer Lecture Notes in Mathematics 890(1980) pp. 115-133, [ 33

[ 4]

DIMITRACOPOULOS, C.

Matijasevic's

FRIEDMAN, H.

On fragments of Peano arithmetic,

[5]

[ 6]

[ 73

JOCKUSCH,C.G. Jr. JOCKUSCH, G. G. Jr. & SOARE, R.I.

[ 8]

KAUFMANN, M.

theorem and fragments of arithmetic,

Ph.D. Dissertation, (1980) o

University of Manchester,

preprint.

Systems of second order arithmetic with restricted induction° (Abstract) J.Symbolic Logic 41(1976) pp. 557-559. Ramsey's theorem and recursion theory. J.Symbolic Logic 37, (1972), pp. 268-280. ~ classes and degrees of theories. Transactions of Am.Math.Soc. 173(1972) pp. 33-56. On existence of

En end extensions.

L_o~ic_Year

1979-80, Springer Lecture Notes in Mathematics, 859(1980) pp. 92-103. [ 9]

MILLS, G. & PARIS, J.

Regularity in models of arithmetic. J. Symbolic Logic, 49, Ne 1 (1984), pp. 272-280.

[I0]

KIRBY, L.

Ultrafilters Preprint.

[Ii]

KIRBY, L. & PARIS, J.

[12]

KRANAKIS,

E.

~n-Collection

and types on models of arithmetic.

schemas in arithmetic.

Logic Collo-

quium 77, pp. 199-209, North-Holland Publishing Co., (1978). Partition and reflection properties of admissible ordinals. Annals of Mathematical Logic 22(1982), pp. 213-242.

[13]

Stepping up lemmas in definable partitions. Symbolic Logic 49,Nel, (1984), pp. 22-31.

J.

68 [14]

PARIS, J.

Some conservation results for fragments of arithmetic, in: Model Theory and Arithmetic, Springer Lecture Notes in Mathematics, N ~ 890 (1980) pp. 251-262.

[15]

PINO, R.

~n-Collection, ssableso

indicatrices et ultrafiltres d~fini-

Th~se de 3°cycle, Universit~ Paris VII

(1983). [16]

PUTNAM, H.

Trial and error predicates and the solution to a problem of Mostowski, J. Symbolic Logic, 30 (1965), pp. 49-57.

[17]

WOODS, A.R.

Some problems in logic and number theory, Ph.D. dissertation, University of Manchester (1981).

[18]

WRATHALL,

C.

Rudimentary predicates and relative computation, SIAM J. Comput. 7(1976), pp. 194-209.

[19]

HAY, L. Small recursive ordinals, many-one degrees, and MANA~TER, A. & the arithmetical difference hierarchy, Annals ROSENSTEIN,J. of Math.Logic, 8(1975), pp. 297-343.

[20]

PARIS, J.

A hierarchy of cuts in models of arithmetic, in Model Theory of Algebra and Arithmetic, Lecture Notes in Mahtematics, 337.

Springer

N ~ 834, pp. 312-

ON THE AXIOMATIZABILITY

OF SETS IN A

CLASS THEORY Manuel Corrada C. Universidad

Cat61ica de Chile

Santiago,Chile.

Among the systems of set theory frequently used in mathematical ce we find Zermelo-Fraenkel's theory

system ZF, Von-Neumann-Bernays-Godel

VNBG, or the Kelley-Morse

difference between these systems ee of very comprehensive "inconsistent

system,

symbolically

multiplicities"-and

or, in Cantor's

others do not.

The

terminology,

The reason to intro-

at the very beginning

ment of set theory as a formalized discipline paradoxes;

denoted KM.

is that some of them admit the existen-

objects-classes

duce such kind of objects appears

However

practi-

of the develop-

in order to avoid the

some authors call this the "limitation of size" doctrine.

this type of question is far from the working mathematician's

interest.

As Kreisel[1971]

says, "day-to-day

does not affect day-to-day practice. hind the mathematician's

reluctance

the mathematicians, are provable

that classes are at least for

but a facon de parler.

This point of view seems

if we call VNBG m the set of formulae referring in VNBG,

then Novak

[1950] proves V ~ G m = ZF.

classes do not add new information of set-theoretic

research

to get involved in foundations...".

During a long time it has been maintained justified;

foundational

This conviction may well be be-

to sets that That is,

about sets to that obtained by means

axioms.

But in the case of the system KM things are different. prove the existence of a universal

class,

Here we can

call it V, and also V ~ ZF.

But then we have proved in KM the consistency

of ZF.

consistent,

Con ZF (C°nzF is the formal

Godel's

second theorem gives ZF ~

(number-theoretic)statement

expressing

the consistency

we obtain COnZF~ KMm - ZF, from which ZF ~ K~. the formula COnzF is not of mathematical infinitely many formulae Kreisel-Levy[!968].

Unless ZF is inof ZF).

Hence

It has been argued that

character,

and neither are the

~cKMm-ZF which are obtained by results

Recent results by Friedman show, however

is not the case;KM proves predicative

formulae of mathematical

which cannot be proved in ZF (see Friedman

[1981]).

in

that this interest

70

V~BGmis

axiomatized

set of sentences, sively

by the axioms

then using

axiomatizable;

set-theoretical The natural

The

character

formalization

and classes,

of ZF.KM m is a recursively

the device

question

language°

be ~m = {~:there

exists

presented

models

as <S,CuS,E> where

binary relation which

we have the characteristic

CoP

(S)

terested

in structures

in which

In these

cases we will write <S,C>

cumulative tructing

hierarchy°

denote

of R B until

level

the interpretability mutually

in the subsequent

THEOREM

0.

PROOF:

Co

We are in-

set and E is standard.

the class which

structure. consists

Y T!R + I and RE is the usual

constructibility

over R B cons-

degree of the theory T, ioe.

interpretable

the

with T.

(private

communication)

is im-

discussion°

For each n there exists

a closed unbounded

n such that for each ~ n there exists an ordinal I=AI comprehension + replacement '

of S) and E is the

~ so L < (RB) i RB+I"

theorem due to W, Reinhardt

portant

class

of the corresponding

where

LK(R B) is relative

class of all theories The following

instead

denote by N

of the form

the subsets

Dg T will

(P(S):power

structural

of KM can be re-

the binary predicate

S is a transitive

For each ordinal number ~ we will of all structures

of KM, A X K M , of the impredica-

of KM, that is every model

interpretates

in

the set of set formulae

(~ZKM~V)}.

in the set of axioms

of possible

of

sets

The set of all formulae

Therefore v~

tive axiom schema of comprehension property

KM m is recur-

of a theory with two kinds of objects,

is in a two-sorted

Due to the presence

[1953],

how to find axioms

for KMm?.

this language will be denoted by ~. of K M w i l l

of Craig

arises,then,

enumerable

6

class of ordinals such that

n

By reflection

principle

type S such that Ls[V]

<

hension + replacement. But there is a closed unbounded then LS~ R ( R )

<

Ls[V].

prehensio~ + replacement ordinal y. Then SPR =6 n As a consequence

we have

(see Chuaqui

I L[V]. An

[19801])

Take the first

class of ordinals

"

there is a well order

Then LS[V]I=

~n such that if ~e~n

Let ~n~ be the least

B such that

(it always

But S I R ~

exists).

A1-compre n

LB(R )l=A~-eom-

y for a certain D

71¸

T H E O R E M i.

For each finite subset of the axioms

a class of structures K, such that KMIof ordinals

for KM,E,

there exists

(KI= z)and a closed unbounded

class

~ such that for every ¥ ~ , N y n K ~ 0 .

The following

immediate

consequence

is obtained by means of Godel's

second theorem: THEOREM 2. (a) (b)

KM is not finitely axiomatizable KM m is not finitely axiomatizable

This can he extended T H E O R E M 3o PROOF:

to

KM is not finitely axiomatizab!e Assume the contrary,

be a closed unbounded

ioe. KM=ZF+z

class of ordinals

over ZF° for some finite

such that for every

E!~. Let r ~cr,

~ZFo By theorem i let ~ be a closed unbounded every ¥c~, < R y , T , E > ~ E where T ~ R ¥ + I. ~ Z F + E = K M w h e r e

T ! R +i.

class of ordinals

Because

such that for

~nF~0 take ~c~nr.

Therefore K M ~

COnKMwhich

Hence

is im-

possible unless KM will be inconsistent. As a trivial T H E O R E M 4.

consequence we obtain the It cannot hold that KMm will be finitely axiomatizable

ZF and KM will be finitely axiomatizable By an argument

similar to those in the proof of theorem 3 it can be

proved that KM m is not finitely axiomatizable technics [1980]

over

over KM m.

in Kreisel

and Levy [1968]

over ZFo

Moreover,

using

this can be extended to (see Chuaqui

):

T H E O R E M 5.

(a) F~ m is not finitely axiomatizable

by formulae of bounded

depth over ZF. (b) K M is not finitely axiomatizable

by formulae of bounded

depth over KM TM. In spite of this unboundness

results

a certain kind of finitization may

be obtained, THEOREM 6. (a) There exists a finite set of formulae KM TM = {~ : El- ~}n~ m (b) There exists a sentence

E, zc~

such that

e ~ m such that DgKM m = Dg(ZF+0).

72 .°

PROOF:

(a) See Krajewski [1974]

;

(b) Similar to Lindstrom [1979] D

The following theorem will be useful for our purposes THEOREM 7. If ~ m and ne~ then KMI- ((@~) (~F) (Ord (~)AF!Re+I^~KMn)A(~<-->~R~)) (KMn is KM with comprehension restricted to ~I formulae) n PROOF: By Reflection Theorem there exists a closed unbounded class of ordinals r such that <--> ~R6 Because

KNn i s

finitely

,

axiomatizable,

8er using

Theorem

1 get

a class

of

structures K such that K D KMn . Again by Theorem I let a be a class of ordinals such that KnN ~0 for Y each y~a. Take e ~ n r o The above theorem is closely connected with the notion of extendability of models of set-theories (see Marek & Mostowski [1974]). In fact (@F) ( F I R + I A I=KMn)

means that R~

is KMn-extendable.

So

we can rewrite the theorem as: THEOREM 7' KMI- (~e)

If ~e~ m and ne~ then,

(Ord(e)^R

e

is KM

n

extendable)A(~<->~R~.

An improvement of Vaught [1967] as presented in Ratajczyk [1979] gives the following THEOREM 8. If j is an interpretation of Peano's arithmetic P in KM and ( V ~ m) (KMI- ~+Conj(A~+A$) for each finite fragment S of KM then the set of sentences ~ (P) + { ~ ÷ C o n j ( A ~ + A S ) : ~ m , s ~

finiteAXKM}

is

an axiomatization of KMm° PROOF:

Ratajczyk [1979], Theorem 0.

By results in Montague so we obtain THEOREM 8'°

[]

[1960] it is known that (V~c~ m) (KM> ~÷Con i (A~+AS))

If j is an interpretation of P in KM, then ~(P)+{#÷Conj

(A~+AS):~e~m,s!finit e AXK M} as an axiomatization of KM m, Using different methods the following theorem has been proved by Ratajczyk [1979]o

73 THEOREM 9. The set ZF+{(~ )(R is KM n extendableA#<->~R~:~m^ne~} KM m

is an axiomatization of

PROOF: Let r={(~ )(R is KMn-extendableA~R~ ÷>~):~eCmAne~} Enumerate F,i.eo r={XCn : ~ m ^ n e ~ } o T h e n for each ~ecm and each ne~, by theorem 7' , KMI- X~n from which rc~M m,_~. ~ hence F~ml - r . Call A={~÷Con~(A~+AS):~mAS!finit e AXKM}O

We will prove ZF+rk~(P)+A.

Let <M,E>I=ZF and suppose for each n, and each ~, <M,E>I=X~, n. ~e~m and let S!finit e AXKM

Assume <M,E>I= ~.

Let

Because S!finit e AXKM

there exist ~ O R M and F~M such that <M,E>I= FiR +I^(l= S)^ ( R <__>~).

As we have assumed <M,E>I= ¢ then ( < F , R c > ~ S ) ^ ~ R ~ . Therefore <M,E>~Con(A~+A S) from

<M,E>~ F i R + I ^

which <M,E>~Conj(A +AS). Although this theorem solves the problem of finding a set-theoretic set theoretic set of axioms for KMm we will give a simpler axiomatization which is based on reflection principles. For each n~w and each p ~ m let F be the following set schemata: ~ ~8 (lim>~(~) ^Iim>~(B)^ < < R ~,L~(RB)>^ ~ R ~ ~) were ~nK is the first B such that I= A~-comprehension, and < is the formal statement that is an elementary submodel of and the relation occurs only with set parameters. THEOREM I0. (W. Reinhardt) Let @ be an instance of F. Then KMI- @, and therefore KMml -@. PROOF: By Theorem 0 take 8e~ n and get a structure . Let P b e t----he s e t o f d e f i n a b l e e l e m e n t s o f R B i n L~n(R 8) u s i n g B p a r a m e t e r s from R~, i . e . tEP<--> t h e r e e x i s t s a f o r m u l a O a n d S p a r a m e t e r s b I , b 2 . . . . . bn i n RB s u c h t h a t t={ xcR~: I= e ( x , b I . . . . . b n ) } . I t i s o b v i o u s that PcL~(Rs) and m o r e o v e r we h ~ v e P
~ . is inmediate.

This

is

the desired

structure.

The r e f l e c t i o n

~P~<--~@ D

74

THEOREM II: If ~e~m and E is a finite subset of the axioms for KM then, ZF+FI-

(~)

(@F) ( ( O r d ( ~ ) ^ F ! R + I ^ < R , F > I = I)^~<-->~P~).

PROOF: Let ~ and Ebe given. Assume the more complex instance of comprehension which occurs in E is A~for a certain n. Then clearly ~R~ Using these results we can give the following axiomatization for KM m using the "reflection principle" F , ~HEOREM 12. The set ZF+F is an axiomatization for KM m. PROOF: Let A {~ ~ Conj (A~+AI)

: ~E~ m, Ea finite subset of the axioms

for KM}. By Theorem I0 it is sufficient to prove ZF+FI-~(P)+A Let <M,E>;= ZF and assume <M,E> satisfies all the instances of F, i.e. <M,E>I= ZF+F. Let ~c~ m and let E be a finite subset of the axioms for KM. Assume <M,E>I =r. By Theorem II, since <M,E>I=F there exists ~ O R M and F~M such that ~ , E > ; =

FsR + I ^ ((R ,F) I=E) ^ R ~ < _ > ~ .

Since we have assumed that <M,E>I= ~, we have <M,E>I= FsR~+ I ^ (I=E) ^ #R , then <M,E>T=Con(A~+A E) from which follows <M,E>I=Conj(A~+AE). NOTE: This paper was partially written while the author was at the Instituto Venezolano de Investigaciones Cientificas supported by a grant from PNUD/UNESCO (Matem~tica). The author thanks Dr. Carlos A. Di Prisco for some very helpful conversations.

REFERENCES Chuaqui, R. [1980].

Internal and forcing models for the impedicative theory of classes, Dissertationes Mathematicae CLXXVI, Warzawa.

Craig, W.

On axiomatizability within a system, J.Symbolic Logic 18,pp. 30-32.

[1953]

Friedman, H.[1981]

On the necessary use of abstract set theory, Advances in Mathematics 41, pp. 209-280.

Krajewski,

Predicative expansions of axiomatic theories, Z.Math.Logik, vol. 20, pp.435-452.

S[1974]

75 Keisel,G.

[1971]

Observations on popular discussions of foundations, Axiomatic Set Theory, ed. by Dana Scott, American Mathematical Society, pp. 189-198.

Kreisel, G. and Levy, A. [1968]

Reflection principles and their use for establishing the complexity of axiomatic systems, Z.Math. Logik, vol. 14, pp. 97-112.

Lindstrom, P [1979]

Some results in interpretability, Proceedings from 5th Scandinavian Logic Symposium, Aalborg University Press, pp. 329-361.

Marek, W and Mostowski, A [1974]

On extendabiiity of models of ZF set theory to the models of KM theory of classes, Lecture Notes in Mathematics, vol~ 499.

Montague, R. [1960]

Semantical closure and non finite axiomatizability, Infinitistic Methods, P.W.N. pp 45-69.

Novak, I.L.

A construction for models of consistent systems, Fundamenta Mathematicae, vol. 37, pp. 310-317.

[1950]

Ratajczyk, Zo[1979]

On sentences provable in impredicative extensions of theories, Dissertationes Mathematicae CLXXVIII, pp 1-44.

Vaught, R.L. [1967]

Axiomatizability by a schema, J. Symbolic Logic 32,pp. 473-479.

APPLICATIONS OF MODEL THEORY TO REAL ALGEBRAIC GEOMETRY A SURVEY. M.A. Dickmann CNRS - University of Paris VII Paris, France

CONTENTS. § i.

Introduction

77

§ 2. § 3.

Real versus complex geometry Quantifier elimination

79 82

§ 4.

Applications of quantifier elimination.

87

A. Elementary properties of semi-algebraic sets and functions. B. The real nullstellensatz. C. The simple point criterion D. Hilbert's 17th problem. E. Further applications. § 5.

The structure of semi-algebraic sets

§ 6.

The real spectrum.

97 105

A. Introduction B. Elementary properties C. Constructible sets § 7.

Affine varieties over real closed fields.

118

§ 8.

Dimension

121

§ 9. §i0.

Continuous semi-algebraic functions Nash functions.

124 131

A. Basic algebraic properties B. Nash functions and real algebraic geometry C. The separation theorem. §II.

The substitution theorem;

"stellensatze".

137

A. The substitution theorem B. Applications:

"stellensatze";

cormnutative algebra

of Nash functions. References.

144

77 §i. INTRODUCTION.

This survey deals with real algebraic geometry, a subject still in its infancy, and with the application of model-theoretic techniques in this area. Let

V

be an algebraic variety defined over a field

ly from standard usage, we will think of finite set

PI ..... P£

of polynomials in

equivalently, by the formal expression a field

F

containing

K

points (which we will call

we let

V(F)

V

K.

Deviating slight-

as being given by a specific

~ ( X I .... ,Xn)

for some n - or

"iA__l__Pi(XI, .... Xn) = 0". denote the set of

For

F-rational

F-points) of V:

V(F) = {aEFn I Pi(a) = 0

for

i = i ..... £}.

Classical algebraic geometry was primarily the study of the complex points V(~), and in spite of the attention payed to rationality questions in the development of the subject, even the general theory initiated by Grotendieck has been set up to deliver results corresponding to the complex case.

Thus, some of the most natural questions connected with the study

of the real points

V(~),

like the number of connected components in the

Euclidean topology, or even the existence of non-singular real points, simply do not arise in that context. Indeed, such questions -which form the proper subject matter of real algebraic geometry have received detailed consideration only relatively recently.

In the last ten years especially,

a body of theory has emerged in

which real closed fields assume their proper role, the real spectrum (with a suitable topology) has been constructed, and the geometry of semi-algebraic sets and rings of Nash functions have been explored

(subjects with

no meaningful parallel in the classical theory).The necessary definitions and background will be given below as the occasion arises Model theoretic techniques - notably a systematic use of quantifier elimination for real closed fields and related ideas- have a role to play in

78 all of this.

The precise purpose of this article is to survey this in-

teraction of model theory with various kinds of geometrical investigations. As we intend to give only an overview of the subject, we omit many proofs, in order to concentrate on those which illustrate the use of an important technique combined with some aspect of mmthematical logic.

At the same

time we have tried to supply many examples and references for further study.

Some familiarity with the elementary algebra of real closed

fields is assumed. We will begin our survey by reminding the reader of some "pathology" associated with the geometry of the sets

V(]R)

(see §2).

The general ana-

lysis of the behavior of such sets depends on a body of theory based on the following four ingredients: (I) Quantifier elimination for the theory of real closed fields (see

§3).

This provides a substitute for the techniques of classical elimination theory.

Quantifier elimination in algebraically closed fields is a rather

trivial affair which was replaced by other devices;

in the real closed

ease it constitutes a powerful tool, at least for the present.

In par-

ticular, the presence of an ordering introduces features having no parallel in the classical case. (2) The "separation theorem" and related tools (see

§5).

These yield

considerable information concerning the topology of real algebraic varieties and, more generally, of "semi-algebraic" sets (arbitrary definable subsets of ~ n ) with respect to the ordinary Euclidean topology, where for example the connected components, set are of interest.

the interior or the closure of a

They are also used in transfering

such notions

to other real closed fields. (3) The real spectrum of a ring (see

§6).

The use of this instrument in

real algebraic geometry is similar to that of the prime spectrum in the classical theory.

It turns out, however,

that its topological properties

are better, yielding information about the sets V ( ~ ) of real points with respect to the Euclidean, rather than the Zariski topology. (4) Nash functions (see

§I0, §II).

This class of functions

(analytic

functions with a definable graph) is the tool which makes possible an explicit description of the "branches" and the connected components of real varieties, in spite of the complexity of the situations arising in this case, without a common measure with those occurring in classical algebraic geometry. Applications of quantifier elimination to real algebra are surveyed in

79

§4, while the study of real algebraic varieties properly is carried out in

§7, §8.

We have also included a section ( § 9) on continuous semi-

algebraic functions, the analysis of which leads naturally to the theory of Nash functions. This article (and the author's interest on real algebraic geometry) owes much to Coste/Coste-Roy

[62],

of the core of the subject.

a clear, deep

and inspiring exposition

Thanks are due to the referee, whose sugges-

tions helped to improve substantially the original manuscript. §2. REAL VERSUS COMPLEX GEOMETRY. We illustrate some of the comments made above by recalling results of complex algebraic geometry which rule out phenomena occurring quite commonly among real algebraic varieties. 2.1 (when

NOTATION. F = k

Let

k

First we must fix some notation.

be a field and

F

a field extension of

k

we will drop it from our notation).

(a) To each subset

X c k n we associate the ideal

~F(X) = {PeF[X I ..... X n] J P(~) = 0

for all

~EX}.

(b) Given a set S ! k[X I ..... X n] we define VF(S)

=

{xeFn

I P(x) = 0

for all

Pc$ }.

Recall that by our earlier convention a variety ed by a

specific finite set of polynomials

(c) We write

IF(V)

for the ideal of

V

over

k

is determin-

PI ..... P£ c k[X I ..... Xn].

F[X I ..... X n] generated by

PI,...,P£. (d) The (affine) coordinate rin$ of a variety k[Xl ..... Xn]/ ~(V(k))' denoted by (e) A variety REMARK.

V

A variety

over

k

is called irreducible if th~ ideal V

is the ring

I(V(k)) is prime.0

is irreducible iff we cannot write

proper union Vl(k ) uV2(k), with k. cf. Hartshorne [3; Ch.l, §i] THEOREM 2.2.

V

k[V].

Vl,V 2 varieties defined over []

(Hilbert's nullstellensatz).

over an algebraically closed field

V(k) as a

k, and

Let

V

be a variety

Pek[X I ..... Xn].

Then the

following are equivalent: (i) P(x) = 0 (ii) There is

for every n ei

xeV(k).

such that

pn~I(V).

In other words: l(V(k)) = Rag (l(V)),

0

80

where

Rad(I)

For a proof, Of course,

= {Pck[X 1 ..... X n ] I see Lang

Since

Let

V

V(~)

(ii) frequently

(or any real closed field):

(i)

~

of 2.2

defined by the polynomial is true for any P, while

fails.

D examples,

we state a fundamental,

and highly

result about complex varieties:

THEOREM 2.4.

Let

V

is a connected subset of Euclidean

~

the variety over

= ~ , then

Turning to more interesting non-trivial,

n e I}.

[8; p. 256].

this theorem fails badly over

EXAMPLE 2.3. X 2 + I.

pn~I for some

be an irreducible variety over ~n (here

~

~.

Then V(~)

is considered with its standard,

topology).

D

The proof is given in Shafarevich

[14; pp. 320-32].

This result also fails over the real field: 2.5.

EXAMPLES.

ly is irreducible, ly, the cubic

(a)

The hyperbola of equation

consists

of two disjoint branches

y2 _ X 3 + X, irreducible

Lang [8; p. 128]), has two connected

These examples remain irreducible

YX-I = 0, which clearin

by Einsenstein's

R2°

(b) Similar-

criterion

over

~.

The examples below require the notion of dimension of a variety. oum present purposes

(cf.

components:

the intuitive notions

suffice;

For

but as we will

develop the analogous

"real" theory later, we recall here the customary

algebraic

corresponding

definitions

DEFINITION (a) n

2.6.

The (Krull)

Let

A

to those intuitive notions.

be a commutative

dimension of

ring with unit.

A, dim A, is the supremum of the integers

such that there is a chain Po ~ PI ~ ''" ~ Pn

of length

n + 1

of prime ideals of

of this type of unbounded Let

V

(b)

The (global)

A (dim A = ~ if there are chains

length).

be a variety over a field dimension

of

k. V, dim (V), is the dimension of the

81

coordinate ring

k[V].

(c) The (local dimension of dimension of ideal

k[V]M_, x

V

at a point

xEV(k), dim (V,x), is the

the localized of the ring

k[V] at the maximal

M~x of the point x : = {P~(v(k)I Pck[Xi'''''Xn]

and

P(x) = 0}.

0

There are several equivalent characterizations of these notions;

we

mention the following: THEOREM 2.7. type and

K

Let

k

be a field,

the field of fractions of

dim In particular,

A

=

if

A an integral A.

transcendence degree of V

K

over

is an irreducible variety over

the field of fractions of

k-algebra of finite

Then: k. k

D and

k(V) denotes

k[V] (called the function field of V), then

we have: dim (V) = transcendence degree of

k(V) over k.

A proof of Theorem 2.7 can be found in Matsumura [9; Ch.5, §14]. The fundamental result on the dimension of varieties in classical algebraic geometry is the following: THEOREM 2.8. closed field

k.

Let

V

be an irreducible variety over an algebraically

Then

dim(V) = dim(V,x)

for all

The proof is given in Atiyah-Macdonald

x~V(k). [i;

0

pp. 124-125].

This result also fails over the field of real numbers: 2.9.

EXAMPLES.

(a)

The irreducible cubic

isolated point at the origin:

I/ T\

y2 _ X 3 + X 2

has an

82 It is (intuitively) obvious that the local dimension at the origin is zero, while the same dimension at any point of the main branch is one. (b) An example of the same phenomenon where the set

V(~)

is, in add-

ition, connected, is obtained by considering Whitney's "umbrella" (or Cartan's, if you are French), given by the irreducible polynomial Y3-(X2 + Y2)Z, whose graph looks as follows:

z

The theory of dimension for real varieties is discussed in §3.

§8 below.

~UANTIFIER ELIMINATION. The theory

RCF

of real closed fields is most simply defined

as the first-order theory of the field

~

of real numbers in the language

L of unitary ordered rings, built from the symbols

+, -, ., 0, I, <

and logical symbols (but no symbol for the multiplicative inverse). In order to derive useful mathematical information it is necessary to have good axioms for the theory and some algebraic machinery, notably the existence and uniqueness of the real closure. of axioms is:

The most efficient set

(i) The axioms for ordered fields. (2) The intermediate value property: variable, and for any points

a
for any polynomial P(X) in one for which

P(a), P(b) are non-zero

and of opposite sign, P(X) has a root in the interval (a,b). Applied to polynomials of the form

X 2 -a, the intermediate value proper-

ty implies that non-negative elements have square roots, and hence that

B3

real closed fields have a unique ordering which is definable in the language of rings.

Nevertheless,

retain the symbol

for reasons which will soon be clear, we

< .

DEFINITION 3.1.

Let k, K be ordered fields,

K an ordered extension of

k. K is a (the) real closure of k if K is real closed andalgebraic over k. 0 The existence of a real closure is straightforward up to (unique)

k-isomorphism

requires

tained by use of Sturm's algorithm. use the intermediate

However,

and an elementary

can be found in basic algebra texts, [5],

Lang [8; Ch.IX] EXAMPLE 3.2.

importance

or Ribenboim

[50]).

Details concern-

information on real close fields

such as Jakobson

[4, Ch.VI],

Jakobson

[13; Ch. IX].

We recall an example of real closed field of great

in real algebraic

series with coefficients Laurent

theorem (cf. Gross-Hafner

(see Knebusch

algebraic

Its uniqueness

touch, usually ob-

there are proofs which instead

value property and Rolle's

[49]), or even other algebraic tools ing this result,

(Zorn).

a more delicate

geometry:

the field P(K) of (formal) Puiseux

in a real closed field K. This is the field of

series with fractional

exponents, i.e. formal expressionsofthetype

a n XnIp

n=N where

p

is an integer ~ i,

N ~

and

a N ~0.

The algebraic operations

are defined as in the case of formal power series; out the explicit

definitions

The real closedness

of P(K) follows easily from that of K and the following:

THEOREM 3.3 (Newton). characteristic

zero,

Walker

If k is an algebraically

then so is

This is the fundamental curves.

the reader can figure

as an exercise.

closed field of

P(k).

0

theorem of the classical

theory of plane algebraic

The standard proof, by the "Newton polygon" method, [15; Ch. IV, §3];

model-theoretic

of this type gives a direct proof of the real closedness The key point here is that P(k) is the algebraic geometric

interpretation

various orders, when K is real closed,

The main result concerning THEOREM 3.4.

(Tarski;

see Brumfiel

§9 below. the theory approx.

of P(K) as well).

closure of k(X).

of the real closure of the field

We will use this example in

is given in

proofs also exist (an argument

K(X)

For a with its

[65; pp. 183-184]. 0

RCF

is:

1930). Let

~(v I .... ,vn) be a first-

84

order formula of the language L having Then there is a quantifier-free free variables,

v I ..... v n as free variables.

formula

~(Vl,...,v n) with the same

such that

RCF ~ Vv I ..... Vn(~(v I ..... v n) <-> ~(v I ..... Vn)). In order to prove this result we will need the following model-theoretic criterion,

which is a simple consequence

first-order

3.5.MODEL-THEORETIC theory formulated

CRITERION.

in a language

Let

L.

ment of Theorem

3.4

holds for

(2) For every pair of models

~

~,~

implies

~=

of

~(v,~)

~

see Chang-Keisler

PROOF OF THEOREM 3.4. (*) when

first-order

KI,

be an arbitrary

in the language

with one free variable

~ ~v~(v,~)

For details

T

T, instead of

and every conjunction

their negations, (*)

theorem of

Then the following are equivalent:

(i) T admits quantifier-elimination

c ~,

of the compactness

logic:

K2

ordered subring of both Trivial manipulations

K1

RCF).

T, every

L-substructure

~ 2 ~,

of atomic formulas of

L

and parameters

~, we have:

~

in

and

~v~(v,~).

[33; 3.1.17]

By the model-theoretic

~=

L (i.e., the state-

0 or

Dickmann

§i].

criterion we need to prove

are real closed fields and and

[72;Ch.II,

~ = A is an

K 2.

show that in the present case a formula

@(v,~)

as

above is of the form m A Pi(v) = 0 i=l with Let

t ~iQj(v) > 0, j

^

Pi,Qj ~A[X]. k

be the field of fractions and

of both K 1 and K2, we have the situation

Fi

of

A, which is an ordered subfield

its real closure in

FI

!

KI

F2

!

K2

Ki(i = 1,2). Thus,

k

By uniqueness

of the real closure

In order to prove

(*)

there is a

it suffices

to show:

k-isomorphism

~ :F I ÷ F 2 ,

85

(**)

If

K I ~ @v ~(v,~), then there is

Indeed, we would have then, valid in

K2,

F 2 ~ ~[o(b),c]

we conclude that

Let us assume

bcF I such that ;

F I ~ ~[b,~].

since this formula is also

K 2 ~ ~v ~(v,~).

K I > ~v ~(v,~) and let

acK I

be such that

K I > AP:(a)= i ±

0 ^ AQ.(a) > 0 j J CASE I. m > 0

and one of the polynomials

is algebraic over CASE 2. Let

Either

R

k, which implies m = 0

is of degree >0.

Then

a

or all the polynomials

Pi

are of degree

0.

be the (finite) set of all roots (in K I) of the non-trivial poly-

nomials amongst and hence

QI,...,Q t.

R ! FI-

Let

The elements of

(e,B)

Q1 .... 'Qt

interval

R

are algebraic over k,

be the interval of

termined by two successive elements of Since

Pi

aeF I,

are positive at

a,

K1

R (e or B may be

3.6. COMMENTS.

choose

-~

or + = ).

they are positive on the whole

(~,B), and hence also positive on (~,B) n F I.

this set is non-empty;

containing a de-

Since

~,B~FI,

b~(~,B) n F I.

0

The foregoing proof of the quantifier elimination

theorem is by far the shortest and most elegant, but it is not constructive.

For most applications, however, the theorem is quite sufficient

as stated; many applications require only the much weaker transfer principles below. Constructive proofs of Theorem 3.4 do exist. proof, Tarski Ch. IV]

[40],

and Collins

is of this type;

Indeed, the first published

see also Kreisel-Krivine [37;

[35].

The primitive recursive algorithms for finding a quantifier-free equivalent of a given formula provided by the constructive proofs cannot be implemented in practice, except possibly for Fischer-Rabin

E36]

very particular formulas.

showed that any such algorithm is necessarily of

exponential complexity:

they construct

L-formulas of arbitrarily large

length n for which no algorithm can compute a quantifier-free equivalent in less than 2n steps. For further information in this direction, cf. Monk [39] Nevertheless,

and Collins

[35].

algorithmic proofs frequently yield additional information;

in our case a useful corollary of such type of proof is the following: PROPOSITION 3.7° equivalent modulo

For any

RCF, ~,

L-formula ~ there is a quantifier free

such that the coefficients of the poly-

nomials occurring in @ are polynomial functions over occurring in ~.

Z of the polynomials D

86

Next we consider

two important consequences

THEOREM 3.8. a formula without

free variables)

real closed fields

L.

(i.e.

For any pair of

K2 ~ ~

The standard name of this result among logicians

of the theory RCF".

The transfer principles ters.

of the language

iff

The proof is trivial.

ing one;

elimination.

Let ~ be a sentence

KI,K2, we have:

KI ~ ~

is "completeness

of quantifier

(First transfer principle).

stated below are far more powerful

they are also more useful,

than the preceed-

as they apply to formulas with parame-

The first is actually a reformulation

of quantifier

elimination

3.5 above), while the second is known as the "model-completeness

(see

of the

theory RCF". THEOREM 3.9.

(a)

ordered extension of

Let

be an ordered field,

.

Then for every

and KI,K 2 real closed

[-formula

~(v I ..... Vn) and

every al,...,anck , we have K I ~ ~[a I .... ,an ]

iff

K 2 ~ ~[a I ..... an].

In particular: (b) (Second transfer principle).

The equivalence

are real closed fields such that K I ! K2, and PROOF,

Since we may take ~ quantifier-free,

The following

is an equivalent

PROPOSITION

3.10.

formulation

f:K I + K2, is an elementary map~

L-formula

~(v I ..... Vn) and

this is clear. of the second

an inclusion of real closed fields is an elementary

COMMENTS.

(i) It is well-known

admits quantifier

elimination

den Dries

[41]

have shown that

commutative)

elimination

in the language

The interest of Theorem

L is intimately elimination

Macintyre-McKenna-van

necessarily

elimination

any theory

connected with

significance.

(ii) Quantifier For example,

For example,

interest.

in that the language

of mathematical

that quantifier

inclusion.

in a suitable language; but, as a rule,

is of no mathematical

3.4 lies precisely

phenomenon.

for every

K 2 ~ ~[f(a I) ..... f(an)].

to the choice of language.

information

transfer principle:

in other words,

iff

is very sensitive such language

0

a I ..... an~K , we have

K I ~= ~[a I ..... a n ]

3.11.

a I ..... an~K I-

Every unitary ring homomorphism between real clos-

ed fields,

In particular,

above holds whenever KI,K 2

RCF

is an exceptional

den Dries

is the only

[38]

theory of

and van (not

ordered rings with unit admitting quantifier L.

87 On the other hand, transfer principles are a much more common occurrence: Mc Kenna has shown the existence of

2 ~ o model-complete theories of

ordered fields. (iii) definable subset of

~

Quantifier elimination implies that any

is a union of intervals.

Van den Dries

[87]

has shown that this latter property alone already has some significant consequences normally associated with quantifier elimination techniques, and conjectures that this property is preserved if the exponential function is adjoined to the language.

The study of properties of this type

arises in the investigation of: (iv) 45]

Tarski's problem.

In his monograph [40; p.

Tarski asked whether quantifier elimination or some of its conse-

quences could be extended to the theory of the structure < ~,+,-,<,exp,0>. This problem has been taken up again in the last few years, and valuable new information has been obtained (we know, for example that quantifier elimination cannot be extended).

Nevertheless,

the problem is incompara-

bly more difficult than its analog for polynomials. and further references,

see van den Dries

For a discussion

[87].

§4. _APPLICATIONS OF QUANTIFIE R EL IMINAT!ON. A. Elementary properties of s emiTalgebraic

sets and functions.

The following notion is generally considered to provide the analog in real algebraic geometry of the notion of algebraic geometry. DEFINITION 4.1. K n,

n e I,

Let

K > RCF.

"constructible set" in ordinary

A semi-algebraic

(s.a.) subset of

is a finite boolean combination of sets of the form

{xcK n I Q(x) > 0 Equivalently,

} ,

where

QcKEX I ..... Xn].

s.a. sets are the subsets of

D K n defined by formulas of

the form V i

(A j

Pij(vi ..... Vn) = 0 A A k

Qik(Vl ..... Vn) > 0)

(finite conjunctions and disjunctions), with

Pij' Qik ~ K[XI ..... Xn], or

in other words, by quantifier-free formulas in the language parameters in

K.

L , with

Invoking the quantifier elimination theorem 3.4 we

obtain at once the fundamental relation: (*)

Semi-algebraic

=

Definable

(i.e. parametrically definable) which establishes the connection between first-order logic and real algebraic geometry (or semi-algebraic geometry, as it is also called). Once this relationship is understood,

a number of fundamental results

88 follow instantaneusly: THEOREM 4.2. the following

(i)

set

operations;

and interior

projection

(iv)

cartesian Immediate

using

the closure

S

of a definable

:

iff

4.3.

K n.

for Kn);

K n, say);

For example,

(*).

the Euclidean

DEFINITION

into

topology

product.

xES

subset of

(in the euclidean

(of K m

S c K n is defined by

(d denotes

s.a. sets is closed under

operations:

closure

(iii)

The class of

K ~ RCF.

finite boolean

(ii)

PROOF.

Let

Vc>0 ~ ycS(d(x,~)<e)

distance).

Let

K

0

be a real closed

A function

F : S ÷ Km

field,

m,n e i, and

is called semi-algebraic

S a (s.a.)

if its graph Gr(F) is a s.a.

subset of

K n+m.

Note that the domain set,

= {<x, F(x)> [x~S}

S = Dom(F)

since it is the projection THEOREM 4.4.

Let

(a) The set of K-valued

s.a.

of s.a.

(c) The image of a s.a. PROOF.

As above;

operations

write

are actually

functions

trivial

proofs

if desired 4.5.

EXAMPLE.

elementary dS : Kn÷

analysis, K

on a s.a.

and use

facts without

definitions

0

quantifier

elimination

since some of these

elimination).

elimination

of the relevant

(*).

surprising,

to quantifier

=

subset of K n

is s.a.

by giving non-trivial

this is done,

sometimes).

As an example

going beyond the standard

consider,

for a fixed s.a. set

the (Euclidean)

distance

from

is facts

One can avoid the proofs,

examples

of

S c K n, the function

defined by:

ds(x)

F.

is s.a.

down the standard

using quantifier

(in fact,

a s.a.

defined operations.

set under a s.a. function

(this is hardly

equivalent

defined

pointwise

(which are first-order),

a headache

is necessarily

K n of the graph of

functions

The proof of these elementary usually

over

function

K ~ RCF.

is a ring under the standard, (b) The composition

of a s.a.

x to S;

89

dS

is a continuous s.a. function.

General properties of these functions,

yielding non-trivial information, will be studied in §9. B.

D

THE REAL NULLSTELLENSATZ. A fundamental feature of algebraic geometry over an algebraical-

ly closed field, k, is the one-one correspondence between a variety

V

k-points of

and (proper) maximal ideals of the coordinate ring

k[V]

established by Hilbert's nullstellensatz. This correspondence fails in the real case : the polynomial X 2 + y2 + I does not have real points, while its coordinate ring is not reduced to

0

and hence does have

(proper) maximal ideals. In order to develop algebraic geometry over

IR, it is of the utmost im-

portance to restore a correspondence of this kind, that is, to determine which type of ideals correspond to points.

This amounts to proving a

"real" version of the nullstellensatz, which we will presently do. The key notion comes from the observation that the ideal ~ (a) of real polynomials vanishing at a point a el~n has the following property: n p2i

i=l

C

I(~)

implies

PI'

.

.

'Pnc . . ~(a)

Generalizing this observation we introduce the following notion: DEFINITION ing

k

and

4.6.

Let



be an ordered field,

A a ring contain-

I c A an ideal.

(a) We say that

I

n 2 i~l Piai el,

is real over k

with

aicA

iff

and

Pick, Pi>0,

implies a I .... ,ancl.

We will also need the following weaker notion: (b)

I

with

is called semi-real over k Pick,

(c) If 4.7.

Pi>0,

I = {0}

we call

REMARKS.

teristic # 2, extending that

and

(a)

iff

-I

2

is not of the form

~piai/l i

aicA. A If

real

(respectively, semi-real) over k.

I is a prime ideal and

A/I is of charac-

then I is real over k iff A/I has~oan ordering of k iff -i is not of the form ~PiX~. with i

Pick, Pi>0, and

xi

in the field of fractions of (b)

I real over k implies

(c)

Let



D

A/I. I radical.

be an ordered extension of the

90

ordered field over

k

,

and

vanishing at

S (cf. (d)

ideal of A, then

I

S ! Fn.

Then the ideal

I k(S) of polynomials

§2.1) is real over k.

If

A

is a noetherian ring and

is real over

k

iff

I

is a proper

I is radical and is the inter-

section of finitely many prime ideals real over k. (e)

The example

I = (X 2 + y2),

A = k[X,Y], shows

that semi-real is a notion strictly weaker than real. THEOREM 4,8.

(Real nullstellensatz).

its real closure, and I PROOF.

Let

be an ordered field,

I ! k[Xl,''-,Xn] an ideal•

is real over

k

iff

I =

D

Then

I k(V~(1)).

The implication from right to left is just the remark 4.7 (C).

For the converse we need only prove the inclusion

Ik(V~(I)) ! I, since

the other inclusion is trivial. Let

I

be generated by

The condition

QI .... 'Qr ~ k[Xl ..... Xn]"

Q E Ik(V~(I))

r (*) ~ ~ V~[iAiQi(v)= = 0

is expressed by

~

:

Q(v-) = 0 ]• l

Using remark ideals

4.7(d) we can find a representation

Pj are prime and real over k.

I = ~ P. j=l J

Then the condition

Qel

where the is equi-

valent to the conjunction of the conditions (**) k[X I ..... Xn]/p j for

j = I ..... £ .

>

Since each of the rings

ing extending that of Lj

Q(Xl/p ,j .... Xn/e.)j = 0

k

k[X I ..... Xn]/p j has an order-

(by 4.7(a)), we may consider the real closure

of its field of fractions with one of these orderings.

we have

k =__ Lj ;

Hence the formula

by Proposition (*) holds in

we get

Lj as well.

Applying (**) to QI ..... Qr

r Lj ~

Now, using

In particular,

3.10 this inclusion is elementary.

(*)

with

A Qi(Xl/p i=l J, ~ =

<

Xl/P

xI

Lj > Q ( which implies that

QcPj.

X

) = 0. n/pj

x J .... , n/pj>

/p , .... J

x/

we conclude that

p ) = 0, J

Since this holds for

j = I ..... £,

we

91

get

Qcl, as contended.

D

Every ideal is contained in a smallest, possibly improper, real ideal; namely: ~{J

I J

is real over

k

and

I ! J ! A}. R

DEFINITION 4.9.

We shall call real radical of

est ideal of A, real over

k

and containing

I, v ~ ,

the small-

I.

The real radical admits a purely algebraic characterization: PROPOSITION 4. i0. a ~ R Iv~-- iff

there are

n,mEN[ ,

m > 0,

Pl .... 'PnCk+

and

b I ..... bneA

such that a 2m

2 ~ Pibi e I. i=l

+

This result is proved in Krivine [51], Dickmann [72;

Dubois-Efroymson [47]

and

Ch.lll].

We have the following consequences of the real nullstellensatz: COROLLARY 4.11. closure and

Let



I ~ k[X I ..... X n]

be an ordered field, an ideal.

k

its real

Then:

R

Ik(V~(1)) =

~

. R

PROOF.

By the nullstellensatz

R

Ik(V~ (IRE-)) =

V~,

and by the

R

preceeding Proposition COROLLARY

4.12.

V~ (I) = V~ (I~Y--). With the notation of the preceeding Corollary we

have: R

I E I~

iff

V~ (I) = ~.

D

We restate this corollary in more geometric language: COROLLARY 4.12 his.

(Weak real nullstellensatz).

variety over an ordered field of





and let

[

Let

V

be a

be the real closure

Then the following are equivalent:

(i)

V

(ii)

The ideal

has a

k-point,

i.e.

V(k) #

~.

l(V) is semi-real over k.

D

R

(Note that an ideal COROLLARY 4.13. let

V

I

is semi-real iff

Let

be a variety over

k.

and

The map

#-~

k

is proper).

be as a Corollary 4.11, and a

~-~

I(~)/i(V(k)),for

a~V(~),

92

establishes a one-one correspondence between of

k-points of

V

and ideals

k[V] which are maximal among ideals real over k.

PROOF. (i)

We need to show: I(~) contains

real over k; (ii)

l(V(k))

and is maximal among ideals of

Every ideal of this type is of the form

PROOF of (i). that

I(a)

m l(V(k))

I(a) is real over k.

then

V~(1) ~ {a},

PROOF of (ii). By 4.8,

i.e.

Let

4.11

M =I(V(k))

k[X I ..... Xn]

and

and

then

because

a~V(k).

Finally, if

V~(1) = ~ ;

I(a) for some Remark

l(a) ~ I with

hence

i~I

a~V(k).

4.7(c) shows I

real over k,

by Corollary

4.12.

M = l(V(k)) be maximal among ideals real over k. 4.12

we get

aEV(k).

Also

V~(M)_# ~

Let

~V~(M).

Since

M c l(a), and maximality implies the

equality.

0

Exercise.

Prove that in a ring of the form

k[V] every ideal which is

maximal among real ideals, is maximal. Historical rehabilitation.

The authorship of the real nullstellensatz is

usually attributed to Dubois proved

[46]

and

Risler [57].

it long before, as well as Proposition 4.10;

C.

However, Krivine see his paper [51].

THE SIMPLE POINT CRITERION. The weak real nullstellensatz

terms the algebraic condition

4.12

bis expresses in geometric

"I(V) is a semi-real ideal".

We are in-

terested in finding a geometric expression for the closely realted notion "I(V) is a real ideal". It turns out that this condition has a very interesting geometric content: it says that ).

V

has a non-singular

When the variety polynomial

V

is a hypersurface

F(X I .... ,Xn) -

(k = the real closure of

~-x.~F (i = i,. .. ,n)

-i.e. it is given by a single

we know from elementary geometry that a point

aeV(k) is called non-sinsular tion

k-point

This is what we will prove below.

(or simple) if at least one of the deriva-

does not vanish at

a.

The correct definitives

i

in the general case is as follows: mials

PI,...,PI~k[XI ..... Xn] ,

assume that

V

is given by polyno-

and consider the Jacobian matrix,

93

/

~PI .....

~P£ \

~X 1

~X 1

~PI .....

~P£

J(PI ..... P£ )=

This is an

nx£

matrix with entries in

I_< s _<min {£,n}, J(PI'''''P£); s

let

we have

culated modulo

r

Let

For a fixed

sxs

r

s,

minors of

denote the largest

is just the rank of

J(PI ..... P£) cal-

I(V).

DEFINITION 4.14. by

M~l ~k[Xl'''''Xn]"

Ms i ~ I(V);

such that some

k[X I ..... Xn].

M si (i = l,...,t i) denote the

P1 ..... P£

A point

a~V(k)

of a variety

is called non-singular

space to the variety at

a

a

over

k

defined

(or simple.) iff the rank of

J(PI ..... P£)(a), the Jacobian matrix at It is easily seen that a point

V

a,

equals r.

is non-singular

has dimension

n-r

D

iff

the tangent

(cf. Shafarevich [14;

pp. 74,77]). Our main result is: THEOREM 4.15. and

I

Let

be an ordered field,

a prime ideal of

has a simple PROOF.

k-point

Let

k[Xl,...,Xn].

iff

the ideal

A = k[Xl,...,Xn]/i and

its real closure,

Then the variety defined by I

K

k

I

is real over k. be the field of fractions of A.

(a) We prove first the implication from right to left, which is a simple consequence of the second transfer principle. By assumption

K

has an order ~ extending the order of k;

the real closure of rate

I.

With

r

.

Assume that the polynomials

denoting the rank of

J(PI .... 'P£)

in

let

L

PI,...,P£

denote gene-

A, as above,

we have: £

X1

K ~ A

Pj(

j=l In particular, (*) holds in

X

/I ....

tr

n/I) = 0 A '

X1

V Mr. (

X

/I''"

i=l i

n/I) # O. "'

the statement

£ tr r @v I ..... @vn ( A P-(v I ..... Vn) = 0 ^ V Mi(v I ..... Vn) # 0), j=l 3 i=l L.

Since the parameters of

morphism) we have

k c L, then

(*)

lie in

(*) also holds in k.

k

and (up to k-isoThis means, precise-

94

ly, that the variety defined by

I

has a simple

k-point.

(b) The proof of the other implication requires some high-powered commutative algebra, we only sketch it. Let ~ V ~ (I) be a simple point and M = I(a-)/I its maximal ideal in A. A (deep) result from c o ~ u t a t i v e algebra tells us that, under the present assumptions, (t)

the following holds:

The localized

AM

of

A

at the ideal

M

is a regular (local)ring.

(For the definition of a regular local ring and the proof of Atiyah-Macdonald

(t), see

[I; Ch. ii].)

Next observe that the evaluation map at

a,

ev~ : A ÷ k, defined by

ev~ (Q/I) = Q(a), is a surjective ring homomorphism with kernel

M; hence

A/M = k , A/M ÷ AM/MA M

On the other hahd, the map

is an isomorphism (easy verification).

AM/MAM

-

A/M

which sends

X/M

into

x /MA m

Hence

_- ~ .

It follows that the residue field of

AM

is orderable.

Now, another basic

result from real commutative algebra tells us: (tt)

If

then

B

B

is a regular local ring whose residue field is orderable,

is real.

For a proof, see

Lam

[54; Prop. 2.7].

From (tt) we conclude that

AM

is real;

this implies that

A

itself

is real, as one can easily verify.

D

The general case, when the variety is not necessarily irreducible, easily derived from the preceeding theorem.

is

We will not prove this, but

in order to state the result we need to know that a proper ideal in a Noetherian ring is contained in only finitely many minimal prime ideals (that is, minimal among the prime ideals containing it); Macdonald

[i; Thm. 7.13

and

Ch. 4].

THEOREM 4.16.

Let

V

its real closure,

and

Jl ..... Jm

k[X 1 ..... X n]

cf. Atiyah-

be a variety over an ordered field ,

containing the ideal

the minimal prime ideals of I(V).

Then, the following are equi-

valent: (i)

The ideal

I(V)

is real over k.

(ii)

I(V) is a radical ideal and each variety

V~(Ji) , i = I ..... m,

gS has a simple point. For details, see D.

0

Lam

[54;

Prop. 2.9 and Thm. 6.10].

HILBERT'S 1 7 ~ h p r o b l e m . In its original formulation this is the problem of knowing

whether a rational function with rational coefficients, such that

R(~) e 0

at every

x c~ n

vanish, is a sum of squares in

~(X I ..... Xn).

A positive answer was given by Artin [19] breaking work of Artin-Schreier [23]

R ~ ( X 1 ..... Xn),

for which the denominator does not

in

[42], [43].

1927, based on the pathIn the 1950's Robinson [22],

gave an alternative (and generalized) proof using the second trans-

fer principle

3.9.

It should be emphasized that the ideas underlying

Robinson's technique apply as well to other situations where the methods of Artin-Schreier are inapplicable; real closed rings orderings

for example, to polynomials over

(see Dickmann [83]), or over fields with higher level

(see Becker - Jacob [81]), or even over p-adically closed

fields (see Prestel-Roquette Robinson's proof

[86]).

"trivializes"

In a certain sense, one may say that

that part of the work of Artin-Schreier

which deals properly with Hilbert's

17 ~h

problem.

Robinson's proof was the first truly mathematical application of quantifier elimination.

The first chapter of Delzell's thesis [20]

extensive historial account of Hilbert's

contains an

17 ~h problem.

Since the subject has been widely treated in the existing literature, we confine ourselves to the essential points. THEOREM 4.17. and

Let

R~k(X I ..... X n)

that Then

R(x-) e 0



for every

x c~ n

R with PROOF.

Pick,

Pi >0

Assume

=

and

R

Thm. IX.2])

extending that of

(*)

L

its real closure, Assume

2 ~ PiRi l Riek(X I ..... Xn).

(cf. Dickmann

The standard

[72; Prop. I, 1.5]

implies at once that

k(X I ..... X n)

k,

Note also that if

for which

P, Q ~ k [ X I ..... Xn], then Let us denote by ment

~

for which its denominator does not vanish.

does not have this form.

criterion of positivity [13;

be an ordered field,

a rational function with coefficients in k.

Q # 0

R<0. and

R< 0

the real closure of

iff

Artin-Schreier or Ribenboim

has an order R = P/Q with

RQ 2 ~0.

.

Iv I ..... @Vn(RQ2(Vl ..... v n) < 0),

The state-

96

of the language

[ , with parameters in

k,

We can assume, without loss of generality, true in ~, McKenna

showing that

[21]

R(x) < 0

is valid in that

for some

L ~ ~.

xE~ n,

L (for v i = Xi)Then (*)

is

a contradiction. N

has done a conclusive study of several questions related

to this result. A useful application of Theorem 4.17

in real commutative algebra is

the following result, first proved by Dubois-Efroymson [47]: THEOREM 4.18. ed field,

k

(Change of sign criterion).

its real closure, and

(b) Each

Qi

(Q)

be an order-

Q' QI ..... Qr E

polynomials such that Q = Q1 .... "Qr Then the following are equivalent: (a) The principal ideal

Let

and the

k[Xl ..... Xn]

!

Qi s

is real over k.

changes sign in

~n

and the

Qi's are pairwise distinct. 0

Theorem 4.17 gives an easy proof of the implication from (a) the argument appears in Dickmann

[72; Thm. II.6.5].

Other proofs can be found in

Lam

[54; Thm. 6.11]

to

(b);

A model-theoretic

proof, due to Prestel, appears in Elman-Lam-Wadsworth

E.

monic

are irreducible.

[48; §4. bis].

and Ribenboim [56].

FURTHER APPLICATIONS. The results proved in the preceeding paragraphs do not exhaust

the list of basic results obtained by direct application of model-theoretic methods.

We metnion here, without proofs, three results original-

ly proved by Lang

[52]

using other methods;

they form the core of the

theory of real algebraic function fields. PROPOSITION 4.19. (The homomorphism theorem). Let be an ordered field, k its real closure, and A an integral domain which is a

k-algebra of finite type.

order extending that of homomorphism f:A ÷ k.

If the field of fractions of

A

has an

k(i.e, is real over k), then there is a k-algebra 0

This result is, in fact, a particular case of the weak real nullstellensatz

4.12 bis, and the reader can prove it as an exercise;

Th~l~ne

[44],

Dickmann [72; Ch.II]

THEOREM 4.20.

Lam [54; §5], for details.

(The embedding theorem).

L an orderable extension of F a real closed extension of L can be K-embedded into F.

K

Let

K

be a real close field,

of (finite) transcendence degree n, and K

of transcendence degree ~ n, over K. Then D

Proof of this theorem can be found in THEOREM 4.21.

or

Lam

[53; §6],

(The homomorphism extension theorem).

be an ordered field,

cf. Colliot-

A a subring of

k,

g:A ÷ L

[54, Thm. 6.15]. Let



a homomorphism of ordered

97

rings with values in a real closed field ordered ring homomorphism

g':B + L',

subring of

A,

k

containing

and

L.

where

Then B

g

extends to an

is a convex valuation

L' a real closed field extending

L.

D

This important result is a consequence of the amalgamation property of real closed fields, a property which follows from Lang

model-completeness.

[52] states the theorem in terms of real places.A corresponding existence

theorem follows as corollary. Dickmann §5.

A proof of Theorem

4.21

is given in

[72; Thm. II 7.10]. THE

STRUCTURE

SEMI-ALGEBRAIC SETS.

OF

The main topic of this section is the separati0n theorem, an important technical tool in real algebraic geometry, mainly used to show that the topological notion of connectedness for

s.a.

sets can be for-

mulated in the first-order language L of real closed fields.

Among its

corollaries one obtains: -

The finiteness of the number of connected components of a s.a. subset of

-

~n

(Theorem 5.8 below).

A refinement of Tarski's theorem elimination"

3.4

known as

"open quantifier

(Theorem 5.9).

DEFINITION 5.1.

Let

k

be an ordered field and

P1 .... 'P£ek[Xl'''''Xn] (a) We call sign condition

(resp.strict sign condition) on

PI'''''P£

any formula of the form: £ /~ Pi (vl ..... Vn) ?i 0 i=l

~(Vl ..... Vn): where each

?i

is one of the signs

(b) Given a sign condition

_>, >, _< ,<

(resp.> ,< ) or

--

o(v I ..... v n) as above, we call enlarged sign

condition associated to o the formula £ A Pi(Vl ..... Vn) ?i 0. o(v I ..... Vn): i=l where ?i is e or ~ if ?i is > or < , respectively, and is in all other cases. A

?.i D

We begin our discussion by proving the easy case of the separation theorem, viz., for one-variable polynomials. 5.2. in

THOM'S LEMMA.

Let

PI .... 'P£

be a finite set of polynomials

~[X] closed under (non-constant) derivation.

condition

o(v)

on

PI ..... P£'

the set

Then for every sign

g8 A o ~ {x~

J ~ ~o[x]}

is either empty or an interval (possibly reduced to one point or infinite). PROOF. £

Induction on

£, the case £ = 1 being trivial.

and suppose, renaming if necessary,

This has the effect of making

that

P1 .... 'P£

Assume true for

P£+I is of maximal degree.

closed under derivation, so

that the induction hypothesis applies and the set £ A' = (-h { x ~ J Pi(x) ?i 0} i=l is

empty or an interval.

The derivative of on

A';

P£+I

P£+I - which is among

PI,...,P£ -

is, then, monotone or constant on

is of constant sign

A', and the lemma

follows easily using the intermediate value property.

D

The foregoing proof clearly applies to any real closed field.

We mention

the following interesting corollaries: COROLLARY sure,

5.3.

Let



be an ordered field,

Pck[X] a one-variable polynomial.

~2' of P in derivatives of

k

its real clo-

Then two distinct roots

~i'

k can be separated by a sign condition on one of the P, i.e. there is i, l~i <deg(P) such that P(i)(el),

P(i)(~ 2) have different signs (one of them can be zero). irreducible over

k,

then

i

can be chosen so that

If

P

is

P(i)(el).P(i)(~2)<0. N

COROLLARY 5.4.

Let



an element algebraic over el<...<er

all the roots of

define an ordering

~i on

0 ~i P(~)

be an ordered field,

k

k

with minimal polynomial

F

in

k(e)

k.

For each

its real closure, F, and

i = l,...,r

let us

by the condition

iff

P(~i ) ~0

(in

k).

Then these orders extend the order of k,

are pairwise distinct, and any

order on

is one of them.

k(~) extending the order of

k

Returning to the separation theorem, the notion appropriate for a generalization of Thom's lemma to polynomials in several variables stems from the following (trivial) observations: -

-

intervals are exactly the connected subsets of The closure of a set of form sign condition, is the set

A = {x~

~;

J ~ ~ o[x]},

^

= {x~m

I ~k

~[x]} A

defined by the enlarged sign condition o associated to o.

where o is a

99

Thus we introduce DEFINITION 5.5.

A finite family PI ..... P£ c ~ [ X I .... ,Xn]

a separatin$ family iff for every sign condition ~ on (i)

is called

PI .... 'P£ :

The set A

= { x c A n I ~ > o[x]}

is either empty or connected. (ii)

If A

# ~ then A-- (=closure of A ) = { x ~ n I ~ >$[x]}. G ' G Thus, Thom's lemma says that any finite family of one-variable polynomials closed under derivation is separating. The announced generalization is this: THEOREM 5.6.

(Separation theorem). Every finite family of polynomials,

PI .... ,P£ c ~ [ X I ..... Xn] , can be completed into a separating family PI .... 'P£' P£+I ..... Pz+s" Furthermore (a)

The number

function of

£, n

s and

of additional polynomials is a primiteve recursive d = maximum of the total degrees of the poly-

nomials PI .... 'P£" (b) There is a primitive recursive function f(n,Z,d) giving an upper bound on the total degrees of the new polynomials Pz+j(j=l,...,s). (c)

The coefficients of the polynomials

mial functions over

~

of the coefficients of

P£+I ..... Pz+s are polynoPI ..... P£"

D

The proof is not easy. A word of caution: adjoining only the partial derivatives of PI ..... P£ family;

counterexample:

does not produce, in general, a separating the polynomial

(X sin ~ - Y cos ~)(X sin ~ - Y cos B) + i, with 0 < ~ < B < ~ (Houdebine[76]). All

known proofs proceed by induction on the number

n

of variables,

using several of the following ingredients: Quantifier elimination; The continuity of the functions of order of

~,

x

which give, in the increasing

the real part of the (complex) roots of a polynomial

equation of the form pn(~ )

yn + ... + po(~ ) = 0,

Po,...,Pn c ~ [ X I .... ,Xn], at values ~ n such that (cf. Gillman-Jerison [85; Thm. 13.3(a)]);

Pn(X) # 0

The following result, known as the "cylindrical decomposition" theorem:

100

THEOREM 5.7. P

Given a s.a. set

U ~

~n

c ~ [ X 1 ..... X n] there is a partition of

A 1 .... ,Am, such that for each

and a polynomial

U

into finitely many s.a. sets

i = 1 ..... m, either one of the following

properties holds: (i) P(x,y) has constant sign (>0, <0

or

=0) for all x~A i and y E~;

or : (2) There are continuous s.a. functions

i ~I'

i , ~ :A.I ÷ ~ • " "

~i

> I) (£i

-

such '

that : (a)

~

(x~ <... < ~ . ( x )

for all

xcAi;

i

(b)

for each

the polynomial

x c A i,

~ ( x ) ..... ~ii(x) are all the real roots of

P(x,Y).

Note that, in particular,

for

xeA i

the polynomial

change sign in any of the intervals i for j = i ..... £i-i , (~£ ( x ) , ~ ) .

(-~

i ~l(X)),

,

P(x,Y) does not i (~j(~), ~ ~ +l(X))

l

Historical remarks (a)

Thom's lemma was, indeed, first noticed by Thom;

it first appeared in ~ojasiewicz

[28; p. 69].

several variables is due to Efroymson [273, incomplete.

Later

Coste/Coste-Roy [69]

gave a correct (and complete)

proof using a technique introduced by Cohen elimination for real closed fields.

The generalization to

where the proof is (at best) [34]

to prove quantifier

The most efficient proof is due

to Houdebine [763• (b)

Cylindrical decomposition is sometimes used, dis-

guised in different ways, as an alternative approach to quantifier elimination; Brumfiel

for example, see Cohen [65;

[34],

Collins

[35],

Coste

[683,

Appendix].

We turn now to some important consequences of the separation theorem. THEOREM 5.8. (Finiteness theorem).

Any s.a. subset

S

of

~ n has

a finite number of connected components, each of which is a s.a. set• PROOF.

Since

S

is a finite union of sets defined by formulas of the

form ~(v I ..... v n) with

:

m £ A Pj(v I ..... Vn) = 0 A A Pk(Vl ..... Vn)>0 j=l k=m+l

PI .... ,P£ c~[Xl,...,Xn],

it suffices to prove the result for sets

of this form. Complete PI ..... P£ to a separating sequence PI ..... P£'P£+I ..... Pz+s" By condition (i) of 5.5, the connected components of S are unions of non-empty sets of the form

101

where

6(v I ..... v n) is a sign condition on P£+I'''''P£+s"

3 s possible

There are

~' s.

D

The next result, known as

"open quantifier elimination" is a particu-

larly important tool for real algebraic geometry, specially for the investigation of the real spectrum

(cf. §§6.C

and 7 below).

Its

significance was pointed out by Brumfiel [65; Unproved Prop. 8.2] even before it was proved, independently, by Coste-Roy

[69]

Recio [78]

The result gives an answer to the following question given an open s.a. subset K ~ RCF); U?;

in 1977, Coste/

in 1979, and Delzell [20] in 1980.

U

of

: assume we are

IRn (or, more generally, of

K n, with

what can we say about the quantifier-free representation of

does there exist, for example, a representation of the form V i

A Pij (Vl ..... Vn) > 0 j

?

Note that irredundant equalities may, a_priori., appear in an specific representantion of an open s.a. set;

for example

:

U = {<x,y> I x 2 + y2 I y = 0 A X 2 < 1/4}.

!Y

..... ~.

The same set,

....X.

U, can also be defined by replacing the equality

by the inequality

y2 < 1/9, for instance;

y = 0

but how can we guarantee that

ad-hoc devices of this type will work in all

cases?.

THEOREM 5.9 .. (Open quantifier e!iminati0n ) . Let U be an open s.a. subset of K n, where K > RCF. Then there are polynomials Pij c K[X I ..... Xn]

such that

102

U = U l P il{~eKn j(x)f~" I i j

>0}.

Let us derive this result from the separation PROOF OF THEOREM 5.9. closed s.a. set

(case

C c

~In iV

By quantifier

K=I~).

theorem in the case K =IR:

We prove the dual statement:

every

has a definition of the form

AP. j lj- (Vl . . .,v.n). >. 0

elimination we know that

C

has a definition of the

form (*)

V (A Pk£(V) = 0 A A Qkm(V) >0). k £ m

By the separation a separating

theorem this family of polynomials

family;

modulo

that the polynomials the subset of

An

Pk£

'

Ak Since

C=

UA k

=

Qkm

form a separating

defined by the corresponding

we may suppose non-empty;

{~e~n

by condition

from the case

to

family.

Let

disjunct of

(ii) of Definition

Ak

be

(*), which 5.5

we have:

I A Pk£(X) = 0 A /~ Qkm(X) _>0}. £ m

and (finite) unions commute with closure, we obtain:

k

which gives the desired representation Coste/Coste-Roy

can be extended

a change of notation we may then assume

[62] K=~

of C.

derived the statement

D for arbitrary

K > RCF

by use of the first transfer principle

(a) and (b) of the separation

: using points

theorem 5.6 they show that the statement

of Theorem 5.9 is first-order. Instead of working

through this transfer argument we outline below a

direct model-theoretic the separation

theorem.

proof due to van den Dries

[79]

This method has the advantage

which bypasses of solving a

number of related problems which cannot be solved by the preceeding technique. Van den Dries remarked

that (the dual of) Theorem 5.9 just says that any

formula defining a closed set tifier-free

equivalent modulo of K).

C ! C n is equivalent

to a positive

L-formula in any real closed field extending the theory

The model-theoretic

derived from the compactness

T = RCF + criterion theorem.

the quantifier-free

quan-

K (i.e., diagram

to be used in this case is easily

103

5.10 L

MODEL-THEORETIC

and

(i)

~(v I .... ,vn)

There is a positive,

that (2)

CRITERION.

and

Let

L-formula.

T

be a theory w i t h language

Then the following are equivalent:

quantifier-free

L-formula

~(Vl,...,Vn)

such

T k ~ <-> 4. Given

phism

9~ , ~

f: ¢

for any

an

~ T,

÷ ~,

L-substructure

¢ c

~

and an L-homomor-

implies

j~ > ~[f(~)]

then

E ~ cn.

D (general case).

PROOF OF THEOREM 5.9. model-theoretic

criterion.

We check condition

(2) of the

Given the situation

F UI A

(*)

with

f =

L

UI

UI

K

K

F, L, K N RCF,

A an ordered subring of

phism of unitary ordered rings

such that

F

and

f ~ K = id,

f

a homomor-

we have to show

that F > ~[~3 where

implies

a ~ A n.

Before proceeding w i t h the proof, additional (i)

assumptions

Replacing,

remark that we can make the following

on the given situation

if necessary,

field, we can assume that

A

the field

is a convex

is just Lang's h o m o m o r p h i s m extension (2)

We can assume that

Otherwise, (3)

replace

By changing

f[K] = f[A] subfield of

. A

A

K, if necessary,

containing

(a)

R is algebraically

(b)

R is real closed.

by a larger real closed

valuation ring of

is a local ring with maximal

K.

F.

This

at

ideal Ker(f) o

Ker(f).

we may further assume that

This is seen as follows.

fiR] = f[A].

A

A

L

(*):

theorem 4.21.

by its localization

and

Since

L > ~[f(~)],

By Zorn,

let

We have to show that

R R

be a maximal is real closed

closed in A (by maximality).

is a convex subring of

F ~ RCF,

the intermediate

value pro-

104

perty holds for polynomials sign between in

A[X],

a

and

P~A[X].

b (a,bER,

Q has a root

ceA,

Now, if

Q~R[X],

Q # 0, changes

a < b), then, viewed as a polynomial a < c < b.

Hence our claim follows from

(a). (c)

fiR]

(d)

fiR] =

Otherwise,

f[A].

let

dental over tible in

is a real closed field.

xeA

be such that

f[R].

A,

f(x)~f[R].

By (c), f(x) is trascen-

It follows at once, using (2), that Q(x) is inver-

whenever

Q~R[X],

Q # 0;

hence

R(x) c A, contradict-

ing the maximality of R. Now we can complete the proof of Theorem 5.9. F p ~[a]. Since

Using (3) choose

f rK

bcK n

aEAn be such that

such that f(b i) = f(a i) for i=l ..... n.

is injective it suffices to show that K > ~[~].

By assumption the first-order statement If

Let

b E(~ #)K, then there is

" ~

is open" holds in K.

EEK, C >0, such that

n

(**)

Vv I ..... v n ( i ~ l ( V i - b i ) 2 < s

holds in K;

by transfer it also holds in F.

....>. ~ ~(v I ..... Vn))

Since A is convex in F, it follows that the maximal ideal Ker(f) of A is convex in A (cf. Cherlin-Dickmann [82; Lemma 4]). As Ker(f) nK = {0} then

we conclude that y < ~ for all y ~Ker(f). Since ai-bieKer(f), n i~I= (a i - bi )2 < e. Condition (**) implies, then, that

F ~ ~ ~[a], contradicting the assumption of the theorem. COMMENTS.

Applying the same model-theoretic principle to algebraically

closed fields, van den Dries

[79]

ness of projective varieties,

gets a simple proof of the complete-

a basic result in classical algebraic

geometry (for the geometric meaning of this result, see Shafarevich [14; Ch. I, §5]).

The author has used it in

[73]

to prove the follow-

ing result, a refinement of Theorem 5.9, which answers a question of BrScker

[59; p. 261]:

PROPOSITION 5.11. closure, and of polynomials

U

Let



be an ordered field,

an open s.a. subset of Pijek[Xl ..... Xn] U = ~ ij

K n.

K its real

Then there is a finite set

such that

{~cKn I P i j ( x ) > 0 } .

D

105

REMARKS.

(a)

This section contains only a few of the most basic

results on the topology of s.a. sets; interested

in pursuing

Prop. 3.5

and (b)

much more is known.

The reader

this line of enquiry may consult Coste

§IV,

§V],

Hardt

[75],

Mather

[68;

[77].

As a part of his research on Tarski's problem

(cf.3.11

(iv)) van den Dries has embarked on the project of showing that the major topological

theorems of real algebraic geometry are consequences

of the structure of the parametrically

definable

has conjectured that similar results hold for

subsets of

<JR, +,-

~

.

He

, <, exp,0 >;

cf. [ 8 7 ] . §6°

THE REAL SPECTRUM. A.

INTRODUCTION. One of the milestones

introduction

of the prime spectrum of a commutative

When this construction

is applied to the coordinate

V over an algebraically the set subset-

of classical algebraic

V(k) of

closed field

k-points

in the disguised

obtain a manageable

of

V

An obstacle

and generalization [14],

Hartshorne

to the development

information

and M.-F.

Thus, we

[3]

about its geometry.

theory of schemes,

of classical or

algebraic

Mumford

of algebraic

real numbers was the lack, until recently, of the prime spectrum.

ideals.

space not too far removed from the ori-

point of view leads to Grothendieck's Shafarevich

ring of a variety Spec (k[Vj) contains

form of the set of maximal

topological

ring by O. Zariski.

with its Zariski topology as a dense

ginal variety which gives valuable reformulation

k, the space

geometry was the

This

a far-reaching geometry.

Cf.

[i0].

geometry over the field of of an appropriate

analogue

The discovery of such an instrument by M. Coste

Coste-Roy meant,

therefore,

a fundamental

step forward in the

study of real varieties. To be sure, notions for quite some time; -

for example:

the space of orderings Prestel

-

akin to the real spectrum of a ring have been known

[55]

of a field with the Harrison

none of these ancestors

tric information

see

and Lam [53];

the subspace of real prime ideals of Spec(A);

However,

topology;

see Dubois

suitably reflected

[45].

the wealth of geome-

contained in the rings arising in real algebraic

geo-

metry. The crucial question

in setting up a theory of the real spectrum is to

decide which objects

are to be its elements.

Through a sequence of

106

attempts initially motivated by topos-theoretic considerations, Coste/ Coste-Roy

[69], [61], [62] realized that the appropriate choice of

objects was a hybrid made of pairs of the form (*)

<prime ideals

P

of

A;

total orderings on the quotient ring

A/p> In order to give a motivation for this choice, recall that the members of the prime spectrum, i.e. the prime ideals of a given ring

A, descri-

be its homomorphisms onto integral domains (exactly the subrings of algebraically closed fields) modulo the natural equivalence relation corresponding to cormmutative diagrams

j

R1

R2 Substructures of real closed fields (in the natural language L

for

RCF) are ordered domains, and a moment's reflection shows that the pair (*) classify the corresponding class of epimorphisms. Let us now show how the presentation of the objects (*) can be simplified in order to facilitate formal work with them. an axiomatic characterization of the set

of a (proper) prime ideal

DEFINITION 6.1. precone (i)

Let

A

is a subset ~ of x,yc~

(ii)

x ~

(iii)

-i~;

(iv)

xy~

for all

iff

of

{a~A I a/p e0} A

for each pair

and an order ~ on

be a commutative ring with unit. A

implies

2

P

This is done by giving

A/p. A prime

with the following properties: x+y~; x~A;

(xe~ A yc~) V ( - X ~ ^ -yc~).

Conversely, each prime precone ~ of A determines a pair

0

as

described above, by setting:

0 ~ a/p Notation.

iff

aE~

Given a prime precone ~ of A

real closure of the fraction field of by ~,

for

and by

~

aEA.

we shall denote by

k(~) the

A/~n_~ with the order determined

: A ~ k(~) the corresponding canonical map.

0

107

We shall now worry about the topology of the real spectrum, The topology of the prime spectrum, {D(a)

I a~A},

Spec

SpecR(A).

(A), is generated by the base

where D(a) = {P e Spec(A)

is the set of all It is natural,

P

such that

then,

a/p # 0.

to give SpecR(A)

H(a) of prime precones strictly positive

I a~P}

~ attributing

sign.

the topology generated by the sets a definite

sign to

n (a), say

Since the family of such sets is not closed

under intersection we are obliged to consider

finite sequences

of a's.

We are now ready for the formal definition of the real spectrum: DEFINITION spectrum of

6.2.

Let

A

be a commutative

A, SpecR(A) ,

ring with unit.

is the set of all prime precones

The real of

A

with

the topology generated by the sets H(a I ..... a n ) = { ~ S p e c R ( A ) for all finite sequences

I -a I ~ ~^...A

-an~}

a I ..... anEA.

Before studying the basic properties

of the real spectrum let us compute

a few examples. EXAMPLE 6.3. and

SpecR(k)

If

A = k

is a field,

EXAMPLE 6.4. the real line

The real spectrum of

in computing

irreducible say

sign.

are

r c~

Now,

{0}

~

D

ring of

4.6) are relevant

or of the form

(Q) with

,( X-r)

each ordering of

{0}

(smaller)

has to be

~r c S p e c R ( ~ [ X ] ) .

The

I F(r) ~0}.

gives rise to one point of the real spectrum for

~[X]. X

Q

is

is

We know (cf. Brumfiel

that the orderings

that the element

then

(Q)

~ ~, which has only one order.

gives rise to one point

prime precone

Thr prime ideal

is real closed,

~[X] l

~r = {F ~ [ X ]

X larger

the coordinate

(cf. Definition

~[X]

Since

Q = X-r.

corresponding

-

~[X],

{0}

above.

Q, the changing sign criterion 4.18 tells us that

real iff changes

[72; §I.5])

k, mentioned

the real spectrum.

Since the prime ideals of

Thus, each

of

~.

Note that only real prime ideals

linear,

the only prime ideal is

is the space of orderings

defines

in

of ~;

~[X]

[65; §7.5]

or Dickmann

are determined by the cut

these are as follows:

than every element of

~,

which gives rise to the

108

following

prime precones:

~+~=

{FE~EX]

I There is

a c~

such that

Fe0

on

(a,=)},

~. ~ =

{F ~ [ X 3

I There is

b c~

such that

F e0

on

(- ~ ,b)}.

For each

r c~,

r < X < (r, ~ ) precones

X is infinitesimally

((- ~ ,r) < X < r,

larger

(smaller)

respectively).

than r, i.e.

The corresponding

are:

~r+ = { F ~ [ X ]

I There

is

E >0

such that

Fe0

on

(r,r +~)},

er- = { F ~ [ X 3

I There is

E >0

such that

Fe0

on

(r-g,r)}.

Observe ~r + ~ ~r

prime

that the only inclusion and

~r- ! ~r

for

relations

r ~.

between

these precones

are

Thus, we can draw the picture

of

SpecR( ~[X]):

r

r-

As a useful ~[X].

comparison,

Here the points

ing to each polynomial nomial {0}.

X 2 + aX + b,

r+

s

s-

s+

let us draw a picture are the prime ideals X-r,

with

for a, b ~ ,

The latter is contained

tion holds otherwise.

OR

r ~;

4b < 0;

in all the others,

The picture

/'

~ (o)

one correspondto each poly-

and the prime but no inclusion

is this:

(o,b) ..," • •

/ t/

~[X]:

one corresponding

a2

• • .// /////• . #~i•. I///°

-"

of the prime spectrum of of

##

ideal rela-

109

Note that in both the prime and the real spectrum, the set theoretic inclusion

~ c B

means that B is in the closure of the singleton

(in geometric jargon of B );

{~}

: B is an specialization of ~ , and ~ a generizaltion

this is easily checked from the definition of the respective

topologies.

In particular,

is dense in Spec ( ~ [ X ] ) . point of the variety

~

{~}

= Spec ( ~[X]),i.eo

the point

{0}

Geometers say that the ideal (0) is a generic (apparently this notion goes back to the Italian

school of 19th century geometers). The real spectrum of

~[X] does not have a generic point.

r , r , which together with ed as a sort of

r, form the closure of

"generic points" of the half-lines

{r}

But the points are interpret-

(r, + ~ ) and (- ~ ,r)

respectively. Exercise.

Compute

EXAMPLE 6.5.

SpecR(~[X](x_r)) , for

r cA.

In order to give an inkling on how the real spectrum

reflects geometric properties of algebraic varieties,

let us look at an

example having more "singularities" than the preceeding one. der the plane curve C of equation y2 _ X 3 _ X 2.

We consi-

~u

==L

As before, each point since ~[C]/Mp ~ and maximal ideal at P).

P ~ C(~) gives rise .to just one point ~pESpecR(l~fc]), R

Since the polynomial defining of

R[C];

furthermore,

has only one order

C

is irreducible,

(~

denotes the

{0}

is a prime ideal

it is easily checked to be a real ideal (exercise).

The orderings of ~ [ C ] are computed as follows. The ring ~[C] is the quadratic extension of ~EX] in which the equation y2 = X 3 + X 2 holds; in particular,

X 3 + X 2 e0,

i.e.

Xe-l.

Hence, each ordering of

~[X] satisfying this restriction gives rise to two orderings by choosing the sign of

Y.

preco es are thus generated:

of

~[C]

In geometric terms, the following prime

110

(A) If

First fix a point

P = (a,b) ~ C ( ~ ) ,

P # (-i,0), P # (0,0).

and there is e <0

such that

b > 0, we have

~p+ = {F/I(C)

I Fc~[X,Y]

F(x,~x3 + x2)e0 on

~p- = {F/I(C)

I F c~[X,Y]

and there is E > 0

(a,a+e)}

such that F(x,Jx 3 + x 2) e 0 on

(a-s,a) }.

In order to check that these sets are in fact prime precones of

~[C],

only the implication from left to right in Definition 6.1 (iv) requires verification, to see that if

as the other conditions are trivial.

For this it suffices

F 4 I(C), then the function F(x, S x 3 + x 2) has finitely

many zeros. Since the elements of ~[C] are polynomials of the form YP(X) + Q(X), our contention is clear as the equation (X3+X 2) P(X)2-Q(X) 2 =0

has to be satisfied.

(Alternatively, Bezout's theorem (cf. Walker

[15 ;p. 59 ff. ) could have been used to check this point). The prime precones c o r r e s p o n d i ~ to the case replacing -~x 3 + x 2 for ~ x 3 + x 2 above. (B)

If

b <0

are obtained by

P = (-i,0), we have

~p+ = {F/I(C) I There is ~ > 0 and similarly for

such that

F(x,/x3 + x 2) e 0

on (-I,-i+~)

- ~ x 3 + x 2.

Thus, at each non-singular point to that of Example 6.4:

PcC(~)

we have a situation similar

P

p-

p+

The points P-, P+ are interpreted as the "generic" points of the halfbranches determined by

P

on the curve

C(~);

see Figure 7.

(C) Next we have two "points at infinity" corresponding to the upper and lower infinite half-branches: ~,u

= {F/I(C)

IThere is

a>0

such that

F ( x , J x 3 + x 2) ~ 0

on ( a , ~ ) } ,

,

111

and similarly

(D)

~ ,£

_~x3

corresponding to

+x 2

P = (0,0) we have:

Finally, at the singular point

ep+ = {F/I(C) r

I There is

~ >0

such that

F(x, ~ x 3 + x 2) e0 on (0,e)},

+ = {F/I(C) P£

I There is

s >0

such that

F(x, ~ x 3 + x 2) ->0 on (-s,0)},

and two more points,

~p_, r

~P£

given by

- ~/x3 + x 2

instead of

~x 3 + x 2 . As an exercise the reader may check that there is no inclusion relation between any two of these sets. Obviously these points specialize on ~p, and they correspond to the four half-branches the origin (see Figure 7).

of the curve

C(~)

through

In order to complete the analysis of this example we would have to show that (i) There are no real prime ideals in maximal ideals

Mp

at each point

~[C] other than

{0}

and the

P ~ C(~);

(ii) for each of these prime ideals there are no prime precones other than those explicitly constructed above. A simple proof of point (i) goes as follows. prime ideal of

~[C] and let

G I .... ,G£

Assume

J # {0} is a real

be its generators, where

Gi(X,Y) = YPi(X) + Qi(X). By Corollary 4.12, V ~ ( J ) # ~. As in (A) above, the points (x,y) c V ~ ( J ) satisfy the equations (x 3 + x 2) Pi(x) 2 - Qi(x) 2 = 0 (i = I ..... £), and hence V ~ ( J ) is finite, say V R ( J ) = {PI .... 'Pk }" By the real nullstellensatz 4.8 J consists of all polynomials vanishing at PI .... 'Pk; but this ideal is never prime if k e2. Hence V ~ ( J ) consists of one point, P, and clearly J = Mp. above.

Finally, point (ii) is clear by the argument preceeding

(A) D

REMARKS. (a) The correspondence between points of the real spectrum and "oriented half-branches" illustrated by the preceeding examples is a general fact, true of any variety (of any dimension) over any real closed field. The proof of this requires a sophisticated algebraic machinery based on the analysis of valuations; cf. Coste/Coste-Roy [62; §7, §8]. (b)

In both the preceeding examples the length of speciali-

zation chains of the real spectrum is at most 2. This is a manifestation of the fact that the (local) dimension at each point of the curves

112

under consideration B.

se in

§8

below.

D

ELEMENTARY PROPERTIES.

PROPOSITION (i)

is i, as we will

6.6.

Let

The basic open sets

quasi-compact,

A

be a conmTutative ring with unit.

H(a I ..... an),

i.e. compact

in the usual

a I ..... aneA, of SpecR(A)

are

sense but not n e c e s s a r i l y

Hausdorff. In particular: (2)

SpecR(A)

is quasi-compact.

(3)

The irreducible

closed subsets of SpecR(A)

are the closure of a

unique point. In particular: (4)

SpecR(A)

(5)

Let

is a

To-space.

~, B,y ~ SpecR(A).

REMARK.

B, Y c {~}, then

A closed set is called irreducible

two closed proper subsets. PROOF.

If

(I)

order theory

if it is not the union of

set of sentences with parameters

in

A

theorem

of the firs-

(in the language of unitary rings plus an additional unary

P) whose axioms are:

The axioms

for commutative

-

The axioms

for

P

rings w i t h unit.

defining a prime precone

We leave the proof as an exercise

(cf. Definition

6.1).

for the reader.

First check that the set

= is a prime precone. show that

{acA

that SpecR(A)

or c-b

B ~ {~}

iff

~ c ~

to

is in

B

and e, and

~ , it is easiest to check

implies

~ % {B}

or

Y j B , get

beB-y

and

~ c B , y,

B % {e} cey-B.

Since one of b-c

we obtain either b=(b-c)+c ey or

a contradiction.

REMARK.

correspondence

of such point

is To:

B i y

c = (c-b)+b~B, 6.7.

= 0}

Then, use the equivalence

# If

I H(-a)nF

F = {~}.

In order to show the uniqueness

(5)

y!B.

The closure of a point is clearly irreducible.

-

(2)

or

This is easily proved by applying the compactness

to an appropriate predicate

B ! Y

(Functorial properties

which assigns

D of the real spectrum).

The

to each commutative unitary ring its real

113

spectrum is a contravariant functor from the category of such rings with homomorphisms into the category of topological spaces (moreover, of spectral spaces, see Definition phisms.

6.9) with continuous functions as mor-

This simply means that to each momomorphism of unitary rings

f:A ÷ B it is

canonically associated a continuous map SpecRf : Spec R (B) ÷ SpecR(A)

defined by (SpecR f)(B) = f-l[~] It is clear that for

for B c SpecR(B), we have:

a I ..... an E A

(SpecR f)-i [H(a I ..... an)]

= H(f(al) ..... f(an)),

which shows that Spec R f is continuous and, moreover,

that the inverse

image of a compact open subset of SpecR(A) is compact open.

D

In this connection note the following: FACT 6.8.

With notation as above, let

B E SpecR(B) and a = (SpeaRO(B). ~:k(a) ÷ k(B) making the

Then there is a (unique) ring monomorphism following diagram commute: f A

, B

k(a)

, k( B )

The map ~ is elementary. The easy proof is left as an exercise. It is useful to recast the content of Proposition 6.6 in the following language: DEFINITION 6,9.

A topological space

X

is called an spectral space

iff (i)

X is quasi-compact.

(ii) X has a base of open quasi-compact sets closed under intersection. (iii) Every irreducible closed subset of unique point.

X

is the closure of a

The Stone duality between Boolean algebras and Boolean (= compact, Hausdorff,

totally disconnected)

spaces can be extended to a duality

between the category of distributive lattices with homomorphisms and

H

114

the category of spectral spaces with continuous maps such that the inverse image of a compact open set is compact,

To each spectral space it

is associated the lattice of its compact open subsets.

Conversely, to

each distributive lattice it is associated the space of its prime filters with the spectral topology (defined exactly as for the spectrum of a ring). The fundamental result about this class of space is: THEOREM 6.10.

(Hochster).

A spectral space is homeomorphic to the

(prime) spectrum of a ring.

D

For a proof see Hochster [63]

or

Laffon [7].

In particular, the real spectrum of a ring A is homeomorphic to the prime spectrum of another ring. In the case where A is the coordinate ring of a variety over a real closed field, the ring

B

can be computed

explicitly, as we will see later (Corollary 9.10). C.

CONSTRUCTIBLE SETS.

DEFINITION 6.11, A subset of SpecR(A) is called constructible if it is a Boolean combination of basic open sets.

D

Quantifier elimination shows at once that the constructible sets coincide with the definable sets in the following sense: PROPOSITION 6.12. is an

A set

C ! SpecR(A) is constructible

L-sentence with parameters in A,

iff

there

~C = ~C(al .... ,an) , such that

C = { ~ S p e c R ( A ) I k(~) > ~C[~ (al) ..... ~ (an) I}.

D

The (easy) proof is left as an exercise for the reader. It is clear that the constructible sets form a basis for a topology on SpecR(A), called the c onstructible topology.

This topology is obviously

finer than the spectral topology and is compact Hansdorff. The main property of construetible sets is: 6.13.

THE REAL CHEVALLEY THEOREM.

a finitely presented

Let

C

where

Then

be a constructible subset of

C = {BcSpecR(B)

A

be a ring, B=A[X I ..... Xn] ~

A-algebra (i.e. the ideal

and f:A ÷ B the canonical morphism. tible sets into constructible sets. PROOF.

Let

I

is finitely generated),

SpecRf transforms construc-

SpecR(B) given by

I k(~) ~ ~C[z~(Ql/l) ..... ~ ( Q m / l ) ] } ,

Q1 ..... Qm c A [ X 1 ..... Xn].

Let I be generated by

P1 ..... P£E

115

A[X I

. . . . .

Xn].

Then we have the equality:

(SpecRf) [C] = {~cSpecR(A)

^ where for

I k(a) > ~Yl ..... ~Yn

£ ~I(~Pi)(Yl

..... Yn )=0 ^

~ C ( ~ Q I ( Y l ..... yn ) ..... ~ Q m ( Y l ..... yn))]},

F ~ A[X 1 .... ,Xn], N

F denotes the polynomial whose coefficients

are the images of the coefficients of F. The inclusion if

!

follows easily from Fact

6.8.

For the other inclusion,

Yl .... 'Yn ~ k(~) satisfy the given formula, then the correspondence

a

I

~

Xi/ll

>

~ (a)

for

Yi

i = i ..... n,

extends to

a ring homomorphism

precone of

B, B = g-I [k(~)+]

such that

~ = (SpecRf)(B)

a ~A,

=

g:B + k(~).

This morphism gives a prime

{F/I I k(~) ~ (~ F)(y I ..... yn ) ~0},

and

BcC.

D

The study of topological properties of the map SpecRf is of central interest in real algebraic geometry. This study is frequently based on an elegant combination of logical and geometrico-topological techniques. For example, if one needs to show that a certain constructible

set

C ! SpecR(A), given by an [-formula ~C(al .... ,an) , is open, logic helps by reducing the problem to showing that the set (,)

{~n

I ~ > ~C [~]}

is open in A n (of course, ~ can be replaced by any other real closed field). The analytic techniques available in the reals often are of help in proofs of this kind. Indeed, if the set (*) is open, the open quantifier elimination theorem 5.9 implies that the formula ~c(vl ..... v n) (without parameters) equivalent in the theory RCF to one of the form ni V A Pij(Vl ..... v n) > 0 i j=l with

Pij C E [X I ..... Xn].

It follows that

is

116

C = ~ IH ( P i. l ( a l and, hence,

that

C

As an illustration result

where

PROOF.

Obviously

SpecR(K) ,

F

CASE i.

K

F = K(a),

f ~ K[X]

Let

extension

K,F

be orderable

of K, and let i:K ÷ F

is an open map.

to show that

(SpecRi)[0]

is a basic open set of

for

is open in

SpecR(F).

Note that

~SpecR(F). where

and a is algebraic

to prove

x I .... ,xn are a transcendence

over

K(x I .... ,Xn).

the theorem in the cases a algebraic

be the minimal

of SpecR(F)

generated

F = K(Xl,...,Xn,a) ,

over

it suffices

0

theorem).

Then SpecRi

it suffices

= ~nK

We may assume base of

map.

whenever

(SpecRi)(~)

Let

is a finitely

the inclusion

the following

[48].

(The open mapping

F

(al ..... an))

is open~

due to Elman-Lam-Wadsworth

fields,

l

of the use of this technique we prove

THEOREM 6.14. denote

..... a n ) ..... Pin

F = K(a)

Obviously

and

F = K(X).

over K.

polynomial

of

a.

A non-empty

basic

subset

is of the form: £

H(PI(a) ..... P£(a)) where

Pi c K[X]

and

= {~SpecR(F)

Pi(a)

# 0.

I >

Then

A Pi(a) > 0} i=l

f f Pi and we may assume

that

deg (Pi) < deg(f). Let X = (SpecRi) are equivalent:

[H(PI(a) ..... P£(a))].

(i)

BeX

(ii)

B extends

Since

K(a)

conditions

F

such that

the sign-changing

criterion

are equivalent

for any order

B extends

to an order of F;

f changes

sign in

By definition have (iii)

the following

f changes

~

A Pi(a) >0. i=l 4.18 tells us that B of

K:

.

= k(B),

! k(B)-

This condition

B~SpecR(K)

£ to an order ~ of

~ K[X]~f),

the following

For

Hence

and if ~ is an order of

(ii) is equivalent

sign in

is first-order.

k(B) Let:

and

k(B)

F

extending

to : £ ~ A Pi(a) >0. i=l

B we

117

~(x,y,a O ..... an_ I ):

x
A f(x) f(y) <0,

£ ~(a,b o ..... bin_I): where

a ° ..... an_ I

of PI .... 'P£"

are the coefficients of

By continuity,

~(x,y,v o ..... Vn_ I)

~(v o

CASE 2.

and

b o ..... bm_ I those

formulas

define open subsets of

IRn+2 and

Hence, the formula

..... Vn_ l)

defines an open subset of the constructible set {BeSpecR(K)

f,

the parameter-free

P(z,w O ..... Wm_ I)

I~n+l, respectively.

is open.

A Pi(a) >0, i=l

~xy ~(x,y,v o ..... Vn_ I)

:

An.

I k(8) ~

By the remark preceeding the theorem,

~(a o ..... an_ 1 ) ^ P(a,b o ..... bin_l)}

By the equivalence shown above this set equals

X.

F = K(X).

By trivial manipulations be written in the form: H(P I ..... P£)

(cf. 4.17) a basic open subset of SpecR(F) can

=

{~SpecR(F)

I ~

l A Pi >0} i=l

for non-zero polynomials Pi eK[X]" Putting X = (SpecRi)[H(P I ..... P£)] we have, as before, the equivalence between : (i') (ii')

BoX, B extends to an order ~ of

where to :

BeSpecR(K ) .

(iii')

F

such that

~

£ A Pi >0' i=l

Below we prove that these conditions are equivalent

> @x

£ A Pi(x) > 0. i=l

With this equivalence established,

the proof is completed as above for

(iii') defines an open condition, hence an open constructible

subset of

SpecR(K). Let

L = .

Conversely,

Clearly (ii') implies

(iii') implies that all the

(iii'), as

L ~.

!

Pi s are positive on an interval

(a,a+c) of L. Hence the set ~a+ defined in Example 6.4 (with L replacing ~) defines an order of L(X) extending B and making the Pi's positive. This order induces an order on F with the properties required in (ii'). D

118

REMARK. The map SpecRi is also closed, since the real spectrum of an orderable field is Hausdorff (exercise). As a matter of fact, something much more general is proved by Coste/Coste-Roy techniques used above:

[62;Thm. 6.2] with the

THEOREM 6.15. (The closed mapping theorem). Let A, B be rings and f:A ÷ B a homomorphism such that B is integral over f[A]. Then SpecRf: SpecR(B) ÷ SpecR(A)

is a closed map.

0

For still another application of the same technique, §7.

see Roy [32;

§2].

AFFINE VARIETIES OVER REAL CLOSED FIELDS. Now we shall study the interplay between the geometry of affine

varieties over real closed fields -in particular, over ~and the topology of the real spectra of their coordinate rings. Throughout this section varieties are equipped with the euclidean topology derived from the order topology in the base field, and spectra are equipped with their spectral topology (cf. Definition 6.2). Observe that for any ordered base

field

and any variety

V

over

K, there is an obvious embedding : V(K)

~ SpecR(K[V])

given by ' " ~x = {Q/I I QeK[X 1 ..... Xn] (we write 7.1

I FACT.

instead of

and

Q(x) e 0}

I(V(K))).

The map ~ is injective and continuous.

PROOF. Injectivity follows easily by considering linear polynomials. Continuity follows from the equality ~-I[H(QI/I ..... QM/I)]

m n IC'~IQ~. I [ . = (0,~) ],

= V(K)

which is checked without problem.

0

m Since the family of sets of the form

V(K) n ~--i Q$1[(0,~)]

for m c ~ and QI ..... Qm ~K[X I ..... x n] clearly is a basis for the topology of V(K), 7.1 says, furthermore, that the image of V(K) is a subspace of SpecR(K[V]); therefore, we may (and will) identify V(K) with its image by ~. Henceforth we also assume that K is real closed. THEOREM 7.2. With the convention above, restriction to S~-~SnV(K), defines a bijeetive map between: (i)

Constructible

subsets of SpecR(K[V])

V(K),

and s.a. subsets of V(K).

119

(ii)

Open constructible sets of V(K).

PROOF.

(i)

Assume

C

subsets of

SpecR(K[V])

is eonstructible,defined

~c(QI/I ..... Qm/l), with

and open s.a. sub-

by the formula

QI ..... Qm ~ K [ X I ..... x n] (see Proposition 6.12).

Then the equality (*)

C nV(K) = {x ~V(K)

shows that of

C

C n V(K)

is s.a.

I K ~ ~c[QI(~) ..... Qm(X)]},

By induction on the (Boolean) structure

one gets reduced to showing

C = H(QI/I ..... Qm/l).

(*) when

C

is basic open,

In this case one may take

and then (*) is just the equality appearing in the proof of It is clear that the map as values.

tructible subsets of SpecR(K[V]) ~c,(FI/ .... ,Fr/I)

>0,

7.1.

C~-~C n V(K) takes on all s.a. subsets of V(K)

In order to see that it is injective,

and

m

~c(vl ..... Vm): i~iv _

assume that

C

defined by formulas

respectively,

and that

and

C' are cons-

~c(QI/I ..... Qm/I)

C nV(K) = C' nV(K).

By

(*) this equality translates as: l K ~ V x[ ~IPj(~) = 0

(**)

j=

+ (~c(QI(~) ..... Qm(X-)) <--> ~c,(FI(~) .....Fr~))],

where PI .... 'P£ are polynomials generating the ideal I of V. This is a formula with parameters in K (the coefficients of the polynomials). On the other hand, for every ~SpecR(K[V]), k(~) is a real closed field containing K, and hence (**) holds in k(=). Specializing (**) to i = i ..... n, we have Pj(~ (XI/I) ..... ~ (Xn/I) = x i = ~ (Xi/l), = ~ (PJ/I) = 0, and hence: k(~) ~ ~C(~ (QI/I) ..... ~ (Qm/l)) In view of Proposition ~cC i.e.,

<

~ ~C,(~ (FI/I) ..... ~ (Fr/l)).

6.12, this means: iff

~C'

for

~ESpecR(KEV]),

C = C'.

(ii) Since the identification map ~ is continuous, it is clear that C n V(K) is open, whenever C is. The fact that every open s.a. subset of V(K) is of the form C n V(K) for some open constructible set C is an immediate consequence of open quantifier elimination (Theorem 5.9). COROLLARY 7.3.

D V(K)

is dense in

SpecR(K[V]).

120

Notation.

Given a s.a. subset

S

of

constructible subset of SpecR(K[V]) COROLLARY 7.4.

The map

SI

, S

SpecR(~[V])

S

the unique

S = S nV(K).

induces a one-one correspondence

between the connected components of In particular,

V(K), we denote by

such that

V(~)

and those of S p e c R ( ~ [ V ] ) .

has a finite number of connected components

and is locally connected. PROOF. 5.8).

Let

U I ..... U r

be the connected components of

We want to show that

SpecR(~[V]).

By Theorem

tion of SpecR(~[V]). Assume

Since the

7.2

C,C'

open (in

Ui ), non-empty and disjoint.

C,C' are clopen (in SpecR( ~[V])); hence

C,C' are finite unions of basic open sets, and

hence constructible.

By Theorem 7.2 again,

U i = (C n V ( ~ ) ) is a partition of

(Theorem

these sets are clopen and form a parti-

Uj's are open, then Therefore

V(~)

are the connected components of

We only need to show that they are connected.

Ui ~ C U C', with

compact.

UI ..... Ur

Ui

u (C' n V ( ~ ) )

in non-empty open subsets, a contradiction.

The proof of local connectedness is left as an exercise. Exercise.

(a)

Prove the statement of Corollary 7.4 with a s.a. set N

S ~ V(~),

and (b)

S

replacing

The operation

V(~), SI

and SpecR(~[V]),

,S

respectively.

commutes

(i)

with the finite Boolean operations;

(ii) (iii)

with closure and interior; with images and inverse images by morphisms of algebraic varieties over

K.

(Cf. Hartshorne

[3; Ch.l]

for the notion of morphism

of algebraic variety). (c)

If

U = V(~)

open subset W of S p e c R ( ~ [ V ] )

is open

s.a., then

U

W nV(~)

= U.

such that

is the largest

Corollary 7.4 depends essentially on the fact that

V(~)

has finite-

lymanyc~-mected components, a property that only the real numbers enjoy amongst real closed fields. In fact: Exercise.

Prove that if

K

is a real closed field ~

connected component of one point of

Kn

~, then the

is the singleton of that

point.

D

However, using transfer on an appropriate for a fixed constructible set ted" -,

Coste/Coste-Roy

C

L-formula - namely one that

expresses the property

[62; Thm. 5.5]

show:

"C

is connec-

121

PROPOSITION over

K.

7.5.

Let

K

be a real closed field and

V

a variety

Then SpecR(K[V] ) has a finite number of connected components

which are constructible

sets.

The same is true of any constructible

set of SpecR(K[V]).

D

What kind of partition on V(K)?.

sub-

do the connected components

of

SpecR(K[V])

induce

It turns out that the members of this partition are precisely

the components

for the following notion,

S is called s.a.-connected into two disjoint,

defined for s.a. sets

S ! V(K):

if it cannot be split

non-empty,

s.a. open sets.

This notion, which clearly coincides with the standard notion of connectedness in the case

K =

~, has a deep geometrical meaning.

see this, let us consider Example

2.5(b)

the restriction

to the field

Q

this is still a two-component ly, the topology of

~

variety,

this property

Clear-

(see Exercise above),

does.

that the notion of path-c0nnectedness

coincides

The first of these notions

ed to any real closed base field (this was done by ably,

cubic of

Manifestly,

although it has many "holes".

while the notion of s.a.-connectedness It is well-known

of the two-component

of real algebraic numbers.

cannot reflect

with that of connectedness.

In order to

in

]Rn

can be generaliz-

Dells [70]). Remark-

this generalized notion turns out to be equivalent

to that of s.a.-

connectedness. Delfs and Knebusch

[71]

have introduced a theory of "restricted

topolo-

gical spaces" intended to provide a frame in which the "semi-algebraic" versions of some topological

notions

(e.g. that of s.a.-connectedness)

may be cast in much the same terms in which the corresponding topological notions are formulated topology. §8.

See also Brocker

in the frame of general,

standard

point-set

[59; § I].

DIMENSION. We give in this section a brief summary, without proofs,

the theory of dimension

for affine varieties

K, and for s.a. subsets of Roy

[62;

§8].

K n.

of

over a real closed base field

This theory was developed by Coste/Coste-

The algebraic notions of dimension used in classical

geometry were briefly reviewed in

§2.

For most of the present section we will assume that the polynomials P1 ..... P£ E K[X 1 .... ,Xn] I(V)

over

K;

determining

our variety

we will say that the variety

trical point of view this is no restriction place

P1 .... 'P£

by a (finite)

which,

by Corollary 4.11,

V

at all:

set of generators

is real;

V

generate a real ideal

is real.

From a geome-

it suffices

of the ideal

to reI (V(K))

the latter obviously generate

the

122

same set of

K-points.

Moreover,

the rings

K[V] and

The notion of prime precone gives,

a priori,

K[X I ..... Xn]/I(V )

have the same real spectrum. A.

GLOBAL DIMENSION. a new way of measu-

ring dimensions: DEFINITION 8.1. Let

A

be a commutative

affine variety over a real closed field (a)

The real dimension of

ring with unit,

and

V

an

K.

A, dimRA , is the supremum of the integers

n

such that there is a strict chain Co

of prime precones length). (b)

of

~ ~i

~

The real dimension of

~n- ~ =

" ~

A(dimRA = ~

Bo- B

if there are such chains of unbounded

K[V].

and ~ ! B

D imply

strict chain of prime precones of prime ideals of

A.

Hence,

these two quantities PROPOSITION

A

~ = B

(e,BESpecR(A)),

every

induces a strict chain of real

dimRA ~ dim A.

However,

for real varieties

are equal:

8.2.

Let

a real closed field of

n

V, dimR(V) , is defined to be the real dimen-

sion of the coordinate ring Since

"

K.

V

be a real irreducible

Them

dim R K[V] equals

affine variety over

the transcendence

degree

K(V) over K.

D

(Cf. Theorem 2.7). In addition it follows COROLLARY quantities

8.3.

that:

For

K

and

V

as in Proposition

8.2, the following

are equal to dimR(V):

(a) The combinatorial

dimension of

V(K).

(b) The supremum of the length of strict chains of real prime ideals in K[V].

D

The combinatorial

dimension of

V(K) is the supremum of the lengths of

strict chains of closed irreducible topology

(cf. Hartshorne

chains of irreducible

[3;Ch.

subvarieties

subsets of

V(K) with the Zariski

i]); or, in other words, of

V(K).

Proposition

proving the equality of this quantity with dimR(V). quantities

(a) and (b) follows

of strict 8.2 is used in

The equality between

from the Real Nullstellensatz

4.8: the

map I~-~VK(I) is a bijective correspondence between real prime ideals of K[V] and irreducible subvarieties of V(K).

123

Note that the preceeding results are false for non-real varieties: if is given by the polynomial dimR(V) = 0 is

X 2 + y2, so that

V(~)

since the only prime precone of

~ = { F/I I F(0,0) ~0}.

However

A =

V

= {<0,0>}, then ~[X,Y]/I , I = (X 2 + y2)

A contains the chain (o) ~ (X/I,Y/I)

of real prime ideals. The results above show that, as far as measuring $loba.l dimensions is concerned, the use of prime precones yields the same results as the tools of classical commratative algebra.

However, prime precones provide the

means of constructing a theory of local dimension capable of explaining the phenomena of

"fall of dimension" observed in the examples of

§2;

this cannot be done with the classical tools. B.

LOCAL DIMENSION. Definition 8.4.

field

K, and

dimR(V,x), chain

Let

x ~ V(K).

V

be an affine variety over a real closed

The (local) real dimension of

is the supremum of the integers

n

=

a o ~al~

of prime precones of ing to

x;

cf.

an

- . .

K[V] ending in

V

at

x,

such that there is a strict

a--

X

a~(= the prime precone correspond-

§7).

Comparing this definition with Definition 2.6(c) one may wonder whether dimR(V,x) coincides with the real dimension of the ring

K[V]M_.

This

x

is not true in general, but we have: PROPOSITION 8.5.

dimR(V,~ )

=

dimR(K[V]M_) h, X

where

Ah

denotes the Henselization of a local ring

A.

[]

The proof of this result requires some non-trivial arguments developed by Coste/Coste -Roy [62].

For the construction of the Henselization of

a local ring, see Lafon [6]

or Nagata [Ii].

Next we state the central geometric theorem on local dimension: THEOREM 8.6.

Let

V

be an affine real irreducible variety over a

real closed field K, and x ~ V(K). Them dimR(V) = dimR(V,x) iff belongs to the closure (in the euclidean topology) of the set of nonsingular points of

V(K).

[]

The proof is done in Coste/Coste-Roy analogue of Theorem 2.8 Examples

2.9

[62;Thm. 8.9].

for real varieties.

This result is the

Looking back at the

we can see now that our notion assigns the correct dimen-

sion to the origin in both cases

: 0 in the first example, 2 in the

124

second. C.

THE DIMENSION OF SEMI-ALGEBRAIC

One of the remarkable

SETS.

features of the local theory of real

dimension is that it assigns a dimension only to varieties but, more generally, provides a notion of dimension DEFINITION and

x e S.

8.7.

Let

K

to s.a. sets.

be a real closed field, S

at

PROPOSITION

8.8.

sup{dimR(X,x)

sion of the closure of

S

in

Kn

I xES}

is the supremum

in S

ending in

~.D

is equal to the (real) dimen-

with the Zariski topology,

It follows that the natural notion of (global)

real dimension

set is that of the real dimension of its Zariski closure; defined,

it

of a variety.

S c K n a s.a. set,

x, dimR(S,x),

of strict chains of prime precones

not

In particular,

for the connected components

The real dimension of

of the length,

(both locally and globally)

D for a s.a.

this is well-

for such a closure is a variety by definition.

PROPOSITION

8.9.

Let E

S c Kn

~

be a s.a. set.

dimR(S,~), For a given integer

is upper semi-continuous.

defined on S,

Then the function

k ~ i, the

set {xe S

I

dimR(S,x) < k}

is s.a., open in So

D

Looking at Example result;

the set

2.9(b) , we see that Proposition {xeV(~)

I dimR(V,x)

should be, since it coincides with COROLLARY neighborhood

8.10.

If

U

x

of

S c Kn in

S

CONTINUOUS

{<x,y,Z>

such that

SEMI-ALGEBRAIC

The study of continuous

U

V(~),

as it

xeS, then there is a s.a. equals

the (real)

K n.

FUNCTIONS.

s.a. functions has only begun recently.

The subject is still largely unexplored, the past concentrated

gives the "right"

I x=Y =0 ^ z ~ 0}.

dimR(S,x) in

8.9

is open in

is sla. and

dimension of the Zariski closure of §9.

= I}

as algebraic

geometers have in

on the study of analytic s.a.(=Nash)

functions,

which lie closer to the geometrical phenomena and have better algebraic properties.

Nevertheless,

line the increasingly Furthermore,

the investigations

carried out so far under-

important role of continuous

the study of these functions

s.a. functions.

leads naturally

to that of

Nash functions. We begin by introducing

a class of functions which,

in most interesting

125

cases,

turns out to coincide w i t h that of (continuous)

DEFINITION We say that

9.1.

f

Let

S c

is globally

£ ~ i and polynomials such that the equation

A n be a s.a. set and

algebraic

is verified

S c

for all

PROPOSITION ~n Then:

(I) If

f

Let

(I)

finition

f:S

then

f

(2) If f is continuous f is s.a. PROOF.

Pi non-zero,

f(~)£-I + ..... + Po(X~

+ ~

be a function

is globally

and globally

By trivial manipulations

V(Pi(~,y) i

= 0

Pi,Qik c ~ [ X I ..... Xn,Y].

junct defines It suffices nomial

iff there is

some

= 0

defined on a s.a.

set

algebraic.

algebraic,

and

the graph of

S

is open,

f, Gr(f),

then

has a de-

of the form:

(*) with

with

a function.

~ ~ S.

9.2.

is s.a.,

functions.

f:S + ~

(over polynomials)

Po,...,P£ ~R[XI,...,Xn],

Pz(x-~ f(~)£ + P£_I(~)

s.a.

a non-empty

Furthermore,

occur.

i.e.

we can assume

that each dis-

set.

to show that each disjunct

equation,

actually

^ A Qik(~,y) > 0) k

of

that a polynomial

For then,

setting

(*) contains Pi

P(X,Y)

of degree

a non-trivial e i in

= ~Pi(X,Y),

Y

poly-

does

we have

1

P(x,f(x)) Assume

= 0

(2)

5.7

at

holds

contains

at

no non-trivial

<Xo,Yo >,

S.

= PZ(X)~

+

... + Po(X)

a partition =

Ao,A 1 .... ,Am of

... = P£(x)

and for each

= 0

i = l,...,m,

(jj) There are continuous

decomposition

a non-zero

Under the present hypothesis,

(j) Po(X)

that

f

equation.

If

is not single-

Xo' a contradiction.

may only occur when the sign of

arrange

polynomial

then (*) shows

We shall use now the cylindrical

P(X,Y) on

xES.

that one disjunct

this disjunct valued

for all

theorem

polynomial

5.7

annihilated

the first alternative is zero.

Therefore,

S

into s.a.

sets,

all

by

f

of Theorem

P

for

with

we can

so that:

XeAo,

we have: s.a.

functions

~,

giving exactly

the real roots of the polynomial

In particular,

P(x,Y)

has constant

sign ~ 0

.... ~ i : A i ÷ P(x,Y),

~' £i ~ i,

for all

x e A i.

in each of the intervals

126

(_~. ~(~)), (~(~),ij+l(X)) . . . ,

.(x)' + ~ ) . By Theorem 5.8 1 in assuming that A 1 ..... A m are connected.

there is no loss of generality Since

f(x) is a real root of

with one of the

~(x-),

and the continuity cise). The foregoing

of

say f

argument

..... (~

P(x,Y),

by (jj) it coincides,

j = s i.

Now,

imply that

shows

si

for

the connectedness

of

is the same for all

that the graph of

xcA i, Ai

xEAi(exer-

f FA 1 o...uA m is defined

by the formula: V ,,y i=l ~i (~) ^ where

~i

interior

(in

interior

~n),

as

P(x,Y)",

of

f fAo.

(j) shows that

Po .... 'P£"

V(~)

in S,

intersects

root of

A i-

In fact,

defined by

has empty

th

about the definability

of continuity.

V is the variety

~ Ao

si

is a formula defining

Next we need to worry argument

is the

A o ! S n V(~),

In particular,

has this property.

and each neighborhood

A 1 u... uA m.

f(x)

By continuity

=

lim

This is just an

Since (in

where

A o has empty S

is open, A

o

S) of a point

we have

f(y)

e AIO. • .uA m Since

f rA I u...

f ~A O

is also s.a.

uA m is s.a.

and the definition

There is a notion of minimal polynomial This is a consequence PROPOSITION of R,

and

beR

I b = {P~A[Y] then

Ib

9.3.

of the following Let

R

an element

J P(b)

= 0}

for globally algebraic

be a ring, algebraic

of limit is first-order,

A a unique

over

is principal.

If

algebraic

functions.

result: factorization

A.

Then the ideal

R

is an integral

subring

domain,

is also prime.

The proof, which

D

is just a variant

Palais

[30;

applies

to the case under

§3]

and in Dickmann

of standard [72;

consideration

Prop. V.3.1].

DEFINITION

appears

in

This result

by setting:

R = the ring of real-valued , continuous, S ! A n (henceforth denoted C(U)); A = the ring

arguments,

s.a.

functions

on a s.a. set

~ [ X I ..... Xn]. 9.4.

The minimal polynomial

of a continuous

s.a.

(or,

127

more generally, a globally algebraic) function

f

on

S

is defined to

be a generator of the ideal If = {P c~[X,Y] such that the

g.c.d,

I P(~,f(~)) = 0

for all

~S}

of its coefficients is i.

The minimal polynomial is, of course, a polynomial of lowest degree in If, and is unique, but not necessarily irredu.c.ible; for example, the minimal polynomial of the absolute value function on

~

is

(Y-X)(Y+X).

However,

it has the following properties: PROPOSITION 9.5. and let

Let

P = P1 ..... P£

P e~[X,Y]

be the minimal polynomial of

be a decomposition of

P

f ~ C(S),

into irreducible factors.

Then: (a)

The Pi are distinct.

(b)

If

F i = P/

and

Pi

U i = {x~S I Fi(x,f(x)) ~0}, then the

pairwise disjoint, open s.a. subsets of on (c)

S, and

Pi

Ui

are

vanishes identically

Ui . S - 6 Ui i=l

open, then

has empty interior £ S ! i__~Jl.=l~i.

For a proof, see

Brumfiel

[65;

(in

~n);

in particular,

if

S

is D

Prop. 8.13.15].

The proofs of some of the basic properties of continuous s.a. functions which we will consider below, use a technique depending on the fact that the

real roots of a polynomial equation

have a convergent

P(X,Y) = 0

in one variable X

Puiseux series expansion (cf. Example

3.2).

Precise-

ly: PROPOSITION 9.6. and

p

Let

a real valued

[Xo,Xo+e),

for some

P c~[X,Y]

be a polynomial in two variables

function defined on an interval e>0 ,

and such that

P(x,p(x)) = 0 Then

o

for all

x~ [Xo,Xo+e).

has an absolutely convergent Puiseux series expansion p(x) =

for all

x

for all

k ~ N, and

kE N

ak(X-xo)k/P

in some interval (Xo,Xo+~) , a N ~ 0.

function p defined on expansion.

0 <~ ~~

Here

p ~ I, N ~

,ak~

In particular, any globally algebraic [Xo,Xo+S )

has such a Puiseux series 0

128

The existence of a formal solution is a consequence Walker

[15; Ch. IV,

§ 3].

It is a remarkable

series which is a solution of an equation convergent. Chenciner

of T h e o r e m 3.3;

P(X,Y)=0

is a u t o m a t i c a l l y

A neat discussion of the question of convergence [2; Ch. VIII,

§8.6].

see

fact that a formal Puiseux

See also Hormander

appears

in

[18; Appendix].

As an application of this technique we prove: THEOREM 9.7 (The Zojasiewicz s.a.

set an f, g

(i) For all

continuous

s >0,

{xEC

(ii) Z(f) ! Z(g)

functions

I Jg(x) l ~

(where

Let

defined on

c,r > 0

such that

~ m2

[ @ xeC

(u = Ig(x) l ^ v =

is given by a disjunction of conjunctions

H

is a s.a. set contained in the positive quadrant of

certain polynomials, e >0

say

Pl(U,V),...,Pt(u,v).

Let us consider Igl ~ s

Puiseux series expansions

the set

on C.

Hn((0,e)

× ~).

In the last case,

Assume,

Lgl

then,

If it is empty,

that

Hn((0,s)

× ~) ~ 9.

that for

x e C

also that

v >0

I Ig(x) l = u}

it follows that

v

g = 0

the m i n i m u m of

Ill

(i)) and the

This set is bounded below on PI,...,Pt,

implies

on

if(x) i~v(]g(x)l),

and

inf (Hn({Ig(x) i} × m ) . u~(o,E),

then

is a compact set (by(i)) on w h i c h

f ~0

Ifl

say v(u).

we have:

v(ig(x) i) =

v(u) ; hence

either

in C.

0 < Ig(x) i < e

Observe

have abso-

(by assumption

(0,e) by one of the roots of one of the polynomials This means

on

9.6 there

(0,e).

C is compact

and the result follows easily by considering m a x i m u m of

in

~.

of sign conditions

By P r o p o s i t i o n

such that all the real roots of these polynomials

lutely convergent

that

on C.

]f(x) I)}.

Then

If

such that:

Ift eclgl r

H

{x~C

C

be a closed

E} is compact.

Clearly

or

~n

Let H = {

is

C c

Z(f) = f-l[0]).

Then there are constants PROOF.

s.a.

inequality).

(0,s);

for if

has a m i n i m u m ~ 0

(by(ii));

on this set, which equals,

v(u) > 0.

is bounded b e l o w by Ifi e clgl

Let us assume that

for some v

~ >0

on (0,e),

then an easy argument

shows

c >0.

is not bounded away from 0, i.e.

Using the Puiseux series expansion of

v

we get:

lim+ u+0

v(u) = 0.

129

(*)

v(u) =

[

akuk/p

=

a N uN/p (i +

[

k=N

for

~(0,~).

Let

T(u) denote the series expnasion in the last term;

since its exponents are positive, then 0 <6 ~ ~ , so that from (*) that Putting

=

c'

I + ~ e 1/2

aN > 0 . aN/2

and

Igl r

{x~C J Jg(x) J ~ 6}.

Put

REMARKS.

(a)

N/p

implies that N > 0 .

_ jr IfJ > c'Jg

on

for some

c">0

on the compact set

c = min {c',c"}.

D

The idea of the proof above goes back to H~rmander

[18;

Len~na 2.1 ]

(b) When the set

for

v(u) >0, it follows

v(U)u+--~0

we have

C

is compact, the statement of Theorem

fact first-order, as one may take 3.4].

Since

Choose 6 ,

The argument used at the beginning of the proof

JfJ ~c"

Appendix;

r =

lim+ y(u) = 0. u*0

on (0,6).

The fact that

{xeC J 0 s Jg(x) t< ~} shows that

ak/aN uk-N/p)

k=N+l

r

9.7

rational, cf. Dickman

is in

[72; Prop.V

By transfer, the inequality is valid in each real closed field, C

closed and bounded. Dells [70;

Lemma

3.23

gives an elementary

proof valid for arbitrary real closed fields.

D

In the remainder of this section we sum up other results obtained by application of the same technique, and study their effect on the structure of the rings C(S). We shall denote set S c K n, topology.

by

cK(s) the ring of

K-valued s.a. functions on a s.a.

K a real closed field, which are continuous in the euclidean

The following result is proved by Carral-Coste then, by transfer, for any real closed field PROPOSITION 9.8. and

gEcK(s-z(f)).

prolonged by

0

COROLLARY Z(g) ! Z(f).

on 9.9.

S.

first for

~

Let

S

such that the s.a. function fmg,

Let

S

be as in 9.8 me i

V

0 and

f,gecK(s)

such that g

be as in 9.8. if

and

be a locally closed s.a. set, fEcK(s) m ~i

Z(f), is continuous.

In particular,

Spec (cK(v(K))) PROOF.

S c Kn

Then there is

COROLLARY 9.10. phic to

Let

Then there is

[603, K.

be such that

divides fm in

Then Spec(cK(s))

cK(s).D is homeomor-

is an affine variety over

K, then

is homeomorphic to SpecR(K[V]).

This is an application of the duality between spectral spaces

and distributive lattices mentioned in

§6.B.

It suffices to prove that

the lattices of compact open subsets of Spec (cK(s)) and of

S

are

130

isomorphic. By Theorem 7.2 subsets of

the latter is isomorphic to the lattice of open s.a.

S.

The former is simply

{D(f)

] fccK(s)}, since Corollary 9.9 implies that

D(f) uD(g) = D(f 2 + g2). The map the required isomorphism

D(f) I • {xcS If(X) ~0}

establishes

: it is injective by Corollary 9.9, and it is

surjective since for a given open s.a. set

U ! S, the function ds_ U

(= distance to S-U) is continuous. COROLLARY 9.1].. Let

S c Kn

be a locally closed s.a. set.

Then

dim (cK(s)) = dimR(S).

D

This corollary shows that the rings cK(s), c(S), are radically different from the rings of arbitrary continuous functions : the Krull dimension of the latter is one or infinite, whatever the underlying space; inclusion chains of prime ideals in this case are of length 1 or at least 2~I (cf. Gillman - Jerison

[85; Thm. 14.19]).

Thus, we see that rings of conti-

nuous s.a. functions are well-behaved objects which reflect geometric properties of the underlying spaces. The results above have an effect on the structure of the ideals of COROLLARY 9.12. an ideal of (i)

Let

cK(s).

S

be as in Proposition 9.8, and let

I

cK(s). be

The following are equivalent:

I is real.

(it) I is radical. (iii)l is a z-ideal (i.e. Z(f) = Z(g) and The residue rings

cK(S)/p,

where

P

g~l

imply

fcl).

is a prime ideal, have the following

properties, similar to those holding in rings of arbitrary continuous: PROPOSITION 9.13. of (i)

Let

S c K n be a

s.a. set, and

P a prime ideal

C~(S). The relation f/p e 0

iff

there is

defines a total ordering on (it) The ring (a)

gccK(s)

such that

g>_0 on S and

f/p = g/p,

cK(s)/p.

cK(S)/p has the following properties:

It is a local ring (i.e.

P

is contained in exactly one

maximal ideal). (b)

Every non-negative element has a square root.

(c)

Every monic polynomial of odd degree has a zero.

131

In particular: (iii)

If

M

The proof,

is a maximal

ideal,

given in Dickmann

then

[84],

cK(S)/M is a real closed field.

is a "definable" version of an ~

argument known in the case of rings of (arbitrary) see Gillman-Jerison

C

is an algebraic

proved in Dickmann closed ring, COMMENT.

[84];

introduced

curve over

the residue rings

~

and

it establishes

P

is a prime ideal, is

[82].

One may consider classes of continuous

type are familiar in analysis.

conditions;

s.a. functions obta-ned

many conditions

condition

ments;

for example,

for

finite may not be a natural

r

differential However,

differentiable

9.14.

Let

f

be a

C ~ (i.e. infinitely

function defined on an open, connected

s.a. subset of

~n D

[65;

Prop.

8.13.16].

NASH FUNCTIONS.

f:U ~ ~

I0.I.

Let

U

be an open s.a. subset of

~n

A function

is called a Nash function if it is s.a. and analytic on U.

denote by

N(U)

the ring of Nash functions

defined on U.

These functions,

first considered by Nash [29],

prime importance

in real algebraic geometry.

Efroymson observe,

[25; p. 214],

algebraic properties

[26]

[25] is a comprehensive subrings.

constitute

but better geometric properties.

also be found in Roy

survey of the algebraic

a wealth of material. [32]

Bochnak-Efroym-

to the subject, while Bochnak-Efroymson theory of

Many of the basic results were first collected

which contains

a tool of

The point is, as Bochnak-

dealing with Nash functions.

is an introduction

We []

that Nash functions have the good

of polynomials,

There is a vast literature

[28],

differentiable)

is analytic.

DEFINITION

son

s.a. functions

class from the point of view of

The proof is implicit in Brumfiel §i0.

in connection with other require-

r-fold continuously

the following is important:

realvalued f

Semi-algebricity

geometry.

PROPOSITION Then

of this

As far as we know, nothing has been done

in this direction beyond the study of Nash functions. not always be a natural

C(C(IR))/p,

a link with the notion of real

in Cherlin-Dickmann

by imposing further "regularity"

my

functions;

[85; Thm. 13.4].

A result of geometric nature concerning where

continuous

and Palais

these two papers are very different

[30];

Valuable

N(U) and its in ~ojasiewicz

information

can

the points of view of

from the one adopted here.

In this survey we shall only consider Nash functions

defined on connect-

132

ed open domains

in

An.

This will be quite sufficient

although much of the theory below applies definition A.

(see Bochnak-Efroymson

for our purposes,

to more general domains of

[25]).

BASIC ALGEBRAIC PROPERTIES. Many of the good algebraic properties

from:

10.2. Fundamental

fact.

If

U

of Nash functions

is open and connected,

follow

then N(U)

is an integral domain. PROOF.

Let

f, g eN(U)

of these sets, say f = 0

on

U

be such that

fg = 0, i.e. Z(f) uZ(g) = U.

Z(f), has non-empty

by the principle

interior.

Since

of analytic continuation

U

One

is connected

(Dieudonn~

[17;

9.4.2])

D

An immediate COROLLARY Nash

iff

consequence

is :

10.3.

f

Let

be an analytic

it is locally algebraic;

neighborhood V of such that P(x,f(x)) By Proposition

function

on

i.e. for every

~oo and a polynomial = 0 for all x ~ V .

9.3 the minimal polynomial

U.

x-~U

Then

f

is

there is a

P ~ [ X I ..... Xn,Y] , P = 0, D of a Nash function is irreducible.

This has a number of simple but important algebraic

consequences;

we

mention the following: COROLLARY g:U ÷ ~

10.4.

Let

an analytic

f:U x ~ ÷ ~

f(x,g(x)) Then

g

Let

Since

f ~ 0,

follows nomial.

for all

P e ~ [ X I .... ,Xn,Xn+I,Y]

x ~ U.

then

for all

and hence

P

is not divisible by

=

P(~,g(~),0)

=

of

Y;

it

P(~,g(~),

f(~,g(~)))

= 0 D

10.5.

function on

COROLLARY i = i. . . If

. .

f

~f satisfies ~x°

f.

0) is not the zero poly-

x c U.

COROLLARY analytic

P ~Y

be the minimal polynomial

that Q(X I ..... Xn+ I) = P(X I ..... Xn+!, We also have: Q(~,g(~))

PROOF.

= 0

is Nash.

PROOF.

and

be a non-zero Nash function and

function satisfying

Let U

F c N(U)[Y] satisfying

10.6. The ring N(U) :N (u). n, then -~f ~x i satisfies

be a non-zero polynomial F(g) = 0.

is differentially

the polynomial

the equation

Then

equation

g

and g an

is Nash. stable:

P(~,f(~))

D

if F E N (U)

= 0, then

133

De

~f - = 0 ~~P (~,f(~)). ~i(x)

(~,f(xD) +

~X i

which has Nash coefficients.

Hence it is Nash by 10.5.

Another consequence of

is:

PROPOSITION 10.7. open

s.a.

and

10.4

(Implicit function theorem).

~e~n

b ~R

a Nash function such that is a s.a. neighborhood such that

g(a) = b

V

and

f(a,b) = 0 of

Let

be such that e U.

a

in

and ~n

f(x,g(x)) = 0

Q

U c ~n+l Let

~f (a,b) ~ 0. ~Xn+ I

be

f:U ÷ ~

be

Then there

and a Nash function g:V + for all

x E V.

PROOF. By the implicit function theorem for analytic functions (see Dieudonn4 [17; 10.2.43) there is V, which we may take s.a., and an analytic solution g as above; Palais

[30; §i]

g

is Nash by 10.4.

D

shows that this statement is equivalent to more general

versions of the implicit function theorem, for a variety of situations including, of course, Nash functions. The following algebraic property is a consequence of Corollary 10.5: PROPOSITION 10.8. a quotient

closed 10.5,

N(U)

is integrally closed;

that is, if

f/g of Nash functions satisfies a monic polynomial equation

with coefficients in PROOF.

The ring

N(U), then g

divides

The ring of analytic functions on (cf. Dickmann

[72; Ch. V]);

f

U

in

N(U).

is known to be integrally

hence f/g is analytic on U.

f/g is Nash.

By 0

Now we mention, without proof, an algebraic property of crucial importance. THEOREM 10.9.

The ring

N(U) is noetherian.

The original proof, due to Risler [31],

D

is basically of algebraic nature.

A proof using complexification techniques is sketched in BochnakEfroymson

[25; Thm. 3.1].

COROLLARY i0.I0.

We shall use later the following consequence:

Let A be a subring of

nomials, and k I an ideal of that Z(1) = £'h Z(fi). i=l PROOF.

The ideal

I.N(U)

A.

N(U) containing the poly-

Then there are fl ..... fk EI

generated by

I

in

such

N(U) is finitely generat-

ed.

Each of these generators is a linear combination of members of

say

fl,.,.,fk.

The conclusion follows at once.

The algebraic properties considered above are valid for all (open) domains U. On the contrary, unique factorization in N(U) is an

I,

134

algebraic property which depends essentially on the geometry of the domain

U;

namely, on the triviality of its first cohomology group

(cf. Bochnak-Efroymson and Risler B.

[31]

[25;

§4],

where further references are given,

for simple examples).

NASH FUNCTIONS AND REAL ALGEBRAIC GEOMETRY. In order to understand the relevance of Nash functions for real

algebraic geometry, we underline the basic fact that in the classical theory of algebraic curves, the notion of a branch of analytic parametrizations;

see Walker

For example, the branches of the curve Example

is defined in terms

[15; Ch. IV, §2].

y2 _ (X 3 + X 2)

considered in

6.5, are given by the functions

b i : (-i, + ~ )

÷~,

i = 1,2,

(-i) i

~

defined as follows:

+ X2

for

-I<X~0

~X 3 + X2

for

0 ~X

bi(x) (-l) i+l

(where ~Fdenotes the positive value of the square root). tions are analytic branches

(exercise), and they are obviously s.a.

I I

The s.a. function

These func-

f:[-l, +~) ÷If

\bz

defined by

f(x) = min {y c]R I y2 = x 3 + x 2} is continuous but not analytic, thus not a branch.

f

135

As a second example (a).

Its only branch b(y)

defined on

consider

the curve

y2 _ X 3 + X 2

of Example

2.5

is given by the map × such that

y

2

= x

3

2 - x , and

=

the unique

x > 1/2

A

(we have to give it in this form in order to have a func-

tion). It is a remarkable by the preceeding

fact that the description examples

in any number of variab!es braic variety. Proposition

and

(i)

f

(ii)

There

nomials

is

q

q ~ n+l,

Let

U

a "branch"

of continuous

of an alge-

s.a.

given by

be an open,

connected,

s.a. sub-

are equivalent:

variables,

is

an irreducible a continuous

P ~A[X1,...,X gtale

over

q]

V

(b)

(pr FV(A))

o

jection

the first

onto

affine variety

s.a.

such

map

V

given by poly-

s = < Sl,...

,

Sq > :U

÷

VOR)

that:

A. s = idu,

where

n

pr

: A q ÷A n

denotes

the

pro-

coordinates.

f = Pos.

The proof is far outside and Roy

[32]

Condition

(a)

see Raynaud sequences

The functions

(jj)

The map

is classical Exercise.

exercise

f

[17; 9.11.1].]

s I ..... Sq

of Theorem

we con-

I0.i!, we have:

are Nash.

is locally

injective.

D

gives a clue as to why Nash functions

do not appear

geometry.

f:C n ÷ C

is globally then

algebra;

(a) - (c) above:

pr r V ( A )

algebraic

Let

but the mere presenta-

from commutative

but point out for later use the following

With the notations

(j)

The following

[24]

Since this condition will not be used explicitly,

explanations,

FACT.

significance,

the use of heavy machinery

of conditions

10.12

see Artin-Mazur

and further uses of this characterization.

has a deep geometric

[12].

omit further

the scope of this survey;

for details

tion of it requires

f

sense,

illustrated

Every Nash function,

is as follows:

The following

(a)

9.1),

form.

is Nash.

in

(c)

result

(Artin-Mazur).

f:U ÷ A .

and a polynomial

If

is, in a suitable

The precise

I0.ii.

An

of Nash functions

their general

(Compare w i t h the description

9.2).

THEOREM set of

gives

be a holomorphic

algebraic

(=complex

over polynomials

is a polynomial.

[Hint:

analytic)

function.

(in the sense of Definition

use Liouville's

theorem,

Dieudonn~

136

C.

THE SEPARATION THEOREMS. Now we will consider

the problem of finding a simple class of

s.a. functions having the following separation property: joint closed s.a. sets CI, C2, that

f ~C I >0

and

given two disin the class such

are not sufficient

to separate closed s.a.

the sets of Figure ii provide a counter-example.

that Nash functions

-

f

f ~C 2 < 0.

It is known that polynomials sets;

there is a function

- even Nash functions of a particularly

have the required DEFINITION

It turns out simple form

separation property.

10.13.

Let

U !

An

be open s.a.

We define

the smallest

subring of

C( IRn) containing

(i)

If

f e ~ R(U) 2

is such that

f >0

on

IRn, then I/f E R(U).

(ii)

If

f ~ ~ R(U) 2

is such that

f >0

on

U, then

The ring

R(U) can be constructed

B ¢_ C ( A n ) , ~,

let

with =

B (n+l) B (~) Let

SB

such that

as follows:

and

f >0

on U.

B

and such that:

~-E

R(U).

for a ring by all functions

Let

B =

=

inductively

B (I) denote the ring generated over

f ~ ~ B2

B (°)

the polynomials

R(U) to be

(B(n)) (i) U B (n) n~

denote the multiplicative f >0

IR[X I ..... Xn](~)

on

I~n.

Then

at the set

subset of

B

of all functions

R(U) is the localization of S

~[x I .....

Xn ](~)

f ~ ~ B2

t37

Observe that the functions condition #f

in

R(U) are continuous

10.13 (ii) is introduced

THEOREM 10.14. Let

CI,

C2

f.

be disjoint

f c R ( ~ n) - even

A n and Nash on U;

in order to make this true, since

is not analytic at any zero of

Then there is

on

closed s.a. subsets o f ~ n.

f E (~[X]s

)(I) _ such that f ~C 1 > 0 REX]

and

f rc 2 < o.

D

PROPOSITION

10.15.

such that

f r u>0

A relative

separation

THEOREM 10.16. subsets of U.

Let

and

U !

~ n be open s.a.

f r(~n-u)

theorem is obtained Let

U _cA n

Then there is f c R(U)

= 0. from 10.14 and 10.15:

be open s.a. and

Then there is f ~ R(U) such that

CI,C 2 be closed s.a.

f FC I > 0

and

f FC 2 <0. D

Various proofs of these results, Efroymson

[27; §i]

due to Mostowski,

and Bochnak - Efroymson

have the following result of Risler COROLLARY is an open,

10.17.

Let

connected

P

As a corollary we

[31]:

be a prime ideal of

s.a. set.

can be found in

[25; §5].

Then

N(U), where

U i

An

Z(P) is connected.

PROOF.

By Corollary i0.I0 there are gl ..... gr eP such that r Z(P) = ~ Z(gi). Hence Z(P) is s.a. and has finitely many connected i=l con~onents

(Theorem 5.8).

s.a. subsets Let

CI,C 2 of

If

Z(P), closed in

f E R(U) be such that

Then

hlh 2

f rC 1 > 0

rE gi2 cP, i=l

=

but

Similarly h 2 ~ P, contradicting REMARK.

f ~C 2 <0.

Let us now define

for

P

CIUC 2 = Z(P).

i = 1,2.

h I l'C1 = 0

and

h I~C 2 >0.

is prime.

D

The proof shows that the result holds for any subring A of

which contains §Ii.

and

h 1 % P because that

there are disjoint

A n, such that

/~ gi2 + f2 + (-l)i f i=l

=

hi

Z(P) is not connected,

~,

whenever

THE SUBSTITUTION

f e A and

THEOREM;

f >0

N(U)

on U.

"STELLENSATZE".

In this paragraph we prove another central result in the theory of Nash functions: a number of A.

the substitution

"stellens~tze"

THE SUBSTITUTION If

K

theorem.

for the rings

We use it later to derive

N(U) and many of its subrings.

THEOREM.

is a real closed field containing

~,

S c ~n

is a s.a.

138

set and

f:S ÷ ~ a s.a. function,

parameters define,

in

~) defining

respectively,

ed function on

S

it is clear that the formulas

and

Gr(f) are interpretable

a s.a. subset of

S K, denoted by

fK.

K n, denoted by

The substitution

any ring homomorphism ~: N(U) + K some point of U K, Precisely:

an

K

Then

K-valu-

U c R n be an open, ~, and

4: N(U) + K

:

<~(~i ) .... , ~(~n)> E U K,

maps,

Let

a real closed field containing

R = a l g e b r a homomorphism.

(i)

S K, and a

theorem says that

is the evaluation homomorphism at

THEOREM II.I (The substitution theorem). connected s.a. set,

(with

in K, and

where

~i ..... ~n denote the projection

~i(Xl ..... x n) = x i.

(ii)

For every f E N(U),

PROOF.

Observe that

~(f) = fK(~(~ I) ..... ~(~n)).

(i) gives a sense to (ii).

The proof proceed in the

following steps: (I)

We prove (ii) for

~n,

f E R(U).

Since any such

this makes sense even in the absence of

(II)

We prove

(III)

We prove

STEP I.

Condition

truction of

(ii) is clear for polynomials.

R(U) it suffices

to prove that for

holds for

h

and

h rU>0,

(b) if

(ii)

holds for

h

and

h >0,

PROOF of (a). Since

e0

(*)

By the inductive consh~ c(~n),

=

then (ii) holds for f = ~ ;

then (ii) holds for f = i/h.

hK(~(~))

~(T) = <~(~i ) ..... ~(~n)>.

= fK(~(~))2,

In order to conclude that ~(f)=fK(~(~))

to show that both sides are non-negative,

for we have assumed that ¢(f) >0

h e0:

f2 = h, we have

~(f)2 = ~(h)

it suffices

(i).

(ii) for arbitrary f e N(U).

(ii)

is

is defined on

(i).

(a) if

where

f

f ~ 0.

for any f E N(U) such that

Indeed, as we have I / ~ e N ( U ) ,

then

The right-hand side

But we also have:

~(f).#(I/~)2

f > 0 on = !;

U.

this implies that

~(f) >0. The proof of STEP II. f ~(~n (**) holds in

(b) is similar.

By Proposition 10.15 let _ U) = 0.

f e R(U) be such that

f ~U > 0

Then the first-order statement Yx (x E U<-->f(x) > 0)

~, and hence in K.

By (*) we have

%(f) > 0, and by (I),

and

139

fK(~(~)) >0, STEP III.

It follows

Now we use the Artin-Mazur

(Theorem i0,ii). polynomials

s.a. function

P

to prove

X = V(~)

s[U] obviously of

s[U] in

continuity

s(pr(y))

and

for the functions

is connected, component

0 c~q

Using 10.11(b) our contention.

Vx~U V ~ in

~.

prK(~)

Put

theorem

if

y

belongs

to the

for some sequence u n ~ U.

Hence ucU,

closed in X. In

and using and

pr

10.12

(jj)

is injective

on

0 n X ! s[U], which

10.16 to get It follows

that

s[U]is

h c R(W)

such that

from this situation

that

q [z ~ V n W

^ h(~) > 0 it holds

and

^ pr(z)

in K.

= ~ ÷ z = s(x)]

Specializing

~ = <~(~i ) ..... ~(nn)>

of (***)

to

= ~(~)

(which is in U K K ~(s i) = si(~(~)),

is the equality It suffices,

then,

to check that

of (***) hold in K.

= p r o s = idu(10.11(b)).

hK(z) > 0.

By (I),

~(hos)

(*) shows

that

-

- and

statement

By transfer

Irmnediate from -

(a)-(c)

s I ..... Sq.

we conclude

at once that

h T (X - s[U]) <0.

the consequent

the premises

Indeed,

s(u) c 0 n X

i = 1 ..... q, which we want to prove.

-

X).

~ = lim s ( u n~

it is checked

= <~(s I) ..... ~(Sq)> by (i)),

continuous

and we prove next that it is clopen

of

so that

the separation

and

the first-order

holds

(ii)

X, then

0 nX.

Now we invoke

I0.II

a

nW.

p r o s = idu(10.11(b))

proves

h F s[U] > 0

P1 .... ,P£~ ~ [ X 1 .... ,Xq] be

so that conditions

s[U] is open in X, let

choose an open set

of Nash functions

s = <s I .... ,Sq >:U + V ( ~ )

= lim s(u n) = y, i.e. 7 ~ s [U].

order to see that

(***)

characterization

(jj) - hold.

X (hence a connected

closure

~(~)E U K.

q ~n+l, V,

a polynomial,

(j),

and

let

a variety

it suffices

The set

By

and

10.12

W = pr-l[u]

in

Therefore,

defining

hence also Clearly,

from (**) that

z

Since

=(hos) K (~(~))

= hK(z).

Since

h >0

on

s[U],

then

~(hos) >0.

e ( V n W) K.

x = prK(z) E U K,

the variety

V

and ~

then

zcW K.

As the polynomials

P1 .... P£

s[U] i V ( ~ ) , we have £ VycU A Pj(s(y)) = O, j=l

that is, the Nash functions

P.os ]

vanish on

U;

therefore

define

140

V

0 = ~(Pj o s) = P~(~(S))

=

The substitution theorem is due to is due to Coste [67]; §7].

B.

for

j = i ..... £.

Efroymson [27].

D

The present proof

for a different proof see Bochnak-Efroymson

As shown in this paper,

of subrings of

P~(z)

[25;

the result applies as well to a wide class

N(U).

APPLICATIONS:

"STELLENSATZE"

;

COMMUTATIVE ALGEBRA OF NASH

FUNCTIONS. Now we show how the substitution theorem can be used to derive, by a uniform method, all the "stellens~tze" holding for rings of Nash functions.

These derivations

are purely algebraic

- in fact, they are

a sophisticated elaboration of Robinson's method to prove Hilbert's nullstellensatz

-

and work for any ring of s.a. functions

for which a

substitution theorem is available. A "stellensatz"

is a result establishing an equivalence between a con-

dition of the form ~

V ~ c U [A gi(~) ~ 0 i

^ A hj(~) > 0 ^ A fk(~) = 0 j k

+ f(~)? 0]

where gl ..... gr' hl'''''hp' fl .... 'fq' f c N(U) and ? is a sign condition (> , ~ or = ) and, on the other hand, an equation P(gl .... 'gr' where of

P

hi .... ,hp,

is a polynomial with coefficients

r,p, q

f) = 0

fl .... 'fq' in

N(U).

Here, one or more

may be zero.

Thus we have: THEOREM 11.2. gl,...,gr, (A)

(General stellensatz

h I .... ,hp,

for Nash functions).

Let

fl .... 'fq' f e N(U).

The following are equivalent: (1) (2)

R > V~U[A There are

i

gi(~) _>0 ^

A hj(~) > 0 ^ j

t,S,Uk~N(U )

and

sj ~{0,i}

A fk (~) = 0 k

÷f(~) >0].

(l_<j _
where s,t are of the form F(g I .... 'gr' hl ..... hp), with F a polinomial whose coefficients are squares in N(U), such that f-t = s + H he.j + j J (B)

Similarly,

f(~) >_ 0

is equivalent

~fk~k"

a condition of type (A.I) with to a polynomial equation

f(x) >0

replaced by

141

E.

f-t = (Hh. j) f2£ + s + ~ fkUk , j J k for some

£->0

and

s,t, u k, sj

A condition of type

(O

as is

(A.I) with

(A). f(x) = 0 instead of f(x) > 0

is

equivalent to ( ~ h~ j ) f2£ with

£ el

and

+ s + kl fkuk = 0

S,Uk, ej as in (A).

Each of the equivalences "formal stellens~tz".

(A), (B), (C) is derived from a corresponding

For example, the result needed for

PROPOSITION 11.3. (Formal positivstellensatz). tive ring with unit, and: (a)

S a sub-semi-ring of

A(= a subset of

Let

A

(A) A

is:

be a commuta-

closed under sum and

(b)

product, but not necessarily under difference) containing the squares. M a multiplicative subset of A containing i.

(c)

I

(d)

f c A.

an ideal of

A.

Then, the following are equivalent: (i') For every ring homomorphism field, we have: ~[S] ~0, (2')

~[M] >0

and

~:A ÷ K, with

~[I] = 0

imply

K

a real closed

~(f) >0.

There are

s,t e S, m £ M and u e I such that f.t=s+m+u PROOF of : Proposition ! 1 . 3 implies Theorem II.2(A). The implication (A.2) implies (A.I) is checked without difficulty. (A.I) implies

(A.2).

As it is obvious, we apply

11.3 with

A = N(U) and:

S = the sub-semi-ring generated by the gi's, the h.' ,~ s s and the squares. M = the multiplicative subset generated by the hj . I = the ideal generated by the

fk's.

It suffices to derive condition (I') of 11.3 from condition (A.I) of Theorem 11.2. This is done using the substitution theorem, as follows: Let ~ be a homomorphism verifying the assumptions of (i'); thus we have: ~(gi ) ~0,

~(hj) >0,

~(fk ) = 0

for all i,j,k.

142

By the substitution theorem we get: giK(*(Y)) e0, The assumption

h~(~(~)) >0,

(A.I) then yields

again, we conclude

f~(~(~)) = 0

fK(~(T)) >0, and by substitution

~(f) >0.

D

The "formal stellens~tze" are, in turn, a consequence of: 11.4 The pivotal lemma. T, N be subsets of

A

Let

A

be a commutative ring with unit and I,

such that :

(a)

T is closed under multiplication.

(b)

I c T is an ideal of A.

(c)

i)

-ioN.

ii) iii)

x,ycN implies -(xy)eN. (-N). Z TA 2 c ETA 2 .

Then, the following are equivalent: (i")

N n ETA 2

#

0.

(2")

For every prime ideal the fraction

field of

N/p n E

P

of

A containing

I, if

L denotes

A/p, we have:

T/p " L 2 # 9.

D

We omit the purely algebraic proof of this result, which can be found in Dickmann

[72; Ch. V].

This is a generalization of Colliot-Th~lene

[44; Lemma l.bis], who proves it in the case

I = (0) and

N = {-i}.

However, we give the PROOF

OF

:

Lemma 11.4 implies Proposition 11.3.

The implication

(2')

implies (I') in 11.3 is evident. (i') implies (2').

Assume not (2'); this just says:

(*)

-M n ((S-f S) + I) = 9.

We apply Lemma

11.4

with

N = -M

and

T = (S=f.S) + I, leaving as

an easy but tedious exercise for the reader the task of checking that the assumptions

(a)-(c) of 11.4 are fulfilled.

The equality (*) amounts to the negation of condition (i") of 11.4. Then, by (2"), there is a prime ideal (**)

containing I such that

N/pn Z T/p .L 2 = 9.

Since ~,

P

fl cM, then

which makes

-i

~ Z ~

T/p e0.

L2 . ~Let K

This means that

L

admits an order,

be the real closure of

~:A ÷ K the canonical homomorphism,

~(a) = a/p.

and

Then we have %IT] e0,

143

which implies: (***) as

~[S] e0,

S ! T,

~(f) ~0

and

-f • T by definition

By (**) we get

~(x) # 0 for

(****)

and

x•M,

M . E T A 2 ! ETA 2 by (c.iii). i.e.

~[M] > 0.

Since

I c p

we have

(*****)

$[I]

Conditions

(***),

Proposition

(****)

and (*****)

PROPOSITION

11.5 I

f vanishes

(ii)

f • ~I.

In particular,

the numerous

N(U)

on

and

(ii) implies

i0.i0 there are

condition

(i) is equivalent

of

PROOF. • Z(M);

that

Let

N(U)

by

Z(I) = N Z(fk); hence k

f(x) = 0]. to

£ _> i,

R~-coincides

with

Rad(I),

U c 1%n be an open connected

is real,

the radical

s.a. set.

Every

and the map:

: M~a = { f ~ N ( U )

a one-one

correspondence

I f(a) = O} between

points

of

U

and maximal

N (U) .

A maximal

also real.

Conversely,

f • I.

we have:

11.6.

a;

ideals

÷

= 0 implies

f c R¢~.

ideal of

establishes

= 0

for some

The proof shows

COROLLARY

such that

(with r,p = 0) this is equivalent

In particular

maximal

are equivalent:

f rZ(I)

to

f2£c I implies

the following

(i) is trivial.

fl .... 'ft ~ I

R ~ V ~ • U [Ifk(~) ii.2(c)

11.2.

for Nash functions).

f ~ N(U),

I is real iff for every f • N(U),

The implication

By Theorem

of Theorem

Z(I).

Corollary

REMARK.

(I'), which proves

corollaries

(Real nullstellensatz

of

(i)

of I.

contradict

D

a few amongst

Given an ideal

PROOF.

= 0

11.3.

We mention

which

~[M] ~0,

Since then

PROPOSITION

ideal is always radical; M

M _c M~ 11.7.

is proper,

by the remark above,

11.5 also shows

and, by maximality, (Solution

of Hilbert's

that

Z(M) # #.

it is Let

M = M~. 17th problem

for Nash

144

functions). Let

U !~ n

on

U, then

PROOF.

be an open, connected s.a. domain and f

f e N(U).

If

is a sum of squares in the fraction field of

By Theorem

II.2(B)

where I is an integer and

(with r = p = q = 0), we get

s,t

sums of squares in

f ~0

N(U).

f.t = f2l + s,

N(U).

D

Further results in the same vein are reported in Bochnak-Efroymson [25]. As remarked earlier, Theorem 11.2 holds for any ring of real-valued s.a. functions for which the substitution theorem holds. the case for the polynomial ring

IR[XI,...,Xn].

Trivially,

this is

Thus we get back, with

a uniform proof, generalized versions of earlier semi-algebraic

stellensatze:

the real nullstellensatz 4.8 (as in Proposition 11.5) when there are no gi ' s or

hj'

s

but we allow the

fk s;

the equivalence (c) of Theorem

11.2, when there are no fk ' s but we allow the

hj' s, is the

gi ' s and the

semi-algebraic real nullstellens~tz of Stengle [58]; fk's or hj's but we allow the gi's, the equivalence

when there are no II.2(B) is Stengle's

nicht-negativstellens~tz.

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(1981) pp.25.

ON

THE

(%)K

Carlos A.Di Prisco

Wiktor Marek

I.V.I.C.

Dept.Computer

Departamento Apartado Caracas

§i.

Lexington,

1827

on

order type of P = ~} appears naturally

and is specifically

As it is shown in [S.R.K.]

target ~ (c.f. also [B.D.T.]) ultrafilter

(%)K

KY 40506-0027

U.S.A.

(%)< = {P i ~ I

in the study of large cardinals

Science

University of Kentucky

de MatemAticas

1010A, Venezuela

The space

huge cardinals.

for

SPACE

if there is a

containing

related to the so called

, a cardinal

K is huge with

K-complete normal non-trivial

the sets of the form

p = {Pc(%)
p ~ PK(~).

Since any ultrafilter

on the set

ties mentioned above

(N-complete,

[%]K = {p ! %1

IPI = K} with the proper-

normal and fine) concentrates

the existence of such ultrafilter

on [l]r is also equivalent

on ( % ) K

to hugness

of K with target ~. In

[DP.M2] we constructed

trivial

filter on [%]K

Large cardinal property

Related combinatorial structure.

This raises

F~,%, the least K-complete,

, which permitted us to complete

Normal Measure on (K measurable)

Normal Measure on

Closed unbounded subsets of

Closed unbounded subsets of

P~(~)

Normal measure on [%]K

(K÷(~))

could the table be also completed putt-

structure on (%)K instead of FK,%?.

"absolute" notion on (%)K

FK,~

PK(X)

this note is to show that there is no such possibility: natorial

the following table

(K %-supercompact)

the following question,

ing a combinatorial

fine, normal non-

approximating

The purpose of there is no combi-

normal measures on (%)K

In this way we present results which stress the differences and

[%]K studied by Mignone in [Mig.].

pointed by Mignone pertains minateness choice.

is accepted,

between

(~)<

One should note that the difference

to the situation in which the axiom of deter-

whereas

in this paper we work under the axiom of

We would like to thank M. Magidor for providing us with the exact

statement of Theorem 4.

152

Let us recall some results and definitions from be regular uncountable cardinals. subset of PK(%), let of elements of X}. of PK(%)}

If

[DP.M2]:

Let

K<%

X ! PK (%) is a closed unbounded

AX = {Pe[k]KIP is the union of an increasing K-chain The sets of the form

{AxIX

closed unbounded subset

generate a K-complete, normal, fine,nontrivial filter on

[%]K

called FK,%" It is interesting to note that if in the definition of sets of elements of ([DP.M2]).

X

instead of

AX

we use directed

K-chains we obtain the same filter

The following propositions are proved in [DP.M2].

PROPOSITION i.

If

V = L

then

(%)K is not ~,~stationary.

PROPOSITION 2.

FK, k is the least

K-complete,

0

fine, normal, non-

trivial filter on [~]~.

0

From these propositions we obtain the following METATHEOREM:

If

ZFC

is consistent then

plete, normal, fine, nontrivial filter on PROOF.

ZFC

"There is a

K-com-

(~)K,,

Assume the contrary, so ZFC + V=L proves as well that there is a

nontrivial, G on

K-complete,

fine, normal filter F on ( % ) K

Define a filter

[%]K as follows XcG

Then G

is a

K-complete,

~=~

Xn(%) K e

F. [ ]K and

fine, normal, nontrivial filter on

therefore by proposition 2,FK,~! G.

But (%)KEG and so (~)K

is

F

stationary contradicting proposition I.

O

The next statement clarifies the situation. PROPOSITION 3.

The following statements are equivalent:

a)

There exists a

b)

FK,% t(k) K

c)

(%)~

K-COmplete, normal,

fine, nontrivial filter on

(~)~. is

is such a filter.

FK,x-stationary.

PROOF. a) ~

b):

The argument of the metatheorem shows that (%)K

stationary and therefore

FK, ~ induces a filter on (~)K

implications follow easily. The results in is

[DP.M2]

FK,%-stationary or

is FK, %-

The other O

suggest that perhaps we have that either (%)K F~,%

is the CLUB filter on PK+(%) restricted to

[ ]K , or equivalently that either (%)< is F K ,~-stationary or ~ belongs to FK, ~ (where ~ = {Pc[%]KI~!P}). (In fact, the first part of the dis-

153

junction holds if K is huge with target % and the second occurs if V = L or, as it will be shown in the next section if ~ = K+). This alternative does not hold in general but we have something close to it: THEOREM 4. there is PROOF.

Given regular cardinals

K<%'~X

such that (%')< v ~ ~ F~,%

Suppose

is

~FK, %

or

FK,X, stationary.

then, its complement in [X]~, i.e. the set

S = {P ! [X]
K<% then either

f(P) = first element of

P

Considering the function f:S ÷ X defined

after the first gap, and using the normali-

ty of FK, % we see that there is an FK, % -stationary S I ! S such that for all PcSl,f(p ) is a fixed value ~. Since F is fine (i e. contains v K,X " all cones p = {PeE%]KIpIP} for pcPm(%)) this ~ cannot belong to K. th Thus ~eK. So, if an element P of S' has order type >K, its element must be above K.

Therefore,

then a stationary subset

S2

greater than

g:S 2 ÷ % by

K.

Define

is a stationary subset g(P) = ~,

S 3 i $2

From this we conclude that given X' =

SI

is not

FK, % stationary

contains only elements of order type g(P) = K th element of P.

and a

(x,)K is

There

X'<% such that for every p~S 3,

FK, ~, stationary.

is closed and unbounded in

S3nA X, is thus

PES3nA X, Pn~'

This is so because

(X)K

is not

PnX'~Ax,n(X) K.

T<X)actually occurs.

<
A = {P~[x]KIy+[Pny]=P-y} "just like"

P-y).

(i.e.

0

FK, % stationary and ~ is not in FK, ~

(but (y)K is FK, Y -stationary for some example due to J. Henle. Consider

PK(X) SO AX,CFK, X.

FK,x-stationary.

has order type K and so

The possibility that

is

(%)<

X ! P~(X') closed and unbounded the set X' ! P<(X) defined by {pePK(x) Ipn%'~X}

The set For

of

if

A is the set of

This set is in

X = {p~pK(%)Iy+(pny)=p-y} is club and stationary. We also have that ~v F < , x

We give an

The set

P~[X] ~ such that

Pny

F<,% because the set A X = A. Thus (x)K is not FK, %since if ~ e F< X then there is

a closed and unbounded set X ! PK (%) such that ~ ~ A X. But since K<X~y there is a bijection f from X to y which is the identity on ~. The set

Y = {f"PlPeX}

is a closed and unbounded subset of

PK(y) such that

Ay is in F~,T and is contained in ~. But this contradicts that in this case (y)< is F<,y-Stationary. §2.

We consider in this section the case in which X = K+.

PROPOSITION i. Then

K+(y) since

A X n Ord

Let

X

be a closed unbounded subset of

is an unbounded subset of

m+

P (~+).

154

PROOF. stages than

Given

y
a directed

let

pEX

subset of

be such that

X

y~p.

We will

whose union is an ordinal

in

construct in + K greater

y. STAGE i: For each ordinal

containing

it.

Put

~ in

up-p

let

be an element

p~

X.

X

Dl={p~l~up-p}u{p}. D2

STAGE 2: Close D 1 to obtain D 2 ~ D 1 such that set of

of

(Note we can obtain

such a directed

set

is a directed

D2

sub-

such that

}D21=ID!I+ N0).

STAGE 2n+l: For e v e r y ~ E sup(uD2n)-UD2n l e t containing it.

Put

STAGE 2n+2: containing

D2n+l=D2nU{P ] ~

Close

It follows

D =

closed and unbounded,

subset of

2.6

AXnK +

of

~ < K +,

PK(K +)

Therefore

if

and

+

a filter on

K-completeness,

the filter,

~, p

XsP

(K+), X

K +, thus we have

subset of

fine filter on

and so

(K+) K

CLUBK+ _c FK,K+ rE +

K + (it is fine because is a closed unbounded

is not

if

X

FK,K+ stationary.

is closed unbounded

i shows

which provides

normality,

or fineness.

that if we want

an "absolute"

The easiest

if we can maintain

sub-

On

then the

to produce

counterpart

to

to give up seems

unboundedness

is that,

with respect

if

K÷(%)

such a filter modifying

X ! PK(%)

if for all

that

cones

to be in

Another

condi-

of the filter

this fact.

a construction

is end-unbounded

P Se q (where p Se q means

to it).

the elements

to any normal measure witnessing

to

of the elements

(That is to say: even if we do not require

they must be stationary

We will define We say that

that for each

On the other hand,

of section

(%)K

tion we wish to maintain must belong

in

we must drop one of the three usual properties:

specially

of our filter.

is an ordinal

(AXo { $}) nK + .

The m e t a t h e o r e m

be fineness,

[DP.M2]

K c FK,K+ rE + because

(%)K,

uD

A^ ford c K+-~). p

set K^ = ~ -K contains

on

such that

pePK(K +) contains

FK,K+ ~ = C L U B

the other hand,

X

is a closed

since this is a ~+-complete normal

such that

X, D2n+2,

0

FK,~+ rK + __c CLUBK+.

given any

(in ZFC)

subset of

u D nn~

from the lemma

shown that

a measure

X

sup(uD2n)-UD2n}.

D2n+l to obtain a directed

Clearly D is a directed + K above y.

§3.

be a set in

D2n+l.

Finally we put

set of

pe

q

from [DP.M2].

p~PK(%)

there is q~X

is an end extension

of p).

155

A

Clearly every end-unbounded set is unbounded. not end-unbounded (with the exception of filter generated on

PK(%)

The cones

$=P<(%)).

p={qJp!q} are

Call SUPERCLUB the

by the end-unbounded and closed subsets of

P<(~). PROPOSITION. The filter SI~ERCLUB is trivial.

<-complete, normal and non-

It is not fine but all its elements are unbounded subsets of

P<(%). PROOF.

Let us first recall a definition from

filter on

P<(%) generated by sets which are

[DP.MI].

with respect to unions of end extension chains of length is

K-complete and normal but not fine

E-CLUB is the

end- unbounded and closed

([DP.MI]).

<<.

This filter

It is easy to verify

that SUPERCLUB=CLUB n E-CLUB and thus we have only to prove that this intersection is non empty. For this, note that the sets = {p~PK(~)lorder type of Since

p>~} are in SUPERCLUB. + SUPERCLUB is not < complete.

n{~:~e<}=~,

D SUPERCLUB is not the

least <-complete, normal filter in P<(%) containing {~I~
peP<(%) there is

~<~

such that

pu(~-up)eX). Using the filter

ideas

on

o f [DP.M2]

(~)<.

Given

we d e f i n e

X ! PK(~),

now a

K-complete

normal non-trivial

let

A"x={PE(%)K I There is an end extension chain {p~}~
If

X

is end-unbounded then

GK, % the the filter on

THEOREM 3.

G <,% is a

up~ =P}

(X) K

A" X #~.

generated by

D

{A"xIX superclub}.

K-complete, normal, non-trivial filter in

(~)K PROOF.

The

K-completeness and normality are easily verified.

To see that

G

is non-trivial, in the sense that it does not contain small subsets K,% of (%)<,we prove the following lemma. LEMMA.

If

B ! (X)K

is such that

[%-uB[=%

then B is not G <,%-

stationary. PROOF.

It is enough to show that there is a superclub set ~

A"~nB=~.

We put

~ = { p E P K ( X ) j for all

PeB(p-P#~)}.

such that

156

Clearly

~

is closed, and since

k-uB

is cofinal in ~, it is super-

unbounded. Therefore, of (~)K

0 if

K
then

of cardinality

GK, ~

contains the complements of all subsets

<~.

0

We will now show that the elements of respect to normal measures on (k) < LEMMA. PROOF.

If

[

Axn(k) K.

elements of X.

are of measure one with

is closed and unbounded then

The inclusion

belongs to

and

X

GK,k

is obvious.

Then

P =

v

A"x--Axn(%) < .

For the other one, assume that

p~

where

Since we can assume that (P~)$
is a continuous chain

P = K it is not difficult to construct a subchain of

$<< which

is an end extension chain. COROLLARY 4. sure, then

If

P

~<< is a chain of

0

A e GK, ~, K ÷ (~) and ~ is a witnessing normal mea-

A e ~.

PROOF.

There is an

(k)K~

SO

X c SUPERCLUB such that

A" X ! A.

But then

AXC~ and

A" X ~ ~.

0

Finally let us mention that if

X~

is closed and unbounded for each ~<~,

A A" X~ = ~<~

A" ~<~ A

X~

Note that both sides could be empty unless the sets

X~

are superclub.

REFERENCES [B.D.T.]

[DP.MI]

Barbanel, J., DiPrisco,C.A.& Tan,I.B. DiPrisco,C.A.& Marek,W.

Many-times huge and superhuge cardinals. J. Symbolic Logic 49(1984),pp 112-122. Some properties of stationary sets. Dissertationes Mathematieae CCXVIII(1982), pp.l-37.

[DP .M2]

Di Prisco,C.A.& A filter on ~rek,W. 591-598.

[Mig]

Mignone, R.J.

[k]K Proc.Amer.Math. Soc. 90(1984), DD.

On the ultrafilter characterization of huge cardinals. Proc.Amer.Math.Soc.

90(1984), pp.

585-590. [S.R.K.]

Solovay,R., Reinhardt,W. Kanamori,A.

Strong axioms of infinity and elementary embeddings.

Annals of Mathematical Logic 13(1978),

pp. 73-116. This work was partially supported by CONICIT grants SI-I129 & Capt 15-84

THE MODEL EXTENSION THEOREMS FOR ~ 3 -THEORIES

Itala M.L. D'Ottaviano Universidade Estadual de Campinas Instituto de Matematica,

Estatistica e Ciencia da Comp.uta$~o

Caixa Postal 6155 13100 Campinas, SP,

Brasil

Two generalized versions of Keisler's classical Model Extension Theorem are obtained for

~ 3 -theories.

These theories are three-valued with

more than one distinguished truth-value, reflect certain aspects of modal logics and can be paraconsistent.

~3-theories

were introduced in

the author's doctoral dissertation. §i. INTRODUCTION.

A theory

T

is said to be inconsistent if it has

as theorems a formula and its negation;

and it is said to be trivial

if every formula of its language is a theorem. A logic is paraconsistent if it can be used as the underlying logic for inconsistent but nontrivial theories

(see [I] and

[2]).

In a previous paper (see [4]) we introduced a three-valued propositional system,

~ 3 ' with two distinguished truth-values, which is para-

consistent and reflects some aspects of certain types of modal logics. In another paper (see [5] ) we axiomatized

~3

and established relations

between this calculus and several known logical systems, as, for example, intuitionism.

We especially emphasized the close analogy between

and Lukasiewicz'

~3

three-valued propositional calculus £3"

We also introduced the corresponding three-valued first-order

~3 -

Theories in whose languages may appear other equalities in addition to ordinary identity, and which extend the first-order predicate calculus , ~ 3 =" We proved the Completeness Theorem and the Compactness Theorem for ~ 3 - t h e o r i e s . The model theory for ~ 3 -theories reflects much of the classical model theory.

We were able to obtain in our doctoral dissertation generalized

versions of the following classical results: Keisler Model Extension Theorem, Lo~-Tarski Theorem, Chang-Eo~-Suszko Theorem, Tarski Cardinality Theorem, LSwenheim - Skolem Theorem, Joint-Consistency Theorem,

158

Craig Interpolation Lemma, Beth-Padoa Definability Theorem, the Quantifier Elimination Theorem and many of the usual theorems on complete theories and categoricity. In some cases, as for example the Model Extension Theorem and the Definability Theorem, we proved more than one generalization of the classical results, all of them compatible both with the many-valued aspects and the modal aspects of ~ 3 - t h e ° r i e s " Our aim here is to present two generalized versions of Keisler's classical Model Extension Theorem, asserting both the conditions to extend a structure ~ for a L3-1anguage In

L

of

T

to a model of the ~ 3 - t h e o r y

§2, after giving some basic notions and the axiomatization for

T.

~3-

theories, we emphasize several results and theorems which are necessary for the development of the last part of the paper. In

§3, after adapting the concepts of isomorphism and elementary equi-

valence, we define

F-substructures and discuss the case of certain

special sets of formulas ro After this, we present the two Model Extension Theorems for ~ 3 -theories. Both theorems generalize the classical Model Extension Theorem of Keisler and their proofs involve specific characteristics of many-valued and modal logics, reflecting the existence of more than one distinguished truth-value in the matrices defining ~ 3 - t h e ° r i e s " Finally, we give a three-valued version of Lo~-Tarski Theorem, which characterizes universal-open ~ 3 -theories; Tarski Lemma, on the union of elementary chains; and Chang-Lo~-Suszko Theorem, characterizing universal-existential ~ 3 - t h e o r i e s . In this paper, definitions, theorems and proofs, when similar to the corresponding classical case, are omitted. For definitions and theorems we use the nomenclature of [13]. We have also extended some of the above results about ~ 3 - t h e ° r i e s ~n-theories,

to

3 ~ n ~ ~o"

The results about Joint non-Trivialization,

Definability Theorems,

Complete Theories, Quantifier Elimination Theorems, Categoricity and the mentioned results about ~ 3 - t h e o r i e s will appear elsewhere. The content of this work is part of the results of our doctoral dissertation.

159

§2.

FIRST ORDER

~3-THEORIES. The symbols of a first-order

guage are the individual symbols, V,

the primitive

variables,

the function

connectives

~ , v

and

symbols,

~3-1an-

the predicate

V, the quantifiers

@

and

and the parentheses.

The identity

= must be among the predicate

In particular

cases,

other predicate

symbols

symbols

of every

L3-1anguage.

can be considered

as equali-

ties. We use f c

x,y,z

and

w

as syntactical

and g, for function for constants.

The definitions

p

of term and atomic

order languages; The definition a formula,

symbols;

variables and

formula

q,

then

VA

variables;

symbols,

variables

for terms.

is also the usual plus the condition:

is a formula;

A,B,C,

and

are the usual ones for first-

a,b ..... etc, are syntactical

of formula

for individual for predicate

etc, are syntactical

if

A

is

variables

for formulas. By a L 3 - 1 a n g u a g e of [13]) whose

we understand

logical

The truth-functions;

symbols Hv, H V

a first-order

language

are the ones mentioned and

H~

(in the sense above.

are defined by the following

tables:

AvB •

0.,

,

A

VA

0

0

0

0

I

½

1

½

½

1

i

i

I

0

{0,½,1}

and the set of distinguished

{½,1} are denoted by "V" and "Vd" respectively.

A

&

B

abbreviations

=def ~ ( T A v

will be used:

-~B)

AA =def -IV ~ B -~*A =def -,VA A >+ B =def V-dAy B A = B =def A ~+ B =def

~A

½

The set of truth-values The following

A

-~VA v B (A >+ B) & (~B >+ -,A)

A = B =def(A = B) & (B = A)

truth-values

160

is called weak negation or simple negation, implication, We present

the table of some of the non primitive A

AA

~*A *A

A

~ * strong negation, = basic

~ basic equivalence.

A

AA

0

0

connectives: ~-~

g

1

0

o

½ I

i

I

0

1

I

I

½I

I ½o

½

A=-B

A=B

_a °

I

o

L

o

0

0

t

1 0

0

0

0

Free occurrence

of a variable,

open formula,

I

closed formula,

variable

free term and closure of a formula are defined as in [13]o We let

b x l , . o o , X n [al,...,a n]

replacing

all occurrences

respectively; ed from

A

and we let

of

be the term obtained x I .... ,xn

by the terms

from

b

by

a I ..... a n

AxI ..... Xn [a I ..... an] be the formula obtain-

by replacing all free occurrences

of

x I ..... x n

by

a I ..... a n respectively. Whenever

either of these is used,

x I ..... x n

are distinct variables

Axl ..... Xn [a I ..... a n ] , a i D E F I N I T I O N 2.1. sists of : i) a nonempty

it will be implicitly and that,

is substitutible

iii)

I ~71 n

to

x i,

i = I .... ,n. ~3-1anguage

con-

f

of

L, an n-ary function f ~

I~I;

for each n-ary predicate

n-ary function

for

I ~ I called the universe of ~ ;

ii) for each n-ary function symbol from

in the case of

A structure ~ for the first-order set

assumed that

p~

from

l~I n

symbol to V.

p

of

L, other than

=, an

161

As in [13] ~f(a) ~ V

, we construct

the language

for each variable

L(~)

free term

a

;

of

and define a value

L(~f),

and

g-instance

of a formula A. We use

£

and

j

as syntactical variables

for the names of individuals

of ~f. DEFINITION in

L(~f) i)

if

A

contrary, ii)

= p~

2.2.

The truth-value

~f(A) for each closed formula

A

is given by: is

~(A)

if

A

is

a = b,

then

~(A)

= I

if

~f(a) =

@£(b);

on the

= 0; p(a I ..... an) , where

( ~ ( a I) ..... ~ ( a n ) )

p

is not

=, then

~f(A) =

;

iii)

if

A

is

~ B, then

~£(A)

is

H (~(B)) ;

iv)

if

A

is

VB,

~(A)

is

HV(~f(B));

v)

if

A

is

B v C,

vi)

if

A

is

~xB,

then

~(A)

= max { ~ f ( B x [ Z ] / Z E L ( ~ f ) } ;

vii)

if

A

is

VxB,

then

$(A)

= min { @f(Bx[Z] Z c L ( ~ ) } .

-7

DEFINITION for every

2.3.

then

A formula

~-instance

DEFINITION

then

2.4.

A' of

~(A)

A

of

is

Hv(~f(B), ~f(L));

L

is valid in ~ i f ,

A, ~(A') belongs

A first-order

~3-the°ry

to

and only if

V d.

is a formal system

T

such

that: i) ii)

the language of the axioms of

T, L(T), T

further axioms, iii)

is a

•3-1anguage;

are the logical axioms of the nonlogical

the logical axioms and rules of

T

AXIOM

i :

A(A ~+ (B ~+ A))

AXIOM

2 :

A((A ~+B) ~+

((B ~+ C) ~+ (A ~+ C)))

AXIOM

3 :

A((~A~+--,B)

~+ (B ~+ A))

AXIOM

4

A(((A ~+ -~ A) ~+ A) ~+ A)

AXIOM

5: :

AXIOM

6

7 •

x

AXIOM

8

Ax[a]

AXIOM

9 :

AXIOM i0

:

A(A(A ~+ B) ~÷ A(AA ~+ AB))

AXIOM

: :

Vx(x = x) =

y ~ (A[x] ~ A[y]) ~ @xA

VxA = A x[a] ~xA - - ~ V x ~ A

and certain

are the following, w i t h the

usual restrictions :

:

L(T)

axioms ;

162

A X I O M ii :

VxA : m ~ x ~ A

A X I O M 12

: -~@xA ~ V x ~ A

A X I O M 13

: ~VxA

- ~x-,A

A X I O M 14 :

V~xA =- ~xVA

A X I O M 15

VVxA E VxVA

RULE

:

A,A(A ~+ B)

RI :

B

RULE

R~

RULE

R3

VA

:

L

T A~

(@-introduction) :

C

@xA~ RULE

R4

(V-Introduction):

C

C = A C ~ VxA

A

is a Theorem of

T, in symbols

:

~ T A,

is defined in the standard

way. Observe that it is possible connectives

7

and

to define

~÷, instead of

L 3 - 1 a n g u a g e s , using the Eukasiewicz' v, V

and

m-

So, there is a close analogy b e t w e e n Lukasiewicz's (see [3],

[7], [8] and

THEOREM 2.1. connectives

7

~3

is a n o n - c o n s e r v a t i v e

and

THEOP~M 2.2. positional

[9]) and the underlying

~3

is a conservative

calculus with connectives

[6]),

the matrices

~3

In the case of classical is not the case for ~T~A

~-

DEFINITION

£3

with

extension of the classical *,

, &, =

and

~ has to be gradually

pro-

~. fortified

(see

have more than one d i s t i n g u i s h e d logic,

the equivalence

to the other logical ~3-the°ries'

~ behaves

symbols°

truth-value.

as a congruence

Unfortunately

since it is possible

to have

this

~T A ~ B

~B.

Hence, we introduce ce in

extension of

in order to obtain a relation compatible w i t h the fact that

defining

relation with respect

and

logic £3

~+.

The initial notion of equivalence [5] and

three-valued

logic for ~ 3 - t h e o r i e s .

theories 2.5.

a stronger equivalence, and A ~

~*, called strong equivalen-

which is a ~ ] - c o n g r u e n c e

relation.

B =def(A ~ B) & (~A z m B ) .

For this strong equivalence we obtain an Equivalence

T h e o r e m for

163

~3-the°ries

(see [5] and

[6]).

The following closure theorem and the theorem on constants are identical

to the classical ones.

THEOREM 2.3. if, If

for ~ 3 -theories

If

A' is the closure of

A,

then

~--TA

if,

and only

~--TA' . T'

is a ~ 3 - t h e o r y

for every formula

A

obtained from

of

T

T

by adding new constants,

and every sequence

then

el,o°.e n of new constants,

~-T A if, and only if, ~-T'Ax I, ... ,xnEel ..... en]" We observe

that if F is a set of formulas

in the theory

T, then

is the theory obtained from F by adding all of the formulas nonlogical

T[F]

in F as new

axioms.

THEOREM 2.4.

(Reduction Theorem for non-Trivialization):

non empty set of formulas

in a

~3-theory.

trivial if, and only if, there is a theorem of tion of negations

Let F be a

Then the extension T

T[F] is

which is a disjunc-

of closures of formulas of type

VA,

with

To finish this section, we discuss how to convert a formula

A A

in

F.

into a

formula in prenex form. DEFINITION operations

either

2.6.

on

If

A

A

is a formula in a

the classical prenex operations,

b)

replacement or

L, the prenex

are:

a) ~x

L3-1anguage

of a part

VQxB

with the usual restrictions;

of

A

by

QxVB, where

Qx

is

Vx.

By the above definition we can obtain the prenex form of every formula A, based on the following result. THEOREM 2.5.

If

prenex operation,

A'

then

is obtained from D-TA

A

with

A

in

L(T), by a

~* A'.

~3-theories

are three-valued

truth-value,

they can be paraconsistent

theories with more than one-distinguished and reflect certain aspects of

modal type logics. The details of a further study, this paragraph [5] and §3.

the proofs of the theorems mentioned

and other results about

~3-theories

in

can be found in

[6]. THE THEORY OF MODELS° The theory of models

of classical model theory.

for

~3-theories

reflects a great deal

164

In some cases, theorems,

there is more than one generalization

all of them compactible

of the classical

with the characteristics

of ~I 3 -theo-

ries. In the following, we give the two versions Model Extension Theorem for dl3 -theories. The proofs

of

many-valued

both theorems

and modal

logics,

reflect

appropriate

and the conventions

3.1.

Let ~f and

such that for

a I .... ,an in l~f I:

ii) p

If

from

u

[13],

w i t h the

~B is a bijective

for the

mapping

•3-1anguage

~ from

I • I to

L. I~ I

P~f (al ..... an) = P~B (~(al) ..... ~(an)) , for every predicate of L.

of I~fl

JB .

of

the existence

~(f~f (al ..... an)) = fdB (#(al) ..... ~(an)) , for every function of L;

If ~ is a mapping a

of

£B be structures

of ~f and

symbol

characteristics

involve

adaptations.

DEFINITION

i) f

Keisler

truth-value.

An isomorphism

symbol

the specific

and they specially

of more than one distinguished We shall use the notations

of the classical

u

from

l~fl

, we use i ~

to

THEOREM 3.1.

and i is the name of an individual

to designate

is an expression

by replacing

IJB I

of

the name of the individual

L(~f),

each name i by

u~

= JB (a ~) for every variable-free

term

for every closed formula

A

L(~f).

The following

are similar

definitions

~(a) of obtained

i#.

Let ~ be an isomorphism of

is the expression

a

of ~f and of

~ .

L(~f),

Then

and

to the classical

~(~f(a))

~f(A) = ~ (A ~)

ones°

DEFINITION 3.2. Two estructures ~ and ~ for L are elementarily equivalent, in symbols ~ z ~, if the same closed formulas of L are valid in

@£ and

~o

As in the classical

case,

then they are models A

is a closed

~f(A)

if ~

of the same

formula of

and

~

are elementarily

313 -theories.

But if

L, it is not possible

equivalent, ~f

to conclude

z

JB and that

= ~B ( A ) .

DEFINITION from

3.3.

l~f I

to

An embedding I~B I

tion 3.1 hold for all nonlogical al,...,a n

in

l~fl .

of

~f in

~

such that conditions symbols

f

is an injective

mapping

(i) and (ii) of Definiand

p

of

L

and all

165

D E F I N I T I O N 3,4. an e x t e n s i o n of m a p p i n g from

We say that ~f is a s u b s t r u c t u r e of ~

l~fl

when

to

I @fl is a subset of

I~ I

I~ I

is an e m b e d d i n g from

~ , or

~f in f;

ii)

p~f (a I ..... a n ) = p ~ (a I ..... a n ), for every

p.

In the conditions of the d e f i n i t i o n

T H E O R E M 3.2.

is

~ , that is:

f ~ (a I ..... a n ) = f ~ (a I ..... a n ), for every

are models of some

~

and the identity

i)

D E F I N I T I O N 3.5. ~B

,

3.4,

if ~f and

dl3 -theory T, we say that ~f is a submodel of ~B .

Let ~f and

~B

I~I.

be structures

for L, and let ~ be a

mapping from

l~fl to

Then ~ is an e m b e d d i n g of ~,f in

~

if,

and only if,

~f(A) = ~ (A~), for every v a r i a b l e free formula

A

of

L(~). Let F be a set of formulas in the ture for set of

L.

~f-instances of formulas in

D E F I N I T I O N 3.6. is a subset of ~f is a if

Let

l~B I,

~

and

implies

T H E O R E M 3.3. formula

A

L

~

~

in

be structures for L, such that

and that

~

is a

F,

-~ VA

is in

F.

If ~

in

A

L

is a

L.

I~I

We say that

r - e x t e n s i o n of

JB (A) e Vd, for every formula

if, and only if,

~f be a struc-

designates the

r.

Let F be a set of formulas

~(A) e V d

and let r(~f)

and let r be a set of formulas in

P - s u b s t r u c t u r e of

~[(A) ~ V d

then

]L3-1anguage

As in the classical model Theory,

in

~f

r(~) .

such that for every

r - s u b s t r u c t u r e of

~B (A) E V d for every formula

A

.~, in

r(~). PROOF.

If

~(A)

= 0, then

If r is a set of formulas b e l o n g s to

A

in

F,-~A

F, it does not seem p o s s i b l e to obtain the result above.

or

if

~f (A) = 0

~B (A) = i

In

we can conclude only that

~ (A) = 0.

On the other hand, we can prove that and

0

such that for every formula

fact, u n d e r such conditions, (A) = ½

~ (-~VA) = i.

implies

~(A)

= ½

implies

~ (A) = ½,

~f(A) = Io

It seems impossible also to prove the above result w h e n r is a set of formulas such that for e v e r y formula But,

if for every formula

A

in

A

F,-~ A

in

F,

and

VA is in VA

r.

b e l o n g to

F,

then

we can prove a stronger result. T H E O R E M 3.4. formula of ~ ,

A then

Let P be a set of formulas in

in

F, -I A

~(A)

and

= ~ (A)

VA

are in

F.

for every formula

L If A

such that for every ~f is a in

P-substructure

r(~f).

166

PROOF.

By T h e o r e m

3.3,

as

sufficient to prove that A

in

implies

~ (A) = ~ , it is

~ (A) = I for every formula D

(I)

If r is the set of all open formulas

are structures for is a substructure of

and ~

is a

and ~ and

If

in

Furthermore,

and

~

L,

~f and

~ , then

~f

~f. L

and

~

is

are e l e m e n t a r i l y equivalent.

If F is the set of all formulas in an L 3 - 1 a n g u a g e

• , or that

T H E O R E M 3.6.

in

P - s u b s t r u c t u r e of

is an e x t e n s i o n of

~ , then ~

~

• , we say that

~

~f is an e l e m e n t a r y

is an e l e m e n t a r y e x t e n s i o n of

is a elementary substructure of

e l e m e n t a r i l y e q u i v a l e n t to A

•

F - s u b s t r u c t u r e of

substructure of

formula

is a

If F is the set of all formulas in

F - s u b s t r u c t u r e of D E F I N I T I O N 3.7.

L ~

(2)

L

= ~ implies

r(~ ) .

T H E O R E M 3.5.

a

~(A)

~f(A) = I

~

and

~(A)

= ~ (A)

~

~ , then

°

~f is

for every closed

L.

in the case of

L 3 - 1 a n g u a g e s , it is convenient to observe

that if ~f is a substructure of

~

and e l e m e n t a r i l y equivalent to

there can be closed formulas

in

L(~)

A

such that

~f(A) and

~ ,

• (A)

are d i s t i n g u i s h e d values, but different ones. T H E O R E M 3.7 ~f is a If ~

If

x = e

belongs

P - s u b s t r u c t u r e of

is a s t r u c t u r e for

to the set F of formulas in

~ , then

L

and

~f(e) = ~ (e).

L, we define the

F - d i a g r a m theory of

~f as

in the classical case° D E F I N I T I O N 3.8. The structure such that,

Let

~f

~,f and

for the

~

be structures for

~3-1anguage

L(~)

if £ is the name for an individual

L

with

~fc ~

.

is an e x p a n s i o n of a

of

I~I

then

(~f) (£) = a. D E F I N I T I O N 3.9.

The

F - d i a g r a m theory of

~ 3 -theory w h o s e language is the formulas

A

LEMMA.

Then

a model of

~

:

~[ is a

DF(~f) , is the

and w h o s e n o n - l o g i c a l axioms are ~f(A)

belongs

to

V d.

Let F be a set of formulas in L

such that

F - s u b s t r u c t u r e of

~

l~fl

L, and

is a subset of

if, and only if,

~f

is

DF(~).

formula of the form A

L(~f)

such that

be structures for

D E F I N I T I O N 3.10.

formula

F(~)

(Diagram Lemma)

let ~f and I ~ I.

in

~f,

in

F,

A set F of formulas of x = y

or

x #

y

L

is in

every formula of the form

is a l m o s t - r e g u l a r F,

if every

and if for every

A [ x I ..... x n]

is in F.

167

THEOREM 3.8. L,

T

a

(Model Extension T h e o r e m I):

~3-theory

formulas

in

L.

w i t h language

Then ~f has a

PROOF.

Suppose

that such a

is not valid in A 1 ..... An,

~f,

fore

~ (~*A~) = 0

what

is impossible,

On the other hand,

T

A 1 .... ,A~ i,

Vd,

for

from

There-

= 0,

T.

T'

as new axioms.

of

1 ~ i ~ n.

~ (~*Aiv...v~*A~)

Hence

An

Let

T'

be the

L(~f)

as new

by adding all the non-

We shall show that

T"

is non-

then by the Reduction Theorem for n o n - T r i v i a l i z a t i o n

*Aiv...v~*A~,

and

~--~*AlV..oV~*A ~ T'

~f(A~)

Hence by the Theorem on Constants, by replacing

~*AlV...v~*An

!

by adding all the names of obtained

such that

of

if,

and ~ * A l V . . . v ~

that the condition holds.

T"

T

If

F,

~f-instances

is a model of

~

in

to

since

~

regularity

exists.

i, 1 ~ i ~ n.

there is a formula of type

from A~

~

formulas

for every

DF(~f)

If not,

F-extension

~f(A~) belongs

from

set of

~f.

then there are

and let be

logical axioms of

and F an a l m o s t - r e g u l a r

for

which is a disjunction of strong nega-

A 1 ..... A n

suppose

obtained

constants; trivial.

with

such that

~3-theory

T

in F is valid in

~-T ~*AlV" " " v ~*An,

@,f be a structure

F-extension which is a model of

and only if, every theorem of tions of formulas

L

Let

with

belongs

AI, ' .. "'A'n to

Vd

L-T~*AlV...v~*An,

the names by new variables

F, we have that is valid in

A 1 ..... A n

~f.

Therefore

~(A~)

for

r(~f),

i = 1 ..... n.

where A i results

Then

belong

in

by the almost-

to F and

= 0

so

for some

i, w h i c h

is a contradiction. By the Completeness proof becomes

Theorem

identical

the r e s t r i c t i o n

~

T"

has a model

to the classical

of

~ '

to

L

~ '

one.

is a

and from here on the

That is, we prove that

F-extension of ~f, w h i c h is

a model of T. COROLLARY. A

D

Let F be an almost regular set of formulas

be a set of formulas

is a disjunction structure

for

F-extension PROOF. If

Let

A 1 ..... A n

Len~na where DA(~F),

~f~ B

L

C of F'

containing every formula

of strong negations and ~

~

is a

of formulas

be the set of formulas are in

F'

is a model of

and

~

F.

~ * A l V . . . v ~*An,

Dr,(~)

Dr(~f). So ~@

then by the D i a g r a m Lemma

~f(B)

If

in

B

~

let

where

A

is a

~f. L(~).

then by the Diagram

(B) c V d and As

and

then there is a

extension of

AE£ 1 ..... £n ]

is the closure of ~ * A l V . . . v ~ * A n .

L

VXl...VXnA , in

A-extension of ~ ,

which is an elementary

in

~f~S)

=

~'(B),

is an axiom of

is distinguished.

Hence,

168

by Closure Theorem Therefore DF,(~)

~*AlV...v~*A

there is a

is valid in

n

r'-extension

6'

of

~

,

~,

which

and the proof follows as in the classical

is a model of

case.

If in the statement of T h e o r e m 3.8, we had "disjunction of negations of formulas

in

F", then we could not prove it by the same method.

Given an almost regular L(T),

set r of formulas

in

if we suppose that there is a model

~

tension of ~f

~

~AlV.

formulas i,

~, •

then we can not prove that .v~A n.

In fact,

A 1 ..... A n in

1 ~ i ~ n,

of the characteristics On the other hand, .~ ~ A l V . . . v ~ A n Theorem 3,8, if

such that

of

~ (A~) belongs

of the basic negation

~

of

By the observations

and if we construct

the theories

and

~ ~

~AlV...V~An,

we cannot conclude

with that

above,

almost regular DEFINITION

~

of

sets w i t h special

3.11.

DEFINITION

T'

and

A

~f(A') i

L

in F and

= 0 for some

theorem

we have to consider

A set F of formulas

is

in

of

and if for every formula

L A

L

if F is al-

VA belongs

is in

of

V-regular

F,

A-regular F, AA

to

F.

if F is

belongs

to

r.

is regular if F is

A-regular.

THEOREM 3.9. L

In fact,

A I ..... A n

~3-theories,

A set r of formulas of

3.13.

and

T", as in

characteristics°

3.12. A set F of formulas

DEFINITION

PROOF.

implies

in order to obtain a model extension

most regular and if for every formula

almost regular

for every on account

classical Keisler Model Extension T h e o r e m and which

uses the primitive n e g a t i o n

valid in ~ ,

Vd

= 0,

~3-the°ries"

~--T~AlV...V~An

F(~f),

w h i c h generalizes

language

to

~(~Aiv.o.wA ~)

that

r-ex-

implies

Ai ..... A 'n of

~f-instances

i,l~i~n,

theorem of

w h i c h is a

then we can not prove that T" is non-trivial•

in

V-regular

T

if we suppose that

~-T ~ A i v ' ' ' V ~ A n

A i ..... An

F,

and a structure $ for

~-T~AlV...V~An

if there are

we cannot conclude

L(T)

Let

~ be a structure

and let r be a T

V-regular

which is a disjunction then ~- has a

We construct

the

for

L, T

a

~3-the°ry

set of formulas of L. of negations

with

If every

of formulas

in r is

P-extension which is a model of T. ~3-theories

T' and

T"

like in the proof

of Theorem 3.8. If

T"

is trivial,

there are formulas

by the Reduction Theorem for non-Trivialization, A',.. ,A' I " n

in

r(~)

such that

~T~*AI v "

'

"

v~*A' n

169

and

~(A~) c V d,

3.8,

i = i .... ,n.

By the Theorem on Constants,

we can obtain formulas A I ..... A n such that

By hypothesis, some

~f~

i i = i,

~VAlv...v ~VA n . n, and

That is a contradiction,

A~

in

so

as in T h e o r e m

~ T ~ *AlV-.-v ~ *An"

Therefore,

we have

~(A~)

= 0 for

F(~)

T"

has a model

~ '

The proof then follows as in Theorem 3.8. THEOREM 3.10. language has a

L

Let

~f be a structure

and let F be a

r-extension w h i c h

If

~(AA~) If

is a

(A~) and

~ ~A~)

~T~AlV...V~An

L, T

a

in L.

and only if, of formulas COROLLARY

belong

~

~

is a

COROLLARY guage L

2o

i, I ~ i s n .

for every

has a

containing

So,

i, 1 ~ i ~ n.

Hence

D

Theorem II). L

Let

T

which

for

set of

is a model of

T

if,

is a disjunction of negations

~.

every formula of

~ be a structure

and let F be a regular

F-extension w h i c h

of formulas

F-extension

Let ~

and

Therefore,

Let F be a regular set of formulas

is an elementary

P,

A i ..... A'n

~AlV...v~A n .

every theorem of

set of formulas and

Vd,

w i t h language

junction of negations which

for every

in

which is a model of T, we have that

to

in r is valid in

I.

~f

~f.

W-instances

which is a contradiction.

Then

If

then every theorem of T which

A I .... ,An

= 0,

with

L.

in F is valid in

then there are

(Model Extension

~3-the°ry

formulas

of

with the formulas

of ~

implies ~

THEOREM 3.11.

~3-the°ry

i, 1 ~ i ~ n .

F-extension

~ A i ~ . . . v ~ A ~) = 0,

a

~f,

~f~A~)

= i, for every

~

of formulas

~T~AlV'''v~An'

A 1 ..... A n such that

L, T

set of formulas

is a model of T,

~ A l V o. .v~A n is not valid in of

for

A-regular

is a disjunction of negations PROOF.

D

~,

in

ro

be a structure

L

VXl.ooVxnA , If

and let A be a where

A

~ is a structure

then there is a

extension of

in

F-extension

is a disfor

L

~ of

~f. for L.

and let F be an almost regular

T a ~ 3-theory with lan-

set of formulas

in L, such that

~A and VA belong to F for every formula A in F. Then, ~f has a F-extension which is a model of T if, and only if, every theorem of T which

is a disjunction of negations

of formulas

The Model Extension Theorem I (Theorem 3.8) ~3-theories,

seems to be natural

in ~f . for

on account of the role played by ~* and the Reduction

Theorem for non T r i v i a l i z a t i o n The Extension

in F is valid

in ~ 3 - t h e ° r i e s "

T h e o r e m II (Theorem 3.11)

seems to be more compatible

170

with the main characteristics for

of

~3-theories,

besides more meaningful

~ 3 -theories and general many-valued theories and model logics.

But it seems not to be possible to obtain each one of the two versions from the other.

That is, it seems that Theorem 3.8, does not imply

Theorem 3.11, and vice-versa. It is convenient to observe that if r is an almost regular set of formulas in F,

L

such that

~

A

and

AA belong to F for every formula A

in

then we can prove a result similar to Corollary 2 to Theorem 3.11,

though F can be not regular. We yet observe that, in the proofs of some of the following theorems, as the case of Lo~-Tarski Theorem, in the Joint non-Trivialization

Chang-Lo~-Suszko

Theorem for

Theorem and also

~ 3 -theories, when we apply

one of the model extension theorems, we need only a sufficient condition to

F-extend a structure

~

for

L(T) to a model ~

of the

~3-theory

T. In these cases, considering the special characteristics of the sets F , we can indistinctly apply Model Extension Theorem I, or Model Extension Theorem II. Now we study three-valued versions of Lo~-Tarski Theorem and Chang-Lo~Suszko Theorem. THEOREM 3.12. to an open T

is a model of

PROOF.

(Lo~-Tarski Theorem): A ~ 3 - t h e ° r y

~3-theory

If

T

T

is equivalent

if, and only if, every substructure of a model of

T.

is equivalent to an open

show that every substructure

~3-the°ry

~X of a model

~

By Theorem 3.2, every open formula valid in

of ~

T', it suffices to T' is a model of T'

is valid in

~f , so

is a model of T'. On the other hand, we first prove that if F is the set of open formulas in T, if

F'

is the set of formulas in r which are theorems of

if every structure for

L(T) in which all the formulas of

is a model of

T

T, then

logical axioms are in theorem of

T

F.

is equivalent to a Then,

is valid in ~ ,

~3-theory

F'

T, and are valid

whose non-

it suffices to show that if every open then

@X is a model of T.

a regular set, by the Model Extension Theorem II, ~f has a

So, as F is F-extension

which is a model of T. DEFINITION 3.13.

D

A sequence

chain if for each n, ~ n + l

@~I' ~ 2 .... of structures

is an extension of

~n"

for

L

is a

171

DEFINITION

3.14.

Given a chain of structures

the chain is the structure verses n

of the

~n;

if

such that all of

~ whose universe

are individuals

V d, for every DEFINITION

3.16. n,

THEOREM 3.13.

A'

~

Lemma):

formula of A.

in

L(~fn),

If

A

is atomic,

If

A

is of type

hypothesis If

A

Let

A

%(VB)

~fn(A)

the result ~B

or

~fl' ~2'''"

~(A).

~xB.

L(~n) ;

is an elementary

extension

BVC,

if

the result

tables

of

= 0

~(~EB) = ½ ;

follows

H

that

and

A

chain, ~fn"

is a closed on the length

from the induction

H

~n(VB)

= 0

{f(B) = i

if, and only if,

or

~(B)

if, and only if,

= ½ , then

such that

~k(Bx[i])

of each

is obvious.

~f(%xB) k

of ~fn"

We use induction

= i if, and only if,

we choose

and we have that

=

VB, we observe

be of type

L(~),

If

~fl' ~f2 ..... such

extension

case, we must show that if

and the definition

for each i in in

then

is of type

~[(B) = 0;

chain is a chain

of the chain is an elementary

As in the classical

.... and belongs to

A, n = 1,2...

is an elementary

PROOF.

is an

..... ak).

of

An elementary ~n+l

(Tarski's

then the union

then there

~fn and we set

If ~ is the union of the chain ~ i , ~ 2 L, then A is valid in ~ if L ( A ' )

~n-instance

that for each

of

(a I ..... a k)

n

P ~ (a I ..... a n ) = p ~ n ( a l DEFINITION 3.15. is a formula of

L, the union of

a I ..... a k are in this union,

a I ..... a k

f ~ ( a I ..... a k) = f~f

A

for

is the union of the uni-

k>n

if

= ~.

@£n(Bx[i])

~[(Bx[i])

= ½ for some i

and i is a name in

~f(@xB)

= I

= 0

we proceed

L(~k) , similar-

ly.

D

THEOREM 3.14. equivalent

to a

(Chang-Lo~-Suszko ~3-theory

whose non-logical

and only if, the union of every PROOF. T'

As in the classical

whose non logical

that the union T'.

Let

L ( ~ k ).

~f of a chain

~Xl...~xnA

For large enough Hence

axioms

k,

Theorem):

A

axioms

chain of models

case,

if

T

~3-theory of

be an ~f-instance

~(qXl...~XnA)

cVd

and

to a

of T'

of a non logical

~fk(A[il ..... £n ]) e V d for some

if,

is a model of T. ~3-The°ry

then if suffices

9,fl, ~2 .... of models

is

are existential T

is equivalent

are existential,

T

axiom of T'

i I ..... i n

~f is a model of

to prove

is a model of

T'.

in

172

Now suppose that every union of a chain of models of T.

It also suffices to show that if ~

T

is a model of

is a structure in which every

existential theorem of T is valid, then ~f is a model of T. as in the classical case by constructing a chain ~[I = ~' ~ n of

~f2n+l"

If

~

valent,

is a model of

T

and

~f2n+3

is the union of the chain, as then ~[ is a model of

~ and

~fl' ~f2 ....

We proceed such that

is an elementary extension

•

are elementarily equi-

T.

The second part of the proof is possible because the set of all universal formulas of

L(T)

is regular, in the sense of Definition 3.13. REFERENCES

[ I] ARRUDA, A.I.

A survey of paraconsistent logic, Mathematical Logic in Latin America, North-Holland, Amsterdam, (1980), pp. 1-41.

[ 2] ARRUDA, A.I.

Aspects of the historical development of paraconsistent logic (To appear)

[ 3] BORKOWSKI,L.

(ed.) Selected Works of J. Lukasiewicz, NorthHolland, Amsterdam (1970).

[ 4 ] D' OTTAVIANO, I.M.L. & da Costa,N.C.A.

Sur un probl~me de Ja~kowski, C.R. Acad.Sc.Paris,

[ 5] D ' O T T A V I ~ O , I.M.L.

The completeness and compactness of a three-valued

270A (1970), pp. 1349-1343.

first-order logic.

To appear in Mathematical Logic,

V Latinamerican Symposium on Mathematical Logic, Marcel-Dekker. [ 6]

Sobre uma teoria de modelos trivalente (Thesis) Universidade Estadual de Campinas, Campinas

[ 7] ~.UKASIEWICZ,J.

On the principle of contradiction in Aristotle, Review of Metaphysics

[

8]

(1982).

XXIV (1971), pp. 485-509.

Philosophische Bermerkungen zu mehrwertigen systemen des Aussagenkalk~lls,

C.R.Soc.Sci.Lett.

Varsovie 23, (1930), pp. 51-57 (Translation to English in [3] , pp. 153-178).

173

[ 9] ~UKASIEWICZ,J. & Tarski, A.

Untersuchungen Nber den Aussagenkalk~ll, C.RoSoc. Sci.Lett.Varsovie 23 (1930) pp. 39-50 (Translation to English in [3] , pp 131-152).

[I0] RASIOWA,H.

An algebraic approach to non-classical logics, North Holland, Amsterdam (1974).

[II] RESCHER, N.

Many-valued logics,

McGraw-Hill, N.York (1969).

[12] ROSSER, J.B. & Turquette,A.

Many-valued logics, (1952).

North-Holland, Amsterdam

[13] SHOENFIELD,J.R. Mathematical Logic. (1967).

Addison Wesley, Reading

COMPLETENESS THEOREMS FOR THE GENERAL THEORY OF STOCHASTIC PROCESSES Sergio Faj ardo ~* University of Wisconsin Department of Mathematics Madison, Wisconsin 53706 U.S.A.

§0.

INTRODUCTION. Adapted Probability Logic, denoted Lad, is a logic adequate for

the study of stochastic processes.

In this paper we show how to axio-

matize in Lad the basic notions of the general theory of stochastic processes

(i.e. stopping time, martingale,

adapted, progressively mea-

surable, optional and predictable stochastic processes) and prove a "completeness" theorem for each such notion. The study of stochastic processes has been for the last forty years one of the main topics of research in probability theory. The development of the theory of martingales and Markov processes has brought many new ideas into this field. Concepts such as filtration of o-algebras and stopping times are now of fundamental importance after the work of Levy, Doob, Ito, Chung, Hunt and others. more advanced tools has appeared.

Along with these ideas the need for The "General theory of Processes" is

the branch of probability that has as its subject of study all these new objects.

Among the many concepts studied by "general theorists"

(besides the basic notions mentioned above) are: stopping times of different types, local martingales,

semimartingales,

generalized stochas-

tic integrals, stochastic differential equations, etc.

Good sources of

information about these topics are [D], [DM!] , [DM2] , IMP] and the Lecture Notes in Mathematics published regularly by the members of the Strasbourg school of probability

~*

Current Address

:

Department of Mathematics University of Colorado Campus Box 426 Boulder, Colorado 80309

(U.S.A.)

175

After the introduction of the Loeb measure in [Lo] and Anderson's nonstandard construction of Brownian motion [An],

nonstandard analysts

have developed a new approach for the study of stochastic integration (see [KI], [Pa], [Li], [HP], [Pe]) ([KI] , [HP], [C])

and stochastic integral equations

that is more intuitive and seems to be very well

suited for direct applications. What does logic have to do here?

Hoover and Keisler realized that many

of the concepts in [K I] could be naturally expressed in a language with integral quantifiers and conditional expectation operators, and that some of the properties of the hyperfinite adapted spaces introduced in that paper were general model theoretic theorems in the logic constructed with this language.

Thus, adapted probability logic was born.

logic first appeared in [K2].

This

Later Rodenhausen in his thesis [R] proved,

among other things, a completeness theorem for Lad. Hoover and Keisler in [HK] introduced different notions of elementary equivalence for Lad and using a model theoretic approach gave direct applications to probability theory. Keisler in [K 3] tied

Lad with the two previously known probability

logics L A P and LAf (see [K3] , [K4] , [HI] , [H2] , [H3] , [H4]) by introducing a new logic, the so called Probability Logic with Conditional Expectation of LAE

LAE.

Lad was then reintroduced as a two sorted form

and in this way Keisler in [K3] , [K 5] gave a new (simpler) proof

of Rodenhausen's completeness theorem for Lad. Keisler [K 3] is an upto-date account of the development of the different probability logics including

Lad.

For more on

LAE

see Fajardo [F].

A description of the contents of this paper is as follows

: Section 1

contains an introduction to adapted probability logic that "almost" does not require a previous knowledge of the other probability logics, but Keisler's survey paper [K 3] is strongly recommended.

In section 2

we show how to axiomatize the concept of stopping time and prove our first completeness theorem.

In section 3 we give a set of axioms and

prove a corresponding completeness theorem that simultaneously covers adapted, progressively measurable and optional stochastic processes. We also give a different set of axioms for predictable processes and prove its completeness.

Finally we present a completeness theorem for martin-

gales. The reader is assumed to be familiar with the basic concepts of probability theory such as random variable and conditional expectation which can be found in any elementary textbook in probability

(for example

176

[As]). We do not expect readers to know the general theory of processes, so all the basic definitions are presented. Elliot [El is a good introductory book covering the concepts studied here.

We do not use nonstan-

dard analysis and only require some familiarity with basic model theory. §i.

DEFINITIONS AND BACKGROUND: We introduce adapted probability Logic

original presentation in [K2].

Lad

following Keisler's

An alternative way of defining

Lad as

a two-sorted form of logic with conditional expectation LAE was given recently in Keisler [K3]. We follow the former approach since it allows us to go directly into the basics of

Lad

without assuming a previous

knowledge about the other probability logics.

Basic notions from adapt-

ed probability logic such as adapted model, interpretation and adapted elementary equivalence axioms for

Lad

(weak and strong) together with a complete set of

are given.

We also present all the basic definitons

from the general theory of processes that are relevant to our exposition. Those readers familiar with Keisler [K 3] will realize that we do not worry about admissible fragments LAa d of Lad. Instead, we just work with A = HC (the hereditarily countable sets), but all results generalize in the obvious way to fragments with A c HC and ~ s A. Throughout this paper we assume that so-called "usual conditions" of a filtration of o-algebras (see definition I.I below). These are, as the name suggests, the kind of assumptions that probabilists working with the type of structures that we are going to be dealing with usually make.

A

theory of processes that does not assume these conditions has been partially developed but it is at a very early stage and it does not yet have important applications.

For more on this subject see [DMI].

DEFINITION i.I (a)

An Adapted Probability Space or Stochastic Base is a structure = (~,(Ft)t~[O,l],P) where: (i) (ii)

P

is a complete probability measure on ~.

(Ft)t~[0,1]is an increasing filtration of

o-algebras of

P-measurable sets satisfying the Usual Conditions, (I)

F0 is Complete: in

(2)

F 0 and

(Ft)t~[0,1]is Right Continuous Ft

i.e.:

all the P-null sets are contained

=

n F s>t S"

: For all t c [0,I]

177

(b)

Let

M

be a Polish

on ~ if

is measurable

X(t)

X

Let

instead of X

and

Y

X(w,t)

(i)

(ii)

processes from

= Y(w,t)

(i)

X(w,t) of

definition

Y

Y

if for almost all

if for all

on

ws~

for

t s [0,i]:

of adapted probability

Symbols

of

logic.

Lad are:

Countably

(2)

One Probabilistic

many time variables:

First Order Function each

(iv)

Intesral

i ~)

listic variable

and from

quantifier

Conditional

The Non-Logical

connectives:

connectives

ne~

functions (iii)

variable:

tl,t2,...,Sl,S2,...

w

Connectives: (2)

~n

:

the function

into

symbol:

Expectation

Sy~ols

~ ,A •

of

for

~.

f

operator

Lad

w i t h two arguments,

symbol:

E[I].

are stochastic

the first argument

and the second for time variables.

seen to w o r k

symbol F

F ~ C ( ~ n) = the set of continuous

carried out using just one stochastic for countably

process

process

symbols

for the probabi-

Most proofs

symbol

many stochastic

are

X, but everything processes.

1.3.

The set of (i)

on ~ taking values

a.s.

(i)

(I)

(a)

we write

Variables:

(ii)

DEFINITION

Sometimes

1.2.

The Logical

is easily

to the product measure

Then we say:

t s [0,I]:

= Y(t)

space).

Process

X = (X(t))ts[0,1 ].

be stochastic

X is a Modification

DEFINITION

(Xi(...):

and

metric

Stochastic

on [0,I].

all

Now we give the formal

(b)

with respect

X is Indistinguishable

X(t)

separable

is an M-valued

~ is the Borel measure

the same space M.

(a)

(i.e. complete

X:~ × [0,I] ÷ M

P x B where

(c)

space

A function

Lad-terms

For each

is defined

r ~ 4 + , [X(w,t)Fr]

inductively

as follows:

is an atomic

term

(see remark

1.9). (ii) (iii) (iv)

(v)

Each time variable Each real

ra~

If T is a term, If

T 1 ..... T n

t

is a term.

is a term. so are are terms

fTdw, and

fTdt

and

F E C ( ~ n)

E[TIs] then

(w,t).

F(T 1 ..... ~n )

178

is a term. (b)

Free and Bound variables the integral

/~dx

in a term are defined

binding

x, in

of s are bound and w and t are free, first occurrence we shorten

of s is bound

this expression

to

as usual, with

E[~Is](w,t) and in

E[~Is](w,s)

and the second E[TIs]

or

all occurrences is free.

E[Tls](s).

the Sometimes

A closed

term is a term with no free variables. (c)

The set of

Lad-Formulas

(i)

For each Lad-term

(ii)

If ~ is a formula

(iii)

is the least set such that ~,[Te0] so is

If ~ is a countable variables

then

is an Atomic

formula

~ ~.

set of formulas

A ~ is a formula.

with finitely many A Sentence

free

is a formu-

la with no free variables. (d)

There

is one natural way of extending

la given in (c). r c ~n[0,1] guageo

We can add probability

and allow formulas

does not increase

of [K 3] for a rigorous things

tifiers

smoother.

in situations

quantifiers

of the form ( P ~ r )

As for the Logic with Integral

addition makes

the definition

the power of

(PEer) with

~ in our lan-

Quantifier Lad

proof of this fact) their presence

L w f, this i (see Section 3.6

but sometimes

We will not hesitate where

of Lad-formu-

it

to use these quan-

helps

the reader

to

have a better understanding. Remark

3°9

suggests

other possible

extension

of the definition

of

Lad-formula. DEFINITION

1.4.

An Adapted Probability A = (~,(Ft) t~[O,I],X,P) process

Structure

for Lad is a structure

= (~,X) where X is a real valued

on the adapted probability

Before we can give the definition Lad in a model,

(Model)

stochastic

space ~. of the notion of interpretation

we need to introduce

the main concepts

of

of the general

theory of processes° DEFINITION

1o5.

Let ~ = (~,(Ft)ts[0,1],P) X a real valued (a)

stochastic

be a fixed adapted probability process

X is Right Continuous all

we~,

space and

on ~.

w i t h Left Limits

the path X(w,-):[0,1]

÷~

(r.c.l.l.)

if for almost

is right continuous

with

left limits. (b)

X is Adapted with respect

to (Ft)t~[0,1 ] if for all t~[0,1]

the

179

random variable X(t) is Ft-measurable. (c)

X is a Martingale

(d)

if the following two conditions

(I)

For each t, X t is integrable

(2)

For all

s ~ t, E(XtlF s) = X s

X is Progressively

the restriction of

measurable,

where

[0,t].

A !

function

B([0,t])

~ x[0,1]

tic process.

(e)

X

to ~ x[0,t]

is the

is F t xB([0,t])-

o-algebra of Borel sets in

is said to be a Progressive

IA(W,t)

is a progressively

The Pr0gressive

(Ft)ts[0,1 ] is the progressive

a.s°

Measurable with respect to (Ft)tg[0,1 ] if for

all ts[0,1]

dicator

hold:

~-algebra

o-algebra on

set if the in-

measurable

M associated

stochas-

to

~ x [0,i] generated by the

sets.

A random variable

S:~ ÷ [0,I] is a S t o ~ w i t h

respect

to (Ft)ts[0,1 ] if for all t s [0,i] {ws~:S(w)< t} ~ F t . ing times are sometimes called Optional Times) o (f)

The

o-algebra

sets to

B

0 on

~ x [0,i] generated by the

such that for each

F t and

Bt =

n

Bs

respect (g)

The

Predictable

P on

A x Is,t] with

G-algebra with respecto

As u Fr is called the r<s to (Ft)te[0,1 ].

if it is measurable with respect

In the probability

literature

the Optional

and Predictable

ing sections

in [D],

o-algebra

to P.

there are other ways of defining

o-algebras.

[DM I] and IMP].

See for example the correspond-

The definition of Optional

that we have given is taken from

[K 3] and we think it is easier

to understand

for the reader not familiar with stochastic

The following

theorem gives an equivalent way of introducing

nal and Predictable them.

with

...................

~ x [0,i] generated by the sets of the form

AsF 0 and

X is Predictable

NOTE:

~

belongs

o-algebra

if it is measurable with respect to 0.

o-algebra

A x {0} with

(i)

t g [0,1]B t = {w:(w,t)sB}

to (Ft)ts[0,1 ]

X is Optional

(h)

p x ~-measurable

is called the Optional

s>t

(Stopp-

o-algebras

A proof can be found in

and establishes

processes. the Optio-

the relationship

between

[DMI].

THEOREM 1.6 Let (a)

~ = (~,(Ft)te[0,1],P) The Predictable

be an adapted probability

(Optional)

o-algebra on

by the real valued continuous stochastic processes

on

~.

space,

then:

~ x [0,i] is generated

(right continuous)

(Ft)-adapted

180

(b)

P ~ 0 ! M. (These inclusions are usually strict).

DEFINITION

1.7.

(Keisler [K3]):

An Interpretation

of Lad i n a n L a d model A is a function

that assigns to each Lad-term T(w,t) a P × Bn -measurable (~n is the Borel measure on [0,I] n) T A such that:

[r (i)

If T is [X(w,t)~r]

then

if if

TA(w,t)=~-r

~ X(w,t) (ii) (iii)

If T is

(v)

/edt

then

LEMMA Lad.

otherwise

For each term

T A = /eAdB.

(~,w,s) the following conditions hold:

(E[T(~,w,s) Is](w,a)) A

is

0 × Bn-measurable,

Borel o-algebra on [0,I] n, n the Optional o-algebra. (b)

X(w,t) e r X(w,t) < - r

If T is a time variable then TA(a) = a(as[0,1]). If T is /@dw then T A = /TAdp.

(iv)

(a)

function

For each n-tuple

b

where B n is the

is the length of

~

and 0 is

we have:

i-

(E[T (~,w, s)] s] (w,a)) A = E [ r A (~b , ' , ' ) 1 0 ] ( w , a )

2-

For all as[0,1], (E[~(~,w,s) Is](w,a)) A = E[TA(b,,,a) IFa](W)(P a.s.)

1.8.

(Keisler

For each term

tions agree at

[K3]): T(w,~)

T(w,~)

~

in

for P-almost all w.

B a.s°

[0,I], any two interpretaIn particular, in

A

if

agree at

w

is T(a)

We can see from the definition of interpretation of

in a model that the interpretation process.

×

Every model A has an interpretation of and all

not free in T(~) then any two interpretations for all ~ in [0,i]. REMA~( 1.9.

P

Lad

of each term is a bounded stochastic

The process X may be unbounded but the atomic terms [X(w,t)Fr]

truncate X at r and consequently all the terms, built from the atomic terms, are bounded. DEFINITION i.i0. (i)

(Keisler [K3]):

The Logic Lad has the following axiom schemes: (0)

Axiom schemes for two-sorted

(I)

Axioms for the conditional a-

LAf(see [K3])

e~pectation operator:

E[T(u,~)Iu](w,s ) = ~[T(v,t) Iv](w,s ) v do not occur in t.

b - ~E[~ I t](w,t)

where

u

and

- Tdw = /E[~It](w,t ) .E[TIt](w,t)dw

181

(2)

Axioms

for the filtration

a - fF(t)dt = r. flF(x)dx 0~

For any

and stochastic F:[0,1] ÷ ~

processes:

continuous

with

= r

b -

s-
÷E[TIs]

= E[E[TIs]It].

C -

A V //flT(W,S) m n

- ~(w,t) l"

max(0,1-1s-tl'n)dsdtdw

1

<

m°n

Lad has the following

(ii)

a -

Modus ponens:

b -

Conjunction:

c -

Generalization:

l-~÷[T(w,s,t)

Rules

of Inference:

~,~ ~ ~ ~ {~ ÷ ~I~ E F} ~ ~ ÷ A

e 0]

I-~÷f~(w,s,t)dwe

0

where w is not free in ~. 2-~÷[T(w,s,t) ~ 0] l-~÷fT(w,s,t)ds e 0 where THEOREM

Ioli.

!Rpd~enhausen's of

s is not free in ~.

c o m p l e t e n e s s a n d soundness, for

Lad-Sentences

PROOF.

See

To conclude

[K 3]

has a model

ed in [K3].

two notions

They were introduced context

Here we just present

and their

of elementary

by Hoover

importance

the definitions

equivalen-

and Keisler for

in

Lad is explain-

and some properties

proofs.

DEFINITION

1.12.

(Hoover

and Keisler

[HK])

Let A and B be models. W

(a)

A is said to be weakly Lad-Equivalent if for every Lad-Sentence

to

B, denoted by

A ~ B,

0:A ~ @ if and only if B ~ @. S

(b)

A is strongly each free:

THEOREM

1.13

~

in

Lad-Equivalent [0,i]

A ~[~] (Hoover

W

(i)

A ~ B

to

and formula

B, denoted by ~(~)

A ~ B, if for

of Lad in which w is not

if and only if B ~ [ ~ ] . and Keisler

Let A and B be models.

set

B

this section we present

in a probabilistic

without

A countable

and [K 5]

ce for adapted models. [HK]

Lad ) .

if and only if ~ is c--onsistent with Lad.

[HK])

The following

are equivalent:

182

(ii)

~±~ere is a set ~

each

(iii)

(b)

If

~

B

A

=>

X

and

s

A ~

-

If

A ~[~]

if and only if

[HK~:

Y

of

B if

B

are stochastic processes X

then

(~,X)

and only if

over time variables (d)

~(~):

W

fication (c)

and formula

(Hoover and Keisler

S

A

T ~ [0,i] of measure one such that for

T

For each L ~-term T(~) with no integrals over time variables • A 'ta) ~ = T B ta) '~ for almost all ~ in [0,I]

THEOREM I. 14. ~a)

in

for

Y

is a modi-

5 (~,Y).

each

and all

Lad-term T(~) with no integrals in [0,I], TA(%) = TB(~).

a

S

and

T c [0,I] has measure one --

and

on

s

A ~ B

with time parameters

W

restricted

to

T

§ 2. A COMPLETENESS

then

A ~ B. "Just as the seemingly

THEOREM FOR STOPPING TIMES.

trivial definition of the derivative

contains

in germ all of the calculus,

and it its discovery may have involved as much genius as the whole development

that followed it, the seemingly

(due to Doob)

is the cornerstone

~ith this statement,

Dellacherie

te the study of stopping times it seems appropriate

trivia± notion of stopping time

of the "General Theory of Processes" and Meyer in [DM I]

(sometimes

called optional

some few lines to this basic concept. and it immediately

suggests

stochastic processes

a natural

on

~ × [0,i].

This is precisely

times take into account

bability

space.

set of axioms.

is very simple

We first have to

defined on ~ and not

We do not work with

L~I,

X(w,t)

the because

the time structure of the adapted pro-

Instead of artificially

using a stochastic process

and giving an axiom that characterizes

processes

that are time independent

primitive

symbol the random variable

Lad-terms

(definition

in place of

the content of

logic designed for the study of random variables,

stopping symbol

Thus

logic, we dedicate

The definition of stopping time (def lo5.d)

observe that stopping times are random variables probability

times).

that in this paper where we intend to study the

general theory of processes using adapted probability this section.

"

(page 115) initia-

those stochastic

(i.e. random variables), symbol

S(w).

we take as

The definition of

1.3) is modified in the obvious way with [S(w)~r]

[X(w,t)~r].

All the results

valid with this modification.

stated in section

Let us first introduce

that we can follow the usual probabilistic

practice.

i remain

some notation so

183

NOTATION. (a)

From now on we are going to shorten the expression [S(w)~r]

(b)

Given

to

Xr(t)

Lad-terms

formula

and

~ and ~ we write

719 - ~Idw

7(4 - min(~,~))dw

~ 0

[X(w,t)rr]

and

S r respectively.

and

~ = ~ a.s. instead of the Lad-

~ ~ ~ a.s.

instead of

~ 0.

DEFINITION 2.1. A model A = (~,S) = (~,S,(Ft)te[0,1],P) is said to be a Stopping Time model if S is a stopping time with respect to (Ft)te[0,1]" DEFINITION 2.2.

Let

ST

be the following set of axioms in

(ST1)

Axioms for

(ST2)

A + (0 ~ S r ~ i a.s.) rs~

(ST3)

A + (Pt e l)(E[min(Sr,t)It](t) re~ (Intuitively, is

Lad:

Lad (see definiton 1.9)

for almost all

= min(Sr,t)a.s.)

t

the random variable

Sr A t

Ft-measurable).

THEOREM 2.3

(Soundness)

Let

A = (~,S) be a stopping time model then

A ~ ST. PROOF.

It follows trivially from the definition of stopping time.

THEOREM 2.4. Lad. PROOF.

(ComRleteness)

Let ~ be a countable set of sentences

in

If ~ uST is Lad-Consistent , then ~ has a stopping time model. If we go back to Keisler's proof of the completeness

theorem

for Lad(See [K 3] and [K5]), it is easy to see that by adjoining axioms (ST 2) and (ST 3) to (ST I) we can get a model A = (~,S) of ~ having the properties written in parentheses.

This model is

"almost" the

one that we want but we have to make sure that these properties hold everywhere and not only almost surely. (ST 2) (I)

we can find For all

weU

and

If we recall that for all function S': ~ ÷ 0,i

We proceed as follows:

From

U i ~ of measure one such that all

r ~ +, Sr(W ) e [0,i].

wen

S(w)= lim Sr(W) r÷~ defined by

S'(w) = I S(w) 1

if

w ~U

if

w ~U

then using

(I) the

184

has the following

properties:

(i)

S' is measurable

(the probability

space is complete)

(2) (ii) CLAIM.

S' = S a.s.

S' is a stopping

order to prove random variable exists (3)

S'A t

t sT,SA

We can strengthen For all

This

the function

is a model of

Ft-measurab!e.

~:

ts[0,!]

In the

By (ST 3) we know that there

one such that:

t is Ft-measurable.

t c [0,I],

SA 0

te [O,I]\T.

A' = (~,S')

(3) to

is easy to see:

find

is

T c [0,I] of measure For all

(4)

time and

this we just have to check that for all

SAt

is Ft-measurable.

First observe is trivially

Since

that we can assume

Fo-measurable.

[O,I]\T has measure

zero,

0 c T, since

Now let's

for each

consider

n ~

we can

m n e T such that (i)

(5)

for each

(ii)

For each

ws~

lim n+~

nc ~ m n < m n + I < t

and

mn = t

we have

S ( w ) A t = lim S ( w ) A m n.

By (3)

we know that

n+~

for each n the function

S A m n is

that for each n,

is

Ft-measurable Finally, usual of S. A'

SAm n

as we wanted

§3.

Ft-measurable ;

we can conclude

This proves

that

COMPLETENESS

predicctable

process

this question;

T on

processes.

predictable

by optional theorems

therefore

SAt

equivalent

holds with

time and by

to and so

(2.ii.)

Similar

we prove questions

or progressively

can be proved.

the model

A' b ~.

PROCESSES.

In this section

the

S' in place D Given a theory

have an adapted model A = (~,X) with ~?

is also

(Ft)te[0,1 ] satisfies

(4)

S' is a stopping

in other words,

predictable completeness

that

T H E O R E M FOR STOCHASTIC

Lad, when does

and by (5.i) we have

to show.

given the fact that the filtration

conditions,

is weakly-elementarily

T in

Fmn-measurable

(theorem

3.12)we

a completeness

X

a

answer

theorem for

can be asked by replacing measurable,

and corresponding

The main difficulty

in finding

185

correct axiomatization notions

of these notions arises from the fact that the

of predictable,

optional

in terms of measurability

and progressive

conditions with respect

product space ~ x [0,i] but the conditional only

"talks"

about

o-algebras

no "obvious" ways of axiomatizing from Probability

processes to

o-algebras

expectation

defined on

~.

these notions

operator

Therefore in

are given

Lad.

gales. the

for Lad

there are Using tools

theory and the model theory of adapted probability

logic we can find the correct axioms and prove the completeness To conclude

on the

this section we present a completeness

In handling

"obvious"

this notion it is, in principle,

axioms but we immediately

tifier over time variables about this problem);

is needed.

nevertheless,

realize

theorems.

theorem for martineasy to write down

that a universal

quan-

(See remark 3.9 for some comments

we are able to avoid this obstacle

by making use of a simple probabilistic

construction.

Here are the

details° DEFINITION

3°1o

structure

and

(M,B(M))

where

Generated

in

completion

X

(Skorokhod

Let

a stochastic process B(M)

is the Borel

(M,B(M))

under

[Ski):

P

if

~

be an adapted probability

taking values in a Polish space

o-algebra on M.

o({X(t):t E [0,i]})

of a countably

Now we state the main results

generated

from probability

S is Countably

is contained

in the

o-algebra. theory that we use in

this section. THEOREM

3.2.

generated (a)

(Skorokhod

M-valued

X has a predictable predictable

(b)

For all

THEOREM 3.3.

[Ski)

stochastic

~

X = (X(t)) t [0,i] be a countably

modification

The following are equivalent: (i.e. there exists

Y

on

such that for all t, Y(t) = X(t)a.s.)

tE[0,1],

X(t) is Ftl measurable,

(Dellacherie

and Mexer

adapted stochastic process on Y on

Let

process.

such that for all

~

[DM2]):

where

Ft_ = o( u Fs). s
If X is a measurable

then there exists an optional process

t~[0,1],

X(t) = Y(t) a.s.

(i.e. X has an

optional modification). REMARK

3.4.

generated

The obvious question is:

stochastic processes?

gave an argument nerated.

are there enough countably

In an earlier version of this paper we

showing that the optional processes

are countably

The referee has pointed out that it is easy to prove that

any measurable

process

is countably generated.

The importance of Skorkhod's

theorem

3.2

for us is that it gives a

ge-

186

a characterization of

"predictable modification"

rability condition with respect to the

in terms of a measu-

o-algebras

(Ft~)ts[O,l ].

The following theorem is going to show us a natural way of expressing this condition in the language of adapted probability logic. THEOREM

3.5.

Let

structure and (i.e. for all

~

= (~, (Ft)t~[O,l]P)

X = (X(t))ts[0,1 ] an integrable stochastic process on te[O,l], / X(w,t)dP < ~). The following are equivalent:

(a)

X(t)

(b)

~ ~ plnlE(X(t) IFt_I/p)

PROOF.

be an adapted probability

is

Ft-measurable X(t) I < I/m

a.s.

Condition (b) is just a "suggestive" way of writing:

(I)

E[X(t) IFt_I/n] n-~X(t)

For each

a.s.

n e ~ let: I

Ft-I/n

if

L

F0 otherwise

t e i/nan d X(n) = E[X(t) IF(n)]

F(n)=

Clearly, the sequence (X(n))ns ~ is a discrete martingale with respect to the increasing family of o-algebras (F(n)) n c~" Since X(t) is integrable, we can apply a well known martingale convergence theorem due to Levy (see for example [DM 2] Thm. 31, page 28). With our notation the theorem says: (3)

X(n ) = EEX(t)]F(n )3 n÷ E[X(t) l O ( n ~ F ( n )]

a.s.

Using this fact our theorem can be easily proved as we now indicate. Suppose

X(t)

is

E[X(t) IFt_] = X(t)

Ft_-measurable. a.s.

with

This by definition, means:

Ft_ = O(s~tF s) = ~( u n ~

F(n)),

therefore by (3) and (a) we have X(n ) = E[X(t) IF(n )] --~ X(t) Suppose (b) holds, then E[X(t) IFt] = X(t) a.s.

a s

and this is

X(n ) ÷ X(t ) a.s.

(b) (i e. (I))

and by (3) we can conclude D

NOTE° Observe the importance of the integrability condition on the stochastic process X = (X(t))te[0,1 ] in the previous theorem, in particular, if X is bounded then it is integrable.

187

DEFINITION 3.6.

A model

A = (~,X) is said to be a Predictable Model

if the stochastic process X is predictable with respect to (Ft)tg[0,1 ]. Similarly we can define adapted, progressively measurable and optional models.

By Theorem 1.6, every predictable model is optional, every

optional model is progressively measurable and every progressively measurable model is adapted. LEMMA 3.7. ture.

(a)

If

Let

~ = (~,(Ft)t~[0,1],P)

be an adapted probability struc-

A = (~,X) is an adapted model then there exists an optional

stochastic process

Y

on

~

such that the optional model B =

(~,Y)

is strongly Lad-equivalent to A. (b)

If X(t) is

Ft_-measurable then

X(t) is

Ft-measurable.

PROOF. (a)

Recall that in our definition of stochastic process only deal with measurable stochastic processes°

(def l.b) we

Then if

adapted, Theorem 3.3 provides an optional modification s B = (~,Y) ~ A = (~,X).

X Y

is of

X

and Thm 1.14ob shows that (b)

Trivial.

By definition,

F t_ ! F t-

We can now present the first completeness theorem.

It basically says:

"If X is adapted this is enough in order to get an optional model". At first sight it may seem strange that from an adapted model we can go to an optional model (compare the definitions), but the force behind this is Theorem 3.3 THEOREM 3.8.

which is a non-trivial probabilistic result. (Soundness and Completeness for Adapted, Progressively Measurable and Optional Processes):

Let

T

set of sentences in Lad. (a)

If T has an adapted model then T U{rAE~+(Pt ~ l) (E[Xr(t) It] (t ) = Xr(t ) a.s.) is consistent in Lad°

(b)

If T u{ A r~+

(et > l)(E[Xr(t)It](t ) = Xr(t ) a.s ) -

is consistent in

Lad

then T has an Optional model.

PROOF. (a)

Immediate from the definition of adapted model.

be a countable

188

(b)

If we add the axiom T u { A + (Pt ~ l) (E[Xr(t) It] (t) = Xr(t) re~

to the list of axioms completeness

for

Lad

theorem for

(definition

For each From

(2) Let

r s2 + ,

r

2+

A =

for almost

(i) we can find

For each

(~,X) all

CLAIM.

X'(w,t)

for each

t

U, Xr(t)

X'

A' = (~,X')

is adapted

rable and this property By the definition for each

lim Xr(W,t)

Then by

t c [0,i]

=

lim

by proposition

lemma 3.7.a

is

Ft-measurable. If

tc[0,1]

is

(Xr(t))tg[0,1 ] we know that

and almost all

1.13.d.

=

w s ~:

lim Xr(w,t ) = X(w,t)

r÷~ w A' ~ A and so modification

A' ~ T.

Finally by

X ~ of

In the above proof we did not immediately

model of T due to the fact that with the integral make sure that a given property

holds

necessarily

A research

[K 3] is to develop universal

the argument perties

t ~ [0,i].

X'

of the previous

of stochastic

naturally

expressed

sec. 4.4 in [K3]).

logic with

theorem

processes

we can only

te[0,1]

but not

in Keisler

the addition

of the

If this were done,

could be simplified

that hold for "all times"

in this extension

is an

get an adapted

problem proposed

ranging over time variables.

and the A"

quantifier

for almost all

adapted probability

quantifier

Ft-measu-

:

we can find an optional

for all

ws

to the limit.

model A" = (~,X") is such that A" ~ A' and consequently optional model of T as we want to show. REMARK 3.9°

and

and 0 other-

It is easy to

r s ~+ E[Xr(t) IF t]

by passing

process

E[Xr(t) IFt](w )

r÷~ Therefore

one such that:

= X(w,t).

(2) and (3) for all t s U

X'(w,t)

Ft-measurable.

if the limit exists

for each

of the truncated

w e ~ and each

(3)

since

is

is an adapted model of T:

is preserved

ex-

such that: Xr(t)

as follows:

= lim E[Xr(t) IFt](w) r-~

The model

see that

t~[0,1],

proof of the

can be naturally

of T,

U c [0,i] of measure

X' = (X'(t))ts[0,1 ] be defined

then let wise.

1.9), Keisler's

Lad (see [K 3] and [K5])

tended in order to get a model (i)

a.s.)

and many procould be

of adapted probability

logic

(see

189

DEFINITION (PI)

3.10.

Axioms

Let

for Lad

P

be the following

(see definition

set of axioms

in Lad.

1.9)

(P2) (et~l)( A rs~ +

A ms~

V nsiN

A (Pu~l)(t-I/p
(intuitively, for almost all t, ~(t) THEOREM 3o11.

(Soundness

of P)

lu](u)-Xr(t)
is F t_-measurable).

The axioms

of P are true in every pre-

dictable model. PROOF.

It follows

from the soundness

theorem for Lad and theorem

3.2

and 3.4. We are now ready for our main completeness THEOREM 3.12.

(Completeness

be a countable

set of sentences

then

T

PROOF.

has a predictable Let's

quantifier)

Theorem

for Predictable

in Lad.

If

T uP

Processes).

Let T

is consistent

in Lad

model.

first make an observation Consider

theorem.

structures

for

about L

•

L i f (Logic with integral

of the form ~I f

A = (~,(Yn)

,Y,P) where

!

the Yn s

and Y are unary random variables

n siN and add to the axioms Then Keisler's

of L

~if

completeness

the axiom

theorem for

^ v A m n p~n

i Y

L w /(see

p

- YI < I/m a . s

[K3])

can be extended

1

to obtain models

A as above where

The idea here is to introduce symbols

(Xn)n e ~

(E[X(t) IF

n

--~ Y a.s. n÷~

in ~.

countably many new stochastic

that are going to represent

]) t-i/n

Y

.

process

the processes

So, for each n let

ts[0,1],n s ~

X n = (X$ (t)) r t~[0,1]

be defined by E[Xr(t) IF

] (w)

if

t -> I/n

if

t-< i/n

t-I/n (I)

X n (w,t) = EEXr(t) IF ](w) 0

As we noticed /l main

/l nslN

xn(t)~-~Xr(t)

in Theorem

3.5,

the expression

/l IE(Xr(t) IF ) - Xr(t) I < i/m a.s. p ->n t-i/p a.s..

Axiom

Just means

(P2) is the formal way of expressing

this

t90

(for almost all t) in the language of Lad. axiom (P2) holds (2)

For all

Therefore

any model in w h i c h

is such that:

rg~ + for almost all

tc[0,1]

the r a n d o m variable Xr(t)

is Ft_-measurable. By the above remarks T

such that

it is clear that we can get a model

(2) is true of X.

Using a similar argument

given in the proof of theorem 8.b. X' = X'(t))t~[0,1 ] on (3)

For all

~

A = (~,X) of to the one

we can find a process

such that:

tE[0,1]the r a n d o m variable X'(t)

is Ft_-measurable

and

w

(4)

A' = (~,X')

~ A.

By lemma 3.7 and theorem 3.3 we can find an optional m o d i f i c a t i o n X" = (X"(t))ts[0,1 ] of X' such that s

(5)

A"=

(~,x")

z A'

By the characterization

(Theorem 1.13.c)

property

in

(3) also holds

A".

of strong equivalence

Then by theorems

and

(P2)

3.2 and 3.4 we can

obtain a predictable m o d i f i c a t i o n X''' = (X'''(t))t~[0,1]of X" such that the model A''' = (~,X''') s~ A . . . Thus . . A _ ' is the desired predictable model of

TuP.

D

If we take a close look at the above proof we will see that the theory T plays a "cosmetic" role because proof of the completeness theorem.

theorem,

its importance

is only felt in Keisler's

for Lad and we are just quoting his

We can isolate the m a i n model

theoretic

feature w i t h the follow-

ing T H E O R E M 3.13.

If A

~ "for almost all t, X t is Ft_-measurable,"

there is a predictable

model B such that

Let's remark that all the other completeness can also be put in this form. reader.

The two previous

then

A ~_ B.

D

theorems

We leave the details

from this section

to the interested

theorems are stated in a form that is familiar

to logicians but we can use the same proof in order to present some results

that are along the lines of the probabilistic work of Hoover

and Keisler

in [HK].

undefined notions T H E O R E M 3.14.

Let

We refer the reader to that paper where all the

that appear in the following X

and

different adapted spaces) (a)

Y

theorem can be found.

be stochastic processes

and suppose

X

(on perhaps

is predictable, then s If X and Y have the same adapted distribution (X ~ Y in our lan-

191

guage) then

Y

has a predictable modification (but

Y

itself is

not necessarily predictable). (b)

We can find a predictable stochastic process

Z

on the adapted

S

Loeb space such that

X ~ Z.

PROOF. (a)

Just read carefully the proof of Theorem 3.12.

(b)

Use (a) and the saturation property of the adapted Loeb space. D

Finally we study the notion of martingale which is of central importance in probability theory and crucial for the development of the general theory of processes.

A thorough exposition of the theory of discrete

and continuous time martingales can be found in [DM 2]

We are going to

prove a completeness theorem that gives us martingales with time parameter [0,I)

instead of

point on the right.

[0,i].

The problem arises because i is an end-

After the proof of the theorem we indicate how to

take care of this problem. DEFINITION 3.15.

Let

M

be the following set of axioms in Lad:

(M I)

Axioms for Lad (see Definition 1.9).

(M2)

(et ~ I) ^ v A (f[Xq(t)-Xp(t)Idw < I/n) n m qepem

(M 3)

A+ r~

(Pst>l)(s~t

÷ (E[Xr(t) Is](s) = Xr(s)a.s.) )

Intuitively, axiom (M2) says that for almost all

t

the random variable

X(t) is integrable. THEOREM 3.16.

Let

T

be a countable set of sentences in Lad°

is consistent in Lad then PROOF.

T

If

has a martingale model.

As in previous theorems the first step is easy:

use

M

to

extend Keisler's proof of the completeness theorem for Lad in order to get a model

A = <~,(Xt)ts[0,1],(Ft)tE[0,1],P>

(i)

For almost all t, X t is integrable

(2)

For almost all (t,s)s[0,1)xE0,1), X s a.s.

We now show how to find a martingale

if

s ~t

of

then

T

such that:

E(XtlF s) =

(Yt)tE[0,1) living on

W

(~,(Ft)t~[0,1],P) From (i) (3)

and

For all

(2)

such that

A ~ B = <~,(Yt)ts[0,1],(Ft)ts[0,1],p>.

we can get G ! [0,I) of measure one such that:

t s G, X t is integrable

T uM

192

(4)

For all

t EG

the following holds

in

A:

(Ps -> l)(s ~ t ÷ E(XtlF s) = X s a.s(3) gives us the

It is easy to verify that

(Xt)ts G

integrability

So we just have to clneck:

For all

condition.

is a martingale.

s,tsG(s-
In order to see this let rsG such that

s -< t < r

Using

s, tsG with s-< t0

(4) we can find an

and

(5)

E(XrIF s) = X s a.s.

(6)

E(XrIF t) = X t a.s-

Thus we have: E(XtlFs)a?sE(E(Xr;Ft) IF s) by a.s.= E(XrIFs) = X s by a.s.

Now the definition s t > to

Let

Yt

of

since

(6), F s _
(5)

(Yt)ts[0,1).

If

ts[0,1) Xt

if

tgG

E(Xst IF t)

if

t ~G

=

It is a simple exercise to show that we leave it to the reader.

(Yt)te[0,1)

Finally observe

and since G has measure one theorem A w= B as we w a n t e d to show. Let's

find s t s G such that

t ~G

is a martingale

that for all

1.14.d.

0

indicate what to do if we want to have a m a r t i n g a l e with

Fortunately

in Lad we can express (Xt)ts G is uniformly

one

from

ts[0,1]. to

[0,I]

the fact that on a set G c_ [0,I) of integrable.

We do it by adding the

following axiom to the list of axioms of the definition A n

[0,i)

integrable martingale.

measure

(M 4)

t s G, X t = Yt

tells us that

The idea is that in order to ensure the transition we have to have a uniformly

and

3.15:

V (Pt e i) (fl t) Xm(t) Idw < I/n)) m (q>Am Xq( -

Once this axioms holds G an increasing

in our model

sequence

(see above proof) we can choose from

(dn) n slq converging

to 1 and define

193

YI = n_~=limXdn.

It is now easy to show that (Yt)te[0,1] is a martingale

and we leave the details to the reader. REFERENCES [An]

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[As]

Ash, R., Real Analysis and Probability, Academic Press (1972)

[C]

Cutland, N., On the Existence of Strong Solutions to Stochastic Differential Equations on Loeb Spaces, Z.W,V.G., 60(1982), pp. 335-357.

[D]

Del!acherie, C., Capacites et Processus Stochastiques, SpringerVerlag, (1972).

[DM I] Dellacherie, C. and Meyer, P.A., Probabilities and Potential, Vol. I, North-Holland, 1978. [DM 2]

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[E]

Elliot, R., Stochastic Calculus and its Applications, SpringerVerlag, (1982).

[F]

Fajardo, S., Probability Logic with Conditional Expectation in: Annals of pure and applied Logics (To appear).

[H I ]

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[H 2]

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[H 3]

, A Normal Form Theorem for Lw, p with Applications, J.S.L. 47(1982), pp. 605-624.

[H4]

, A Probabilistic Interpolation Theorem.

(To appear)

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[HP]

Hoover, D. and Perkins, E., Nonstandard Construction of the Stochastic Integral and Applications to SDE's. I and II, T.A.M.S. 275(1983) ,pp. 1-36 and 37-58.

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Keisler, H.J., An Infinitesimal Approach to Stochastic Analysis. Memoirs Amer.Math.Soc. 297, Vol. 48(1984).

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[K 23

Keisler, H.J., Hyperfinite Probability Theory and Probability Logic.

[K 33

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Lecture Notes, Univ. of Wisconsin

(Unpublished)(1982).

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Lindstr6m,

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I,II, and III.

Math. Scando 46(1980), pp 265-333. ELo]

Loeb, Po, Conversion from Nonstandard to Standard Measure Space and Applications to Probability Theory, T.A.M.S. 211(1975), pp. 113-122.

[MP]

Metiver, M. and Pellaumail, Press, 1980o

[Pal

Panetta, L., Hyperreal Probability Spaces: Some Applications of the Loeb Construction, Ph.D. Thesis, Univ. of Wisconsin (1978)

[Pe]

Perkins, E., Stochastic Processes and Nonstandard Analysis. L ~ 983.

[R]

Rodenhausen, H., The Completeness Theorem for Adapted Probability Logic, Ph.D. Thesis, Heidelberg University (1982).

[Ski

Skorokhod, A.V., On Measurability of Stochastic Processes, Theory of Probability and Its Applications, Vol. XXV,(1980), pp.139-141.

J., Stochastic Integration,

Academic

I wish to thank my advisor, Professor H.J. Keisler for suggesting the subject of this paper and for his invaluable cormments. This research was partially supported by a University of Wisconsin Research Assistantship.

A BARREN EXTENSION J.M.Henle

A.R.D. Mathias

Smith College

Peterhouse

Northampton,

Cambridge,

Mass.

England

U.S.A. W o Hugh Woodin California Institute of Technology Pasadena

California U.S.A.

ABSTRACT.

It is shown that provided w ÷ (w) m,

extension adds no new sets of ordinals. the same extension preserves elucidates

a well-known Boolean

Under an additional assumption,

all strong partition cardinals.

the role of the hypothesis

This fact

V = L[R] in the Kechris-Woodin

characterization of the axiom of determinacy. §0.

INTRODUCTION. Let

B = Power(m)/Fin be the quotient of the Boolean algebra

of all subsets of m by the ideal

Fin

the set of non-zero elements of

of finite sets, and P = [m]m/Fin

B, with the induced

partial ordering.

We shall study the Boolean extension that results from using notion of forcing: with a famous theorem AC

of (ml,ml*)

gaps in

P

[5]

P

as a

about the existence under

in mind, we shall call this the Hausdorff

extension. Our underlying set theory is Zermelo-Fraenkel.

We shall be working in

contexts where the full axiom of choice is false; use DC, the axiom of dependent choices, states that if

Q

then there is a map

is a relation on f:m÷Power(m)

notation of Boolean extensions Mathias

[15],

3.7

of < of order type

and %,

3.8.

at times, we shall

or DCR, its weaker form, which

Power(n)

such that

such that Vp~ q Q(p,q), Vn Q(f(n),f(n+l)).

Our

and forcing follows that described in We write

[<]%

for the set of subset

and follow Supercontinuity

[8]

in our notation

of partition relations. It has long been known that under

DCR

the Hausdorff extension adds no

new sets of integers: what it does add is a Ramsey ultrafilter on ~. (Cf [15],

Theorem

4.2).

For a recent discussion under

AC

of

P,

see Dordal [3]. Since without

AC

there are difficulties

in the simultaneous

choice of

196

representatives

of equivalence classes

, it will be convenient to take

as our forcing conditions

infinite subsets

standing that if

q

p

and

p~q, they force the same statements, tion than

q

if

p\q

p,q

of

~, with the under-

have finite symmetric difference, written and that

p

is a stronger condi-

is finite.

In section i we prove in the theory

ZF + w ÷ (m)m

that the Hausdorff

extension is barren in the sense that every map in the extension from an ordinal into the ground model lies in the ground model: no new sets of ordinals are added.

in particular,

We characterize this latter proper-

ty in terms of the Galvin-Prikry notion

[63

of completely Ramsey

families. In section 2, LSU and EP.

we consider three set-theoretic principles, LU and

LSU are the weak and strong uniformisation princi-

ples discussed in Mathias [16], valent in

ZF + DCR

holds and

V = L[R] or if

to

where it is shown that

m ÷ (m)m + LU, ADR

and that

holds, or if

V

"all sets of reals are Lebesgue measurable". this section from

called LU,

LSU

in the theory

logical terminology of Ellentuck

LSU

is equi-

LSU is true if

AD

is Solovay's model for

EP, which we derive in

ZF + DCR, is, to use the topo-

[4] ,

the assertion that the inter-

section of any well-ordered collection of co-meagre sets is co-meagre. In section 3,

we prove in the theory

ZF + LU + EP that the Hausdorff

extension preserves every partition property of the form K ÷ (~)%~, where

K > ~,

Finally,

2 ~

< K, and

0 < ~ = ~

~ ~.

in section 4, we comment on the implications of the results

of section

3

for the recent characterization

by Kechris and Woodin,

[123

of

AD

in

L[R]

and discuss some problems related to our work:

reference to this discussion is made

in the text by the string

[PROB]. Assumptions are given in full in the statements of theorems, but may be omitted in Lemmata and Propositions when the flow of the narrative demans it. §i.

The end of a proof is signalled by

THE BARRENNESS OF THE HAUSDORFF EXTENSION.

THEOREM 1.0. N

ordinals;

M

be a transitive model of

P0

Then

M

moreover every sequence in

and N

N

ZF + ~ ÷ (~)~ and

have the same sets of

of elements of

It will be sufficient to prove in the theory ^

if

Let

its Hausdorff extension.

PROOF.

4.

^

l~ f:~ ÷ V, then for some

Fix then such

M

ZF +

lies in M. m ÷ (~)~that

^

q0 ! Po'

qo

P0' f' ~ and suppose that no such

T~ fcV. q0 exists.

Then for

197

each

p c [P0

there will be at least one ordinal

xcV

exists with

For

P ! ~, define

enumeration Now define ing as

of

p

I~ f(~)

P~\{k}

p~(n)

for

equal ~.

Q = P\{k},

~(P) = 0,

ty, so

~(P)

Put now

we have

~(Q) = 2;

can only equal

v = ~(P).

= p(2n),

that

x # y, q

choosing

q,

but

is compatible

Since

sr

Since

s~ ~ q,

forces

the value

As in Happy Families sup{n+lJn ~ s},

DEFINITION (p e A

and for

defined

of

q ~ [p]~ The

CR

tement

1.2.

The

r,

~(s~) > ~(s r)

there is a

always

Following

r,

q ! P

P, ~(s r) =~. and

q

and thus,

Isl ~ np,

Isl for

we write

Is,p]

for

of

[~]~ will be called p' ~ A;

(p' ~ domain(F)

Galvin and Prikry

invariant

and

if

F(p)=F(p')).

[6] we call a subset

CR, if for all or

a function

[m]m will be called

implies

or

invariant

similarly

<s,p>

there

is a

[s,q]nP = 0. of

to the assertion

[15],

1.3.

The sta-

that all subsets

of

Ramsey. We shall

is a

call a subset

q~[p]~with

q c [p]~ with

sets are those in

in the topology

from

s~ ! q,

absurdity.

subset of)

is equivalent

1.3.

x,y c V

se[P] ~ by

and extends

since

sets are chosen in ~H in the notation

there CR

and

implies

Ramsey,

<s,p>

<s,p>

q

s ~ [m]<~, we write

A

always

Is,q] ! P

DEFINITION

Define

q, the fourth

and so on, so that

with

with either

m * (m)m

~;

[0,p] = [p]~.

[~]m completely

[~]~ are completely

all

for

A subset

p~p')

and

q,r e [P]m and

~ y.

from

~(q) >~, So

p ~]~

and p~p')

DEFINITION

s

an evident

Thus

on (an invariant

(p ~ domain(F)

P

i.i.

and

I~ f(~)

of

but

[15],

{x~[~]~l s ! x ! sup}.

F

r

f(~).

~(s) = 2 # 1 = ~(P) = ~(s),

if

and

with both

for

P~

^

~(s~) e ~(q); x

~(pr ), and let

~(P) = ~(Q) by homogenei-

of ~ there are

then one from r,

(st) r c r "

than

I.

~ x

(Sr) ~ c q,

~(p) = 0, 1 or 2 accord-

Q~ = Pr' Qr = P~\{k}; 2, 0;

the first three elements

then three from

= p(2n+l).

and so have the same value under

if

By definition

I~ f(~)

Pr(n)

~(P) = I: for putting

^

with

q for the monotonic

to, or greater

Notice

force the same statements,

thus if

to be the least such ~.

~:[p0 ]~ ÷ 3 by setting

is less than,

and

~(p)

by writing

q i m and setting

PE[p0]~ be homogeneous k = min P

define

P~ ! m, Pr ! ~

a partition

~(p~)

~ x:

~ < ~ for which no

I H in

P

of

Is,q] c P, and [s,q]nP [15],

defined by Ellentuck

[~]m CR + CR-

if

for

if for all

= 0. 1.4, and are the meagre

in [4].

sets

198

1.4

In a manner

tivized versions is

CR+ on

Is,p]

there is an

r

familiar

to readers

of these concepts: if for all

in



[q]~ with

We are now in a position

of

[15],

we shall want rela-

we shall for example with

s ! t

and

say that

P

(t\s)uq ! P,

It,r] ! P-

to reformulate

a weaker

form of the conclusion

of 1.0 as PROPOSITION

CR

1.5

(ZF)

The following

(1.6)

The Hausdorff

(1.7)

Let ~ be an ordinal

subsets

of

[~]0J

extension

adds no new sets of ordinals.

and



Then there is a

< ~, C~ is, relative

to

[p]~,

either

CR +

(1.6) ÷ (1.7)

:

filter

in

added by the Hausdorff

B = {v < ~ IP c G(Cj,



relative

to

a sequence

p~[~]~

PROOF.

Power(u)

Fix

are equivalent:

or

CR-.

and let

G

be the generic

extension.

[p]~,

of invariant

such that for each

is

In

V[G] put

CR+)} ^

By (1.6), shall

there is an

show that for all %# in

that for all%#not contrapositives Let

v < ~.

CR

there is C , ^

~

If

in

A,

of these

If

of q

A c ~, and a

such that

C%# is

is CR-

CR +

q

I~- B - A.

relative

relative

to

to

We

[q]m,

and

[q]~, by proving

the

statements.

C%# is not

Is r ] c _

A,

C %#

q

CR +

[0,q]

[r] ~ nC%# is empty,

relative

with

to

Is,r] ^n C

and so r

%#

i~v~

[q]W,

then since

empty,

so by the invariance

B;

hence

q

~

Cv

~e ?B,

is

so

^

%#eA, and thus %# ~ A. C v is not

CR-

relative

to

[q]~,

there

is

Is,r] ! [0,q]

with

^

Is,r] c C , q

~v~

so by invariance

B, so

q

[r]~i C%#,

~ A, and so

and so

r

i~- v c B;

hence

%#~A. A

(1.7) ÷ (1.6):

Suppose

Then each iff a

C is invariant. ^ %# l~%#eB, and C%# is

p p

satisfying REMARK

for

n~, 1.9

1.8. let

Notice

(1.7)

that

CR- relative

the conclusion

of

Set C

%# to

(1.7) will

is false without

C

= {ql q

I~

~ c B}.

is

CR + relative to [p]m ]~ ^ [p iff ^p ] ~ v ~ B: so

force

B c V.

the hypothesis

of invariance:

C n = {p I n ~ inf p}.

The conclusion

square-bracket

I~ B ! ~"

partition

of Theorem relations:

1.0 may be derived write

m ÷ [~]~,

from certain where

2 ~ % ~ w,

199

to mean that for any

~:[~]m + %

some

y ~ [x] ~, ~(y) # v.

v < %

and all

from a sequence n, ~n

<~n I 2 ~ n

families,

~ ÷[~]~

not known whether n e 3 [PROB]

0 ~ i < B,

to m ÷ [m]m~

for countable

~ ÷[~]~n+2.

It is

~ ÷ [~]~n

for

{p(mB+i) Im~ ~}.

and

application

than n,

Thus if

(q)B-i = (P)0 \{~(0)}"

of

1.0.

Vj
~((p)i)}.

w + [~]~n,

a

is of cardinality

for some

i,j

Set

p c [p0 ]~ may be found for

less than n. p e [p]~,

But then since if

less than B , by the primality

~((p)0 ) = ~((p)j)

< ~((p)i )

of B and the invariance

the B values

*(p),

,(p\{p(0) }),

,(p\{p(O) ,p(1) }) .

,(p\{p(0),p(1) ..... p(B-l)}) Now put

~ = ~(p).

q I~ f(v) (s)0n q

that

for each

1.0 may be derived as follows.

let (P)i be

... = ~((p)B_l ) for any

~,

AC

from any

of theorem

~((p)l ) =

of

n < ~,

f' ~' ~ be as in the proof of Theorem

@[[p]~]

where

Let B be a prime number very much larger

~(p) = { i l i < B

which

,

a counterexample

that for some

, (q)i = (P)i+l'

By (repeated)

has observed

of [m]~

DC, or at least

m ÷ (~)~ can be derived

p ~ [~]~,

P0'

implies

m ÷ [~]mn.

q = p\{p(0)} Let

m ÷ [mien'

but the conclusion

Suppose and for

to

thus assuming

x e [m]m such that for

Kleinberg

<~> of partitions

is a counterexample

may be constructed:

there is an

is infinite,

$((s) 0) = v

while

established REMARK

two transitive

they might

differ

an extension

(s) 0 n r

q, r e [~]m, x,

x # y.

Find

is infinite,

~((s) I) z ~(q)

.

> 9,

y e V,

s c qur

(s) I i q-

with

so that

Then

contradicting

the property

of

paragraph. V

It is a theorem of Vopenka and Balcar

models,

axiom of choice,

there are

~ y, and

in the previous

i. I0.

.

will be distinct.

As before,

E x, r I~ f(~)

.

of which at least one is known

have the same sets of ordinals, if neither

satisfies

of that result,

see Monro

AC [17].

[21]

that if

to satisfy

they coincide.

is due to Jech

the That

[9]: for

200

2.

THE L A R G E N E S S OF THE INTERSECTION

DEFINITION continuous

on

2.0. A function [s,p]

F(q) n(k+l)

formised on

2.1.

[s,p]

DEFINITION R

as above,

on

[0,x]) is

2.2.

a sequence of

CR +

([15];

D

n C

[6].

uniformised

I~ < e>

CR +. ZF + DCR, and is due

A proof may be given by the methods

suppose

Proposition

of Happy

I.i0).

the theorem true for all sequences

of

e , and that it fails at O for the sequence of e and the first sentence of this proof

cofinality.

(for the case D

n C }.

Then by Proposition

A = [m]~, B = [m]m\( ~<~ n C ))

~ D~ for

and, by r e l a t i v i z i n g

~ < ~ < e,

to some

[s,S]

q ! P ~D

if necessary,

2.8 of D

is CR +

,

implies

q ~D ,

we may further assume

is empty.

Thus we may define

X(P),

The function X:[~] m + e

for pe[~] m

V~<8 ~(p)

to be the least ~ < e w i t h p ~ D

will have these properties: q i P ÷ X(P)

Define now

is

this is a theorem of

= df {p E [~]~l[p] ~ c

for each ~ < e ,

v~eD

F

that for any relation

{xlR is (strongly)

Then

By the m i n i m a l i t y

Happy Families

that

uni-

function

Let e be any ordinal and.
see in particular

e must be of uncountable Now put

(strongly continuous)

Vp] y R(p,y),

families.

Let us therefore length less than J~ < e>

is (strongly)

R(q,F(q)).

(ZF + DCR + LSU)

to Galvin and Prikry


R c [m]~ x Power(u)

LU (LSU) is the assertion

For e countable,

Families

k e q,

CR +.

T H E O R E M 2.3.

PROOF.

and all

if there is some

such that

is called strongly

of finite sub-

= tqn(k+l).

A relation

such that for all q ~ [s,p],

CR + FAMILIES.

F:[~] ~ + Power(m)

if there is a tree

sets of ~ such that for all q ~ [p]~

DEFINITION

OF

~ x(q),

V <s,q> ]p~[s,q]

= u{x(q)[q=p}

and

(X(P) >v).

¢(P) =n{x(q)]q=P}.

We assert

that V qc[~] To see that, put q = P0 Z Pl ~ P2 ~

v = ~(q),

w

]p c [q]~(~(q)

construct

< ~(p)).

sequences

"'" such that for each

n o < n l < n 2 < ... and

i,n i = min Pi' and for each

201

s _c n.+l~ , [s,Pi+l ] ~ D v, and If

p'z p, there is an

Pk+l ], By

so

p'\n k = Pink,

X(P') >v;

R

and

on

is continuous, [18],

12), if we define

since if

ml

so

<s,p>

with

R(sup, q), and so

Thus if for

~ < n to

we set

[0,p]

[0,p];

]q'~q ]p'=p(F(p')

and so by the Kunen-D~rtin

q E [~]w

To see that, take

CR- relative relative to

and all

iff

= ~(p) ->X(P)-

page 99, Theorem 2G.2;

Notice that for any p R(q) > PR(SUp).

R is Borel,

R(p,q)

R(p,q),

theorem

see also page 114, footnote

pR(q) = U{PR(P)+lJp c [p]~ and

some n less than

q~F(p),

function

$(F(p)) >~(p)).

x(q) e ep(q) = qb(q') > q ~ ( F ( p ' ) ) > ~ ( p ' )

(Moschovakis

consequently ~(p) >v.

[p]m by setting

will be well-founded

F

p' e [p'n(nk+l),

p ~ [m]m and a strongly continuous

Define the relation

Since

so

such that

Vp e [p]m(F (p) ! P

= q'):R

p = {niJi < m}.

n k such that

p' c D , and

LSU, there is a

F:[p] ~ + [ p ] ~

set

F(p) = q}, then for

pR(q) < n.

sup ! p, there is a q = su(F(p)\Jsl):

q c Is,p]

with

then sup~p

and

pR(q) > PR(SUp)-

E~ = {p c [p]~JpR(p)

= ~}, each

But then since ~ is countable

• but that is absurd,

E~

u E ~
as

[0,p] =

will be is

CR-

~

'

u E . ~
DEFINITION 2.4. EP is the statement of Theorem 2.3, that the intersection of any well-ordered collection of CR + sets if CR +. PROPOSITION PROOF. or

If



[s,x]nC~ = 0}

Let

P §3.

2.5 (ZF)

•

be a member:

EP implies

(1.7).

is as in (1.7), let

Then each

D~

is

then P satisfies

D

= {xJVs ! n x

CR+: by

EP, v<~ n D

is not empty.

the conclusion of (1.7).

THE PERSISTENCE OF PARTITION PROPERTIES.

PROPOSITION 3.0 (ZF + EP)

Let ~ be an ordinal and

an invariant function.

Then ~ is constant on some

PROOF.

C

so by

(Is,x] i C

For

~
= {pjv ~ ~(p)}.

1.7, which by 2.5 follows from

relative to

[p]m,

each

C

is either

[p]~.

Then each

EP, there is a CR +

or

~:[~]m ÷ [K]K

CR-

C

is invariant, p ~ [m] ~ such that,

:

put

202

D

= {qe [p]~l[q] ~ ! C } in the first case and

in the second. Let

p

Then each

Dv

is

be in the intersection,

for each

qe[p]m

and ~ <~,

~ ~(P) ÷ P e C

CR +

on

invariant

The above Proposition

: for example,

PROPOSITION

3.2

< K, K ÷ (K)%~, [~]w>

£

is a

p* ~ [~]m

for each

(ZF + LU)

that

PROOF.

For each

as in

that

0 < ~ = ~X ~ K,

to be

~(p)

define

~p:[K] % + p.

function

pp:[~]~

for

÷ ~

and that Then there

~:[p,]m + [<]K

is homogeneous

2 ~

~:[w] ~ ÷ [K] <

of partitions

pp(X) where

and

may fail if ~ is not required

there is a surjection

p,

Then

÷ ~(q)

Suppose

and an invariant

p ~ [p*]~,

{Dvlv
if ~ is an injection.

is a collection

<~plP

EP, of

~ ~ ~(p),

÷ ~ ~ ~(q),

~ ~(p) ÷ p ¢ C~ + D n C ~ = 0 + q ~ C REMARK 3.1.

by

iff

+ Dv -c C v + q e C

= 0}

[p]~.

non-empty

~ e ~(q)

= {q~ [ p ] m I [ q ] ~ C

such that

#p.

by

= ~p( x ),

Supercontinuity

[8]

,

x

={ o

x(m~+n) l~ < %}:

note

nE~

that as

w% = k, ~x

is in

[K] ~

Hp = {qc [~]~I~(q) By

LU

there

is a

q ~ [p*]~,

~(F(q))

For each

q ~ [p*]~,

ordinal

greater

least ordinal [q,~)

p*

and an

define

than all

The regularity

x

is homogeneous

is. for

for

pp.

B(q')(O) setting

for ~

at least one element

q' ~ q;

of each

the soundness

B(q')

C(q) = C ( q ' ) w h e n e v e r

q ~ q'

Now put

~(q) = ~C(q).

form

for some y E [B(q)] %, and hence

Hence

~(q)

of

Then ~ is invariant; B(q),

is homogeneous

~q

C(q)(~)
for

of any

be the

the interval

q' ~ q.

of

Note that

{q'lq'

~ q},

and

: so that C is invariant. any

~q(X)

is independent for

be the least

of this definition.

therefore

~y

let

all

= ~(F(q)).

C(q)(0)

= v{C(q)(~')I~'

does not rely on an enumeration

by the homogeneity

B(q)

let

this definition that

pp}.

Write

C(q) ¢ [K] < thus:

of < ensures

Set

F:[p*] ~ ÷ [~]~ such that for

is homogeneous

~ such that

contains

whenever

x ~ [~(q)]~

= ~q~y) y

as required.

is of the

= pq~),

and therefore

which, of

x.

203

THEOREM

3.3

(ZF + EP + LU)

Suppose

and that there is a surjection extension, PROOF.

0 <~ = ~

~:[m]m ÷ ÷

[<]<

~ <, 2~p<~,<

÷ (K)l~,

Then in the Hausdorff

K ÷ (<)~.

Note first that by

quently

2.5

no new subsets

[<]% is the same whether

the extension,

interpreted

that

conse-

in the ground model or in

and thus may be written without

Note secondly

of < are added; ambiguity.

~ ÷ (<)kp+l. ^

Suppose

P0

such that

I~ f:[<]% ÷ PP2

For each

I~ f

We shall find a

is constant

p e [p0 ]~,

define

~p(A)

= ~

3.2,

there is a

assert

that

If not,

^

I~ A

^

~ g and

~q. §4.

otherwise.

#(p)

^

~ ~): ~q.

on

for

by

function

~:[pl ]m ÷ [<3 K

for

[p2 ]~.

Zp.

Set

By 3.1

A = ~(p2 ). We

f.

q e [p2 ]m, ~ < ~ < p

so

But

A ~ [<]<

~

is homogeneous

~ constant

D,E c [A]~, (~)

for

~q(D)

= $, ~q(E)

A = ~(p2 ) = ~(q)

with

= ~, which

and thus is homogeneous

-[ PARTITION

CARDINALS

An important ordinal

= p

is homogeneous

f

A is inhomogeneous for

P l~ f(A)

p e [pl ]~,

there will be

q l~ (f(D)

if

P2 E [pl ]~ with P2

~ :[<]~ ÷ ~+I P

P l e [P0 ]~ and an invariant

such that for each there is some

P2 ~ [P0 ]~ and an

[A]~.

a partition

~p(A) By

on

> 0

LEMMA

ordinal

onto which 4.0.

same whether

WITHOUT

in the study of

Power(m)

AD

is

~,

the least

cannot be mapped.

If the Hausdorff

calculated

DETERMINACY.

extension

is barren,

then 6 is the

in the ground model or in the extension. ^

PROOF.

Suppose

PIP

f:[~]~ ÷ ÷ 8. ^

put ~(q,r)

= ~ if

q I~ f(r)

is a surjection PROPOSITION K ÷ (<)Ip, PROOF. ments

then

of

4.1

For each pair

~ ~,

and

~(q,r)

[~]~ x [~]0,i onto

(AD

+

V

=

L[R])

If

e, 0

By a theorem of Kleinberg [13],

< < 8 ;

[14],

<

hence,

= 0

otherwise.

Then

a contradiction.

< ~l=

K + (K)%p in the Hausdorff

to be found in

(q,r) with q\p finite,

^

~ ~<

,

2_<~

< <

and

extension. is measurable,

so by argu-

by a theorem of Moschovakis

204

([18], 7D.19, page 442), there is a surjection

~:[m]m ÷ +[K] < .

DC is provable in

ZF + AD + V = L[R], by Kechris

by Theorem 2.2 of

[16];

by Theorem

Thus all the hypothesis of THEOREM 4.2

If

AD

V~<@ ~ K(~ ~ m < @

If

AD

Again, by As

holds,

[i0],

m ÷ (m)m,

hold,

so is

LSU,

EP then holds.

so the conclusion follows.

is consistent,

so is

DC +

and K is a strong partition cardinal) + "there is

a Ramsey ultrafilter on PROOF.

3.3

[i0];

2.3 above,

m". it stays true in

DC holds,

so

DC

(cf Theorem 2.2 of

the results of Kechris,

L[R], as strategies are reals.

holds in the Hausdorff extension. [16]),

Kleinberg,

the extension is barren. By

Moschovakis and Woodin

Eli],

there

are arbitrarily large strong partition cardinals below @ in the ground model,

so by

4.0

and

4.1, the same holds in the extension.

A remark

in the introduction completes the proof. REMARK 4.3.

The significance of

4.2

is that as the existence of an

ultrafilter on ~ implies the failure of

AD, the hypothesis

is an essential ingredient in the Kechris-Woodin

[12]

V = L[R]

derivation of

AD

from the existence of arbitrary large strong partition cardinals below 6. REMARK 4.4.

The arguments of section 1 generalize with little change

to obtain a new proof of Henle's result

[7]

that Spector forcing

[20]

at a strong partition cardinal K is barren. REMARK 4.5. measurable,

If and

AD

holds and

~ + (~)m,

Ramsey ultrafilter on

V = L[R],

then

~i

and

so in the Hausdorff extension,

~, and

ml

and

~2

~2

are

there is a

are still measurable: we

may say that there are in this model two and a half contiguous measurables. A model for that statement may also be found assuming something presumably much weaker than model of

ZFC

Con(AD): by an unpublished result of Woodin, a

in which < is

%-supercompact and

admits a Boolean extension in which measurable,

and

~ ÷ (~)~;

K = el'

~ > < is measurable

~ = ~2' <

and ~ are still

in the Hausdorff extension of that model,

there will be a Ramsey ultrafilter on

~,

while the measures on K and

will remain measures as no new subsets of either will be added. Several open problems are related to PROBLEM 4.6.

Can

m ÷ (m)m

[16]

and the present work:

be deduced from

m ÷ [mien

for any

n ~ 3? PROBLEM 4.7.

Does

AD

imply that

ml

is huge?

Is there a huge

205

measure on

[~2]~I?

PROBLEM 4.8. DC + ~ + (~)~

Is it a theorem of

that there are no

PROBLEM 4.9.

Is there a

that there cannot exist

k

k

k c ~ such that it is a theorem of

if

AD

is consistent,

might be quite small, though:

guous large cardinals, PROBLEM 4.10.

see Apter

Is

EP

The least 8 for which PROBLEM 4.11.

[I]

and

a theorem of

2.3

k

ZF + DC

must be greater

Bull

[2].

ZF + DC + ~ ÷ (~)~ ?

fails must be measurable.

If < and % are strong partition cardinals with

which are studied in

[83

,

~ < %,

~, or vice versa?

The similarities between strong partition cardinals and ÷ (~)~,

ZF +

for limitations on conti-

will < remain strong in the Spector extension for

above, suggest

or of

~?

contiguous measurables?

By a result of Kechris, than three,

ZF + DC + ~ i ~ 2 ~ °

MAD families coded on

w,

when

and the result mentioned in 4.4

that SDector forcin~ should preserve somethin~ more than

plain measurability.

But there are limits:

is replaced by a strong partition cardinal PROBLEM 4.12.

Does

ADR

the analogue of 2.3, when + fails for e = < .

K,

imply that every set of reals is Souslin?

A theorem of Woodin states that A D R is provable in

ZF + AD + "every

set of reals in Souslin". PROBLEM 4.13.

How strong is the theory

"8 is a regular limit cardinal"? every real

ZF + DC + V = L[R] +

Does it prove the existence of

The challenge in this question is to get 8 a limit cardinal: result of the Cabal that In O

[133,

~#

for

~?

ZF + DC + V = L[R]

Kechris and Woodin show that if

it is a

proves that 6 is regular. AD + V = L[R] holds,

then

cannot be weakly compact. PROBLEM 4.14.

It follows from the results of

is a strong partition cardinal)

follows from,

[I03,

that Con(~ I

e.g., Con(there are

arbitrarily large strong partition cardinals), but the proof goes via determinacy.

Is there a direct proof?

206

REFERENCES [ 13

Apter, A.W.

Some results on consecutive large cardinals, Annals of Pure and Applied Logic, 25(1983), 1-17.

[ 23

Bull, E.L.

Successive large cardinals, Annals of Mathematical Logic, 15(1978), 161-191.

[ 33

Dordal, P.L.

Independence Results concerning Combinatorial Properties of the Continuum, Thesis, Harvard, (1982).

[ 4]

Ellentuck, E.

A new proof that analytic sets are Ramsey, J. Symbolic Logic 39(1974), 163-165.

[ 5]

Hausdorff,F.

Summen von ~ i Mengen, Fundamenta Mathematicae, 26(1936), 241-255.

[ 63

Galvin,F. & Prikry,K.

Borel sets and Ramsey's theorem, J.Symbolic Logic 38(1973), 193-198.

[ 7]

Henle, J.M.

Spector forcing,

[ 8]

Hen!e, J.M. & Supercontinuity, Mathematical Proc. Cambridge ~thias,A.R.D. Philosophical Society 92(1982), 1-15.

[ 93

Jech, T.

On models for set theory without AC, Mathematical Reviews, 43 # 6078.

[I03

Kechris, A.S.

The axiom of determinacy implies dependent choices in L[R], J. Symbolic Logic 49(1984), 161-173.

[i13

Kechris, A.S., The axiom of determinacy, strong Kleinberg'E'M''ties and non-singular measures, Moschovakis,Y. N. & Woodin,W. 77-79, Springer Lecture Notes in H. Volume 839, ed. by A.S. Kechris,

J.Symbolic Logic 49(1984), 542-554.

partition properCabal Seminar Mathematics, D.A.Martin and

Y.N. Moschovakis, 75-99. [123

Kechris,A.S. & Equivalence of partition properties and deterWoodin,W.H. minacy, Proc. Natl.Acad. Sci. United States of America, 80(1983), 1783-1786.

[133

Kechris,A.S.& Woodin,W.H.

On the size of 0 in

L[R], (To appear).

[143

Kleinberg,E.M. Infinitary Combinatorics and the Axiom of Determinateness, Springer-Verlag Lectures Notes in Mathematics, Vol. 612(1977).

[15]

Mathias,A.R.D. Happy Families, Annals of Mathematical Logic, 12(1977),59-111.

207

[16]

Mathias,A.R.D. A notion of forcing: some history and some applications,(To appear).

[17]

Monro, G. P.

[18]

Mos chovakis, Y.N.

[19]

Oxtoby, J. C.

Models of ZF with the same sets of sets of ordinals, Fundamenta Mathematicae, 80(1973),105-110. Descriptive Set Theory, North Holland, Studies in Logic, vol. 100(1980). Measure and Category. Heidelberg,

Springer-Verlag,

New York,

Graduate Texts in Mathematics, vol. 2

Second Edition (1971). [20]

Spector,M.

[21]

Vopenka,P. & Balcar,K.

V

A measurable cardinal with a non-well-founded ultrapower, J.Symbolic Logic 45(1980),623-628. On complete models, Bull. l'Acad~mie Polonaise des Sciences, S~r.Sci.Math.Astron. et Phys. 15(1967), 839-841.

PROOF FUNCTIONAL E.G.K.

CONNECTIVES

Lopez-Escobar

University of Maryland Department

of Mathematics

College Park, MD. 20742 U.S.A. A characteristic

of the classical propositional

are truth-functional, tic connectives

is that they

that is,the truth of a sentence depends only on

the truth of its prime components. notions.

connectives

On the other hand the intuitionis-

are supposed to be much less dependent on semantical

Consequently,

one avoids

saying that a "sentence

is true",

rather one tends to say: "the construction Nevertheless,

c

proves

some aspects

(or justifies)

the sentence A".

of truth-functionality

can still be found in

that: (I) a construction

c

(2) the conditions

either proves a sentence

for a construction

given in terms of the conditions for example,

c

proves

of constructions

or it does not

,

to prove a compound sentence is for the proof of the components;

the conjunction



A

such that

(A 0 A Ai)

co

proves

iff A0

C

is a pair

and

cI

proves

Ai• Thus for a conjunction

to be provable

that the conjuncts be provable. different

constructions

traditional

it is necessary

In other words,

(and sufficient)

the fact that quite

may prove a given formula is not exploited

in

intuitionism.

G. Pottinger conjunction",

[1980]

introduced a conjunction,

w h i c h he called "strong

w h i c h requires more than the existence of constructions

proving the conjuncts.

According

to Pottinger:

^

"The intuitive meaning of ^ assert

A ~ B

~

can be explained by saying that to

is to assert that one has a reason for asserting

A which

is also a reason for asserting B"° Hence ~ is one of the first, ly proof-functional. a proof-functional

if not the first,

connective w h i c h

This paper is an introduction connective;

is tru-

to the "logic" of

in fact for the most part we shall con-

sider the extension of the positive

intuitionistic

calculus

of impli-

209

A

cation obtained by the addition of strong-conjunction

~.

As with any "new" logic, there are five questions that immediately come to mind: (I) Is there a reasonable,

intuitive concept of validity for the senten-

ces of the language? (2) Is there a formal concept of validity for the sentences of the language? (3) Is there a formal concept of derivation for the sentences of the language? (4) What are the relations between (I), (2)and (3)? (5) How does the new connective affect familiar mathematical theories (for example

: elementary number theory)?

The paper is broken down into 5 sections corresponding to the above 5 questions. §1.INTUITIVE VALIDITY. I.i THE LANGUAGE ~.

The sentential language ~

is to have the

following symbols: p,q,r ....

for the sentential variables, for the conditional connective, for Pottinger's strong conjunction,

( , )

for auxiliary symbols.

1.2 AN INTUITIVE INTERPRETATION FOR THE INTUITIONISTIC CONNECTIVES. Let

HA(C)

express the (decidable) predicate

proves the sentence

A"

dicate "the construction Finally if to

c,d

and c

~(c,O(x))

"the construction

c

express the (decidable) pre-

proves the free-variable formula e(x)"

are constructions then

d'c

is result of applying

d

c,

The Brouwer-Kreisel interpretation of the intuitionistic conditional is that: (*)

~A=B()

And although

(*)

iff

~(c,(~A(X)

~ ~B(d'x))).

is not universally accepted, it certainly is, and was,

a good point of departure for an intuitive interpretation for the intuitionistic conditional. 1.3 AN INTUITIVE INTERPRETATION FOR STRONG CONJUNCTION. Kreisel interpretation for ordinary conjunction a is ~AAB()

iff

~A(C) A ~B(d).

The Brouwer-

210

The latter

suggests

the following

~A~B() The following

iff

~A(C)

are equivalent

interpretation

for strong conjunction

^ ~B(C).

interpretations

(in an appropriate

theory

of constructions) ~A~B(C)

iff

~A~B() where

~

iff

~A(C)

is an equality

1.3 AN INTUITIVE A

of

~

that

~A(C)

A ~B(d)

relation

CONCEPT

is intuitively

^ 7~B(C) , ^ c ~ d,

on constructions.

OF VALIDITY

valid

FOR SENTENCES

OF ~.

iff there is a construction

A sentence c

such

~A(C).

The following

sentences

(I)

A ~ (A~B) ~ B

(2)

(A=B)

(3)

(A=B~C)

(4)

(A=C)

(5) (6) (7) (8) (9) (lO) (ll)

~ (A=C)

are easily

seen to be intuitively

valid:

= (A=B~C)

= (A=B)

~ (A~C)

= (A~B=C)

A ~ B = (A=B) A = (B=C) = (A~B=C) A~B=A A=A~A A ~ B = B ~ A ((A~B)~C) {(A=B~C)

We let

IVAL~

§2.FORMAL

= A ~ (BgC) = (A=B)

Z

~ {(A=B)

be the set of intuitively

~ (A=C)

valid sentences

ALGEBRAS

AND CURRY ALGEBRAS.

is to be of the algebraic

type.

However

algebras

of open sets, we plan to use algebras

binatory

logic.

2.1~i where

DEFINITION~

o is a binary

Given an applicative we will write

= (A=B~C)} of ~.

VALINITY.

2.1 APPLICATIVE for

~ (A=C)}

operation

on M.

algebra

<M,o>,

"(a,b)".

from indeterminates

An applicative

The formal instead

related

algebra

of using

to Curry's

A polynomial

in

<M,o> of

M

the com-

is an algebra M = <M,o>,

then instead of writing

x,y,z,...,elements

semantics

"o(a,b)",

is a term built up and the application

operator. 2.1.2

DEFINITION.

A combinatory

algebra

is an applicative

alge-

211

bra

<M,o>

which

and which

is non-trivial

is combinatory

h(x,y,...,z)

in

<M,o>

2.1.3

DEFINITION. <M,o> #

..... z)

k

VxVy[((k,x),y)

(3)

VxYyVz[(((s,x),y),z)

=

are combinatory

assignment

is an algebra

algebra

such that:

C

<M,o,k,s>

=

and

x], =

((x,z),(y,z))]. on combinatory

SATISFACTION

IN CURRY ALGEBRAS. C

that

=

is a function We extend

logic

that the Curry

complete.

Assume sets of M.

a ~M

s,

It was one of the first theorems

2.2

an element

h(x,y ..... z)]

A curry slgebra

(2)

algebras

=

is an applicative

(I)

at least two elements),

that is, to each polynomial

there corresponds

VxVy...Vz[(..~(a,x),y)

such that

(i e. contains

complete;

<M,o,k,s>

pa

pa

is a Curry algebra.

which maps

the sentential

to act on all sentences

of

A

C-proof

variables ~

to sub-

by requiring

that: pa(A=B)

=

{m ~M!

pa(A~B)

=

pa(A)

The elements

in

Curry algebra Loosely pa(A)

#

~

logic,

n pa(B). will be called

the "pa-proofs

of

A

~

is to be formally

for every Curry algebra

C

and

because

quired uniformity

a sentence

A

of the constructive

C-proof

(intuitionistic)

degree of uniformity

(in the

is required.

valid iff

assignment

pa.

charscter

of the

We achieve

by using terms of a first order language

for both the Curry algebras ~C

c pa(B)]}

of

a certain

cally,

÷ (m,n)

C)".

spesking,

However,

pa(A)

Vn[n Epa(A)

and the proof-assignments.

is the first-order

language

~C

Or more

(with equality:

the resuitable specifi-

m ) which also

contains the individual the binary

Next, pa

constants: function

the unary relation

symbols:

given a Curry algebra

we form the first-order

C[pa]

K, S,

(infix)

=

<M,o,k,s,pa(p),pa(q)

associated

to the language

C

symbol:

=

,

P,Q,R .... <M,o,k,s>

and a

C-proof

assignment

C[pa]

is a structure

structure: .... >. ~C"

Clearly

212

In order to relate each formula

A

pa(A)

of

~

to satisfaction in

a formula

~A[X]

if

A

is the sentential variable

if

A

=

(B~C), then

if

A

=

(B=C),

~A[X] =

then ~A[X] =

of

C[pa], we associate to ~C as follows:

p,

then

~A[X] = P(x),~..,

~B[X] ^~c[X], VY[~B[y] =~c[X-y]]-

A routine induction then gives us that for every sentence (*)

m

satisfies

~A[X]

in the structure

Finally, given a closed term a Curry algebra

C,

tC

t

C[pa]

iff

A

t

~:

mepa(A).

of the first-order language

is the denotation of

of

~C

and

in the algebra

C.

We are now ready for the definition of formal validity: 2.2.1

DEFINITION.

A sentence

A

of

~

is formally valid

there is a closed term

t

of the first-order language

for all Curry algebras

C

and all

iff

L~ such that proof-assignments ~ pa:t C e pa(A).

In view of the remark (*), the condition "t C ~pa(A)" can be replaced by: (**)

The first-order sentence

We let FVAL~ 2.3

~Art] is true in the structure

be the set of sentences of

~

which are formally valid.

AXIOMATIZABILITY OF THE FORMALLY V A L I D S E N T E N C E S Let CA

be the following set of sentences of

-IK ~

C[pa].

OF

~.

~C:

S,

VxVy[((K-x)-y) ~ x],

~x~y~zE(((S-x) .y) .z) ~ ((x-z)- (y.z)) ] and

THMcA

be the set of logical consequences of

CA.

G@del'a completeness theorem and (**) gives us that: 2.3.1

THEOREM.

is a closed term

t

A formula of

~C

A

of

such that

~

is formally valid iff there ~A It] E THMcAo

An immediate consequence of the above theorem is that recursively enumerable set. following: 2.3.3

THEOREM.

FVAL~

is a

A variant of Craig's lenmm gives us the

There is a recursive axiomatization for

FVAL~.

PROOF. Let S0,S 1 ..... Sn,... be an enumeration of FVAL~ by a (primitive) recursive function. Then for axioms take the following

213

set of sentences

of ~:

{(S0&S0),((SI~SI)&SI),(((S2~S2)&S2)~S2) Clearly allows

the set is recursive. one to conclude

An application conjunction

A

As rule'of

from

of the concept of formal validity

is indeed different pa(A)

p = (q=p~q)

(2)

(p~q) ~ ((p~r)

(3)

(p~q=r)

(4)

(p=p)

3.1

A

requirements

we place on

~

A, for all sentences

of ~, then

A

and

(M)

if

r

~

A

then

in F also occurs

A,

F ~

A ~

B

A

whenever

~

F ~

sequence

of formal

r of sentences

are:

B,

every

sentence

that occurs

in A.

since the conditional

we also require

be a relation

and a finite

A

~

is the intuitionistic

conditional

that

F, A I" B 3.2

Let r ~ A

a sentence

F

validity

pa

A:

~ (p~q&r))

REQUIREMENTS.

between

if

(D)

sentences

DERIVABILITY.

(T)

Furthermore,

and proof-assignments

it can

& (p=(q=p)).

of ~.Then the minimal (R)

C

For example

= (p=(q=r))

MINIMAL

derivability

take the rule that

is to show that strong

from conjunction.

is empty for the following

(i)

§3. FORMAL

inference

((--(A~A)~...)&A).

be shown that there are Curry algebras such that

.... }

iff

SEMANTICAL

(see 2.2.1),

r ~ (A=B).

CONSEQUENCE. we propose

In view of the definition the following

definition

of formal

for semanti-

cal consequence: 3.2.1

DEFINITION.

of the sequence iffthere C

if

B I .... ,B r

is a term

and all

The sentence

blC pa(B I)

of

assignments

'''"

~C

BI,..o,B r ~ A,

such that for all Curry algebras

then '

t

tC bl'"

validates Xl,...,x r

<,A>.

in symbols:

consequence

pa:

,b r Epa(B r)

We shall say that the term

is a semantical

of sentences,

txl ..... Xr

C-proof

A

.. b c pa(A). ' r

the pair

214

Call a pair



a sequent.

Using the formulae 3.2.2 3.3 Let

LEMMA.

A RECURSIVE AXIOMATIZATION

by a (primitive)

following

VALS = {IP >

VALS is a recursively

<,A0 > ,

VALS

Define

~A[X] of section 2.2

A}.

one can then show that: enumerable

set.

FOR DERIVABILITY.

< ,AI> .... be an enumeration

recursive

function.

Then let

AXMS

of

be the

set of sequents:

{<<(B00~B00) ..... (B0r0~B0r0)>,

(A0~A0)>,<<((BI0gBI0)~BI0)

....

((B!r ~Blr )~Blr )>,((AI~AI)~AI )> .... } Again,

it is clear that

Then let

~

AX~S

is a recursive

set of sequents.

be the smallest relation containing

a sentence of

~}

and closed under

(T),

AXMS

u

{<,A> IA

(M),(D),(~ ~ ) * and ( ~ ) * ,

where the last two rules are: (~ 3)*

F., (... (((.B~B)

~B) . . . ~.B..)..... ~... A

r, B ~ A, (~),

F~

(...(((A~A)~A).o.~A.~

.......

F~A, respectively. After

verifying

sequences

one

3.3.1

that easily

the

above

obtains

THEOREM.

the

A

rules

following

For any sequent r ~

3.4

mentioned

iff

preserve

completeness

F ~

which is closer to Gentzen's sentential Gentzen's

calculus.

FOR DERIVABILITY.

the conditional of

Sequent calculus

schema for the rule.

Unfortunately

although

it may appear as simply sentential

The modification

is in the applicability

calculus of

(for the introduction

of the rule, not in the

Some persons may find the restriction of interest

since it is a global restriction of the inference,

§3.3,

for the intuitionistic

for the intuitionistic

and conjunction.

~ in the succedent)

given in

We now present another axiomatization

In fact, at first sight,

axiomatization

theorem:

A.

The set of axioms of the axiomatization is of no practical use.

con-

:

A SOUND AND NATURAL AXIOMATIZATION

recursive,

semantical

involving all the nodes above the node

and not just those immediately

above.

a price has to be paid for such naturalness

and although

215

we can show that the a x i o m a t i z a t i o n ed in showing

In order to further

emphasize

write

in the form

the sequents 3.4.1

the relation

to Gentzen's

3.4.2

~A.

CUT RULE OF INFERENCE: F,A~

B r

3.4.3

F ~ A

~

B

STRUCTURAL RULES OF INFERENCE r~A

(M3N)

5=>A provided

every

A r~ ~ A

r 3.4.4

, sentence

r~ (REP)

occuring

in

r

also

(= ~)

r, A ~ B

3.4.5

r~

r

~ A

r~

B

(g~)

the rule

(--> g)

speaking

(~

F

and

B look the same.

PRE-DERIVATIONS.

~

C

C

in obtaining

F,B ~

C

~

C

r,B g A

is not universally

A pre-derivation

N, while $N;

$ ~ N

and

~

~

C

applicable. of

in other words

sentential

consists of a finite tree

defined on T.

SN

is the

is the name of the rule schema a pre-derivation

is to all intent and purposes

of the intuitionistic

C

the latter statement.

and two functions

sequent occuring at the node Z

B

~

g) may be applied when the (sub) derivations

We now proceed to make precise

T (of "nodes")

~

F,A g B

Loosely

r ~

r,B

~ A g B

As already remarked

language

F,B

r ,An

A r

followed

5.

RULES FOR STRONG CONJUNCTION.

g)

3.4.7

in

RULES OF INFERENCE FOR THE CONDITIONAL.

=)

~ A

occurs

A

r~A=B

(~

systems we will

"r ~ A".

A X I O M A SCHEMA: A

(~

is sound, we have not yet succeed-

it is complete.

calculus

a derivation

in the

(with analysis)

of the conditional

and usual

conj unc t ion. Given a p r e - d e r i v a t i o n structure of

P.

P

=

The reduced

,

then



logical structure of

is called the logical P

is the pair

216



where

unless

~N

R

is the function defined on

is either

(MON),

(REP),

(~)

T

such that

or

(~)

=

in w h i c h case

A

RN

=

0.

Then two pre-derivations equivalent,

in symbols:

PI = <TI'$1'~I> PI ~ P2'

and

P2 =



iff their reduced logical

are

structures

are isomorphic. 3.4.8

DERIVATIONS.

A pre-derivation

w h e n the following condition At each node

N

of

tions immediately

T,

T

above



at which N

~

=

(~),

D

=

is a derivation and

then

D

is a derivation of the sequent

in case that there is a derivation F ~

is a derivation

the two sub-pre-deriva-

are equivalent.

If

If F is empty and



is met:

A, then

~

is the root of the tree

$~.

We write

of the sequent

A

"F ~ A" just

F ~ A.

is a (formally)

derivable

sentence

of Z . 3.4.9

AN EXAMPLE OF A DERIVATION.

First consider the following

two derivations: A = B DI

(A=B)

--> A = B

~ (A=C)

D2

(A~B)

~ A

~ A = B

(A=B)

A = C

A

~ (A~C),A

~ A ~ C

g (A~C)

~

B

A,A = B

~

B

C

~

C

A,A ~ C

~

C

~

A

B

~ A

~ A ~ C

(A=B)

B

g (A=C),A

~

C.

A moments reflexion

shows that they are equivalent.

is a derivation of

(A=B) ~ (A=C)

DI (A=B) ~

Thus the following

= (A~B~C): D2

(A=C),A .~ B (A=B)

(A=B)

~ (A=C),A

(A=B) ~ (A=C)

~ (A=C),A

~

C

B ~ C A=B

~ C

(A~B) ~ (A=C) = (A=B~C) 3.4.10 derivation n

SOUNDNESS THEOREM.

An induction on the length of the

gives us that to each derivation

we can associate

a term

tD x0,-.-,Xn_ I

D

and each natural number

such that

:

217

(i)

if

D

is a derivation of the sequent

then

(2)

if

tD Xo,--.,Xn_ I DI,D 2

validate

are equivalent

DI =

S

derivations,

then

t

x0,..-,Xn_ I term

tD

3.4. Ii

x0,-..,Xn_ I.

can then be used to show that:

THEOREM.:.... If

F ~

A,

then

F

~

A.

A NORMAL FORM THEOREM FOR DERIVABILITY.

of the natural a x i o m a t i z a t i o n

given in

normal

t h e o r e m for it.

form (cut-elimination)

the proof of n o r m a l i z a t i o n (the structural

(A~B)

§3.4

and

An interesting aspect of

D

in w h i c h

t-cut-free

the usual rule of repetition).

there are no cut-formulae

equivalent

N

of

T

w i t h cut-formula

(A~B),

derivation with cut-formula

that around the node

D

F, (A~B)

have

[D 2]

is



A

or

B.

and a D

~ C

F ~

--> (A~B)

C

case is w h e n the last rule of inference applied in (~ 3)

[(~g)

respec.]

In such a situation we w o u l d

(for example): 2

Di F,A

~

_F.., (A~B)

D2 C

F~A ~

C

. F....-~

F

F

~

B

(A~B)

-~ C

Then we transform the above derivation

to an

Thus assume

is as follows: D2

F

D1

D =

one can transform

either

N, the derivation D1

The critical

of the

derivation.

The first thing we show is how given a derivation cut-node

Another advantage

is that we can prove a

is the essential use of the rules of repetition

rule (FDN)also includes

Let us call a derivation form

~ A,

D2

t

The

S = B 0 .... ,Bn_ I

into the derivation:

218

Di F,A

-~C

F,A

~

F~A

C

F r

~

r ~

~A

A

C

Iterating the above p r o c e d u r e one can then t r a n s f o r m a given d e r i v a t i o n into a d e r i v a t i o n w h i c h is

~-cut-free.

Then traditional methods of r e d u c i n g a cut-formula of the form (A=B) can be m o d i f i e d so as to apply to our axiomatization.

I n t e r t w i n i n g the two

reductions one can then obtain the n o r m a l i z a t i o n theorem. §4.

R E L A T I O N B E T W E E N THE CONCEPTS INTRODUCED. 4.1

THE C L A S S I C A L CASE.

of i n t u i t i v e l y valid sentences, provable sentences respectively°

Let

IVALc,

SVALc,

THM C

be the sets

s e t - t h e o r e t i c a l l y valid sentences and

(in some natural axiomatization)

of classical logic

Then one usually argues as follows:

(.i)

IVAL C

!

SVAL C,

because if a sentence is valid in all possible structures then it certainly is valid in all s e t - t h e o r e t i c a l structures. (.2)

THM C

~

IVALc,

b e c a u s e the axioms and rules w e r e chose so as to be correct• Combining

(.i)

and

(.3)

THM C

(.2) !

Then GSdel's completeness

one then obtains IVAL C

!

SVAL C •

theorem, using a rather w e a k set-theory,

gives

the m a t h e m a t i c a l result: SVAL C Combining the latter w i t h (.4)

THM C

=

(.3) IVAL C

i

THM C •

then gives us: =

SVAL C,

We remark once again that in order to derive assumptions 4.2

(.4) certain existential

in set-theory were required. THE CASE F O R THE (CONSTRUCTIVE)

instead of 4o1.1 (.i)

LANGUAGE

~.

First of all,

we have FVAL~

~

IVAL~,

b e c a u s e if we have a term (of c o m b i n a t o r y logic) w h i c h validates a sen-

219

tence

A, then we do have an intuitive construction that proves A.

The converse is by no means obvious. The soundness theorem, combined with (.I) then gives us (.2)

THM~

!

FVAL~

2

IVAn.

Now a completeness theorem for the axiomatization of know that F V A ~

§3.4 (we already

is recursively axiomatizable) would give us only the

mathematical result that TH~

=

FVAL~.

Unfortunately the above result does not produce the analog of the classical 4.1.4.

In order to obtain such a result (even in the presence of

a mathematical completeness theorem) we need first to justify that (.3)

IVAL~

2

FVAL~

The latter could be immediately obtained if one could show that any intuitive construction could be represented by a term of combinatory logic; in other words, 4.2.3 Church's thesis.

would be a consequence of (some form of)

If Church's thesis is to be involved, then it is probably advisable to consider number theory in some more detail. §5.

NUMBER THEORY AND STRONG CONJUNCTION. 5.5

A CONCRETE MODEL FOR THE SENTENTIAL LANGUAGE ~.

Using

the techniques of Troelstra [1979] one can show that there are primitive recursive terms each sentence

where

PRA

E, p, %, A

of

~

primitive recursive predicates a primitive recursive predicate

P, ~ PA

(.i)

PRA

~

PB~C(~) ~ P(~,FPB(X)~Pc(~(~,x))]),

(.2)

PRA

~

PBEC(~) ~ P(n,FPB(%n)^Pc(Pn)^%ngpnT),

(.3)

PRA

~

n ~ ~

(.4)

PRA

~

PA(~) ^

(.5)

if

and for

such that:

A (n-mm=m~n) ^ (n~m^m~r=n~r) n ~ m = PA(m) ,

A c FVAL~, then for some n,

PRA ~ PA(n),

is primitive recursive arithmetic and F ] gives the numeral

corresponding to the G~del number.

Let Then

5.2 A FORMAL SYSTEM OF NUMBER THEORY FOR STRONG CONJUNCTION. HA be intuitionistic number theory formulated in a sequent calculus ~ HA(E) is the extension of

HA

obtained by

220

(I)

Enlarging

(2)

Add the inference

(3)

Add the (binary)

(4)

Define pre-derivations ly to

(5)

schemas corresponding

(& ~)

and ( ~ & ) ,

and equivalent pre-derivations

analogous-

§3.4.7. in

HA(&)

FORMAL REALIZABILITY

complexity

is a formula

xrA

lizability

A.

of

to

&.

rule of repetition,

Define derivations

5.2.1 logical

the class of formulae so as to include

FOR

of the formula of

HA

analogously HA(&).

A

of

§3.4.8.

Using an induction on the

HA(&)

one can show that there

which formally expresses

The only addition required

to

the recursive rea-

to Troelstra

[1973]

is the

is an almost-negative

formula

clause: xrCA&B) For each formula of ~ .

of

(.i) Now let

then give us that:

HA(&)

[1973]

(.2)

~ (.i)

xrA

[1973] )•

~ A, then

HA

ECT 0 be the schema of the

Troelstra

Combining

if

xrA ^ xrB

HA(~),

(see page 193 of Troelstra

The usual techniques

over

A

=

~

~x[xrA]

"extended Church's

~

~x[xrB]

and

HA + ECT 0

(.2)

iff

HA + ECT 0

(with respect

connectlve

)•

of

§3°4

HA)

However

that the completeness

will involve some form of Church's

thesis.

that the results obtained in this

connectives

do not find that strong conjunction

are viable concepts,

candidates would be given by: A ~A~B(C)

~-> B

=

(A~B)

we

is, per se, of great interest.

much more interest would be some kind of strong equivalence;

(2)

HA

for & to be considered as

the contention

Although we feel confident

(I)

of

over

CONCLUSION.

note show that proof-functional

possible

HA

In any case, the above conservative

extension result further supports of the axiomatization

In

of

is conservative

is conservative

(which in our opinion is a minimum requirement an "intuitionistic

HA(&)

to the formulae of ~(&)

B

~ B.

we then obtain that

more interest would be to show that

§6.

thesis".

, page 196 it is shown that for formulae

~ (B~A),

~ ~(c,F~A(X ) ~ ~B(X)7).

two

Of

221

But of course, it was strong conjunction that led us to the concept of a proof-functional connective. It perhaps should be remarked that accepting proof-functional connectives, such as strong conjunction, requires rejecting the assumption that a construction proves a unique sentence and thus forces us to distinguish between a construction as an object and a construction as a method. REFERENCES Pottinger, G.

[1980]

A Type Assignment for Strongly Normalizable %-Terms, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Edited by J.P. Seldin and J.R. Hindley, Academic Press, N.Y.

Troelstra,A.S.

[1979]

The interplay between logic and Mathematics: Intuitionism.

Report 79-01 of the Mathematisch

Instituut, University of Amsterdam. Troelstra,A.S.

[1973]

Mathematical Investigation of Intuitionistic Arithmetic and Analysis.

Lecture Notes in Mathe-

matics, vol. 344, Springer-Verlag Publishing Co.

ULTRAPRODUCTS

AND CATEGORICNL

LOGIC

M. Makkai* McGill University Department of Mathematics and Statistics 805 Sherbroo~e ~t. West Montreal H3A 2K6 Quebec, Canada Introduction In categorical

logic, there is a natural way of considering

idea of a logical operation construed

as an operation

a given category,

in general.

A logical operation

acting on finite diagrams

yielding

finite diagrams

the

can be

of a given type in

in the same category

as

values. In symbolic certain standard symbolic

a standard

development category,

in categorical

most frequently

The next step is to consider

disembodied

the original abstract

become formulas

in symbolic

In symbolic life situation

the

and morphisms,

including sentence

only has to codify in precise

In an attempt

universe,

just as sets and relations

in choosing

from the real the rules of

and inference.

in natural

languages;

In one

terms what is already given intuitively

form.

be hampered

or the "true",

nature of

rather than helped by our linguis-

This background may have been shaped in an arbitrary

manner by an evolutionary mathematical

in arbitrary

of the diagrams become

formation,

to clarify the general,

logic, we may, however,

in

logic.

this we are uniquely helped by experience

tic background.

logic starts by

'same' operations

to the symbolic one consists

complete

the

Set, the category of

logic the main step of the transition

formal manipulation,

in a practically

to considering

acting on finite diagrams

sets and functions

objects

having

governed by formal rules of mani-

certain selected operations,

categories;

expressions

and one proceeds

uninterpreted,

The parallel

considering Set.

interpretations,

expressions

pulation. sets, and

logic, one starts with symbolic

process.

logic is essentially

If one believes,

as I do, that

a part of the makeup of the physical

rather than being an addition

life, then a theory of logic independent

to it contributed of concrete

by intelligent

aspects

of language

is desirable. A particular

difficulty

arising in symbolic

encountered when trying to coach in symbolic * The author's

research

logic is the one

terms a general

concept of

is supported by NSERC Canada and FCAC Quebec.

223

logical operation.

In abstract model theory,

at such a general concept,

one considers

operations on strings of variables,

in an attempt at arriving

generalized quantifiers,

and strings of formulas.

It is

strikingly clear that the formal aspects of the concept are too closely modelled on previous abstract integrity,

linguistic patterns;

the concept does not have an

a good chance of being comprehensive.

In categorical

logic, the main step of the transition to the

abstract situation is to decide what constitutes, arbitrary category, standard category, natural one.

in a more or less

the same operation as the one we start with in the say Set.

One considers

The general shape of the procedure those properties,

as many as one can find,

or maybe, only some of them, of the given concrete operation combinations

of various given concrete operations)

in the language of categories "theories",

(or

that can be expressed

(i.e. in terms of composition of arrows),

and one simply imposes these properties categories;

is a

as requirements

on the kind of

that serve to embody the abstract essence of

the logic under consideration.

The crucial point is that the "theory"

is a structure of the same kind as the standard structure we start with: they are both categories.

In the symbolic context,

a structure radically different

the formulas

form

from the entities making up the standard

interpretation. Although the choice of the properties may not be straightforward, and a priori

it is not clear if the proposed procedure

in capturing at least as much as symbolic

can be successful

logic does, the proposal at

least has a clear enough outline to constitute a program for an investigation of logic in a general manner. This is not the place to give a historical categorical

logic.

that categorical

Nevertheless,

introduction to

we may make the blanket statement

logic has been successfully developped along the lines

indicated above to cover at least first order, and also, higher order logic, in both the so-called classical and intuitionistic versions. particularly attractive character.

E.g., the definable concepts

order theory on the one hand,

(roughly,

but those two structures construction,

are connected,

functor, mapping the theory-category

a category.

Net only that

the so-called evaluation

into the category of functors from

This, or other similar constructions,

ultimately, be explained without so gotten are clumsy,

of a first

as a result of a general cat-

by a crucial functor,

the model-category to Set.

formulas)

and the models of the theory on the other

hand, both form the same kind of structure: egorical

A

feature of the resulting theory is its unified

could,

category theory, but the explanations

and they lack the coherence of a good theory.

224

In first order logic egorical

(our sole interest

logic has revealed

in this paper),

cat-

that, in order to have a good general

of logic, one should consider

theory

certain kinds of 'definable concepts'

associated with any given first order theory that do not fall under the usual definable

sets or

relations

of symbolic

able sets arise either as a disjoint set of equivalence definable

set.

has become [CHL].

important

Those new defin-

sum of definable sets, or as the

classes of a definable

Recently,

logic.

equivalence

relation on a

the use of the more general

in stability

(These developments

theory;

c.f.

definable

sets

ESh] and especially

have been independent,

so far, of categorical

logic proper). The central

concept of first order categorical

the notion of first order theory, notion that underlies, stability abstract logic.

implicitly,

theory mentioned algebraic

above.

geometry

view of logic,

has its central

since it enjoys completeness

that the concept without

the generalized

definable

The concept originally

(sheaf theory), without

The notion of pretopos

logic, replacing

is the notion of pretopos.

cannot be made to comprise

role,

sets of

arose in

any connection

each indicating

logical operations

causing the notion of theory to lose essential properties.

The subject matter of this paper is one such completeness It is based on a very general operation on diagrams, operations) complement.

~in fact, too general)

of pretopos

operations

functors

is shown to be not too general).

follows a familiar pattern:

logical concepts

in Set

on Set are

'algebraic'

the definition The result,

characterizations

a stronger

Gaifman's

form of Gaifman's

theorem.

again in essence,

theorem

The proof of the theorem of the arguments.

The comparison with

result is made harde~ although only in a superficial

the fact that Gaifman's

terminology

Part 3 of the paper contains The exposition

(not using categories)

the details of the comparison.

to categorical

in the categorical

w a ~ by

is different.

of the theorem is used as an opportunity logic.

to give

The methods used here differ from

those in [MR] inasmuch as I have tried to use, wherever inherent

of

consist of a proof of Gaifman's

theorem followed by additional

an introduction

of

The main result of the present paper is, in essence,

present paper will, original

of

via ultraproducts.

The main result is closely related to an unpublished of Haim Gaifman.

('logical'

augmented with Boolean

(Thus, after all, in the given context,

a logical operation

property.

concept of

and it says that the operations

that commute with the ultraproduct

exactly the composites

course,

to

from the point of

properties, further

It is the

formulations.

I could, methods

All unexplained

category

225

theoretical terminology can be found in ECWM]. I owe special thanks to Andrew Pitts,

from whom I have learned

much on conservative and quotient morphism.

I also thank Michael Barr

who pointed out an embarassing error in an earlier version of the paper. Part 1. I.i

Logical Operations

Operations o ~ Diagrams

Logical operations are seen, on the simplest on sets:

the operation,

level, as operations

applied to one or more sets

arity of the operation),

yields another set.

(depending on the

On the second level, the

operation on sets is abstracted, with retaining its essential properties, and one ends up with an algebraic operation, the customary sense: ations union, algebras,

or a logical operation in

an operation acting on formulas.

intersection,

The set oper-

etc. giving rise to abstract Boolean

and giving rise to the logical connectives,

are the obvious

illustration of this state of affairs. In fact, not only the connectives, be so construed.

but also the quantifiers

To do so, one should, however,

idea of an operation on sets.

can

slightly refine the

Instead of talking about operations

acting on ordered finite tuples of sets, one has to consider ones operating on finite systems of a slightly more elaborate kind.

It is

a basic insight of category theory that, by talking about systems of sets and functions,

each mapping one of the given sets into another one,

complex ideas concerning sets can be elegantly expressed.

A system of

the kind we need to have is usually called a diagram of sets and

functions.

In this paper, we are interested mainly in finite diagrams.

For a while,

a diagram will always mean a diagram of sets and functions.

A finite diagram is a system of finitely many sets and functions given shape'.

'of a

An example would be the diagram f > B

A

h

~C

g consisting f is

A,

of three and

f

sets

maps

and t h r e e S

into

functions,

B, e t c .

The

such that 'shape'

the

of this

domain of diagram

is this:

÷ -

such a thing is called a graph.

;

(I)

226

Before we make these concepts present

the existential

a finite diagram,

quantifier

general

and precise,

as an operation

let us re-

that yields,

from

more sets and functions making up, together with the

given one, a larger finite diagram. One starts with a relation over a set

X,

its second-place

can also write of

X.

Now,

five sets:

Rxy R

for

R.

variable

itself is a subset of R

i: R -+ XxY.

j: ~yRxy --~ X.

The fact that

'explained',

#2: X×Y --~ Y.

Thus,

first-place on

Y

XxY.

variable

(usually,

ranges

Y = X).

~yRxy,

We

a subset

So, we have the following

These sets are connected by maps.

being a subset of

inclusion map: will be

whose

We want to talk about

X, Y, X×Y, R, 3yRxy.

First of all,

Y

R

X×Y

means

the presence

of the

We will have another inclusion map: X×Y

is the Cartesian product

by two projection

the operation

gives, i

R

>

maps:

X

and

from the diagram XxY

X

one further set and one further

of

#i: X×Y -+ X,

Y

function,

extending

the given diagram

to i

ayRxy

> XxY

~ X

We have not mentioned yet how actually we only w a n t e d to make clear its operations

yields

Y

the operation

'shape', or arity.

from each of certain diagrams

7\ a diagram of the shape

is defined:

In fact, the

of the shape

227

/\ extending the first diagram in the obvious sense. The above example following way.

can be streamlined,

We start with a single

in the above situation, clearly,

the subset

of the function with any

f.

f

~ X

;

is the composite

3yRxy

of

of

i

and

~I"

is the same as the range

X

So, we consider

Then (or image)

the following operation:

f

R

in the

function

f

R

and generalized,

~ X

we associate f

R

with S the range of latter operation

f,

and

has the

j

~ X

the inclusion map of it into

7

It is the next step in category

theoretical

that not only the form, but also the content quantifier

language of diagrams,

~and many others)

theory.

For an indication

the idea of

added:

in the

the concept

of

Doing so, we are right in the middle of

look at the above simplified explicate

insight to realize

of the operation of the

can be expressed

with one new element

composition of functions. category

The

'shape'

° ~

existential

X.

of what would be going on,

operation.

'inclusion'.

let us

The first thing would be to

This is done by the notion of

monomorphism, whose definition I will not repeat here (c.f. ECWMI). To be sure,

a monomorphism

in Set is not necessarily

an inclusion

~the

228

converse

is true, of course) but at least,

category of sets) R

) U

inclusion,

it is the case that

any m o n o m o r p h i s m is i s o m o r p h i c

is a m o n o m o r p h i s m ,

and i s o m o r p h i s m s

R

then there are ~ R',

U

(in the

to an inclusion: R'

* U'

if

) U', an such that the

square

In fact,

~ommut~8.

U

~ U'

U'

the i d e n t i t y

R

+U

R'

+ U'

can be chosen to be equal to

U,

with

function.

S t a r t i n g w i t h the d i a g r a m f

R

+ X

we consider an e x t e n s i o n diagram f

R

/, ,

X

S with the following properties: q: R

~ S

j

is a mono;

R

commute

moreover,

there is

making

[since

j

following p r o p e r t y

is a mono, (q

is an

whenever

..............l. X

q

is n e c e s s a r i l y unique] and having the

'extremal epi'):

R

q

) S

q' S' commutes,

and

k

is a monomorphism,

then

k

must be an isomorphism.

229

(We have described phism

f).

the extremal

epi-mono

factorization

It turns out that the property

uniquely up to a unique isomorphism: S

(with the same isomorphism S

defines

j

whenever

.................... J

S'

described

of the mor-

~ X

............ ~ '

,~ X

then there is a unique

X) both answer the description, ~ S' making

"X

commute.

In fact, the inclusion

does answer the description; tion, we have described, 'range ' What is important ultimately,

of the range of

thus, by the above

f

into the set

'categorical'

up to a unique

isomorphism,

the concept of

about the above

'definition'

of range,

that of the existential

quantifier,

elements of the sets.

mentioning

that:

their elements about the

as objects

'category of formulas'

existential sets.

quantifier

In the foregoing terms of category

Therefore,

in a category.

discussion,

theory:

without

abstract,

it is natural

commute,

that

to try to talk

In fact, we will give in

the same definition

(or 'range')

We will have to continue for explanations,

representing

their being unspecified means in particular

are not given.

'formulas'

abstractly,

Now, it is clear that in logic we do

we talk about formulas

"unspecified", sets;

and

is that it makes sense

in a context where we talk about sets and functions precisely

X

defini-

of the operation

of

as we did above in the category of

I already mentioned category,

some technical

monomorphism,

to use such elementary

categorical

isomorphism. concepts;

I refer to [CWM].

A graph (see [CWM~), is like a category but with reference to composition and identity morphisms removed. The appropriate structure preserving maps between graphs are called diagrams. Since every category is a graph

of type

G

(has an underlying

in a category

C:

graph), we may speak of a diagram

a diagram

D: G

~ C.

230

Given a graph C

form the objects

transformations type

G

and a category

of a category

(see [CW~]).

(G,C)

E.g.,

(I) is a system of arrows

C,

f

in

between

C

G

in

are natural

two diagrams

of

as in the picture:

h

B

of type

whose morphisms

a morphism

~, B, y

A

the diagrams

, C

f'

A' satisfying the square

-+ B'

g,

three commutativity

A

conditions,

f

.

A' should be commutative;

one for each arrow

~ B'

and two more similar

conditions.

is an invertible

it is one in which every component

In particular,

natural

(in the example,

transformation;

~, 8 or ~)

We are ready to give a very general should be.

Upon reading

definition

of what

the definition,

it turns out that what one would

are not sufficient

Let

term C

in the important 'operation'

be a category,

C

of type

C

(K' c Ob(G',C))

(i)

K'

diagrams D~I c K'

is closed under in

(G',C),

(i = 1,2),

and

(G,G')

of first order

logic.

Thus

of G'.

An

~:

G a subgraph

is given by a class K' of diagrams satisfying

Di c K',

that restricts

analogy

context

(if then

Di = D!IG1 (restriction then there

K'

operations'

isomorphisms

is an isomorphism, to

the following Di

and

Di c K') to G), and

is a unique isomorphism

are isomorphic

and

(ii)

~: D 1

if + D2

~': Di -----+ D½

~'IG = ~.

algebraic

operations).

of

conditions:

D½

may be called the graph of the given operation to ordinary

How-

'fully defined

G and G' graphs,

type

in

instead.

call

is preferable.

operation in G'

an operation

the reader may

feel that we should have talked about partial operations

the shorter

is,

an isomorphism.

in a category ever,

in (I):

~ B

f,

an isomorphism of two diagrams individually,

h'

(in obvious

231

An operation

is finitary

(having finitely many objects

if

G'

The domain of the operation (G,G)

which are restrictions

It is easy to see that

K

(hence

G

too)

is a finite graph

and arrows). is the class

of ones in

K

of those diagrams

K': K = {D'IG:

D': G'

is closed under isomorphisms.

We are going to call the operation fully defined if We will soon see an important

example

When we have two categories, so that the two operations consider

in algebra.

of a non-full

with an operation

to be the

specified

In particular,

"operation-preserving"

functor of any diagram in the graph of the operation

Before we come to examples, satisfy a stronger

arbitrary morphisms,

in fact,

the modified

Let us call such operations important

operations

which

of the

done with ordinary operations (with re-

this is the case if the image under the

gory is in the graph of the operation

that,

to

we have the important notion of a functor

from one category to the other being

operations

in each

'same', or "realizations

as it is frequently

spect to the given operations):

meaning

K = (G,C).

operation.

are of the same type, we might want

the two operations

same operation-symbol",

in

~ C}.

let us mention

condition, not just

condition

in the domain cate-

in the codomain

namely

category.

that the most

important

(ii) above with ¢ and ~'

isomorphisms is stronger

strong operations.

(it is easy to see than the original

Nevertheless,

do not satisfy the stronger

one).

there are

condition.

Logical operations are operations in the category of sets, and, more generally,

operations

operations

in Set.

less, e.g.

our description

in other categories

Of course,

the designation

of pretoposes

The reader

(operations

as operations

should be like

(or look up, say in [CWM])

in categories. in a different

on models).

in the above sense,

With any given category, limit

operations

Finite

for us as logical operations;

and colimits will be important operations

neverthe-

operations.

is now asked to recall

of limit and colimit

will be important

is imprecise;

in Set. 1.2. Pretopos

concepts

'behave like'

in the next section will be

explicitly based on the idea that pretopos certain operations

that

Limits

also,

role,

the

limits and colimits as

and colimits

infinite

limits

'semantic' can be construed

in the obvious way.

we obtain one

(partial)

operation

of

for each graph serving as the type of the diagram the limit of

which we take; the operation,

this graph is the type of the diagrams denoted

G

above.

E.g.,

in the domain of

in any category,

the operation

232

of product of two objects is given by the class

K'

of product-

diagrams C

of type the graph

G':

the domain of the operation consists of those pairs of objects (diagrams of type G: • .)

which have a product in

colimit operations are, in fact, obviously,

C.

A, B

The limit and

strong operations.

A limit,

or colimit, operation of a particular kind is fully defined in a given category just in case that category "has all limits given kind", in the usual terminology.

(colimits) of the

In Set, the category of sets and functions, all small limits and colimits are fully defined, and they have meanings which are familiar from many contexts.

E.g., any diagram of the form B

Ax

Ad

with

A × B

~l() Set:

the

= a,

Cartesian ~2()

B

product: = b,

is

A × B = {: a product

aeA,

diagram.

beB},

and

The diagram

in

C

is a copreduct diagram if and only if both C

i, i'

is the disjoint union of range(i) and range(i').

are one-to-one, (Therefore,

and

in Set,

233

coproducts

are also called disjoint sums).

The limit of the empty diagram is called the terminal object; Set, any one-element

set serves as such.

diagram is the initial object; It is a well-known terminal object

fact

it is the empty set in Set. (see [CWM])

G

is now the empty graph],

and has equalizers of parallel pairs of morphisms,

then it has all finite limits. consequence,

that if a category has a

[this is a ~-ary operation:

has binary products

in

The colimit of the empty

for colimits.

An exactly dual statement holds, as a

The notion of pretopos will be based on

finite limits and finite colimits,

the latter suitably restricted.

A diagram q

A

, B

(I)

q' in Set will be called an equivalence relation if the map a l

~

is a bijection of

in the usual sense, on (2) is, by definition, F: C

~ Set

B.

onto an equivalence

relation,

C, a diagram

an equivalence relation iff for all functors

preserving

valance relation,

A

In an arbitrary small category finite limits,

in the previous sense,

F

takes

(2) into an equi-

in Set:

F(q) F(A)

:> F(B)

F(q') is an equivalence relation in Set. It turns out finite limits,

(see the next section)

the notion of equivalence

that, in a category with relation can be

defined by referring to finite limit diagrams;

'internally'

in particular,

between categories with finite limits, preserving

a functor

finite limits, will

take an equivalence relation in the domain category into one in the codomain category. Let (I) be an equivalence (ordinary)

equivalence B/R

relation in Set, let R c B×B

induced by

tion above.

Let

B ~ B/R the diagram

be the map that takes

(I) on

B

be the set of equivalence classes of

.........

A

is a coequalizer diagram,

q

b

be the

as explained in the definiR,

and let

into its equivalence class.

Then

)

q'

) B

,+ B/R

as it is easily checked.

For this reason,

a

234

coequalizer

of a pair of morphisms

(in any category)

that form an equivalence

is called a quotient of the equivalence

Now, part of the definition

of 'pretopos'

relation

relation.

can be put as follows.

A pretcpcs is a category which has finite

limits,

(colimit of the empty diagram),

of any two objects,

finally,

coequalizers

coproduct

of equivalence

completed by imposing conditions have as basic operations operations: particular

relations;

the fully defined finite

product operations,

above),the

operations.

limit operations object,

of coequalizer

and

~.

G'

logical

is:-

~ .).

diagrams

the three

relations

(in this operation,

A morphism of pretoposes,

functor in EMR], and an elementary

functor that preserves

(or:

and the binary co-

and finally the non-full

operation whose domain is the class of equivalence graph consists

is Thus we

the following pretopos

initial

both fully defined,

and

the definition

on the mentioned

in any pretopos

ones mentioned

an initial object

functor

and whose G is ---+*,

also called a

in [MI],

all the listed pretopos-operations

[M2],

is a

(the meaning

of a functor preserving

an operation was explained

embedding of pretoposes

is a pretopos morphism which is conservative:

if it takes a morphism

in the domain-pretopos

the original morphism is an isomorphism The rest of the definition tions put on the pretopos the pretopos

operations

than

of some condi-

in properties

of

of the notion of pretopos will

theorem will be proved.

(GSdel-Deli~ne-Joyal

Every small pretopos has a pretopos of

consists

all originating

the definition

and the following

Theorem 1.2.1.

into an isomorphism,

in Set.

In the next section, be completed

A pretopos

too.

of 'pretopos'

operations,

above).

representation

theorem).

embedding into a Cartesian power

Set. This theorem can be read as a definition:

egory having the pretopos

operations

into a Cartesian power of Set. theorem because essentially

is a catembedding

G~del's name is in the name of the

the theorem is closely related to, and in fact,

equivalent

ship is explained equivalent

a pretopos

and having a pretopos

to, G6del's

in detail

completeness

in [MR~.

instead of pretoposes. (as a set of exercises)

This relation-

Deligne proved an essentially

theorem in ESGA4J, Exp. VI.,

formulated

A suitably modified

be given in the next section.

theorem.

for coherent

form of Deligne's

The concept of pretopos

in ESGA4, Exp. VI.~.

first advocated the concept of pretopos cal first order logic.

appears

toposes

proof will in passing

It was Andr~ Joyal who

as the basic notion of categori-

235

We now consider the non-strong Let

T

be a pretopos.

the partial

ordering

For

A

of subobjects

will see in the next section, Boolean

complement

~' c Sub(A) minimal

elements,

then

F(¢')

~

in

and

A Boolean pretopos object

A.

i.e.,

appropriate properties;

Properties are freely

is the Boolean in

morphism,

T'

in which Boolean

of pretoposes

complements

algebra for every

between

Boolean pretoposes

B

in our general setting:

+ A

a diagram

for suitable

categories

up to a unique

Semantical

B

~ A ~

is clearly

assured.

in Set.

of certain

in any category

with

(such as pretoposes)

isomorphism

operations

it

B'

limits and colimits

to much of the theory of logical operations.

interchangeable,

an example,

~'

is a pretopos

is a Boolean

of interchangeability

Set are fundamental

The Boolean

it is p r e s e r v e d by

F(~)

is an operation

of uniqueness 1.3.

are the

Sub(A)).

that there is no new notion of m o r p h i s m of

to any m o n o m o r p h i s m

the condition

of

The

is an element

structure.

"Boolean complement" assigns

Sub(A)

morphisms

the additional

if

+ T'

is a pretopos

It is important

Boolean pretoposes: preserve

F: T

in w h i c h

of

denote As we

lattice.

if exists,

moreover,

is the Boolean complement

always exist,

Sub(A)

~ v 9' = 1A (0A, 1A

respectively,

As a consequence, T,

let

(see V.7 in [CWM]).

is a distributive

(if it exists);

lattice-homomorphisms. of

A

of Boolean complement. T,

~ ~ Sub(A),

such that ~ ^ #' = 0A, is unique

complement

of

Sub(A)

of an element

and maximal

complement

operation

an object of

(c.f.

in

Limits

IX. 2 in [CWM]).

As

if

ei

f. i

__

Ai

.

, ~ Ci

Bi

(I)

gi in an equalizer

diagram for

i = 1 and 2, then

elx e 2

fl x f2

AIXA 2

> BI×B 2

>

> CI×C 2 gl × g2

is again one. The interchangeability (c.f.

loc. cit.)

equalizer

of finite

limits and filtered colimits

is an important property

diagrams

a directed partial

of Set.

(I) in Set are given for all order,

connecting morphisms

and if, moreover,

~..: A. iJ

i

~ A. j

with

Thus, i~I,

we have,

if the

and

I

for each

is, say, isj,

236

ajkOaij and

similar o n e s

8ij' Yij'

= aik

~ j ~ k)

(i

for the B's and C's such that the diagram ei

A.

+ B. 1

c~ij

iBij

A. j

and

the

similar

ones

with

is again an equalizer Finite colimits Of course,

the

f's

and

products

all

colim C. 1 i

relations

with colimits.

are interchangeable

but not n e c e s s a r i l y

the only positive

features.

Coequalizers with

with more general in general.

(finite

limits;

even

For binary co-

fact we can state is that they are interof products

and directed colimits,

ultraproducts.

i¢I. which way.

1

colim gi i

in Set offer less interchangeability

changeable with certain combinations the

then

~i

1

this much cannot be said about coequalizers products,

commute,

diagram.

of equivalence

or infinite)

g's

~ colim B. i 1

they are interchangeable

(quotients)

j

colim

colim e. l 1

colim A. i 1

+ B.

e. J

Let

U

Let

U °p

be an ultrafilter

P ~ Q

be the partial iff

Consider

Q c p,

on a set I, let

and consider

be a set for each

U °p

a category

set

U,

in

in the usual

the diagram U °p

P

Q = P

+ Set

r

' ~ A. ieP z

A.

canon.pro~.

~icQ 1 the colimit

Ai

ordering with underlying

of this diagram

is the ultraproduct

In short-hand:

H Ai/U = colim ~ A.. i~l p~uOP icP z

-~ ~ A i , ; , iEP "

of the sets

Ai, iEI.

237

When we choose the standard representation limits, we obtain the usual definition maybe)

of ultraproduct

for the fact that under the present

consider vectors and not just of the

Ai

Moreover, product

i~ P of elements

P = I;

of products

of sets

definition,

aicA i

and co(except,

one has to

indexed by arbitrary

but, when one thinks of the possibility

P~U,

of some

being empty, one sees that ours is the right definitionl). with the standard choices made, we may consider

the ultra-

(with given I and U) a functor Set I

+ Set

where the action of the functor on arrows properties we put

of products

and colimits:

is given by the universal

given

fi: Ai

~ Bi

for

i~I,

H fi/U = colim ~ f.. iEI pcuOP icP 1 It is a basic fact that coproducts products:

(disjoint

sums) commute with ultra-

if

C. 1

AY 1

is a coproduct

1

diagram in Set for all

icI,

then

~c./u i l

~A./U i i is again one. ultraproduct ultraproducts

~ ~ B i / U I

The reader is invited to check this directly. of copies of the empty set is again empty.

In other words,

commute with finite coproducts.

Since ultraproducts

are combinations

limits, what we said above implies and quotients

Also, the

of equivalence

of products

and directed

co-

that they commute with finite limits,

relations.

We therefore

conclude:

238

Theorem 1.3.1.

(Los's theorem)

Any u l t r a p r o d u c t f u n c t o r

(-)/U: Set I

~ Set

icl

i s a morphism of p r e t o p o s e s . The relation of the last theorem to the 'fundamental property of ultraproducts' Part 2. 1.4.

(Los's theorem in the usual sense) will be explained in

Generating diagrams, and the statement of the main theorem.

Composites of pretopos operations will commute with ultraproducts as a direct consequence of 1.3.1. operation on Set of taking

Such a composite is, e.g., the

(A×B)~ (C×D), whose graph can be construed

as the class of all diagrams

EilY F G

(1)

2

B

D

with (~I,~2), ( ~ , ~ ½ ) being products, (il,i 2) being a coproduct; the domain of the operation is the collection of all tuples (A,B,C,D) of objects. The general concept of a composite operation will be approached in a somewhat abstract manner.

There is a free Boolean pretopos F(G)

on any graph G; e.g. on four objects, A0, B0, CO, DO, as generators. This is defined by a universal property similar (but not identical) to the one used for free algebras in ordinary algebraic situations.

The

above composite operation can be performed in every pretopos, hence in particular in the free Boolean pretopos on

(A0,B0,C0,D0) , and we can

consider its result on the particular argument

having

say E0,F0,G 0 as additional objects and appropriate arrows. Then, the operation in question can be performed in any Boolean pretopos T in the following manner.

Say, the arguments are

the essentially unique pretopos-morphism A 0 to A, etc;

~

~:

A, B, C, D; F(G)

~ T

one finds that takes

will pick out the desired diagram(l) as the values

F(E0), F(F0) , F(G0) and appropriate ones for arrows. In other words, the operation is entirely described by two diagrams, DO: G ~ F(G),

239

and an extension domain,

G'

of it,

D~: G'

> F(G);

here

G

is the type of the

the type of the graph of the operation.

Moreover,

conversely,

every object and morphism

tained as part of a "composite be made clear by explicitly

Boolean pretopos

constructing

F(G)

creasing union of larger and larger diagrams every operation obtained Boolean pretopos

operation",

meaning of the latter. Boolean pretopos

from a

DO

and a

in

F(G)

operation";

as made up of an in-

of such composites. D~

Thus,

as above is a "composite

if we have in mind some direct,

Instead, we take the description

as the first approximation

is ob-

this could

syntactical

using the free

definition of composite

of a

operation. However,

the free pretopos

the non-full pretopos

operation

The domain of this operation

construction of quotient

consists

additional

The free Boolean pretopos,

form (2) will not

conditions,

'contain' on

F that

operation will be available. of a pretopos

T

by imposing

a class of morphisms procedure

in

of identifying

the form of passing relation of

T.

relation

morphisms

in it.

conditions

elements

But, if we

'impose'

relation,

the

the

of forming a

analogous

in a (universal) ~

simply because

quotient

on it in the form of inverting

This is entirely to

~/E,

to the algebraic

algebra,

which takes

with

a congruence

E

~.

We will adopt the definition through'

in question,

There is a process

that a composite pretopos

with domain of type the finite graph

G,

a pretopos which is a quotient

pretoposes

to (2) being an equivalence

say F, on a fixed diagram of the

(2) be an equivalence

from the algebra

relation.

(2)

amounting

the operation

(2) will not be an equivalence further condition

of an equivalence

B

)

satisfying

Consider

of diagrams

A

relation.

is not enough.

T, T', Pretop(T,T') T ---+ T';

denotes

means one that

of the free pretopos

to

on

G.

For

the category of pretopos

it is the full subcategory

of all functors from T Mod(T) = Pretop(T,Set).

operation,

'factors

T', with objects

of

(T,T')

the category

the pretopos morphisms.

240

Definition-Proposition Let

G

(i)

1.4.1.

be a small graph.

The free pretopos

over

G

together with a diagram

Fpt (G))

the following

universal

is a pretopos

9: G

property:

+ T

T (= F(G) =

(~ = ~G = ~ t )

for any pretopos

()o~: Pretop (T,T')

T'

with

the functor

.........(G,T')

defined by composition: F: T

> T' }

+ Fo~: G

is an equivalence

of categories.

and is determined

up to an equivalence

(ii)

* T'

The free pretopos

The free Boolean pretopos

over over

G

T',

is faithful,

full on isomorphisms,

free Boolean pretopos equivalence

over

isomorphism

F

over

F: C

g: FC

Ff = g.

~ FC'

surjective.

The

and is determined up to an

is full on isomorphisms there is an isomorphism

if for every f: C

such that we'll

FC

is isomorphic

indicate

to

in part

~ C'

such

D ~ Ob(D) D).

the proof of the proposition;

of the free objects will use formulas.

the weaker statement

(ii) is the presence

The reason

for

of a non-strong

Boolean complement.

These definitions well-known

of free objects

diagrammatical structures.

Boolean pretopos, as a consequence there is

and essentially

exists,

~ D

In the next section,

operation,

G

+ (G,T')

is essentially surjective if for all

C c Oh(C)

the construction

algebraic

for any Boolean pre-

G.

(A functor

there is

exists,

the functor ()o~: Pretop(T,T')

that

G

is a Boolean pretopos

T (= F(G) = FBpt(G)) together with a diagram Bpt~ = ~G ) with the following universal property: topos

over

G.

M

definitions

of the definition:

of the

of free groups and other free

In either the case of pretopos

we have the following

such that

are natural versions

or that of

form of the universal

given

T'

and any

F: G

property ~ T',

241

G

~

,

F(c)

v T'

commutes up to isomorphism:

there is an i s o m o r p h i s m

~: F

.......

Mo~:

~

and if

then there is a unique i s o m o r p h i s m

v: M .... ~

~ M'

such that

/vocp

M'o~

commutes.

The e s s e n t i a l l y unique

be induced by F,

in

O.

E.g.,

all iso's in

f F

T.

F: C ...... ~ D, in

C

for which if

1.4.2

(i)

Let

~

Ff

for any pretopos

is an i s o m o r p h i s m

Inv(F)

~ T",

Q: T

= Inv(Idc)

(invertible)

= the class of

is said to be obtained

if we have the following universal

~ Pretop (T ,T")

induces an e q u i v a l e n c e of Pretop(T',T") c o n s i s t i n g of those

~ T'

the functor

()oO: Pretop(T',T")

Pretop(T,T")

denotes the c o l l e c t i o n of all

be a c o l l e c t i o n of arrows in the

A pretopos m o r p h i s m

by inverting the morphism8 in property:

M F-

Inv(F)

is conservative

is said to

C.

Definition pretopos

with the said p r o p e r t y

and may be denoted by

For a functor those morphisms

M

G: T

onto the full s u b c a t e ~ o r y of ~ r"

for w h i c h

~ c Inv(G).

242

(ii)

~ T', a pretopos morphism, is a quotient morphism

Q: T

(a quotient) if it is obtained by inverting the morphisms in Q

is a finite quotient if there is a finite subset

such that

Q

is obtained by inverting the morphisms in

(iii) Let

T

be a Boolean pretopos.

(finitely) generates morphism

E

T

of

Inv(Q). Inv(Q)

E.

A diagram F: G

~ T

(as a Boolean pretopos) if the pretopos-

MF: FNp t(G)

,,~ T

induced by

F

is a finite quotient

morphism. If and

M

is a (universal) algebra, in a given equational class, say,

is an arbitrary subset of IMIxlMI, then there is a universal

E

property similar to 1.4.2 (i) of a morphism

q: M

~ M' being

"obtained by identifying a with b", for all pairs c E: for any M", the map Hom(M',M") ~ Hom(M,M") defined by composition with q

induces a bijection onto the set of those

identify all pairs in quotient M/[E] of M E, with

q

the map taking an element of

class in [E].

M

~ M"

which do

E. Of course, such M' is nothing but the by the congruence relation [E] generated by IMI into its equivalence

(In equational classes, quotient morphism (those that

are "obtained by identifying those elements identified by the morphisms") are the same as regular epimorphisms.) Concerning (iii), note that in the context of an equational class, a map f: X > M generates M, i.e. f(X) generates M in the usual sense, iff the map by

f

F(X)

~ M

from the free algebra on

X

induced

is a quotient. Definition 1.4.3.

operation (ACBPO)

An abstract composite Boolean pretopos

is given by a commutative triangle G

Do

i=inclusion~

> T

/ ~

G' where

G, G'

are finite graphs,

T

is a Boolean pretopos, and both

D O and D~ finitely generate T (see 1.4.2). In any pretopos S~ e.g. S = Set, an ACBPU as shown defines the composite Boolean pretopos

operation of type

(G,G')

whose graph is

K' = Iso{MoD~: M c Pretop(T,S)}.

243

Remark. operation with

C

S

C, and whose

notation

of 1.4.3,

objects

define

in 1.4.3 from an ACBPU

(G,G').

To see this,

morphisms

let

of

is indeed

let us write

C

whose

are the isomorphisms

in

similarly.

the functor

i*:

K'.

Let

(K') is°

is the graph of an operation

i

are With the

of

(G',S) is°

D' e K'},

induces

> K is°.

of type

C.

K = {D'oi:

The functor

an C is°,

objects in

(K') is° be the full subcategory

are the diagrams

K is° c (G,S)

(restriction) K'

as defined of type

any category, for the subcategory

those of whose

K'

in

and

by composition

Note that to say that

(G,G')

is to say that

i

is

full^ and faithful. With F = FBpt(G), F' = FBpt(G') , D = MD0' ~' = MDu" i = M we have that the given ACBPU induces the diagram ( Bpt oi)' F

~ T

F'

commuting

up to an isomorphism.

Passing

to the induced

diagram

^.

D

Pretop(F,S) Is° .

Pretop(T,s)iSo

Pretop (F' ,S) is° we note that ~.

D

^

and

being

(K') is°

image of

(D')*

Since both ^

are full and faithful

quotients,

(K') Is°,

the full replete

.

(D')

it follows

that

is full and faithful.

Definition

1.4.3'

one of the same type

(i)

as a direct

consequence

of

D, D'

i*I(K') is°, the restriction

of

i* to

It follows

that

An operation

i

is full and faithful.

is a restriction of another

if the graph of the first

is contained

in the

graph of the second. (ii)

An operation

in Set of type

commutes w£th ultraproducts filter on I, D.: G' D'

i where

[U]

and Set I

if

the

D': ~' .+ S e t I P r ° ] i > Set (iel)

is the ultraproduct n (-)/U: ieI

(G,G')

following has

in

having

when

its

components

all

K',

functor Set I

the graph

holds~

> Set.

then

U

is

K' an ultra-

[U~ o D' e K',

244

Theorem

1.4.4.

ultraproducts

Every

Part 2. 2.1.

finitary

is a restriction

operation

of a composite

in Set commuting Boolean

Categories,,,,,for the working

pretopos

with operation.

logicli,an

Sites Let

C

be a category with

finite

topology on

C, J, is a collection

(abbreviated

as (A i

domain

~ A)i)

(such a family

to satisfy

Every

of families

of morphisms

closure

isomorphism

{A i -----+ A: i~I} C with a fixed co-

in

a J-covering of A), with

is called

the following

(i)

A (Grothendieck)

limits.

J

required

conditions:

A' --=---+ A

is a 1-element

covering:

{A' -~--+ A} ~ J. (ii)

(Stability

under pullbacks).

Whenever

A.×A iB for all (Closure

(Aik

Aik

.....

then

B

(Ai×A

for all

> Ai

) A

i,

then,

If

(A i

i

through

(A i ..... > A)j

then

It is important of one object, the codomain collection precise

A.

definition

With

If

X

with

The reading

> A

j

with

and

denoting

the

we have

Ai

(A i ~ ~ A

A)j

is

factoring

c J.

A

since

more precisely, a (possibly

of the closure

in the empty set have

empty)

conditions

J

as a

set of morphisms under

the more

should be clear.

of topologies

is jointly

is regarded

Aik

arrows,

Therefore,

we should,



C = Set,

the family

with

> A) i e J

that we may allow the empty family as a covering

is not given,

Examples

is

but not of another.

of pairs

with codomain

there

(A i

> A) i ~ J, and

a family so that for all > A,

If

of the given

~ A)i,k c J. (iv) (Monotonicity). A~j

) B)i E J.

under composition).

(Aik ---+ Ai)ke K E J composite

i,

and

[

I

(iii)

> A) i ~ J

A

A.

is a pullback

(A i

let

J

are the following. be defined by:

surjective,

a site, we always is a topological

(fi: A

i.e. U{Im(fi):

~ A)i~i

ieI} = A.

~ J

When

iff Set

mean this eanonival topology. space,

ordering

(under inclusion)

category

in the usual way, we let

and

C = 0(X)

of the open sets of J

X,

is the partial regarded

be the set of all

as a

(U i ~ U)i such

245

that

U U. = U. i i

The Grothendieck ordinary

A site a topolegy functor

topology

topology

on

(C,J) J

J

X

obtained

in contexts

is a category

on

C

with finite

A morphism of sites

C.

F: C --+ P

in this way may replace

preserving

the

like sheaf theory.

finite

limits,

(C,J)

limits

together with

~ ~D,J')

is a

and taking J-covers

into

J'-covers: (fi: Ai --+ A)i c J implies (Ffi: FA i - + FA)i a J'. We also talk about a (J,J')-continuous functor, or even a J-continuous one, if

J'

is understood. Given any collection

with a fixed codomain a least topology

generated by is a site, covers

J

on

J0'

J0

[such C

or by

J'-covers

then the functor

[C,J0).

in

C,

J0"

J, or

(C,J)

each one has

is said to be

Also note that if, in addition,

C --+ ~

preserving

[we may consider

is a morphism

of morphisms

may be called a pre-topology],

containing

and the functor

into

of families

J0

finite

limits

it a morphism

of sites:

(P,J')

takes

(C,J0) --+ (D,J')],

(C,J) -+ [D,J').

A finitary topology, or site, is one which is generated by a finitary pre-topology, i.e. one in which every covering is finite. that a topology there

J

is finitary

is a finite

Theorem

2.1.1

(Deligne's

small

finitary

there

is a J-continuous

Set

which

J0 J

site,

and

Suppose

A = ~A i -+ A)i

C

by all morphisms J

is a small

C

on

which

using

Let

taking

A

Then in

topology).

One may consider

limits,

J0 o

and

the collection

into canonical

The completeness by

J.

be a

into a family

category with finite

generated

(C,J)

theorem

coverings asserts

that

The theorem will be

the following

Proposition

2.1.2.

I, the terminal

the empty family (C,J) -+ Set.

C.

c J.

any family not in

are carried

[C,J0) --+ Set.

is the same as the topology

proved

that

in

C -+ Set

Note

[A i --+ A)i~ I ~ J

theorem).

(in the canonical

is a finitary pretopology of all coverings

(A i --+ A)ici,

completeness

functor

is not a covering

Remark.

just in case for all

such that

I' a I

J0-

in

Suppose

object J.

of

(C,J) C,

is a small

is not empty,

finitary

site such

i.e. not covered by

Then there is at least one morphism

246

P~oq,~,:

Let

preserving

LexCC,Set) finite

denote

limits;

the c a t e g o r y of all functors M e Lex(C,Set) coverings.

which,

category is

C --+ Set.

in addition,

able functor is in

functors

functors

M

of

(C,Set),

into canonical

as a suitable directed co-

(C c C).

Since each represent-

and directed colimits

Lex(C,Set)

C --+ S e t

subcategory

carries J-coverings

¢(C,-)

Lex(C,Set),

of members of

of all

a full

We have to construct

The construction will give

limit of r e p r e s e n t a b l e

(C,Set))

the

Lex(C,Set)

are again in

(in the sense of

Lex(C,Set)

[this is a

consequence of d i r e c t e d colimits commuting w i t h finite limits in Set], the c o n s t r u c t i o n will,

at least, ensure that

a separate matter to ensure that Let

S = (S,~)

and let

> CoP

S~

+D

s ~ t 1

....

be a diagram of type

in

C °p .

the Yoneda embedding:

Dt

~ Ds

Let us consider the composite

~ coP

D

M(D)

S ~ts:

~

S

S .....

y

It will be

the coverings as well.

be a directed partial ordering,

D: S .....

(with

M e Lex(C,Set).

M 'respect'

Y

C[

~ (C,Set)

~ C(C,-)),

and its colimit:

= colim C(Ds,- ) e Lex(C,Set). seS

fi Let

,4 = (A i

~ A)ie i e J.

For

M

to carry

A

into a canonical

covering in Set, it is n e c e s s a r y and sufficient that for every and every

g: D s

~ A,

(*) there are

i ~ I, t ~ s

D

and an arrow

g

commute.

is aaptured for

Let us call a d i a g r a m of the objects diagram

Ds

D': S'

restriction of

D'

to

with S.

A

When condition

in the diagram

(*) holds,

D

by

of the above type consistent

D

A continuation of

is empty. , C

making the diagram

-, A i

(This is a matter of inspection.) g

g'

÷A

D t ..... g '

we will say that

s c S

the following hold:

S

a sub-poset of

D S',

g'

if none

is a d i r e c t e d and

D

the

The main step in the proof is the following

247

Lemma

2.1.3.

Suppose

A = (fi: Ai consistent

D

) A)iE I

is a consistent

is a finite J-covering

continuation

of

D

in which

Once the lemma is proved, straightforward. in a diagram,

Note that,

it remains D

g

g: D

of

> A, for

A.

the proof of the proposition

diagrams,

is a continuation

with

~

ordinal,

D ,

is

for a covering

in any continuation.

of

and

A. Then there is a

is captured

once an arrow is captured

so captured

is a system of consistent < ~' < B,

diagram,

If

(D)~< B

such that for

then we have an obvious

colimit, U D, ~<~ a consistent start

diagram,

with,

we h a v e

( D O ) ( * ) = 1 e C. the

lemma,

construct

a continuation the

Now,

and using

the

diagram

each

diagram

any

colimit

D . DO ,

consistent

continuation

with

of

m o r p h i s m of sites It remains

to

S = (*), by repeatedly

using

we c a n

in which every arrow with domain in

for every J-covering.

As we said above,

(C,J)

~ti' gti

that,

An u-type co-

so constructed will give us a continuation in

this means

D, that

D of

is captured M(D)

is a

~ Set.

to prove

Let us choose, morphisms

also

diagrams,

in which every arrow, with domain any object

for all J-coverings.

Note

diagram,

construction

is captured

limit of continuations DO

consistent given

a consistent

the original

of

the ]emma.

for any

t e s,

and

to form a pullback ~ts

Dt

.....

) Ds

i e I,

the object

Dti

and

diagram g

~A

T, Dti t

and

*" A .

gti

Then,

for any

u

~uti

making the following

such that commute:

i

s ~ t ~ u

we have a unique

arrow

248 ~LIS J Du

6u t

6t s ' ~

) Dt

)

D

)

S

A

(1)

'If i Duz ~

i

~" A .

Dti

gti

z

j

Ul

If

Dti

is

i e I

all

is n o n - e m p t y , such that

i ~ I

there

is

finite,

and s i n c e

S

for all

i ~ I.

Thus,

(~ti:

Dti

by a x i o m

consistent.

Dt

so is

u ~ t,

is n o n - e m p t y

t I• > s

Dti

is e m p t y

is empty,

Dti i

there

is a c o v e r i n g ,

Du~'z

We c l a i m that

for all

such that

is d i r e c t e d ,

) Dt)ic i (iii),

and Dti

is

t ~ s. is empty.

t ~ s

there

Otherwise, Since

s u c h that

I

is

t ~ ti

for all

i e I.

by a x i o m

(ii) of t o p o l o g y .

contradicting

for

But,

the a s s u m p t i o n

Hence,

that

D

is

The c l a i m is thus shown.

Let us fix

i

as in the claim.

We are r e a d y to c o n s t r u c t

the d e s i r e d

continuation

D'.

The

^

underlying D t' = D t we have

p o s e t w i l l be

for

t e S,

ensured

that

is g i v e n as follows:

S' = S ~

and D'

{t: t~s}

D t' = Dti

for

is c o n s i s t e n t .

for any

= S u {t: t~s}.

t -> s.

Thus,

The p a r t i a l

We put

by the claim,

ordering

on

S'

x,y e S',

x -< y in S' ~=* e i t h e r

x, y c S and x -< y in S ^

One v e r i f i e s The r e m a i n i n g

easily

or

x = t, y = fl and t -< u in S

or

x E S, y = t and x -< t in S.

that this

is,

data of the d i a g r a m

indeed, D'

6 '^ = fit ~uti ~t

= ~ti

a directed partial

ordering.

are g i v e n as follows:

(t < u) ° ~uti

(t ~ u)

(see d i a g r a m ( 1 ) ) . The c o m m u t a t i v i t y o f t h e l e f t s q u a r e in (1) i s used t o show t h a t D' so c o n s t i t u t e d i s , i n f a c t , a f u n c t o r . Now, l o o k i n g a t ( 1 ) , we see t h a t g i s c a p t u r e d f o r A in D' by

gti'

c o m p l e t i n g t h e p r o o f o f t h e 1emma.

• 2.1.3. •

2.1.2.

249

Theorem Lemma 2.1.3. collection

2.1.1 will now be reduced to Proposition Let

J

be a topology on

of objects of

C.

Let

J'

for which there in

X.

Then

Let

X'

and an extension

A' = (A i i e I' - I.

1

e X'

Paoof:

for all

It is immediate

Conversely,

it suffices

satisfying

the condition

A ~ X. A'

+ A) iE I e J'

A.

X

A

~ A

generated by J'

of

A

into an object

that

such

satisfying

is a topology;

A' A I' = I

A' e J

the condition

to show that the collection

J

can he

of all objects

if and only if there is

÷ A)ie I

that any

2.1.2.

an arbitrary

Then

be the collection

is at least one arrow

A = (A i

and

be the topology

together with the empty cover for each described as follows.

C,

and

is in

J'

of the families

this is an easy verification. m 2.1.3.

Lemma 2.1.4. domain

Suppose

the family

I, the terminal

topology

J'

(Ai N

object of

generated by

J

÷ l)iE I

with the joint co-

C, is not a J-covering.

and the empty coverings

Then in the

for the objects

A i, 1 is non-empty.

Proo,,f:

Suppose

J-covering

Bj

I is empty in

(Bj~

~ Ai. J

> l)j

to some

J'.

Then, by Lemma 2.1.3,

such that for each

i. e I. J

j,

there

is a

there is an arrow

Since

A.

Bj

I

commutes

(I being terminal),

covering,

contrary

by axiom

(iv)

(A i

>1) i

isaJ-

to the assumption. 12.1.4.

Proof of Theo,rem 2.1.1.: We consider the comma category C/A (denoted in [CWM]). C/A has finite limits; the terminal object in C/A

C+A

is the arrow

id: A

~ A.

We have the functor

250

F:

C/A CxA

C Ct-

1~2 A C×A_

C fll

+ fxid A

C' C'xA

embedding

C

into

C/A;

topology J/A on C/A

Bk

F

F

preserves

"~A

kcK

is a morphism of sites

always a pullback

finite limits.

We define the

as follows:

(C,J)

diagram in

'~ B)kc K c J

e J/A ~=~ (Bk

~ (C/A, J/A).

The following

is

J/A:

A' .,

+

A'×A

(z) + AxA ~= All the above facts can be verified easily,

if not already known.

Now, assume the hypotheses

The family

of the theorem. fi

I is not a covering J' on fi: Ai

C/A

on

IC/A

generated by ~ A e Ob(C/A),

Ai

.~ A - - A

in J/A

J/A.

/ie I By Lemma 2.1.4,

in the topology

and the empty covering of each

the terminal object

Ic/A

is not empty.

By

251

Proposition

2.1.2, there is a m o r p h i s m of sites M':

By the definition M = M'oF: fixed

C

of

M

A' = Ai,

into a pullback

~ Set.

J', M'(f i) = 0,

+ Set.

i • I,

(¢/A,J')

for all

is a m o r p h i s m f = fi'

in Set by M',

M' (fi)

i c I.

(C,J)

Now,

~ Set.

the pullback

diagram

let

With any

(2) is taken

i.e. the diagram

,

M(A i)

,

j

M(f i)

-, M (A)

1 = M' (1C/A) M'(~)

is a pullback of

M(A)

in Set.

Since

picked out by

i e I, i.e.,

M'(f i) = 0, this means

M'(4)

that the element

is not in the image of any

the image of the family

A

under

M

M(fi) ,

is not a canonical

covering. 2.1.1. 2.2.

Coherent Let

C

categories. be a category with finite

Then the subobjects denoted

Sub(A).

(see [CWM])

In fact,

meet of the subobjects being represented by

Sub(A)

represented D .... ~ S

f: A'

f*FB

A

an object

in

(^-) semilattice, B

C.

ordered set, > A,

C

the ~ A

diagram

~ A

~ A,

f*: S u b A is defined by pullback:

is a meet

by the monos

from a pullback

B

Given an arrow

limits,

of A form a partially

the map

(i)

~ Sub A' ~ A] = [B'

~ A']

where

252

is a pullback,

f*

operation, and each

~ A'

that

"C

Sub A (A e Oh(C))

other words,

element

~f(~)

~f(~)

preserving

for

~ ~

iff

~ ~ f*(~)

extremal

the minimal denoted

of

for all

is the least subobject

~

Sub A'

3f(iA, ).

element

of

To say that

this circumstance

epimorphism.

element.

3f,

of

, e Sub A; A

Im(f)

= 1A

A

unless

of

~f,

in

such that by

IA,,

means

let us

that

f

for all

f,

f

is an

is equivalent

to saying that every m o r p h i s m can be factored as the composition extremal

epi followed by a mono.

isomorphism any

g: B

Beak-condition ÷ A

for

~

means

of an

that a m o r p h i s m

just in case it is both a mono and an extremal

The so-called above,

It is immediate

does

the mono is an

is referred to by saying that

The existence

means

A, mf(~), n e c e s s a r i l y

the maximal

not factor through any mono with codomain isomorphism;

(having a join

(sup of empty family of subobjects),

~ e Sub A' there is a subobject

Denoting

Im(f)

finite sups of subobjects"

is a lattice

(I) of posets having a left adjoint,

such that

~ f*(~).

of meet semilattices.

has stable

is a lattice h o m o m o r p h i s m

The map

write

B'

v) with a minimal f*

that for every unique,

~A

is a h o m o m o r p h i s m

By the condition we mean that each

B

is an

epi.

that for any

f

as

and a pullback A'

f

+ A

(2) B' the

+B

h

diagram Sub A'

~f

+ Sub A

Sub B'

,~ S u b

B

3h

commutes. f

This

is equivalent

is an extremal

epi,

the Beck-condition,

to saying

then so is

we say that

h. C

that whenever If

3f

"has stable

in the pullback

always exists images".

(2)

and satisfies

253

A category suitable for finitary coherent

simply: T

(geometric) logic,

a coherent category (in [MR]: "logical category")

or

is a category

with finite limits having stable finite sups of subobjects and stable

images. Let

T

be a coherent category.

collection of all families that some finite

I c I'

Define

(fi: Ai

÷ A)iei

is a topology on

T.

to be the

of morphisms of

T such

we have V Im(f i) i~I

J

J (= JT)

=

1 A.

Indeed, axiom (ii) follows by the stability

conditions on sups and images. following two formulas:

Axiom (iii) easily follows from the

for morphisms

A"

g

A'

f

+ A

we have

~fg (¢) = ~f(~g(¢))

(~ c Sub A")

and for f

A'

+A

we have

3f(iYl~i) = iYl3f(~i )

(I finite, ~i ~ Sub A').

The two formulas are, in turn, easily checked.

Axioms

(i) and (iv) are

immediate. Note that a monomorphism is a covering in

J

(as a singleton)

iff

it is an isomorphism. For a functor

F: T

~ T'

serving finite limits, saying that (T,J T)

~ (T',JT,)

between coherent F

T

and

F

preserves finite

sups and images (this latter condition means, naturally,

the order preserving map induced by

that for

~ SUbT,(FA) F, FA

is a lattice homomorphism

preserving minimal elements, and that whenever then

pre-

is morphism of sites

is equivalent to saying that

FA: Sub T A

T'

f: A'

~ A

is in

T,

254 Fa (3f(q~))

= 3Ff(FA(¢) )

(~ e Sub A ) ) .

The truth of this assertion is quite clear. naturally,

a coherent functor,

in ~MR]).

The category of all coherent

Coh(T,T');

it is a full subcategory of

T

, Set is a model of A functor

T;

if

Ff

have finite limits and

or coherent morphism T

, T'

(T,T').

Coh(T,Set)

is denoted

being an isomorphism implies that

implies that

F

F

preserves

f

F

may be called,

("logical functor", is denoted

A coherent functor

reflects isomorphisms

F: T ---+ T'

T'

Such an

Mod(T). (or is conservative)

is an isomorphism.

them, then

If T,

F being conservative

is faithful.

One can apply Theorem 2.1.1 to conclude that for any small coherent category isomorphism,

T,

and any monomorphism in

there is a coherent

T

T

which is not an

, Set which takes the given mono-

morphism into a mono which is, again, not an isomorphism. several such coherent

functors together,

Putting

one for each mono in

T

that

is not an isomorphism, we obtain Theorem 2.2.2.

(Completeness

for finitary coherent logic).

Any small

coherent category has a conservative coherent embedding into a small Cartesian power of Set. • 2.2.2. One should note the easily seen facts that if coherent and conservative, T

as well;

T',

moreover,

T,

is

if its

then the original diagram is a limit similar reflection properties hold

for sups and images.

This means that any

objects and morphisms

involving commutation of diagrams,

sups of subobjects

+ T'

then, for any finite diagram in

F-image is a limit diagram in diagram in

F: T

'diagrammatic property'

of

finite limits,

and images that holds throughout in Set, and hence

in any Cartesian power of Set, will hold in any coherent category as well.

This fact expresses

'coherent category' Definition

2.2.3.

(fi: Ai ---+ A)iei

for the

the completeness

of the defining axioms for

'standard coherent logic', that of Set.

In a category with finite limits, a family is said to be an effective epimorphic family

following holds, with

if the

255

A Ai A J a pullback diagram for all for any object

i,j e I:

and any system (gi: Ai , B)ie i of morphisms, if gifij = gj f!. z3 for all i and j in I, then there is a unique morphism h: A + B such that hfi = gi for all i in

B

I.

pr0position 2.2.4.

The topology

JT

defined above in a coherent

T

coincides with the so-called precanonical topology, i.e. the one in which a family is a covering iff it contains a finite effective epimorphic family. The fact that any family in for

T = Set,

general

T,

JT

is effective epimorphic is seen,

by a careful inspection.

Then the same fact, for a

is essentially a consequence of completeness,

2.2.2.

The

detailed proof of 2.2.4 is postponed until Section 2.5. 2.3.

The quotient-conservative

factorization.

The definition of a quotient-morphism is given in Section 1.4 in the context of pretoposes.

The definition has a general character;

can be repeated in other, similar, situations.

Such a situation is

given by a (concrete)

certain categories

2-category

"objects" of the 2-category),

(c.f. [CWM]):

it (the

certain functors between them (the

"morphisms" of the 2-category), and certain natural transformations between those functors

(the "2-cells" of the 2-category).

is the 2-category of pretoposes, pretopos-morphisms, transformations between such.

E.g., Pretop

and all natural

Similarly, we may talk about Lex, the

2-category of categories with finite limits, functors preserving them ("Lex-morphisms") and all natural transformations between such. The 2-category of coherent categories is

Coh.

The 'standard' 2-category is Cat, the 2-category of categories, functors and natural transformations. We want also of

Set,

the category of sets to be an object of

Lex, Pretop, etc.

theoretical universes:

V@. !

Cat, and

Therefore, we have in mind three setfor

i = ~, i, 2,

with

@0 < @i < 82

256

inaccessible

cardinals.

Ob(Set)

= V@0 , Set e Ob(Cat)

c V@I,

Cat e V@

. Small categories are those in V@0. 2 Note the easily seen but important fact that a m o r p h i s m which

a quotient

and conservative

at the same time is an equivalence;

is true as a direct consequence

of the definition,

and holds

is

this

in any of

our 2-categories. Another I: T any

obvious

~ T'

of the definition

is a quotient morphism

T" ~ D,

()oi: D(T',T")

result essentially T" = Set,

consequence

if

I: T

in [MR]

+ T'

is full and faithful,

in the 2-category

+~(T,T")

contained I

is that for

placing

then,

for

D = Pretop,

A and

I* = ()of: Mod T' ----+ Mod T

is a quotient

First, we will be concerned with Lex; concept of a q u o t i e n t - m o r p h i s m

D,

is full and faithful.

is such that

then

is that if

(cf.

§ 3.1

below).

as we said above,

the

is defined as in 1.4.2, with Lex re-

Pretop.

Proposition

2.3.1.

Any Lex-morphism

> T'

F: T

can be factorized

into Lex-morphisms: F

T

+T'

T"

F = FoQ,

so that

Paoo~:

Let

objects

as

Q

is a quotient,

z = Inv(F) T.

c Morph(T).

and

r

is conservative.

We define

T"

to have

the same

The morphisms A ..........÷ B

of

T"

will be equivalence

classes

of pairs

(f,s) as shown:

C

with varying relation:

C

and with

s ~ Z, under the following

equivalence

257

(f,s) ~

(f',s') ~=~ (Ff)(Fs) -I = (Ff')(Fs') -I

(remember

that for

s e E, Fs

is an isomorphism).

To have a simpler

notation,

for the right hand side of the last defining

equivalence

we

rather write fs -I = T' and read:

"fs -I equals

f's '-I

f,(s,) -I,

in

T'".

Thus,

(f,s) ~ (f',s') ~=~ fs -I = T' This clearly defines f e T(C,A), For

(f,s)/~,

The letters arrows

in

an equivalence

s ~ T(C,B)

n ~,

the m o r p h i s m s, t, u always

the definition

f's '-I

relation on the pairs

with fixed

A

represented by

is:

and

B

(f,s) with

but varying

(f,s), we'll write

stand for m o r p h i s m in

E;

C.

rfs-l~.

in diagrams,

E are indicated by double arrows.

To define

composition,

what we need to show is that,

in the

situation

the morphism Consider

gt-lfs -I

is equal,

in T',

to one of the form

hu -I.

the extension

with the pullback

indicated;

this diagram is in

T,

except

for the

258

dotted arrows.

Reading

it in

in

follows

is an iso in

that

t'

T',

T',

it is commutative

we also have the dotted arrows;

and the pullback T',

since

remains

a pullback.

t

thus,

is;

It

t' e Z.

Now, it is clear that

(gt -1)(fs -1) = ( g f ' ) ( s t ' ) - i T'

Therefore,

we define: rgt-l~

o

rfs-ln

= r(gf,) (st,)-l~.

T" This definition equivalence

is legitimate;

classes

again be equivalent, in

if different

are chosen,

simply because

of the

right-hand-sides

will

they will be the same morphism

T'

The identity morphisms

in

A

T'

are given in the obvious way:

,~----°°°°-~

A

So far, we have defined the category as the identity on objects, The functor F takes objects), and rfs-l~ into F

representatives

the resulting

is well-defined,

and takes

it is faithful,

if

fs -I

becomes

is an iso in

T',

then

f

f ¢ Z;

the inverse of

f: A

The functor ~ B

and

F = FoQ.

Q

T'

we can form

acts

Q, i.e.

(since rsf-l~,

on clearly,

is also

an isomorphism under

is an iso in

therefore,

Q

into rf(idA )-I~

A into F(A) (so that F = FoQ fs -I in T', i.e. (Ff)(Fs)-I;

conservative: T'), hence

T".

s

fs -I

is an iso in

and it is clearly

rfs-l'.

Let us check that

T"

has equalizers,

First of all, we observe that,

and that

given any two parallel

F

preserves

them.

morphisms

,.ooo,oo°o~ ,oo°°°o°°°~

in with

T", the

we may represent same

s.

In

fact,

them as consider

rfs-ln

and

rgs-l~

simultaneously

259

t'

S

It is clear that, with rgt-l~ = r(gs,)u -I".

u = st' = ts',

Given two parallel let

e

morphisms

be the equalizer

equalizers•

and

s

of

f

~

and

is an iso in

=

we have

rfs-l~

rfs-l~

~

,

g (in T);

T',

es

=

since

= r(ft')u -I~,

rgs-ll F

is the equalizer

in

T"

,

preserves of

fs -I

and gs -I in T' Therefore, we are left with the task of showing that se = rse(id) -I~ is an equalizer of rfs-l~, rgs-l~ in T". To do this, Consider

we consider

the following

a morphism diagram

in

y = rhu -I~

satisfying

~¥ = ~¥.

T:

j\

(the dotted pullback

arrows

with

the composite

s ~

appear only in and is

that they are equal

h;

e'

T'

r(fh')(us') -I~,

that

T").

~y

is

s' of

e'

gh'

and fh'

h' and

r(gh')(us') -I~.

is to say that they are equal in fh' = T'

in other words,

and

is the equalizer

form a gh'

T', i.e. that

;

is an isomorphism

in

T'

i.e

that

Now,

TO say

e' ~

260

There is a unique arrow commute,

i

by the definition

is commutative.

in

T

of

making the square containing

e.

The whole diagram,

se

is a monomorphism

is easily seen), completes

in

the uniqueness

the verification

The verification

T',

T'

o se.

and hence in

T"

of the factorization

of equalizers

of the terminal

in

as well

(as it

also follows.

This

T".

object and products

well as the remaining parts of the facts that morphisms

when read in

Hence = rhu -I~ = ri(us'e')-l'~

Since

it

Q

and

Y

in

T",

as

are Lex-

are left to the reader.

It is clear that

Inv(Q)

= Inv(F).

Since

doing the bare uinimum to invert the morphisms intuitively rigorously

clear that

Q

is a quotient;

T" in

is constructed by Inv(Q),

it is

it is not hard to prove it

either. •

Proposition

2.3.1,.

essentially

unique.

The quotient-conservative More particularly,

factorization

in

2.3.1.

Lex

is

if in

T'

Q, Q

are quotients,

r, r

are conservative, ^

and we have an isomorphism

A

r Q-= r Q, then there are an equivalence

functor

r~rE

whose composite

E

(i) and isomorphisms

261

is

t h e one g i v e n i n

Proof:

(I).

Straightforward. •

Let us work in a fixed category

graph of a morphism

f: A ~

B

T

with finite

is the subobject

limits.

2.3.1'

The

represented

by the

monomorphism :

A functional subobject monomorphism

X

~

~ AxB

A

of

A×B

is an isomorphism. in

by a

~ AxB

, A

(Verify that these concepts

mean the expected

Set.)

Mapping morphism,

T(A,B)

into

Sub(AxB)

we obtain a bijection

subobjects by

is one that is represented

for which the composite X

things

> AxB.

of

i: X

AxB.

> A×B,

In fact,

> B: the canonical

Proposition

2.3.2.

g: A'

~ A

in

T

graph is

subobject i

is

represented

~2i(~li)-i

projection). A Lex-morphism

if and only if the following f: X

onto the set of functional

given a functional

(Andrew Pitts)

morphism of the form

by taking the graph of the

T(A,B)

the m o r p h i s m whose

(~2: AxB

a quotient

of

> FA

in

and an isomorphism f

X

F: T

condition holds: T',

there

X m FA'

> T'

is

for any

is a m o r p h i s m such that

-~ FA

commutes. Paoo~:

Note that the particular

Q: T

+ T",

the first

factor

ization of any Lex-morphism, uniqueness

in 2.3.1 of the quotient

in the quotient-conservative

satisfies

of this factorization

the condition holds

construction

the condition.

(2.3.1'.),

factor-

Now, by the

one easily deduces

for any quotient-morphism.

that

262

Conversely,

assume

quotient-conservative

that the condition holds,

and consider the

factorization F

T

+T'

K/ T"

First, we may easily verify that as the one assumed for an equivalence.

such that

of

f.

+ rB

Let

h: A f

rh = f,

is that

T'.

is a monomorphism,

and

hence

in

the same condition

rg g

+ B

there

and

represents

r

rg

g: C

have the same graph,

in

T

is

Suppose

> A×B

represents

hence,

subobject

whose

in

the graph

subobject,

a functional

be the m o r p h i s m

r

its graph

is

a functional

represents

is full.

Consider

By the condition,

r is conservative,

A×B.

But then rh rg;

£g

In particular,

since of

f: rA

~ FA×rB ~ £(A×B).

T"

£ satisfies

Next, we use this fact to show that

The only thing to prove

we have a morphism £A

F.

[g]

graph is

[g].

namely the one represented by

as required. • 2.3.2.

Proposition a coherent

2.3.3.

Suppose

functor.

Then,

ization of

F

in

Lex

T, T'

T"

It follows

categories,

F

and the functors

~ T'

Q

and r

are,

that the Lex quotient-conservative

coherent m o r p h i s m

~ T'

F: T factor-

from 2.3.1: T

we find that

are coherent

taking the quotient-conservative

is already a q u o t i e n t - c o n s e r v a t i v e

in fact,

coherent.

factorization

of a

factorization

in Coh.

Proof:

Let us note first that

QA: Sub(A)

> Sub(QA)

every subobject Qf: QA'

> QA.

~

of

Q

induced by QA

Consider

is full on subobjects: Q

is surjective.

is represented the image-mono

the function

Indeed, by 2.3.2,

by a mono of the form factorization

of

f

in

263

T: A'

q ~ A"

i ÷ A,

mono as well.

Since also,

morphism.

Since

morphism.

But then

thus

F

f: A

preserves

Fq

Qi

Ff m FQf

is an extremal

represents

and

is a mono, epi,

Fq

is a

Fq

is an iso-

Fq m FQq, Qq

is an iso-

the same subobject

of

QA

as

Qf;

as required. ~ B,

extremal

Since

is conservative,

# = QA([i]), Let

iq = f.

and

~ ~ Sub(A),

be in

T.

epis, by the subobject-fullness

To verify

of

Q,

that

Q

it suffices

to show the equivalence

Q(~f(¢)) But, applying preserves

r

Since that

to both sides,

extremal

the equivalence T"

Q

~ Q(~) "=~ Q(¢)

epis;

~ Q(f*(~))

(~ E Sub B).

the equivalence

since

F

becomes

is conservative,

true since

it follows

F

that

itself holds. is, also,

has images

heSS of

F,

pullback

of images

essentially

and that

it easily

F

follows

them.

that the property

is reflected

Similar arguments

surjective

preserves from

T'

to

(see 2.3.2)

it follows

By the conservativeof stability under

T".

apply to sups of subobjects. • 2.3.3.

Corollar Z 2.3.4.

If a C o h - m o r p h i s m

is a Lex-quotient,

then it is a

Coh-quotient. m 2.3.4. Corollar[

2.3.5.

on subobjects

Proof:

The

'if' part, 2.3.4.

A Coh-morphism

and essentially

'only if' part assume that

it suffices

satisfies Let X m FA';

f: X

~ FA

follows

Coh

iff it is full

from the proof of 2.3.3.

F: T --+ T' F

satisfies

For the

the conditions.

is a Lex-quotient,

i.e.,

be given.

Find

A'

and an isomorphism

the composite FA' m X

f', the subobject

:

FA'

~ FA'xFA

FA'×FA,

for some mono

of

f ~ FA.

FA'×FA

represented by

is represented by i: A" ........~..A'xA.

By

that it

of 2.3.2.

f': The graph of

in

surjective.

to show that

the condition consider

is a quotient

Fi: FA" ---+ F(A'xA)

But then the diagram

264

Fi

F~ 2

FA"

÷ F(A'xA)

~ FA

F~ 1

X commutes,

and

so does +

FA

FA" as

required. • 2.3.5.

Corollary

2.3.6.

then

is Boolean as well.

T'

Proof:

If

F: T ---+ T'

Straightforward

is a Coh-quotient,

and

T

is Boolean,

from 2.3.5. i 2.3.6.

2.4.

Pretoposes. Every category

the context

in this section has

is not explicitly mentioned,

(at least)

finite

limits.

If

we work in a fixed category

with finite limits. The ith projection

AlX'''XAn ---+ Ai

is denoted by

~i'

and

for the morphism

<~il' we write Definition

• ..

, ~ i k >: A l X ' ' ' x A n

~ Ail

×.

"'×Aik

~il...ik" 2.4.1.

An equivalence relation is a pair of parallel

morphisms )

R

such that

i = :

R

+ AxA

+

A

is a monomorphism,

and such that if

265

we also write

R

for the subobject

of

A×A

represented by

i,

then

we have the following: AA s R

(reflexivity)

=21*(R)

= R

(~21: AxA

=I2*(R)^~23*(R)

> A;

symmetry)

~ ~I3*(R)

(~12' ~23' ~13: A x A x A -- -- ~ One immediately in 1.2.

notes that in

A×A;

Set,

It is clear that an equivalence

transitivity).

this is the definition relation

given

in the sense of the

last definition

is taken into another one by any functor preserving

finite limits;

hence the condition of 2.4.1 implies

condition

in 1.2.

conservative relation

if a pair of parallel

functor preserving

according

relation too. limits',

Also,

to 2.4.1,

It follows

the special

two definitions

the defining

morphisms

is taken by a

finite limits into an equivalence

then the original pair is an equivalence

(by the

'completeness

of the logic of finite

case of 2.1.1 for the trivial

topology)

that the

are equivalent.

Next note that the kernel pair of any morphism is always an equivalence

relation,

equivalence

relation

equivalence

relations

since this is true in Set.

could be defined by the condition

taken into a kernel pair by some conservative limits. It is easy to see that if a morphism pair of morphisms, effective 2.2.3.

it is a coequalizer

is called an effective epi:

morphism

epi iff

{f}

2.4.2.

an initial object, equivalence (i) codomain (2)

a morphism

epimorphic

finite

of some f

Such a is an

family according

epis are extremal

to

epis.

A pretopos

is a category with finite limits having

coproducts

of pairs of objects,

relations,

and satisfying

The initial object O

that they are

is a coequalizer

clearly,

is an effective

(checkl),

functor preserving

of its own kernel pair.

It is also easily seen that effective

Definition

Since in Set, any

is the kernel pair of its own coequalizer

O

the following

is strict,

is an isomorphism; If

C

A

B

and coequalizers additional

of

conditions:

i.e. every morphism with

266

is a coproduct [i], by

diagram,

then

[i'] they represent, O

are

i, i'

monos,

[i] ^ [i'3 = O C

and

for

the

(0 c = subobject

subobjects represented

* C); If i n

(3)

C A

B

p.b.

p.b.

^

A

i)

i I ^

Lemma Let

form a coproduct

i, i'

then

also

diagram,

Any pullback

(5)

Any equivalence

2.4.3.

universal

A

and

B

are pullbacks

as shown,

form a coproduct;

(4)

of an effective relation

In a pretopos,

s: A

B

C

let a morphism

be the coequalizer

property

epi is again

the coequalizer,

an effective

epi,

is a kernel pair of its coequalizer. f: A

~ B

be given.

of the kernel pair of we have a unique

i: C

f; ~

by the B such

that A

f

,,), B

C

commutes.

Then

effective

s

and

(hence extremal)

We know that mono

is postponed

Proposition (ii)

i

s

A functor

epi followed

is an effective

until

2.4.4.

give a factorization

(i)

between

of

f

into an

by a mono. epi.

The proof

that

i

is a

later. Every pretopos pretoposes

is a coherent

is a Coh-morphism

category. iff it is a

Pretop-morphism.

Proof:

(i).

the supposed

Let

T

stability

It is immediate

be a pretopos. of effective that

OA,

T

has stable

images by 2.4.3 and

epis under pullback

represented

by

O

(2.4.2.(4)). ) A,

is the

267

minimal

subobject

of

A;

its p u l l b a c k

along

any

A'

-

~ A

is 0A,,

Let

two s u b -

by 2.4.2(1). Here objects

is

of

A

the

construction

be represented

f o r m the c o p r o d u c t C

~ B~C.

of

B~C,

We h a v e

joins

by the

with

of

subobjects.

monos

B ---+ A,

canonical

a unique

arrow

C

injections

Bn C

. + A

~ A.

L e t us

B ....., B ~ C, making

the

following

commute:

B

A.

Now,

let us

f o r m the

effective

epi

- mono

B~C

We

claim

that

E ......... ~ A it,

then

D

, A

is any m o n o there

tive

diagram

Now,

if we

of

B~C

~ A:

÷ A

represents

such

that

is a u n i q u e

B~C

f o r m the

factorization

effective

both

epi

the

desired

B ----+ A,

...... ~ E

- mono

B~C

\o/

such

join. C

that we h a v e

factorization

~ E

Indeed,

.~ A

of

if

factor

through

the c o m m u t a -

B~C

.....+ E,

268

then,

since

E

D'

> E

) A

B~C

, A,

means that

) A

is a mono,

B~C

form an effective

-~

D

~ > A

factors

The stability

2.4.2(3)

and

and the composite

epi - mono factorization

which has to be isomorphic

proved.

D'

through

to the one under

E

of

(I).

This

) A, which was to he

of sups under p u l l b a c k

is a consequence

of

(4).

(ii).

The construction

of images and joins

it clear that any P r e t o p - m o r p h i s m For the converse,

what we need is a C o h - c h a r a c t e r i z a t i o n

effective

epis and coproducts

category,

and hence

in a pretopos.

in a pretopos,

makes of

Note that in a coherent

a m o r p h i s m is an effective

it is an extremal

epi, as a special

in a pretopos

a diagram

T

in a pretopos

is a Coh-morphism.

case of 2.2.4.

epi iff

Also, by 2.4.2(2),

C

A is a coproduct

diagram iff

meet of the subobjects morphisms

(a) A

~ C, B ---+ C

represented by them is

form an effective

same diagram is a coproduct B ---+ C

B

form a covering

epimorphic

family.

diagram iff

in

JT"

are monos, OC,

and

Hence,

(a) and

(b)

(c)

by 2.2.4,

(b) hold,

and

Thus, we have the required

the

the two the

A ---+ C, Coh-

characterizations. Given a coherent m o r p h i s m T'

we now see that

it remains

F

to see that

relations.

Let,

F: T --+ T'

preserves F

finite

preserves

between pretoposes

T

and

limits and finite coproducts-

coequalizers

of equivalence

in f

R

A

h

÷B,

g h

be a coequalizer

of the pair

(f,g), an equivalence

by the above, F preserves effective epis, Fh hence a coequalizer of any of its kernel pairs. of the definition

of pretopos,

(Ff, Fg)

is a kernel pair of

equalizer

of

(Ff, Fg)

relation.

(f,g) is a kernel pair of

h;

Fh.

is a co-

We conclude

that

Fh

hence

as desired. •

Proposition

2.4.5.

For any coherent

pretopo8 aompletion of

Since,

is an effective epi, By the last clause(5)

T

category,

we have the

2.4.4.

(free)

269

YT: T ~ + with

P(T)

universal

a pretopos, property:

¥ = ¥T

P(T)

a Coh-morphism,

for any pretopos

T',

the functor

) Coh(T,T')

( ) o ¥: Pretop(P(T),T') is an equivalence Moreover, full),

of categories. y

is conservative,

and "every object of

object

X

of

the form Proe~:

~(r)

(Y(Ai)

P(T)

full on subobjects

is covered by

there is a finite effective

It resembles

It is convenient relations

that the final result

of the quotient-field sums;

(Cohds for short),

satisfying

conditions

is a Coh-morphism

(4) in 2.4.2 holds.

(1),

Cohqe(T)

as follows.

A Cohds-cat-

preserving

for short,

is a coherent

We prove two propositions.

is also a Cohds-category,

Let us put ourselves

and

One is our

the other with Pretop has the addendum that

provided

T

is a Cohds-category.

imply the one to be proved.

To carry out the first construction,

let us make some preliminary

into a Cohds-category.

Morphisms

the form

between

finite disjoint

tuples of the form

a

the initial

relation has a coequalizer,

The second proposition

~A. i i

of

A coherent category with quotients of

It is clear that the two propositions observations.

of

It turns out

(2) and (3) in 2.4.2;

with Pretop replaced by Cohds,

replaced by Cohqe.

In the

completion.

(automatically

in which every equivalence

proposition,

domain.

into two steps.

category having an initial object and coproducts

equivalence relations, a Cohqe-category condition

of an integral

we define coherent categories with disjoint

object and binary coproducts). category

in algebra of the

in the second, we adjoin quotients

is the desired pretopos

sums, and their 2-category

Cohds-morphism

family of

long when written

to the result of the first step.

To make this precise, egory is a coherent

although

to break the construction

first, we adjoin disjoint

any two objects,

epimorphic

any one of a number of proofs

type of the construction

equivalence

(hence also

¥", i.e. for every

~ X)i~i.

The proof is quite straightforward,

out fully.

and the following

÷ ~ B. j j

sums are in one-to-one

correspondence

with

of

270


* ~. B j> i •

1

Moreover,

J

a morphism of the form

gives rise to

f: A

fj,

j

from the pullback f

A

B.

J

i. J

J f.

J

A.

+B.

3 and in fact, the ~j disjoint fj: Aj for all

form a disjoint

sum representation ~ Bj,

3 sum diagram,

(Aj --~ A)j

one gets a unique

f

of

Conversely,

A,

given a

and morphisms

making the last diagram commute

j.

This suggests the following definition are. finite (possibly empty) a tuple is denoted =~A." i

tuples (Ai) i A morphism

Cohds(T). Ai

Its objects

of

T;

such

1 r~A

i

is given by:

for

of objects

a disjoint

~

-

-

+

i

r~B j

sum representation

~ j

(Aij

....~.. Ai) j

of each

Ai, and a morphism Aij ~ Bj for each i and j. Two such morphisms, one as above, the other with primed items, are identified if and only if there is a system of isomorphisms

A i m A!I, A.Ij ~ A~.Ij,

such that all of the following commute:

1

1

T

T

A..

~

1J

A.~ .

13

B

To define composition, a pair of morphisms ~A.

i

I

we return to a Cohds category,

, LB.

j

J

~ ~C

k

k.

and in it,

271

Consider the following rfA. i i

~ T[B. 1

IS - - +

Bj T 'B

,

~

k Ck

p.b. j

k

~

f

+ Ck •

Aik

We are interested in deducing, for the sake of a later definition, the data A. 1

Ck for the composite morphism, out of the ones A.

A i j ~ ~

Bj

B.

Bj k

Ck

for the factors. We form the pullbacks Aij k. It is clear that the arrows (Aijk , Aik)j form a disjoint sum diagram. The desired morphism Aik , Ck is, therefore, given by the universal property

272 of this last disjoint Now,

sum,

from the components

it is clear how to formulate

The functor

y~ohds:- T

~ Cohds(T)

We leave the details construction

the subobject

R

of verifying

one.

the required properties

(p,q)

of morphisms,

represented by the mono

relation.

~ A/R.

analyze

is the obvious

> C k.

of the

Cohqe(T).

Instead of a pair

PR: A

> Bjk

of composition.

to the reader.

Let us turn now to

equivalence

Aij k

the definition

For the quotient

Let us put ourselves

an arbitrary m o r p h i s m PR

A

let us refer rather to

:

of

R,

> AxA

into a Cohqe category,

f: A/R

> B/S. f

÷ A/R

as the

Consider

and let us

the pullback

B/S

X and the subobject

R

let us write

B

represented

<~,~>:

by

AxB;

X

let us denote the

^

latter subobject AxB

consisting

the following

by

X.

If our category

of those



implications

aRa'

is Set,

for which

X

f(a/R)

is the subset of Therefore,

= b/S.

are true:

& aXb & a ' X b '

~ bSb'

(2)

T r u e ~ 3b aRb with

a, a' ranging

subset

of

AxA

over

A, b, b'

over

B.

such that the implications

Conversely,

(2) are true,

if

X

is a

then there

is

^

a unique

f: A/R

> B/S

the same as the given In an arbitrary

such that

X

deduced from

Cob-category

T,

1A

with the projections

in the first inequality

AxAxBxB.

(2) and

in

T

< 3#1 (~.) referring

(3) mean the same in Set;

iff for all Coh-morphisms

A, B, R, etc. meaning

as above

is

we write

"~12"(R) ^ ~13"(X) ^ "~24"(:X) < #34"(S)

Certainly,

f

X.

M: T

M(A), M(B), M(R),

> Set, etc.

l

(3)

]

to the product and

(3) holds

(2) holds with all of

273

We make, Cohqe(T)

therefore,

the following

are given by pairs

equivalence

relation on

A;

definition.

(A,R), with we write

A

The objects

an object of

rA/R"

for

T,

(A,R).

of

R

an

A morphism

A

rA/R"

~ rB/S ~

B') satisfying

is given by an

X ~ Sub(AxB)

(a 'relation

from A to

(3).

As for the composite

of two morphisms,

note that if we have a

pair of morphisms A/R - - . and

X, Y

deduced and

are the relations

from their composite

Y,

which,

B/S

C/U

+

deduced from them, then the relation is the relational

product

X~

of

in Set, is defined as

aCX[~)c ~=~ ~bcB (aXb & bYc). The official

definition

is

~ with the projections

=

3

713 (~ 1 2 *(~)

referring

^ ~2 3 *(Y))

to the product

Notice that we have to show that if appropriate

data),

so, it suffices

then

X[ST satisfies

A×B×C.

X, Y satisfy

(3) with

R

(3) (with the

and

U.

For doing

to verify the same fact in Set, and appeal to

completeness. We leave the remaining

details

to the reader. • 2.4.5.

Corollary

2.4.5'

If

T

is a Boolean category,

then

P(T)

is a

Boolean pretopos. The proof is an instructive of 2.4.5.

exercise,

using the

'moreover'

part

• 2.4.5'. Let us call a Coh-morphism objects, statement that,

quotient-like if it is full on sub-

and every object in its codomain of the

'moreover'

for a Pretop-morphism,

a quotient.

is covered by it (see the

part of the last proposition). being quotient-like

But first, we prove

We will show

is equivalent

to being

274

Lemma 2.4.6.

A quotient-like

conservative

Pretop-morphism

is an

equivalence.

Proof:

Let

F: T ~

T'

We have an effective

satisfy the conditions,

epi family

(F(Ai)

A

=

X ~ 0b(T'). with

~A.

i we therefore have a single

and let

~ X) i (one in JT);

1

effective

epi

f: F(A) .........~..... X.

Let

P .....

Y

,, + F (A) p'

be the kernel pair of by

:

Y

ed by : equivalence

f;

let

, FAxFA. B

~ AxA,

relation,

Y

be the subobject

By assumption, say, whose

and

F

F-image

+

g'

relation.

AxA

represented

is

Y.

Since

represent-

Y

is an

is conservative,

B

is an equivalence

of

we have a subobject,

Let

A

h: A

C

be its quotient.

But

then the two diagrams P

.4~

Y p'

)

Fg

.

F(B)

f

F (A)

F(A)

Fh

~ X

~ F(C)

Fg'

are isomorphic

since each of

and second diagram, that

F

is essentially

consequence

f

and

respectively: surjective.

of being conservative

Fh

is a coequalizer

in particular F

in the first

X -~ F(C).

is full and faithful

This shows as a

and subobject-full. • 2.4.6.

Proposition

2.4.7.

an isomorphism, a conservative

(i) Every Pretop m o r p h i s m can be factorized,

as the composition (Pretop-)morphism.

of a (Pretop-)quotient

up to

followed by

275

(ii)

A Pretop-morphism

on subobjects,

is a quotient

and every object

if and only if it is full

in its codomain

is covered by it (it

is quotient-like). (iii)

If the domain

of a Pretop

quotient

is Boolean,

so is its

codomain.

Proof:

(i)

it being

Given the morphism

in Coh,

and factorize

F: T

÷ T'

in Pretop,

it in a quotient

we consider

and a conservative

one

in Coh:

T

F

,,

~ T'

\/ T"

Now,

form the pretopos

clearly

completion

P (T")

y: T"

of

T".

We

have a diagram F

T'

T"

\

~-

F

P (T")

in which

Fy m r.

The universal

=T'"

properties

of

Q

and

y

immediately

^

imply that

Q = Qy

is a quotient

like, by 2.3.5 applied to

y.

Using

to

Q,

in Pretop.

Moreover,

and the "moreover"

the latter property

of

it is quotient-

part of 2.4.5 applied

Q, we now show that

r

is

conservative. Let

X ----+ Y

isomorphism.

be a monomorphism

Consider

the pullbacks,

a finite

one for each

in

covering

T'"

taken by

family

(QA i

r

into an

~ Y)i'

and

i: X

~

T

Z. i

-- . . . . . . .

Y

I

QA.. 1 ^

^By

Q

being subobject-full,

Zi ^

> QA i is isomorphic

to some

^

Qfi: QBi ----+ QAi' with fi a mono. r phism (as a pullback of an isomorphism).

takes Since

Qfi

into an isomor-

F m pQ,

and

F

is

276

conservative,^ (Zi X

Qfi

itself is an isomorphism, for every

i.

But then

> QAi .. ~.. Y)i is a covering, hence, since it 'factors through' ~ Y, the monomorphism X ~ Y must be an isomorphism. It is a simple general fact that a Coh-morphism conservative with

respect to monomorphisms in the domain category is conservative. follows immediately when one reflects that a morphism an isomorphism iff the mono representing the subobject mono

A

) C,

f: A

This ) B

3f(l A)

is

and the

in a factorization A

> C


/p,q>

A×A with

(p,q) (ii)

the kernel pair of

f,

are both isomorphisms.

By the proof of (i), and 2.4.6.

(iii) By 2.3.6. and 2.4.5'. 112.4.7.

The final group of results in this section should, logically, be the first ones in the study of concepts related to the quotient-conservative factorization. Pff0position 2.4.8. In any one of the 2-categories Lex, Coh, Pretop, the following is true. If T is any object of the 2-category, N is any set of morphisms in

T,

then there is an arrow Q: T

~ T[~ -I ]

in the 2-category so that "Q is obtained by inverting the morphisms in ~" in the sense of Definition 1.4.2.(i) (with Pretop replaced by the relevant 2-category).

T[~-I]

and

Q

are determined up to equivalence,

resp. up to isomorphism. The proof will be discussed (although not given) in the next section. 2.5.

Relations with symbolic logic. Let us start with a fundamental observation that goes a long way

towards explaining the relation of model theory and categories. be a small graph) or category, and consider a diagram (functor)

Let

C

277

M: C

> Set,

language

Then

given by

each object of prets

each

morphism >

C:

interprets

C,

say

operation

f: A , In fact,

are precisely

empty partial

> Set

each sort of the language,

say A, as a 'partial

> M(B).

Set

possibly

C,

M

(sorted unary)

of

M(f): M(A) C

is the same as a structure for the m a n y - s o r t e d

M

C.

domain'

symbol

as a corresponding

if

C

is a graph,

domains.

If

are those L-structures,

then the diagrams

is a category,

for

L

that satisfy a certain set of identities: ............ h

i.e.,

kind of operation

for the language

C

i.e.

and it inter-

of the language,

B,

the structures

A

M(A);

C,

with

the functors

the underlying

graph of C

whenever

> C

B

commutes,

we have the identity Va ~ A g(f(a))

= h(a);

also,

(i)

Va e A idA(a) for the identity m o r p h i s m Moreover,

natural

idA: A

= a

> A.

transformations

C

between

diagrams

(functors)

+ Set

are seen as the same as homomorphisms in the usual sense between structures. In first order logic over the language of which has a definite 3x

is meant

(¥x c X), that

sort,

an object of

to range over a fixed sort;

(~x e X)

with

X

we have variables A quantifier

to emphasize

the sort of

x.

Yx,

each or

this, we write

We have an equality

sign

is sorted in the sense that only terms of the same sort are allowed

to fill in the places equality

of it (equivalently,

sign for each sort).

unary operation

symbols

we have all the Boolean

there could be a separate

The only nonlogical

(the arrows

the natural way as illustrated L,

L, L.

under

in

L);

(i).

connectives,

symbols

are sorted

their use is regulated In full first order

in

logic over

and the two quantifiers.

278

Although many-sorted seem to lack expressive ical logic, attention

languages with unary operation

power,

it turns out,

existential Let

by using

~

be a coherent

category,

a string of variables ÷x.

If

is a functor

~ Set,

{a c M(X): M If

M: T ~ ~[a]},

~

subobject,

M

i.e.

a subset,

of

of

formulas.

Proposition

(i)

M c Mod(T).

~

For

T

M(X).

of

graph,

be the sort of M~(~)

~(~)

in

of and

if

M

=

M,

a subset of

then

The next proposition

M(~)

is a

tells us that

logic is at least as great

formula

category,

~ a

x i,

in particular

a Coh-functor,

categorical

= M~(~)

~(~)

there

over

(the

is a unique

subobject

of

(

~ Sub M(X))

is called the canonical interpretation of [~:~].

~

over

The proof of the existence the complexity

M

a Boolean category,

any first order formula

sections,

and

such that

it may be denoted by (ii)

X.i

For any coherent

M(¢)

~(~);

X,

graph of) T, a coherent

[5] = X

let

then we may consider

as that of coherent

for any

disjunction

its underlying

the interpretation

power of coherent

2.5.1.

L

is any L-structure,

is a subobject

the expressive

underlying

formula is one that is built conjunction,

such that each free variable

~ = <Xl''" . 'Xn> ,

If

X = [5] = XIX...×X n.

of

(finite)

quantification. T

in

M(X).

of categor-

to such languages.

up from atomic formulas

formula,

as a biproduct

may

that in fact there is no loss of generality when we restrict

A coherent, or positive existential,

occurs

symbols

~,

there were

the same conclusion

for

T. of

~,

is an instructive implicit

holds

examples

proceeding

by an induction

exercise.

In fact,

of constructions

on

in previous

of canonical

interpretations. • 2.5.1.

The last p r o p o s i t i o n Proposition

2.2.4.

can p r o f i t a b l y

Suppose

show that it is an effective 2.2.3,

assume we have

gjf'ij

(i,j

c I).

gi: Ai

Consider

be used in the proof of

(fi: Ai

÷ A)ic I c JT;

epimorphic

family.

~ B

(i c I)

the following

such that

formula:

V (3x c A i ) ( f i ( x ) = a A gi(x) i

we would

like to

Using the notation

= b)

gifij

=

of

279

with two free variables is that,

a,b of respective

function

A

Now,

~ B, let

¢ = [ab:v]. morphisms

as immediately

# e Sub(A×B)

If

in Set for which

requirements; infer,

in fact,

at least,

that

M e Mod(T).

object of is

~.

A×B

M ~ Mod(T):

then

M

in

M(h)

v, and

Let

hM

h

hold,

satisfying

Mab(~) , i.e.

subobject ¢

of

the

M(¢).

We

M(A)×M(B)

is a functional

for

sub-

be the m o r p h i s m whose

follow,

hMOM(fi)

and as we know

> M(B)

is

h: A -----+ B

and

of

interpretation

hM: M(A)

clearly,

hfi = gi

= hM,

The uniqueness

of

M(¢) is a functional

T.

The idea

takes the data to objects

the same hypotheses function

By completeness,

B.

seen.

the graph of

The equalities

A and

the graph of the desired

be the canonical

M e Mod(T),

already there is a unique

all

sorts

if we are in Set, then this defines

graph

since they are true in any

= M(gi).

can be deduced in the same manner. • 2.2.4.

Note that for any subobject is a formula canonical

;(x)

¢ ~ Sub(X),

with one free variable

interpretation

is

~.

If

¢

in any fixed

x

of sort

X

T,

there

whose

is represented by

m: A

÷ X,

^

then not

~(x)

can be taken to be

uniquely

determined

by

(3a ~ A)(m(a)

¢,

we w i l l

= x).

Although

;(x)

is

the formula representing

call

it

Set

are

E.g.,

it

¢. O n c e we a c c e p t directly

interface

a natural T

with

functors model

transformation

h:

Set

is

an elementary

consequence

of

2.5.1.:

h:

to

that

M

~ N

doxical

as Let

remember

topos

is it

us

at

is a coherent

prove that such that,

i

T

proof

stage,

holds

in

Boolean,

of

makes

to

two the

sense

functors following this

say

that

M, N f r o m

M, N e M o d ( T ) ,

M(s)

In

not

yet

immediate then

not

this

proof,

established

any

that

para-

we h a v e that

to

a pre-

With the notation of 2.4.3, we want to Let

M: T

is surjective.

÷

Set

Then,

be a Lex-functor

in Set, M(f)

and by taking into account

and M(s)

the meaning

of

in Set, we see that

M.

(Va,

a'~A)

If(a)

This

immediately

= f(a')

¢=~ s ( a )

implies

that

= s(a')]

M(i)

In order to be able to apply completeness, J

we c a n

(Why' i s

2.4.3.

we h a v e

category.

have the same kernel-pair, pullbacks

is

Note

embedding.

is a monomorphism.

in addition,

between

structures,

first?)

the

this

........ ~ N

embedding.

if

complete

that,

theory. M

an elementary

may s o u n d

into

generated by the single

covering

{s}

is a mono in Set. consider

of the object

the topology C.

Since

s

280

is an effective

epi,

effective

(a condition on pretoposes),

epis are stable under pullback

effective

epis are closed under composition and since the pullback composite contains which

epis are extremal,

(exercise; or see 3.1.5 in EMR]),

it follows

that every covering

a morphism that is an extremal

is not an isomorphism

general

completeness

T

of a composite m o r p h i s m can be formed as the

of two pullbacks,

A ----+ C

in

extremal

epi.

It follows

in

that a mono

is not a covering by itself in

theorem

2.1.1,

if

i

J

is not a mono,

J.

By the

i.e.

the mono

in the factorization

A

.....

>

C

A×A

with

(p,q)

M: T

the kernel pair of

~ Set,

contradiction. Although

M J-continuous,

i,

is not an iso, then there

such that

M(i)

is

is not a mono,

a

this proof is not even shorter than the direct proof

(see the proof of Theorem 1.52 in ETTJ),

I find it preferable

because

it is more comprehensible. Let us now look at the interaction egorical

formulations

categorical

conditions

in the opposite

is a class of structures well-known diagrams this:

in

sense.

the Cob-operations)

elementary

classes.

E.g.,

are strict

type;

of the particular

Set

is a product

our

for the operation

limit and colimit

of binary products,

a diagram

A

thus,

is verified by writing out the

C

in

of

Note that each such graph

over a finite similarity The assertion

characterizations Set.

the expressibility

is that the graphs of all the Boolean pre-

(and of course,

(finitely axiomatizable) statement makes

direction:

and cat-

by symbolic means.

The first observation topos operations

of the symbolical

B

diagram iff the following

is true:

we have

281

(VaEA)(VbeB)(~ccC)(fc=a (Vc~C)(Vc'eC)((fc=fc' It is instructive diagrams

Next,

disjunction;

in fact,

the underlying diagram in

that expresses

T).

with symbols (2) in

see (i) above.

a Pretop-morphism

T

~ Set

we write

in that diagram,

is a product

diagram, we

We also add axioms ensuring By the definition

of what

> Set, it is clear that the resultexactly the class

Let us discuss ultraproducts the categorical

(c.f.V.3

for this

and for each in-

appearing

T

(3) to our axiom system.

ing axiom system will axiomatize

C

operation

T,

(over the

which is in the graph of that operation,

sentence,

that we have a functor;

C

class

In fact, the axioms

for each pretopos

E.g., if we find that the diagram

With

class.

that the diagram would be the given kind were it in Set.

add the sentence constitutes

it is not an elementary

is an elementary

graph of

T

down the first-order

is defined

we see that, for any (small) pretopos

Mod(T)

class are the following: dividual

(3)

+ c=c')

on the domain of the operation)

as a consequence,

the class of objects of language

gc=gc')

A

to see that the class of all coequalizer

(without restriction

by an infinitary

A

fc=b)

A

of structures,

Ob(Mod T). i.e. functors,

from

point of view. a (small)

graph,

the category

has limits and colimits in [CWM]).

Therefore,

(C,Set)

of all diagrams

that can be computed pointwise

if we define ultraproducts

in (C,Set)

by the same formula as in Set: n Mi/U = colim n Mi , icl peuOP icP and similarly

for morphisms

in

(C,Set),

then,

for

A

c Ob(T),

we

have ( H i~I and

a similar

tion

given

C

> Set

formula

here

for

coincides

Mi/U )(A) = morphisms with

the

~ ieI in

usual

M i(A)/U T.

This

one,

(4) shows

considering

that

defini-

diagrams

as structures.

Now, let

T

be a small pretopos,

and let

C

be its underlying

graph. Then, if in the above the M i again in Mod T. This is an immediate

are in Mod(T) then consequence of 1.3.1

called Los's theorem)

(4).

from the fact that theory,

the

the

and the formulas

Mod(T)

is an elementary

and the usual formulation

Of course,

~Mi/U is (which we

it also follows

class in the sense of model

of Los's theorem.

282

Let us turn to the construction free Boolean pretopos,

of the free pretopos,

needed for proving the assertions

fact, the constructions

pretopos

structures.

operations,

finite diagrams.

proof of the existence

composites

It would also be possible

to give the

versions

of the adjoint

functor theorem.

symbolic

logic provides

a simple and intuitive procedure.

Boolean

However,

the free coherent

category on a graph

analogous

G.

of the

and they would take the form

of the free objects by appropriate

First, we construct

In

of the

free groups and other free

The "words" would be literal

(Boolean pretopos)

of labelled

in 1.4.1.

could be given by a direct application

"method of words" used for constructing algebraic

and the

(2-categorical)

in these two cases

category,

and the free

They satisfy universal properties

to (i) and (ii), respectively,

in 1.4.1. +

For FCoh(G) , we take as objects all the pairs (x,~) with a finite tuple of distinct variables, ~ a coherent formula with free variables we mean,

all included really,

in

x.

"the set of

We write ~

E~:~] with

~

and

~

disjoint, ++

such that and

among

with

meaning unique existence

first order logic.

(x,~), since

Given two objects

formulas

~

with free

and s a t i s f y i n g

and

Such formulas

~

between the same objects

equivalent

formulas.

functional

relations.

are not necessarily

A morphism

,,[2, referring

disjoint

Coh

~G this defines

to provability

Now, two provably

functional

if they are provably

is an equivalence

class of provably

tuples,

in the given objects,

and

is left to the reader to form-

of composition

: G - - +

~

of morphisms,

and that

FCoh(G]...

the free Coh-category

over

G

completely,

of course more work is to be done to verify the required properties. This will not be done here; The construction

in

are called provably functional

are identified

The case when,

ulate, as well as the definition of the G-diagram

Although

~".

coherent

relations from the one object to the other. relations

instead of

[~:¢]

we consider

variables

3~

xy,

[~:~]

for details,

see [MR].

of the free Boolean category

over

G,

283

Boole ~G : G ulas over

G

+ FBoole(G )

is similar,

as language,

in defining both the objects

but uses all first order formand morphisms.

The free pretopos, Pt = ~G : G

~ Fpt(G)

is given as the composite

G

Fcoh(G ) - -

........

* P(Fcoh(G)) Y

with the pretopos

completion

taken from 2.4.5.

property

is straightforward,

required universal of

~

and

~oole

y.

The free Boolean pretopos

in place of Finally,

~;

one uses

the proposition egories

of category-based

conditions".

Since

(see however

some more concrete lectures

I am not quite

category theoretical

e.g.

case of Lex, with 2.3.1 for Lex,

terminology.

the internal

Let me first

described,

of fraction"),

are closely but unless

even for the case of Lex,

is not

Let me add that, of course,

the

case of what we want for the

with the notation of that proposition. of the factorization

(Gentzen-sequents)

R e Coh,

and

are not.)

Z c Morph(R),

R (see p. 128 in [MR]).

expressing

that each

o ~ Z

= b

(o: A ÷ B

in

in a consider

Add the following should be

invertible: True ~ (3~a e A)o(a)

in

con-

the special properties

for our purposes,

for Coh and Pretop are given in [MR] Given

in

for the case of Lex and Coh

facts alluded to above,

theory of

facts

I point out

These constructions

the mere existence

crucial

The constructions

axioms

these general

and the ones for Coh and Pretop are, of course,

of the general

in

"2-catalgebraic

Kelly and myself),

("calculus

is a special

z = Inv(F),

of the factorizations, different

terms.

in [GZ].

in 2.3.1

(Note also that although sequences

ready to describe

it, the construction,

found in full generality construction

asserted

facts concerning

of the desired quotients.

in [GZ]

with

just as in the

the mere existence

detailed constructions

related to constructions I have missed

Again,

that Professor Jean Benabou recently

in Montreal,

in elegant

2.4.8.

defined by essentially

paper by G.M.

constructions

on the basis of those

is obtained similarly,

of general

structures

a forthcoming

of all mention

TR,

just discussed,

are consequences

of the

2.4.5'.

let us turn to Proposition

case of the free objects

The verification

Z).

284

Let the theory so obtained be denoted by

Consider

T'.

the categor-

of T' (see Theorem 8.1.3., p. 239 in [MR]), and call ization RT, it R[~-I]. The canonical interpretation of Y R in T', and that of T'

in

induces

RT,

the required Coh-functor

R Given the properties

of

TR

have the required universal

~ R[~-I].

and

RT, ,

it is easy to verify that we

property.

The case of Pretop can be handled as it was with respect to free objects:

first, we carry out the construction

in Coh, then we take the pretopos

Part 3. 3.1.

The statement Consider

any coherent

Conceptual c0mpleteness.

the notion of the pretopos T,

morphism

T ÷ P(T)

S = Set)

an equivalence

in particular,

completion.

and proof of conceptual

category

on the given pretopos

one associated

that induces,

completion

(see 2.4.5.):

a pretopos

Coh(P(T),S)

of the categories

with

P(T), and a Coh-

for any other pretopos

of categories

an equivalence

completenpss.

S (e.g., for

--+ Coh(T,S),

hence,

of models:

Mod P(T) --+ Mod T. We interpret category

a 'theory', pretopos

this situation by saying that the notion of coherent

is conceptually T

by the concepts

completion

of concepts extension

in

incomplete:

T

'theory'

when extending

(abstract

by definable

equivalence

"behave~semantically

for any (pre-)topos

By contrast,

relations),

or even,

More precisely,

Theorem 3.1.1. I: T --+ T'

the following completeness

of small pretoposes

is sufficient categories

(Conceptual

in its

in T, quotients the resulting T

itself",

the same category of S-

S.

the notion of pretopos

is conceptually

the sense that it does not admit any extension to models.

category,

sets) inherent

'concepts'

in the same way as

by having the same category of models, valued models,

definable

(finite disjoint sums of

a coherent

in

with respect

theorem holds.

of pretoposes).

to be an equivalence

that I induce an equivalence

complete

conservative

For a morphism

of categories,

I*: Mod T' --+ Mod T

it

on the

of models.

This is Theorem an essentially

7.1.8 of [MR].

different

version of the theorem.

Recently,

proof yielding

Andrew Pitts has found

a stronger result,

a constructive

285

On the basis of this theorem, "real" or "complete" P(T).

E.g.,

if we ever

try to recover

Mod T, we should expect to get not be any difference P(T)

it seems natural

form of the theory P(T)

in principle

is its pretopos

T

in an abstract way from

instead of

T

in this area

completion

since there should

between concepts

from the point of view of the category

ation is borne out by results

to say that the

T

in

T

of models.

and those in This expect-

(see [M3],

[M4],

[L], [MI],

[M2]). The proof of Theorem so stated in [MR].

3.1.1 in [MR]

Note that

I: T ÷ T'

means two things:

first, that

full and faithful,

and second,

those

T --+ S

that

tions being required condition,

and even that for

morphism

(Strong conceptual

I: T --+ T'

"I* is faithful",

I;

The strong form of the first

for pretoposes).

to be a quotient,

it is

I*: Mod T' --+ Mod T be full and faith-

7.1.4.

in [MR] says that

I* being essentially

is conservative,

"I* is full" implies

Theorem 7.1.6 says that

surjective

on objects

clearly

the strong version of conceptual

completeness

analog of) conceptual

to note that in the "doctrine"

completeness

holds,

(a classical

Lex,

specific

the 2-category

of all pretoposes.

for Boolean pretoposes

in the common sense of model theory, more natural

see [GU])

them as

Being interested

(corresponding

to theories

in full first order logic),

it is

to consider the category BPretop of Boolean pretoposes,

with morphisms to consider

(the obvious

result,

but its strong form does not (see [MP]). Speaking about Boolean pretoposes, we have considered in Pretop,

implies

version.

It is interesting

in results

For a

implies that "every object in T' is subcovered by I".

implies the original

objects

onto

both condi-

3.1.2 follows by 2.4.7(ii).)

Since I

is

surjective

to require

completeness

"I is full with respect to subobjects",

Therefore,

not

only.

of small pretoposes

(In fact, Theorem

that

inverted by

S ~ Pretop.

S = Set

result,

-~ Pretop(T,S)

is essentially

says that it suffices

sufficient that the induced ful.

that

I*

to hold for all

completeness

Theorem 3.1.2.

I*: Pretop(T',S) that

invert all morphisms

conceptual

gives a stronger

(in Pretop) being a quotient

the Pretop morphisms.

BPretop

Furthermore,

it is more natural

to be Groupoid-enriched rather than Cat-enriched;

286

this means

that BPretop(T,T')

morphisms

T -+ T',

isomorphisms. are those of

Writing C

should be the groupoid of all BPretop

with only those natural C is°

and whose morphisms

are now saying that we should put

without

understanding (ii)).

Obviously,

concepts

involved,

(see Section

Therefore,

we

Gpd-category), Even

the reader can see here is intimately

defining Boolean pretoposes

I.i).

in the context of BPretop,

a quotient-morphism

C,

had to be defined

the fact we are considering

related to the fact that not all operations are strong

of

as 2-cells.

this plainly on the way the free Boolean pretopos (1.4.1

that are

whose objects

= Pretop(T,T') is°

algebraic'

transformations

the general

C

in this way has good general

(it is an 'essentially

but not if we put in all natural

of

are the isomorphisms

BPretop(T,T')

The reason is that BPretop construed algebraic properties

transformations

for the subcategory

the natural definition

of

is the following

Definition 3.1.3. A morphism I: T --+ T' in BPretop is a quotient if Iis°: (Mod T') is° --+ (Mod T) is° is full and faithful and essentially surjective by I.

onto those models of

Note that, by 2.4.7 Pretop

is automatically

I: T --+ T'

in BPretop

which

invert all morphisms

(iii), any quotient

Boolean.

of a Boolean pretopos

completeness

holds

specified.

the question

if an

for Boolean pretoposes.

answer is "yes" with a curious qualification: involved should be countable objects and arrows).

in

in the sense of 3.1.3 is the

in Pretop in the sense previously

3.1.3 is still useful since it suggests

analog of conceptual

inverted

It follows easily that, for a morphism

to be a quotient

same as to be a quotient Definition

T

The

the Boolean pretoposes

(meaning that they have countably many

Theorem 3 . 1 . 4 . ( S t r o n g c o n c e p t u a l c o m p l e t e n e s s f q r c o u n t a b l e Bo,olean pret0poses) For a P r e t o p - m o r p h i s m I : T --+ T' o f c o u n t a b l e Boolean pretoposes T, T' to be a quotient, it is sufficient that Iis°: (Mod T') is° --~ (Mod T) is° induced by I be full and faithful. This result is essentially Abraham Robinson memorial although

say that a Coh-morphism (see before

In 1975, at the

Gaifman announced a result which,

stated in a language not employing

related to Theorem 3.1.4. like

due to Haim Gaifman.

conference,

categories,

A version closer to Gaif~an's I

between Boolean

2.4.6) provided

I is°

is closely result would

(Coh-)categories

is full and faithful.

is quotient-

287

In this section, will

we will

apply it to prove

applicable

to the proof

The proofs

first prove

a stronger

Theorem

3.1.4,

and then we

form of it, which will be directly

of the main result,

Theorem

in this section will use various

1.4.4.

methods

of model

theory. The first

lemma,

Lemma

It is analogous to Theorem different. As above, with denotes the functor (Mod T') is°. Lemma 3 . 1 . 5 .

3.1.5.,

obtained

Assume

from

I:

that

able Boolean categories

of Gaifman's

theorem•

I*: Mod T' --+ Mod T restricted

T --+ T '

X

in

T'

to

is a Coh-morphism between count-

I is°

such that

"subcovers" every object

is the heart

7.1.6 in EMR], but its proof is very I: T -~ T', Iis°: (Mod T') is° -~ (Mod T) is°

is full and faithful.

Then I

in the sense that there is a

finite family of diagrams m. 1

X. I

IA.

1

(i c I)

T',

in

(pi)ie

I

Proof:

Recall

language

~(~)

L

c ~(~)

of types each x

finite ¢(~)

M

¢(~)

and

means

that would

then we have an obvious

realized

in all models

countable model

in

of

"M

simply

A type

T

(i.e.

for all (~ i(xi):

has a model in

~(~)

realizes

i~I}

M ~

omitting

(matching

simultan-

~(~):.

consistent

If

implies

any isolated

type must be

T.

to the notation Mod T'

with

is no tuple

s(~)

T u (@}

converse:

is called

family

T

all

case, we may say

T ~ @),

~

of L-formulas

appropriate).

that there

satisfy @,

~(~)

÷ o(~))

then

Given a

in a countable

is consistent

T ~V~(¢(~)

(for all L-sentence

Let us return

(see [CK]).

a set

given a countable

~(~)"

in the opposite

is complete

if

is non-isolated,

"M omits in

set),

type" may be more

says that,

each of which

for sorts)

(OTT)

tuple of free variables

"incomplete

OTT

theorem

is a countable

is consistent) •

~i(~i).

eously;

types

(having at least one model)

is isolated by

T w {~x¢(~)} a(~)

T

(Morph(L)

(although

such that the family

epimorphic.

the omitting

theory

with a fixed

a type

being monomorphisms,

i

is effective

consistent o(~)

m.

with the

(each

of the Lemma. M(X),

Let

X ~ Ob(T'),

M

be any

is countable);

T

288

let us fix L'

M

for a while.

(= the underlying



(written

graph of

Consider

graph of

simply

a)

the language

T') by adding

for each

T) and for each

A e Ob(L)

a e M(IA).

Let

L

obtained

a new individual

from

constant

(L = the underlying

i: L' ÷ L

denote

the

^

inclusion. We have the structure language L whose L'-reduct i*M be the theory in

M;

T Now,

various single

of

M

let us consider

element

b

such that

isolated. ZA(XA)

L:

the set of all sentences

complete

MX, X e 0b(L')

any finite

~

in

M, i.e.

We now also consider, the requirement

ZA(XA)

in

M;

L

hence,

of elements

T',

product

the set

true

~(~)

to a

We claim that the

sort).

of all L-formulas Z~(~)

A < 0b(L),

is a variable

is non-

the type

of sort

"x A ¢ A" implicitly). by a remark

of the

we could pass

Suppose

for each

(x A

ZX(XA) contains

~

in

is isolated.

= {-x A = a: a ~ M(A)} omitted

tuple

(using products

M ~ s[~],

in

Let

(see above).

of an appropriate

(complete) type of c(~)

in

is clearly

M = (M,a)aciA,A~Ob(L) of the is M and for which (a)M = a.

above,

IA:

hence,

Of course,

each

ZA(XA)

each is non-

^

isolated.

Therefore,

by 0TT, we have a model

÷ ~(~) and every ZA(XA) as well. Consider of M', and consider the L-structures I'M,

hA:

a ~

make up an isomorphism omits

each

a model that

~A'

of

h

hA

However,

I*h' = h,

in particular, h

cannot

same type in there

M'

as in

~

does in

M'

having

We have constructed the claim is, therefore, We conclude finite

tuple

say that show) if

~

that,

that any countable

~

and

~

~

reason

have

(I*M')(A); of

would

~

M'

is

implying

preserving

to an isomorphism

h', h'(~)

M'

since

I*M"

is structure

clearly

and

h': M ÷ M' have the

A

since

M'

the same type as

M,

but,

~.

omits

~(~),

to the fullness

of

Iis°;

proved by contradiction. still with the fixed countable has an isolated

It is well-known atomic model

type.

(c.f.

to

c.

M

of

In model

[CK])

T, each theory,

In our case,

we

(and not hard to

is homogeneous in the sense

have the same type in it, then there

of the model mapping and

h

a counter-example

of elements

is atomic.

M

onto

^

is nothing

omitting

by the fact that

on elements

be extended

since with any such

^

T

M' = i*M' , the L'-reduct I*M'. Then the maps

indeed,

is surjective

is "L'-elementary

is elementary,

reflecting. with

hA

of

(a ~ M(IA))

h: I*M ÷ I*M':

each

T,

~ M'(a)

M'

that

is an automorphism

this implies

the same type in

M,

they are identical:

is that an automorphism

of

M

that if

~ = ~.

The

is the same as an automorphism

h

289

and by the faithfulness of of M such that I*h = Idi,M, therefore, the only automorphism of M is the identity. X ¢ Ob(L')

We now see that for any L'-formula various

~(x,y)

sorts

matching

~

with

IA

with

x

of sort

A e Oh(L),

such that

and

X,

~

I is°,

b c M(X),

a tuple

there

and there are elements

M ~ ¢[b,~]

and

is an

of variables a

of

of

M

M = Vxx'(¢(x,a)A¢(x',a)+x=x

).

^

Namely, i f

~(x,$)

M ~ ¢[b,a], in

M,

isolates

and i f

hence

M ~ ~[b',~],

in fact,

an a p p r o p r i a t e

make

product

We c a n r e - w r i t e

b

then

b

in

M,

then of course

and

b'

have the

same t y p e

~

sort

and

in

~

into

singletons

by p a s s i n g

to

L.

our last

conclusion

this

way:

for the

fixed

M

X, M ~ ¥x

V

~y(¢(x,y)

Here in the infinite form

type of

b = b'

We c a n ,

and

the

¢(x,y)

with

of the form formula

IA,

holds

whose models

^ Yxx,(¢(x,y)

disjunction x

ranges

a free variable

with

A e Oh(L)

in all countable are exactly

L~wenheim-Skolem

~

^ ¢(x',y)

of

over all formulas X,

depending

models

the objects

M

y a variable on

of

of

and the compactness

r~ vx V

~.

T,

Since

with

Mod T',

theorems,

that there are finitely many formulas the ¢ above such that

+ x = x')).

(1)

of the

of a sort the displayed

T

the L'-theory

by the downward

it is easy to conclude

¢o(X,Yo),...,¢n_l(X,Yn_l)

~i(x)

as

(2)

i
~i(x)

from

~

formed

from

~i

as the formula

"V"

is formed

in (i).

It should now be clear that I.

after

Namely,

let

i < n

(2) implies

and consider

that

the canonical

X

is subcovered

interpretation

by

in T'

of the formula

~(yi ) (remember,

T'

the sort of Pi

= ¥xx'(¢i(x,Yi) df

is Boolean).

yi);

let

be the morphism

[yi: ~(yi)]

mi: X i ÷ IA i

Xi + X

in

(it is clear from the definitions is indeed Pi

functional).

form an effective

the relation

of

T

to

^ ¢i(x',Yi)

Then

T'

is a subobject

represent whose

graph

family

is

T', ensuring,

translates in

T'

of

IA i (IA i

this subobject. [xix:

that the last subobject

(2) clearly

epimorphic

÷ x = x')

of

Xi x X

into saying

(see 2.2.4)

by completeness,

Let ~

¢(x,mi(xi))] that the

(here we use

that sentences

290

provable from

~ 'become true' in

T'). •

Lemma 3.1.6.

Suppose

categories such that

P~oof:

I: T ÷ T' I is°

3.1.5.

is a Coh-morphism between Boolean

is full.

Then

I

is full on subobjects.

This is essentially a direct consequence of Beth's definability

theorem.

Let us state Beth's theorem in a notation convenient for our

purposes. Assume a graph-map and let sort

L

~(a)

I(A).

models

and

L'

(diagram).

are graphs Let

~

be a formula of

L'

> I*(N),

M, N e Mod T,

Mfm)

incl.

Then there is an L-formula

, M(IA)

incl.

(= I*(M)(A))

~(a)

(: I*(N) (A)).

* N(IA)

such that

M(m) = I*(M)(~) for all

T.

Assume the hypotheses of the lemma. L

and

and let

T

L'

To apply Beth's theorem,

be the underlying graphs of

be the theory axiomatizing

rary subobject of ~. of

and any isomorphism

IhA

N(~)

let

is

with a single free variable a of

[f

~ Mod

> L'

A E Ob(L),

we have a commutative diagram

({acM(IA): MI= mEaD} =)

M

I: L

L', let

Assume that "~(a) is preserved by all L-isomorphisms of

T", i.e. that for any

h: I*(M)

(languages), and

be a theory over

I(A), and let

¢(a)

T

Mod T'

and

T',

Let

~

respectively, be an arbit-

be the L'-formula representing

We claim that ~(a) is preserved by all L-isomorphisms of models ~. indeed, given ~i,N and h as above, by I is° being full,

there is where

h: M ÷ N

such that

~ = [a: ~(a)]T, ,

I*h = h.

We can now take

the canonical iterpretation of

f = h~ ~

in

T'

Thus, the claim is shown.

By Beth's theorem, we have I([a: ~(a)]T) = ~

~

as stated;

it is clear that

as required. •

P~oo~ of 3.1.4: 2.4.7 (ii).

3.1.6.

The theorem follows directly from 3.1.5, 3.1.6, and • 3.1.4.

291

With

T

a pretopos,

Mod(T), we call

dense

~

~

a (not necessarily

if the evaluation

T --

full)

subcategory

of

functor

OM, Set)

(defined by

M AI-

÷

M(A) M(A)

M

lhA N (A)

N

f

is conservative. objects and

Let's say that

'ultraclosed'

Theorem

3.1.7.

depend only on

Let

I: T -+ T'

able Boolean pretoposes, subcategory

of

Then

I

in

if the class of its

Mod(T).

Both

'dense'

ObOM). be a pretopos

Assume

(Mod T') is°

and faithful.

Lemma

is ultraclosed

~

is closed under taking ultraproducts

that

~

such that

is a quotient

m o r p h i s m between

is a dense Iis°~:

~

count-

full ultraclosed

÷ (Mod T) is°

is full

morphism.

The proof will obviously be accomplished

by 3.1.4 and the following

3.1.8.

Then

Assume

the hypotheses

of 3.1.7.

I is°

is full and

faithful.

Proof:

Recall

that every

underlying

graph of

T',

underlying

graph of

T.

M ~ Mod T' and

The fact that the ultraclosed implies that for every

M ~ Mod T'

(elementary equivalence). without Eel,

free variables)

is a subobject

implies there

is

M c Mod T'

that if N ~ ~

of

Indeed,

~

N ~ ~

consider

I,

the terminal

object of

= 1(Is)

theorem",

Mod T'

4.1.11

T'.

subobject

i.e.

N ~¢.

the L,

M ~ N

(formula

(cf. 2.5.1(ii)), The density of of i), then Now,

let

and by the "ultraproduct in [CK],

the

is dense

such that

any L'-sentence

(01 = the minimal N([¢~)

of

interpretation

by the last sentence

version of the compactness

L', over

there is

its canonical

[¢~ ¢ 01

over

is a structure

subclass

¢;

such that

be given;

is a structure

I*M e Mod T

since

~

is

292

ultraclosed, i.e.

we have

N • ~

satisfying

all

~

satisfied by

M,

N ~ M. Next,

recall

(small) families for all i • I, such that

the Keisler-Shelah

<Mi>ici, then there

(Mi)U m (Ni)U

with exponent

U,

for all

family.

The isomorphism theorem is a singleton.

by considering disjoint ever,

the composite

of the

<''" Mi M i.

of

M

of the constant from the special case

"''>i•I

over the

Another remark is that the logic;

how-

and it can also be deduced

case.)

to the full subcategory

of

(Mod T') is°

to a member of

is closed under taking isomorphic

copies

in

~,

whose objects

we may assume

(Mod T') is°

OM

that

is

in (Mod T')is°).

As a consequence ever

the ultraproduct

in [CK], only for one-sorted

are all those that are isomorphic 'replete'

Given two

in [CK] stated for the case

case has the same proof,

from the one-sorted By passing

(M U,

case follows

structure

theorem is proved,

the general

theorem.

~<M: j•J>/U

is 6.1.15

The general

sum of the languages

isomorphism

i ~ I

is the ultraproduct

when

I

isomorphism

i• I of structures such that Mi ~ Ni is an ultrafilter U (over some set J)

<Mi>i• I

ultrafilter

of the preceding,

is a small U

we have the following.

family of models

(on some set)

such that

in

MU

Mod T', belongs

there

to

~

When-

is an

for all

1

i • I. that

Namely, MU = NU i

for all

choose

i

Next,

'

consider

such that ~

M i ~ Ni;

is replete

the language

structure

morphism

I*M 1 ÷ I*M 2.

of

L',

A • Ob(L);

here

down a set structure I~N

T" N

consisting

(graph)

L",

In detail,

let

L"

XI, X 2

U

such

MY e 1

is a model of

in the language T"

L'" X e S

i: L" ÷ L'" L'"

in

which

M I ~f I~i*P,

whose models

is an isomorphism for

for each

denote the inclusion.

theory

hiA = P(×A )

L"

h: M 1 + M 2

in

L"

such that an L"-

of

L"

are in

such that

(Of course,

of an

obtained by adding an

= Ob(L') - {IA: A • Oh(L)}. df Furthermore, let T'" be the

are exactly those L'"-structures

M 2 ~f I~i*P

A • Oh(L)

X • Ob(L')

It is easy to write

M 1 = I~N and df are the components hA

edge

to

L"

for each

if and only if

Hod T', and the N(XA)

XX: X 1 + X 2

sum of two

(IA)I ÷ (IA) 2

If, 12: L' ÷ L".

type of a

MI, M2, and a homo-

denote the two copies of

injections

of sentences

are in

the similarity

be the disjoint XA:

isomorphism h: I*M I ÷ I*M 2. Let us consider the extension

Let

then choose

and ultraclosed,

of two L'-structures

together with a new edge

under the canonical

M 2 ~f

since

i • I.

composite copies

Ni c ~

(i e I)-

Mod T',

h X = P(XX) T'"

P

and for which for

includes,

X E S,

for there and

among others,

293

axioms

saying that

×A = XA'

whenever

Now we see that the conclusion following

(*)

IA = IA').

of the lemma is equivalent

For every model P

of

T'"

Consider model of

~"

N

such that

the class

and

of

I~N,

T"

there

is exactZy one model

i*P = N.

~

of L"-structures

I~N

belong to

of the lemma imply the following

~.

N

such that

Notice

For every T"

N

in

such that

that the hypotheses

~,

there

is exactly one model

definability

(*) from

(***), obtained by replacing let

N

theorem

such that

p

'exactly one' by

be a model of

(i*P 1 = i*P 2 = N)

T",

(I~N) U

are both expansions

by the uniqueness

part of

being distinct

Pl' P2

be expansions

to

of

which are models of

NU e ~

such that

L"

~.

Hence

assignment that if in

Suppose

is contained to

N

L".

L"

in

is a model of

Suppose

P (i.e.

~f(x,y) T",

P(f)(a)

of

N)

P(f) ~f

However,

N,

T'"

is automatically

of Beth's

the reduct

a model of

theorem are satisfied.

T", Thus

T'"

is a theory in

In this case,

there is an

f: X ÷ Y

L'"-L"

in

the L"-reduct

for

a ~ NX, L'"-L"

in

(Usually,

contains

only a

too, and can

case.)

N = i*P

(***)

such

of ~f

of P);

b E NY.

case is w e l l - k n o w n

be proved in much the same way as the special Since in our situation,

does not contain

is the interpretation

in

the general

We use the

there is at most one

theorem is stated for the situation when

single new symbol.

Hence,

that

are two languages

L'"

also that

T'".

= b ~-~ N ~ ~ f E a , b l

theorem.

L'"

but

to each

then

the interpretation

and

L'"

that is a model of

of an L"-formula

P

other words, Beth's

of

Then

T'".

But then it follows

and we have that for any L"-structure P

N U c ~.

N U

Pl(XX), P2(×X); N(XI) + N(X2) different, be (if you like because two parallel maps

form.

any new sort with respect expansion

of

is a first order property).

in the following

(graphs)

To show

Find an ultrafilter

It is at this point that we can invoke Beth's theorem

of (*), called

'at most one'

T'".

(**)' PIU = p~.

To this end,

(cf ~CK~).

both belong

P1 = P2: were the two maps so w o u l d their U-ultrapowers Set

and let

which are models of

(I~N) U'

of

(**).

The first remark is that we have the w e a k e n i n g this,

P

i*P = N.

we will make use of Beth's

L'"

is a

statement:

We are left with the task of deducing

in

N

A

(**)

Pl'

to the

statement:

of any model

expresses

for each

X ~ S,

P

of

that the hypotheses we have a

294

~x[xl,x2),

an L"-formula,

interpretation

of

~X

We can now prove Remember

that each

claim that each XX'

XX

N;

the proof.)

P

of

pU

P(×X)

CX

XX

of

T'",

is a model of N[XI)

N U.

T'"

P'

T'",

and

This completes

the proof.

NU

p __ pU = p,,

U

which

P

of course,

such that is a model

is the same as the P'

is nothing but the

is an expansion

is the interpretation

is a model of

also denoted

[This will,

of

CX' P'(Xx) which

We

obtained by letting

This means that P

~".

× N(X2).

there is an ultrafilter

of the

in

the

hence we can consider

of

N

is a model of

of a structure (X e S)

P

is an expansion

By the properties

interpretation

N

is an L"-formula,

As before,

(* *), there

ultrapower

Suppose

it is a subset

we have that

By

T'"

of

is the graph of a map N(X I) ÷ N(X2),

complete Mo

P

P(×X ).

(*) as follows.

in

NU

which

is

and for the L"'-expansion

P[×X ) = XX'

of

P

~x(xl,x2)

its interpretation by

such that for any model

in

of

~X

in

of

N,

N.

Since

is a model of

T"'

and for P'

as well. • 3.1.8. R 3.1.7.

Theorem

3.1.4.

throws some light on a situation

this discussion,

we use the terminology

be a locally finitely presentable a faithful

functor,

filtered

colimits

case

is l.f.p,

A

presentable then

F

A.

Proposition

itself.

objects

induces

in

8

pointed out in [MP] that F

is full.

I

I

To formulate T = F(C)

considerations 1.4.1.

induced by

F

Returning assertion

limits

the category of finitely

8 ~ LEX(C°P,Set)) (so that

is not necessarily

8

and

in [MPJ says that in this

this statement

over

C,

and

~ = Af.p

F m I*).

,

It is

a Lex-quotient

we can still say that

precisely,

With

with

property

even if C

+ F(C)

generates

T

some analogs

a canonical

Lex-morphism,

that for any Boolean pretopos

The existence F: C

we introduce

in Lex, the free Boolean pre-

is faithful,

like those in Section

is a quotient

C

~: C

+ Lex(C,r')

surjective.

F

which creates

Let

F: A ....... + 8

2.3.

situation,

For a Lex-morphism

we may say that

and

(for

on the level of Boolean pretoposes".

is defined by the universal and essentially

of [MP~).

category,

I: C --+~

in Section 1.4.

()o~: Pretop(T,T')

in EMPI

C = 8f.p.,

(hence,

In the general

of definitions topos

If

a Lex-morphism

"generates D via

and notation

(l.f.p.)

full on isomorphisms,

in

considered

of

F(C)

can be proved by

2.5 used to prove ÷ T,

with

T',

full on isomorphisms,

T

the assertions

a Boolean pretopos,

if the P r e t o p - m o r p h i s m

F(C)

+ T

in Pretop.

to our situation

is that the composite

above,

the precise

formulation

of the

in

295

D

generates

F(C)

is easily

verified

provided

out that

definability

section,

is

treatment,

that

a consequence) in fact,

by A.M. P i t t s

is essentially

Craig interpolation

a result

has recently

and [P2].

It

the methods

i n [ J T ] he u s e s ,

ical"

proof

of all

3.2.

The pro o f of the main result.

results

In p r e p a r a t i o n fundamental, first,

S

and

that

will

theorem

played

context

(of which

a key r o l e

in this categorical

of intuitionistic

is possible

eventually

of this

that

lead

Pitts'

logic,

methods,

t o a (more)

and

"categor-

paper.

for the proof of 1.4.4,

'trivial',

This claim

been given a purely

i n t h e more g e n e r a l

in [Pl]

countable.

3.1.4.

the

theorem,

~ V(C)

¢

by a p p l y i n g

L e t me p o i n t Beth's

C

÷

Z-categorical

I remind the reader of a

adjunction.

in fact could be any category.

There

Let

S

= Set;

at

is an adjunction between

the 2-functors

Cat ° p ÷ making

•

C -

a left adjoint of

-

F

+

~ Cat

~=(-,S) g.

In detail:

CI

(C,S)

= Cat(C,S)

C'l

(-,F)

=

C'[

(H 1-

()

t h e effect of

o F:

6

is this:

•, ( c , s)

(c',s)

F C

If

~ foid H

(H ¢ O b ( C ' , S ) ) .

,,)

F'

All

arrows

functor

identical

are meant in Cat,

Cat to

) Cat.

~,

and n o t

we h a v e a c o n t r a v a r i a n t functor

Cat

~ Cat,

is

g.

The adjunction

consists

of an isomorphism between two categories:

ec,v: ( c , ( ~ , s ) ) - , which is natural the following

i n Cat °p-,

as a contravariant

Y:

in both

C

and

+

D.

( v , ( c , s~)

In detail:

OC, D

maps

X

into

296

C

x

+ (D,S)

D

y

, (c,s)

)

(

,÷ X ( C ) ( D )

C

D I "'

YI With

f

C

D

g

a morphism

natural

>

, D ' I .....

C'l

> X'

in

, X(f) D

-> x ( c ) ( g )

> [ C

~: X ..

....

Oc, D

(C,(D,S)),

l

].

associates

the

transformation

D I (of

course,

in

C, N

D

ranges

means

that

F

~ (CI over

., CC )

Ob(O),

C

over

Ob(C),

etc).

Naturality

given

~

F'

X

>

( H , S)

_~

x'

'

(H,,s)

+

D'

Sh

,

D

H'

under

the

isomorphism

corresponding H

D'

composites >

$h H' Having

between

the a p p r o p r i a t e

@C',O'

composites

>

the above

to the

in

(F, ~ . . . . .

aCiD (X)

P

go o v e r

(C,S)

~eC,D(~)

,.

l(f,S)

(c',S).

(F' ,S)

OC,D(X')

adjunction,

we can c o n s t r u c t

another,

this time

2-functors G P r e t o p °p

Now,

G

(as a c o n t r a v a r i a n t

Cat

+

functor

Pretop

.

....... > Cat)

is g i v e n by:

297 G(T) = Mod(T) and otherwise

similar

(a full subcategory

formulas

as before.

m(c)

It is essential, topos,

of course,

for any category

=

of

(T,S))

Also,

(c,s).

that the functor-category

C,

with operations

(C,S)

'inherited'

is a special case of the fact that limits and colimits egories

are computed pointwise,

We have, the

provided

for the rest of the effect of

'pointwise'

character

will ensure that

F(F),

the codomain ~,

of the pretopos for arbitrary

similar

F: C

S.

This

in functor catcategory has them.

formulas

operations

is a pre-

from

as before;

in (C,S),

~ C',

(C',S)

as defined above,

will indeed be a Pretop-morphism. The adjunction-isomorphisms Below,

referring

certain restrictions appropriate (0 = @C,N

are defined as before.

to the last adjunction,

of

2-category,

~(F),

or

G(F),

and LH~, ~H l

for some appropriate

for

we will write F* for

for a morphism

F

@(H),

respectively

e-I(H),

in the

C,D).

We turn to the proof of 1.4.4. Let 0

0

be an operation

be of type

(G,G'),

denote the full subcategory similarly With

for

of

K',

and domain

(G',S) is°

K.

Below,

with the objects

in

let

K'

will

K';

K c (G,S) is°.

i: G -

topos completion, F

in Set commuting with ultraproducts;

with graph

* G'

the inclusion,

~ = Bpt,

F(G)

and similarly

denoting

the Boolean pre-

for the primed items, we have

making G

~

, F (G)

(1) G' commute up to an isomorphism.

~'

,F(G')

We can now construct

the following

diagram:

298

K

incl

* (G,S) is°

p

÷-

~

(Mod F(G)) is°

i*

K'

incl

, (G',S) is° +

F*

~

(23

(Mod F(G')) is°

Here, all the starred functors are defined by composition with ~; F* is in fact F(F), properly restricted. The right-hand-side square commutes up to an isomorphism inherited from (i). Moreover, by the universal properties of F(G), F(G'), ~* and ~'* are equivalences of categories. The left-hand-side square commutes when p is defined as the restriction of i* to K'; in fact, Oh(K) is the image of Ob(K') under i*. By the definition of "operation in a category", p is full and faithful. Taking quasi-inverses of ~*, ~'*, and composing them with the inclusion, we obtain the upper square in the left-hand-side one of the following two diagrams: K

H

~ (Mod FG) is°

Pl K'

Note that

-=IF* H'

~ (Mod FG') is°

H, H'

(K,S)

÷

LHJ

p * l _~ (K',S) ÷

F(G)

F LH'.J

F(G')

are full and faithful.

The construction of the rest of the diagrams proceeds as follows. First, we pass to the square on the right-hand-side, by the help of our adjunction. Second, Q and r form the quotient-conservative factorization of LH'J. Finally, again by the adjunction, we return to the triangle on the left (and take appropriate restrictions.)

299

Claim

F(G)

3.2.1. To deduce K'

rr~

QF

this from 3.1.8,

is full and faithful

faithful,

and

is a quotient. let's

~ (Mod T) is°

rr~

quotient),

~ T

since

Q*rr~ m H'

consider

(QF)*

~ (Mod F(G)) is°

H', Q*

are

since it is isomorphic

to

Hp,

(Q*

is because

(QF)*rr 7

The composite

and both

Q

is a

is full and H

and

p

are full

and faithful. Let whose

be the image of rr~;

~

objects

morphisms

are of the form

rr~(6),

with

image is well-defined,

the subcategory

rr~(D'),

6: D'

~ D"

with in

K'.

(Mod T) is°

Since rr~

and we have an equivalence K'

of

D' ~ Ob(K'), K'

and if full, the

~~

such that

+M

(Mod T) iso commutes.

Under the adjunction,

ing to the inclusion definition

of

conservative, that

rr~

'dense'

orization,

above.

we conclude

and

that

is full and faithful. and quotients, G

and

G'

is dense in

~

is full in T(G)

and

F,

and

F

is

By the facts

and by the last fact-

(Mod T) is°

By the constructions

correspond-

in the

to

Mod T.

are full and faithful,

to show that

KI, K 2

or Mod T above) X: K 1

X: K 1

.....

~

and that (QF)*~

of the free Boolean pretopos T

are both countable,

is ultraclosed

~ K2

Thus, ÷ K2,

ing of an isomorphism

X

in

since

Mod T.

('pre-ultrafunctor'

in which ultraproducts

is a functor

fied isomorphisms.

as follows:

~

it is clear that

A u-functor

itself,

~ OM, ~ appearing

are finite.

It remains

categories

T

functor

Since it is isomorphic

that

(QF)*rF ~

we have

the functor

is the evaluation

that preserves consists

in [MI])

have been defined ultraproducts

of a functor,

with

(e.g.,

K'

up to speci-

denoted by

X

together with a transition-structure consist-

[X,U]

for every ultrafilter

U

on any set

I

300

(KI)I

[U]

~ K1

(K2)I

[U]

~ K2

IX,U]: X o [U]KI

,~ [U]K2 o X I

(here the [U] are the ultraproduct functors on KI, K2). u-functor is one in which the [X,U]'s are all identities. The u-functors K1 ÷ K 2 form a category, u-transformations as morphisms. A u-transformation natural additional

transformation condition

between of

the

compatibility

functor-parts with

of

IX,U],

(Kl)I

[U]

÷ K1

(K2)I

[U]

÷ K2

A strict

u(KI,K2),

with the

o:

X

÷ Y

X

and

Y

[Y,U]:

the

is

with

a the

diagram

gives a commutative diagram of 1-cells and 2-cells: X[U]

+ Y[U]

IX,U] 1

flY,U]

[U]X I

+ [U]Y I .

If, in particular, K 1 = K, K 2 = S, and in fact, the forgetful functor u ( K , S)

qK

then

u(K,S)

is a pretopos,

+ (K,S)

(forgetting the [X,U]) is a conservative Pretop-morphism; easy to see (using Los's theorem).

this is

301

If I: T 1 ~ T 2 is a Pretop-morphism, then I*: Mod T 2 --~ Mod T 1 is a strict u-functor. Similarly, for a diagram DO: G , T, (Do)*: Mod T

, (G, ~

is a strict u-functor.

Now, let us turn to the diagram under (2) and (3). (~')*

is a strict u-functor;

is one.

also, the inclusion of

As we said,

K'

in (G', @ i s °

It is easy to see that the quasi-inverse of the functor-part of

a u-functor whose functor-part

is an equivalence can be made into a u-

functor by endowing it with a transition structure

(such that, in fact

it will be a quasi-inverse with isomorphisms at head and tail that are u-transformations,

but we don't need this additional fact).

in (3) can be made into a u-functor, also denoted by readily seen that

LH'J

F(G')

factors

..........

H'.

Thus,

H'

It is then

(exactly) in the form qK' ....... ~ ( K ' , 9 .

~ u(K',~

Let u(K',S)

~

F(6')

T be the quotient-conservative

factorization of

conservative, and hence

is conservative, by the uniqueness of the

qr

H.

Since

q = qK'

is

q-e factorization, we have (K',S)

÷

q,

u(K',S)

*

H

F(G')

T with rqF~ ^

L an equivalence of categories. Now, we verify directly that is a u-functor. In fact, we may put

=

([q~] (Di)icI)A

:

[~ (A) 'U] (Di)i¢I

Remember that each is defined.

((qr)(A))(HD~/U) --+ H(q~(A))(D~)/U. i i

r(A) c Ob(u(K',S))

is a u-functor, hence [~(A),U]

302

We have

that

rr~ m

L*=qr ~.

Thus,

up to isomorphisms,

the image

^

of

vr ~

is the same as that of

ultraclosed serves

since

them. This

Thus

is closed

~

argument

by replacing c.f.

K'

rq~.

The image

under ultraproducts

rqr~ and

is clearly rq~

pre-

is ultraclosed.

could have been subsumed

Cat

of

by a 2-category

in the basic

of categories

adjunction,

with ultraproducts;

[MI]. We have shown that

3.1.8.

is applicable

to conclude

the Claim. •

3.2.1.

Let us define D~ = Q ~':

G' ~ +

T

D O = D~ o i: = Q o ~' o i: G Since and

Q QF

is a quotient,

D~

is a quotient,

In other words,

T.

is a generating

DO

is a generating

we have the commutative DO

G

diagram.

Since

D O m QF~,

diagram. triangle

~T

(1)

i=inclusion~ G'

in which both D' ~ K',

DO

and

we have

that

MoD~ m D'.

It follows

1.4.4 would be proved (and

(i) gives

repair Lemma

3.2.2.

and

generate

that

G

Also,

K' ~ Iso(MoD~: composite

notice

that for every

and, as is easily

M ~ Mod T}.

DO, D~

Thus,

Theorem

finitely generate

Boolean pretopos

seen,

T

operation).

by proving

Given a commutative DO, D~

T.

= ~r~(D ') ~ Mod T, df

if we had that

an abstract

the situation

graphs,

D~ M

triangle

both generating D1

T,

[I), with

F +T I

G, G'

finite

we have a factorization > T

We

303

in which

each of

Note that, M • Mod T,

DI, Di

we have

K' c Iso{NoDl:

finitely

MoD~ m (MoF)

operation

Proof of 3.2.2:

The

We s t a r t

mimicked T

subobjects inverting

T,

of

and

S,

with

domain

follows

of

of

A

T in

the

has

the

flavor

£ c Morph(T).

T,

then

the

is

for

any

is

r

,

be

if can

R

and

is an arrow

r

and

morphism In

S

be mimicked s F:

fact,

~ in

for

R

is

the

T ~

~ T'

the assertion T,

A

T

Rr~S

in

can

constructing

there

monos Pretop

a theory

E.g.,

invertible.

T ¢ • Sub(A×B)

'abstract somewhat

pullback R

as shown;

to

R ~ S

that

representing Fo

i.e.

"axiom"

The reason that

being but

new axioms

morphisms,

iff

of

the Theorem.

remarks.

adding

from

of the abstract

proving

straightforward,

some

such

FR ~ FS

in

Di,

inverting

arrow.

respectively,

monomorphism

and

and

constructed T,

is a restriction D1

is

(universally)

out

a single

explicitly

T I.

hence

some preliminary

(universally)

~ T[£ -1]

are by

by

that

with

The process

K'

given by

lemma certainly

The proof

tedious.

o D 1';

N • Mod TI} , i.e.

Boolean pretopos

nonsense'.

generates

if the assertion of the lemma is true, we get that for

~ S

is clear.

To give another

then the axiom

"¢ is a functional

(universally) imposed on T by (universally) 91 ~ A ¢, ~ A×B (see before 2.3.2). To start the proof of the lemma, F

example,

D

if

subobject"

is

inverting the composite

let us consider

the triangle

~T

F'

induced by

(I)

(here we use the same notation

as after Definition

1.4.3).

^

We pretend that

i

is an inclusion

(i.e., we also write

simply

R

for

^

iR

with

R

in

F),

and similarly

for the canonical

~: G

, F,

^

~': G' ---+ F'.

We will write

R

for

D(R)

^

D(R)

(R

in

F)

and for

^

(R

in

F').

Remember

that

D

and

D'

are quotients

in Pretop.

304

This means

that they are full on subobjects,

there are

A

2.4.7

in

F

(in

F')

(ii) and also by taking

mentioned

by 2.4.7

To each

an object

in

G',

we assign

~X ~ Sub(Ax×X )

-~ ~

Rx e Sub(AxxAx)

for any

T (by

many objects

an object

AX

of

F

in

F'

in

( i n T).

be such that

F

(in T).

PX

be such t h a t

RX = (the s u b o b j e c t of Ix×A x the k e r n e l p a i r of PX Let,

in X

epi

$X = graph of

Let

X

p: A . ~ +

the sum of the finitely

PX: iX Let

epi

(ii)).

X,

and an effective

and for every

and an effective

g: X ......+ Y

in

G',

induced by) (in T).

S$ ~ Sub(AxxAy)

in

F

be such that

S~ = (PxXPy)-l(graph(g)), i.e.,

there

is a pullback

A:xX?y A

Px×PY

Sg In addition, RX

is the trivial

were

introduced.

Since

G'

^ " XxY

+ graph(g).

let us make sure that

and

+

equivalence

is a finite

graph,

in case

X

relation

is in

G,

in the above

finitely many

We are going to define T 1 as a finite quotient universally impose on F' the following conditions: AI. A2.

"~X

is functional"

(X c Ob(G')).

"R X

is the kernel

pair of the morphism

graph is A3.

~X'"

then

AX = X

(equality).

of

qx: AX

F'

items

We

> X

whose

(X c Ob(G')).

"S~ is the pullback in A2." (~: X --+ Y

(qxxqy)-l(graph(g)) in G').

with

qx' qY

as

305

Some explanations specification in

F'

whose

A2, A3. that ~X

~i

qx

of

by remarks

Let AxA

[aeA,

= : df

a'eA:

= R

is

under

Note

concerning

As promised,

are

R'

<- R ,

R'

of

z3

the subobject

-< R)

F: T -+ T'

FR' = FR.

how to impose

~ = ~X

The set Z2

an inequality

corresponding

in Pretop

q: A --+ X is obtained

of subobjects.

to A3 is left to the reader.

we define

QI: T -+ T 1

the canonical

are all satisfied

the statement

is done as

^ ¢(a',x)]

A2

T 1 = F ' E ( ~ 1 u ~2

A2, A3

Z2

and construct

that for any

if and only if

AI,

~2' note

exist only when

is the kernel pair of the morphism

F~

above

above,

(3xcX)(o(a,x)

(i.e.,

X c Ob(G').

The description

with

(it will

Consider

specified

The axioms

AI, FR

graph

F'

the subobjects

R'

satisfying

in

"axioms"

A2 and

of

F'

one for each

as an arrow

Concerning

The specification

R':

in

to the respective

above.

of arrows

A = A X.

R = RX

AxA

Zl' ~2' ~3

functional).

of

whose

is equivalent

sets

X c G'.

Fix

AxX,

finite

from remarks

does not exist

has become

First of all, we intend here the

definite

inversion is clear

follows.

of

are in order.

of certain

by

u ~3)

-i

quotient.

D',

]

Since

clearly,

of the lemma exists.

F

It remains

the conditions

AI,

as in the diagram

in

to show that the

composite ^

F

is a finite quotient Let F

T2

I

quotient

"R X

A3'.

"The subobject

is an equivalence S~

A X /R X ---+ Ay/Ry"

clear

finite

of

F

obtained by imposing

on

conditions:

A2'.

We intend

T1

as well.

be the finite

the following

Q1

~ F'

sets

(see 2.4.1).

the proof of 2.4.5

~2''

~3'

Concerning

relation"

of

AX×A Y

(X ~ Ob(G')).

gives

(~: X -+ Y

in

of morphisms ~3''

rise to a morphism G').

of

F.

the inequalities

show what we need here.

We let

~2' (2)

is pretty (or (3)) in

306

T 2 = FE(E2, with

Q2:

F --+ T 2

Next,

note

the canonical

u ~3,)

-i ]

quotient.

that we have a triangle ^

F

i

~

F'

T 2

commuting

up

isomorphism,

to

G' given

with

Qi

induced by

~"

~' T 2

as

X

Q2Ax/Q2R X

............. la r°w' duc d Y Then,

there

Q2Ay/Q2Ry •

is an essentially

F

G: T 2 --+ T

unique

~

F'

Q2 -=

in Pretop

such that in

Q1

Q½

(2)

m G

2 the right-hand-side

triangle

commutes

up to an isomorphism.

To obtain

^

G,

note

solution

that is

Q1 i Q2"

satisfies This gives

the conditions an essentially

A2', A3' unique

G

whose universal with an

isomorphism ^

GQ 2 m Q1 i . The concrete

definition

of

Qi

gives

GQ½~'

(3)

an isomorphism

m Q1 ~'

(4)

307

for

9'

the canonical

the isomorphism left-hand-side

triangle

We claim that obtained by imposing AI'

diagram

GQ~ ~ Q1

G' -~ F'

induced

by

G is a finite quotient, on T 2 the conditions

property.

that composition

Let

with

G

T3

(2), Q1 and Q2 are quotients. H: T 2 --+ T 3 satisfies

in Pretop

hence

G

satisfies

G

certainly AI',

there

then

is

IG m H,

the correspond-

pretopos.

a full and faithful

is a consequence

satisfies

AI, A2, A3, hence

IGQ2 m HQ2 '

arrow A X --+ Ax/Rx"

be an arbitrary

induces

Pretop(Ti,T 3) --+ Pretop(T2,T3)

that

(3).

in fact that it is

"~X is the graph of the canonical (X ~ O b ( G ' ) ) .

ing universal

of

in the

in (2) is the same as the isomorphism

To show the claim, we have to verify

Then

such that the pasting

(4) and the isomorphism

functor

of the facts

satisfies

AI'

(as inspection

I: T 1 --+ T 3

as required.

The fact that in

Finally, shows)

such that This

shows

if

HQ½

IQI ~ HQ~. the claim.

Q2 and G being finite quotients, their composite, is easily seen to he a finite quotient as well. But then iQl , being isomorphic to GQ2 , is a finite quotient as well. This

completes

the proof of the lemma. m3.2.2. Ii.4.4.

Let us note that the same proof, establishes Theorem

the following

3.2.3.

ultraproduct operation.

Any strong

functors

"Composite "composite replaced

Let me point

a set.

The versions

with

instead versions,

of the general

with

pretopos

in the same way as

the free Boolean

pretopos

1.4.4 and 3.2.3 express

morphisms

of the theorem

Set I --+ Set,

of 3.1.4,

(over a graph).

In fact, of the form

talking still

a

ultraproduct Set I --+ Set,

I

about aZZ Pretop morphisms

of just ultraproducts,

phenomenon

are 1.4.4 and 3.2.3 requires

2-categories"

of a composite

of Set in Pretop".

than the original

formulation which

operation"

Pretop

instead

in Set commuting

is defined

out that Theorems

property

are particular

of the form weaker

operation"

pretopos

3.1.2

operation

is the restriction

pretopos

Boolean

using

of 1.4.4.

finitary

by the free pretopos

"completeness functors

version

although

seem to be non-trivial.

of completeness

the context

The

two instances

of "essentially

of

algebraic

308

References EMRI Lecture

M. Makkai

and G.E. Reyes,

Notes in Mathematics

EMPI

M. Makkai

First order categorical

no. 611, Springer Verlag,

and A. Pitts,

Some results

logic.

1977.

on finitely present-

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P. Gabriel

topy Theory, ECKI

and M. Zisman,

Springer-Verlag,

Calculus

of Fractions

and Homo-

1967.

C.C. Chang and H.J. Keisler,,

Model Theory,

North-Holland,

1973. ESGA41

Theorie

des Topos et Cohomologie

M. Artin, A. Grothendieck no.'s

269 and 270. ECHL~

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G. Cherlin,

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L. Harrington

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EShl

S. Shelah,

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EMIl

M. Makkai,

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North-Holland,

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1982, pp. 217-

232. EM2~ Advances

M. Makkai,

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in

in Mathematics.

ETTI

P.T. Johnstone,

EMRII

Topos Theory,

Academic

Press,

M. Makkai and G.E. Reyes, Model theoretic

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methods

Polon.

in the

Sci.

24,

pp. 379-392. ECWM~

S. MacLane,

Springer-Verlag, EPI]

A.M.

Pitts, Amalgamation

of Heyting algebras. EP21

A.M.

Categories

of

Pitts, An application

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29(1983),

A. Joyal and M. Tierney,

Grothendieck,

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for the Working Mathematician.

1971. in the category

29(1983),

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of open maps to categorical

logic.

313-326. An extension

of the Galois

theory

Memoirs A.M.S.

M. Makkai,

859, Springer-Verlag,

The topos of types, pp. 157-201.

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309

EM4] A.M.S.

M. Makkai,

269(1982), pp.

[LJ

D. Lascar,

Full continuous

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of toposes,

Trans.

in Math., no.

of a complete theory,

249-266.

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Kategorien.

221, Springer-Verlag.

"Added in proof. A.M. Pitts has recently

proved an important result which is a strong conceptual

completeness theorem for Heyting pretoposes (conceptualizing full first order intuitionistic logic). His proof uses results of [PIJ, [P2J and [JT]. Theorem 3.1.2 is essentially a special case of Pitts' theorem, in fact with the countability condition removed. After the fact, it is not hard to see that one can deduce the stronger version of 3.1.2, i.e. strong conceptual completeness for small Boolean pretoposes, from the countable case, i.e. 3.1.2 itself.

One uses a forcing argument

and an absoluteness argument, after suitable preparation."

PROBLEMS

IN TAXONOMY,

A FLOATING LOG

Jerome I. Malitz University

of Colorado,

Department Boulder,

Boulder

of Mathematics Colorado

80309

U.S.A.

ABSTRACT. Given a region partition of

R R

in an n-dimensional into

k

pieces

Cl,C2,...,c k respectively we have

Ix-cil

normed linear space,

RI,R j .... ,Rk with centers of mass

such that for each

~ Ix-cjl

is there a

for all

j ~k.

i ~k

and each

xc Ri

The problem is open even for

n = 2 and k = 2 but some partial results are known. The problem is highly relevant diagnosis §i.

and classification INTRODUCTION.

to the foundations

The problem of classifying

diagnosis.

medical

in general.

ing to certain characteristics and medical

of taxonomy,

arises in several

individuals

accord-

fields such as taxonomy

This problem has been treated by different

authors (see [4],[5],[6],[7] ). If the characteristics can be measured by some metric then the individuals are represented by points in some product

space.

A classification

a partition

{XI,X 2 ..... X k}

reflect relationships

of

Xi

We call such a partition

We consider two notions by A. Ehrenfeucht

Of course,

X

of these points is

the partition

A point

X E Xi

should

should be as

as it is to the members of

Xj

for

'good'

of closeness.

of the computer

biology department nition measures

X.

between the points.

close to the other members j ~i.

of

of the set

The first arose in a seminar led

science department

at the University

of Colorado

and L. Gold of the

at Boulder.

This defi-

closeness by average distance and is discussed

in

§3.

The second definition

suggested by Matt Foreman and the author, measures

closeness by distance

from the centers of mass of the

paradigm method of classification definitions

an d is discussed

the definition based on average distances

§5

§4.

This is the For both

good partitions

For

need not exist.

from paradigms we have only partial

and the existence problem is open.

The notation and basic definitions In

X i s.

the main problem is the existence of good partitions.

For the definition based on distance results

in

a heuristic

argument

are presented

in

is given which suggests

§2. that good partitions

311

exist. §6 presents §2.

some open problems°

DEFINITIONS

space with set

Y

AND NOTATIONS.

the usual norm.

will be discussed

continuous

version°

D(p,Y)

ii)

= Ip-cl

D#(p,Y)

=

iii)

D#(p,Y)

where

of

P(X) = {XI,X 2 ..... X k}

let

x E Xio

C(x),

in

version

Y

n

to a and a

is given

has uniform density.

is the center of mass of Y. :

= [/ylp-yldy]/fy

Then

is a discrete

p

y cY}

for

Y

finite with n

the

Y - {p}.

Let

The goodness

c

Y

in Euclidian

of a point

case each point

case

~ E{Ip-yl n

cardinality

of distance

and for each there

In the discrete

unit mass and in the continuous i)

We will be working

Two notions

dy for

Y

be a partition

the contentment

of the partition

P,

G(P)

bounded

of

of

X

and measurable.

with each

X i~0

x, is minj~i D(x,Xj)

is min C(x). xEX

and

- D(x,X i).

P is good

if

G(P) e 0. If we use the

D#

distance

instead of the

D

distance,

C # and

we get

G#o §3.

GOODNESS

that for

IN THE SENSE OF

n = I and

k arbitrary

be obtained

in polynomial

n = 2

k = 2

3.1.

and

EXAMPLE.Let

matter

(0,0)},

P

for arbitrary

(0,0),

and show

{(0,I),

We do not have a general points

and raised

a

proved

#-good

the question

(unpublished)

P that can

for

n > i.

For

is no.

X = {(0,-i),

to check each

P = {{(0,-i),

time,

the answer

D #. A. Ehrenfeucht

there is always

(0,i),

G (P) < 0o

(i.i,0)}},

method

(i.I,0)}.

For example,

then

ing theorem and its proof indicate

in

if

C#((I.I,0))<0.

for constructing

4, or for partitions

It is an easy

counterexamples

k > 2 pieces.

the difficulty

with

The follow-

of generalizing

the

above example. 3.2.

THEOP~M.

for any

X

For every

contained

there is a partition

P

PROOF.

Let ~ > ~ > 0o

X

X

into

satisfying

E > 0 and every

k

there is an

in the unit disc and having at least into Let

k f

parts with be a

N

such that N

elements

G(P) > -s.

i-I function

on a subset

A

of

312

and Let

a)

Ix-f(x)[ < ~ for all

b)

ix-yj e s

B = X - A.

be the number Notice

that

of radius contained For

x ~ A,

for all

x,y ~ A u F[A].

B = f[A] u K where

of points

in

A

and

Ix-yj e ~ for

all x , y • K.

Let

k

of elements

in K.

k is bounded, in fact 6 - centered at the members 2

in the disk of radius

the number

n

k(!)2~ < H(I + ~)2 since the disk 2 of x ~ K are pairwise disjoint and

1 + 2

x • B - K, =

C#(x)

1

~ I y~A

i

Jx-yJ

~ n+k-I y~B

_ 1 ~ jx_y I ny•A

>

1 n+k-i

Hence

for K

[Ix-y~

~ y~A

- Ix-f(y) 1]

k n+k--+-k~-f -e - ~ n

or

x ~ K,

> -~ for large enough

Hence

for

N

C#(x)

or

C#(x)

large enough

If the set

K

non-empty.

and easier argument

shows

n. (since

k

is bounded)

we have

good partition

an undesirable

G#(A,B)

-~.

by separating

approach

from the

of taxonomy. X

of points

is not required

can be modified so that for each G(P) < k for all P. In the proof of 3.2 points cells of the partition. diagnosis

x • B - K and

a similar

that are close to each other,

standpoint

Jx-yl

yEK

> -¢ for

x cA

Note that the proof gives an approximately points

~

1___!__ ~ jx_y I n+k-I y~K

k n

large enough,

empty,

ix_f(y) i

1 n+k-I

y•A

-n~ n+k-I

>

For

~

I n+k-i

[x-yJ

this makes

to be bounded,

then example

that are close together sense.

example.

3.1

we get a four point example with

are placed in different

From the point of view of taxonomy

little

focus by the following

k

This shortcoming

or medical

is brought

into

313

3.3

EXAMPLE.

Let

X 1 = {(d,ei)

i~ n} u {(l,l+s i) X ~

: i ~ n} u {(l,ci)

: i ~ n}

: i~ n}

>0

where

and let

X 2 = {(d,l+~i): Ee l• < ~ .

for each i and

For

Xi, C(x)

I-~ + --/l+(d-e) 2

>

nd+e

2 Hence

for any

d

partition of 1
2n-I and large enough

X = X 1 u X 2.

P is

n,

In fact,

P =

for

{XI,X 2}

n = I,

is a

# - good

s I = 0 and

# - good

3 §4.

GOODNESS

measurable

sets

IN THE SENSE OF D.

X.

A cut

C

in

Here w e consider only bounded

n

space is a plane of dimension

n - I, and so for n = 2 a cut is a line. For

k = 2,

a good p a r t i t i o n has a very simple geometric

characteriza-

tion. 4.1 THEOREM° and

c I ~ c 2.

Let Let

P = {XI,X 2} ClC 2

with

ci

the center of mass of

be the line segment joining

C be the cut that is the perpendicular bisector closed half space determined by ing statements i)

P

ii)

C

of

that contains

cI

and

ClC2 . c i.

c 2.

Let

Xi Let

Pi be the

Then the follow-

are equivalent:

is good.

Xi ~ P i

The restriction

for that

i = 1,2. c I ~ c2

is needed.

unit disk centered at the origin and ed at the origin,

then

X1

For example,

the disk of radius

P = {XI,X - X l} is good.

of taxonomy or c l a s s i f i c a t i o n

theory,

if

X

is the 1/2 center-

From the point of view

good partitions with

c I = c2

are of no interest. Let

C

be a cut that partitions

of mass of I II So

X i.

Say that

if C bisects

Property

I

into

{XI,X 2} with

has property:

ClC 2

if C perpendicular

{XI,X 2} is good if

L

X

C

to

ClC 2.

has properties

is easily satisfied.

I and

II.

ci

the center

314

4.2.

THEOREM.

that satisfies PROOF. If

For every line

L

there is a cut

C

normal

to

L

I.

Consider

L

to be the

C(x) partitions

X

x-axis and let C(x) be the cut at x.

into X I and

X 2 with centers of mass c I and c 2

we let Ul(X) be the distance between c I and C(x) and u2(x) be the distance b e t w e e n

c 2 and

as x increases some

x,f(x)

In general, property

C(x).

Let

f(x) = Ul(X)/U2(X).

over the domain of f,

is continuous

I.

We conjectured

one cut perpendicular

that if

X

ly proved by M. Elgueta

(unpublished). L

and a line

that satisfies

to a given line w h i c h have

is convex,

then there is only

to a given line that satisfies

bounded

I.

This was recent-

I do not have an example of a

with infinitely many cuts perpendicular

tained by Matt Foreman.

His beautiful

only in odd dimensional

L

Euclidean

space since it depends on the fact that

function

fixed point

x = f(x) or an antipodal point x

THEOREM

f

II was ob-

proof, which we give below, works

a continuous reflection of

to

I.

The next result giving the existence of some cut satisfying

4.3.

and

Hence for

= I as needed. there may be many cuts normal

X

f

f(x) goes from 0 to ~.

on the surface of the sphere

S 2n-I

-x = f(x)

has a

(here -x is the

through the origin). (Matt Foreman).

in an odd dimensional

Euclidean

Let

X

space.

be a bounded measurable Then

X

region

has a cut with property

II. PROOF.

We may suppose that the center of mass

of a sphere c

normal

S 2n-I = X.

to the vector

w i t h centers of mass

For each c,s.

Cl(S),

So

S 2n-I

the same side of s*

C

C

and

C(s), =

f

as

divide s.

of

X

is the center

C(s) be the cut through X

into regions

XI,X 2

One of the

c's, say

The line from

c

through Cl(S)

f(s).

is continuous.

f cannot map

f(s*).

c

let

respectively.

at the point we call

f : s2n-I ÷ s2n-I

fixed point

Let c2(s)

Cl(S ) is on the same side of intersects

soS 2n-I

s

But this means

Since

onto that

-s.

s

Hence

C(s*)

f(s) are on f

has a

is perpendicular

to the line extending

C,Cl(S)

since c = mlc I + m2c 2

where

mi

argument

for

2-space giving a cut with property II

than

two

We have a different but for

n

even and greater

and so is perpendicular

and

is the mass of

to

Cl(S),C2(S)

Xi.

we do not know how to get such a

cut. Of course,

for a cut to be good. it has to have both properties

! and II.

315

As yet we do not see how to prove the existence of such cuts even for regions

X

§5.

in

2-space.

A FLOATING LOG, HEURISTICS.

strongly suggests X in the plane,

Here is a physical model which

to us that good cuts exist.

We will consider a region

although the argument can be given for 3-space and is a

bit simpler. Cylindrify levels.

X

in

3-space and truncate

This results

the cylinder at two different

in a 3-dimensional

object like a section cut from

a tree limb, a wooden log.

Float the log in liquid with its faces

a copy of

to the level of the liquid.

X) perpendicular

until stable position surface of the water, C with centers of mass

is reached. XI

the region above

c I and

By increasing or decreasing log so as to maintain where

C

At such a position,

c2,

C

we have

and

(each

Rotate the log with

X2

C

the

the region below

c I c 2 perpendicular

to C.

the density of the liquid while rotating

stability

(c-~

satisfies both conditions

the

i C) a position will be attained I and

II.

We have not been able to convert this heuristic

argument into a mathe-

matical proof. §6. PROBLEMS. Of course, the main problem is to show that a bound..... ed measurable region in Euclidean n-space has at least one good cut. In fact, we believe

that there must be several.

At the moment, we do not even see that regions one good cut.

Nor do we see that regions

in

2-space have at least

in even dimensions

2n > 2 have

cuts with property II. For discrete point masses the questions C

are also open even if we allow

to cut through some points partitioning

the mass of these points

as convenient. The proof of 3.2 suggests and of measure > 0.

the following problem.

Is there a partition

{XI,X 2}

X I and X 2 have the same measure and i)

IXllp-qld q ~ Ix21P-rldr

for all

p c X I.

ii)

IXI IP-qldq ~ I X21p-qldq

for all

p c X 2.

Let

X

be bounded

of

X

such that

316

Postcript.

Shortly after the conference we obtained a proof for

k = 2

and all odd n. Fixed point theorems are used in this proof. Recently, A. Ehrenfeucht settled the question positively for all n and all k using methods that are not only simpler but more constructive.

However,

related problems in taxonomy that are settled by the fixed point methods do not yield to his methods.

REFERENCES [i]

Ehrenfeucht, A.

Classification Theory

[2]

Ehrenfeucht, A. & Malitz, J.

Problems in mathematical taxonomy (in preparation).

[3]

Elgueta, M.

(unpublished).

Unicity of cuts equalizing distances to centers of mass in convex regions

[4]

Chen, C.

Statistical pattern recognition. (197~ee

[5]

Murtagh, A.

Mauldin, R.D.

Hayden Books,

Chapter VIII in particularS.

Survey of recent advances in hierarchical clustering algorithms,

[6]

(unpublished).

Comp.J., 26,n ~ 4 (1983) pp.354-359.

ed., The Scottish Book.

Birkhauser (1981)(See

problem 19, p. 90 in particular). [7]

Steinhaus, H.

Mathematical Snapshots. (1969)(See p. 50).

Oxford University Press

COUNTING PROBLEMS J.Paris Department Manchester

IN BOUNDED ARITHMETIC and A.Wilkie of Mathematics University

Manchester MI3 9PL, England

§

INTRODUCTION. In this paper we shall consider

P R O B L E M I.

Let

A

be a

the following problems.

A 0 subset of

~.

Then does

~.

Then for

{ I m = IA n n i} have a

A0

definition?.

P R O B L E M 2.

Let

A

{ i have a

A IAnn i =

Does

IA 0 ~ AoPHP

does

i mod k}

where

AoPHP

(A 0 pigeon hole prin-

~x

~z <x

0(Y'z)AVYI>Y2~xVz < X(0(Y2'Z)

^ 0(Y2'Z)

÷ Yl = Y2 )

0A0?

There are several reasons is possible, ningful,

for studying these problems.

even tempting,

feasible

to view the

collections

induction)

as the self evident

standpoint

it is very natural

the

A0

~

as the mea-

fragment of Peano's

axioms.

IA 0 (A 0From this sets

the size of sets is unique. reasons

for w a n t i n g

to know how

sets are and in how weak a fragment of Peano's Axioms

we can still prove the A0

One is that it

subsets of

to enquire w h e t h e r we can enumerate

And apart from this there are practical extensive

A0

of natural numbers and to view

in increasing order and whether

every

k ~

is the schema

@x [ V y for

i
A 0 subset of

A 0 definition?.

P R O B L E M 3. ciple)

be a

subset of

~

PHP.

For example Ritchie

is in Linear Space.

[9] has shown that

The general

feeling how-

ever is that these classes are not equal and since A E Lin. Space

~

{

i m =

I A n n l } c Lin.Space

a way to confirm this belief w o u l d be to give a negative

answer to

p r o b l e m I. Problems

1,2,3 are all open and our intention in this paper will be

318

to survey

the area

(as we know it) and to prove

some new results

of

our o~v~fl. NOTATION AND DISCUSSION= in [7], the

[8].

A0

Most of our notation

When we employ expressions

definition

of exponentiation

will be to base 2 and in expressions be assumed

Throughout

of

A ~A~

xy = z

like

13].

log(x),

xe

as

we shall be using All logarithms it will always

that we mean the integer part of these quantities. this paper

and we identify subsets

like

given in

is standard,

M

a cM

M

with

defined by

will

stand for a countable A M0

{xIM ~ x < a } . A0

formulae,

model

of

is the collection

with parameters

I^ 0

of

from M.

For

let A (~) = { I m = I A n n l } , A (k) = {n[

]A nn]

It is easy to see that {[i
= 0 mod k}.

A(k)¢A~

A I A n n I = i mod k} ~ A ~

1 (2) is equivalent A e A~

is

CkX < t

(where

CkX
additional

t

0(x) ~

CkA 0

problem

ed by such formulae.

meaning

by in

M

PROBLEM Vx,y[<x,

M.

Let

Then problem A~

However

=

y> e B

Let

is

A~

in problems

i, 2 when we replace

these problems

Is there

counting,

~

defin-

to:

B c A0

IA n n i = m

such that,

in

M

> 0 ^ <x-l,y-l> ~ B)

~' A ^ <x-l, y> c B ) } ] ? or,

the standard may have no

to:

~--> {(x = 0 ^ y = 0) v (x > 0 ^ x - i ~ A A y

closed under

of

this

?

since the expression

A ~A .

but with

be the subsets

2 is equivalent

v(x > 0 ^ x - I i.e.

CkA ~

CkA~

we must generalize I'.

LA) be the quantifier

formed as for A 0 formulae

adjoined.

We shall also be interested ~

2 is as follows.

] { x < t i 0 ( x ) } i = 0 mod k,

Is

model

(A (k) ~A~ ?)

is a term of

be the formulae

quantifier

to :

A ( ~ ) ¢ A~ ?

Another way of expressing Let

just if

(*)

to put it another way,

can

319

we count in

M?

PROBLEM Vx,y[<x,

2' . Let

y> ~ B < - - > { ( x

= 0Ay

Is there

such that,

V(X>0AX-I~AAy

k-l>

M A0

=

0^<X-I,

~' A A < x - l , y >

tive answer for all such can exhibit a replaced by

40

M

cB)}]?

theorem p r o b l e m i' (say) has an affirma-

just if given any

formula

~(x,y)

A0

and prove (*)

formula in

14 0

0(x)

we

(with x-I ~ A

e (x-l) etc) .

We remark here that the general solutions.

feeling is that problems

Indeed we suspect that problems

A ~ h~ , for example

Finally,

~ B)

closed under counting mod k, or, can we count mod k in M?

Notice that by the compactness

negative

in M,

= 0) y-l> cB)

is

simple

B •h

V(X>0AX-I~AA0
V(X>0AX-I ioe.

A• 40 .

i, 2 fail for quite

for the set of primes.

in connection w i t h p r o b l e m 3, notice 1h 0 + V x, 2 x exist

so the p r o b l e m amounts

1,2,3 all have

that by the standard proof

j- AoPHP,

to w h e t h e r we can remove e x p o n e n t i a t i o n

in this

proof. SECTION

i.

In this section we mention some simple connections

between these problems. mod 4 since given (i = 0 ^ n E A

Clearly

M A~A0,

0<-i-<3,1Ann

(2) n ( A n A ( 2 ) () 2 )-) -

^ -~ @z < n ( x < z A z ^ -~@z
~A))v(i

if we can count mod 2 then we can count I = i mod 4

v (i = i ^ - ] x < n ( x ~ A ^ x c A

= 2^@x
(2)" - n (A hA(2)) ( 2") " " " (2) n ( A n A ( 2 ) ) (2)

+ yAzeA))v

(i = 3 A ~ x < y < w < n ( x , y , w e A ^ x ~ A

(2) a ( A o A ( 2 ) ) (2) A - ~ z < n ( x < z ^ z + y^z

In general we can count mod k Pl .... 'Pj

<~=>

4 WAzcA))

just if we can count mod

are all the prime divisors

of

k.

However

Pl .... 'Pj

where

the following

p r o b l e m is open° PROBLEM.

Let

p, q be distinct

(standard)

primes.

mod p in M can we count mod q? In this connection we do have the following result.

If we can count

820

THEOREM p

then in

PROOF.

I. If M

in

M

we can uniformly

By the hypothesis we mean that given

cB

<~

n,m,p c M, In = 0 A m

for all primes

p

prime

A ~ A0

there is a

(in M) and

m
v[n>0An-iEAAm>0A
m-l, p> E

v[n>0An-l~A^m=0^
p-l, p> e B]

" ~ B

~

B eA

then

= 0]

v[n>0An-i

Then we can

p

we can count.

such that whenever

i.e.

count mod

~'A^
B]

m, p> e B ]

IA nn I = m mod p".

'define'

IA n n I = j

~=~ j is minimal

such that for all primes

I An~ To show that such a

j

p_
= j mod po

exists we show that by induction on n, for fixed

m,

Vn ~j _
p-<m,

IAnnl

to show that as defined above

= j mod p. IA n n I = j satisfies

the required counting conditions. A close connection b e t w e e n Woods

[II].

Woods'

counting and the

AoPHP

result will be discussed

was discovered by

in section 4 (Theorem 20)

but for the moment we content ourselves w i t h the following special case. T H E O R E M 2. the

Woods

[ii].

If in

M

we can count then

M

satisfies

AoPHP.

Woods'

proof

(of Theorem 20) is quite long.

Perhaps

to prove the theorem is to show by induction on F : a

I~-> a-I

(i.e.

F maps a i-I

into

IF n (yx(a-l))l

= Y

IF n (a×y) I <- Y

y

the simplest way that if

a-l)

then

for

y-
for

y<_ a-I

F c A~

and

w h i c h gives the r e q u i r e d c o n t r a d i c t i o n a = IFn (ax(a-l))l Notice

that this only requires

Of course this result a set has a reasonable

-< a-l.

a single counting,

is not at all surprising,

of

F n (Y0 x yl)-

it simply says that if

definition of its size then there cannot be another

321

such definition

of its size i.e.

Further results

connecting

example,

assuming

i-i onto a-l)

its size is unique.

theseproblems

counting mod 2, if

can be proved similarly.

F : a ~->>(a-l)

(i.e. F maps a

then by induction on y IFn (y×(a-l))l

= y rood 2

IF n (axy) l = Y mod 2 which gives a contradiction T H E O R E M 3. mod k.

Let

Then for

as above.

0
M F~A0,

~x,

for

y~a

for

y ~ a-i

Similarly

and suppose F : X

Returning for a mo~mnt to T h e o r e m 1

that in

l~->>x-p,

M

we can count

for all

0
b o u n d e d l y many primes.

~,

we w o u l d of course have had a much

easier proof of this result if we could have shown that due to Woods

For

In fact by Theorems

M

had un-

i, 2 and the following result

this is the case although to prove it we seem to need

T h e o r e m I. T H E O R E M 4. [Woods,

[ii]0] If

M

has the

AoPHP

then

M

has arbi-

trarily large primes. SECTION

2.

our knowledge)

0

]in this section we shall give the best results

concerning

the problems

(to

in section 0.

Of course the obvious way in arithmetic

of stating that

IA n nl = m

is

simply ~f,

f codes a

Unfortunately f

I-i

map from

m

onto

Ann.

this will give a best upper bound for

cannot in general be b o u n d e d by a term in LA.

suitable basic

conditions

on

A

we can do this.

idea (a trick due to Nepomnyascii,

IA n n I

= m<=~ ~k@g c (k+l)(n+l), increasing,

f

However

see [6]) by disecting

f ~ (k+l)(m+l)

f so that

such that g, f are

f(0) = 0, f(k) = m, g(0) = 0, g(k)=n [g(i), g(i+l)).

this device gives

T H E O R E M 5.

Let

ke~,

0 <e < 1

and

A = {m Im
if we put

We can now refine this

and Vi ~ A n Iterating

of around n m so

A

n

~ 2 l°g(n)~ -

for

all

n ~.

AcA~.

For

e A}, Then

n~

let

322

B k = {lj -
By induction

For

k = i

on

k.

(log(n)T)k+l)}

Clearly we may assume

n

~A~.~

is large.

we have

m = min(IAn c [i, J)l,

log(n)

-2--+1) I-C~

<=> ~f(f:m

IF-->> A n n[i,

v no such

f

exists

Then with the given condition

j) ^m-
and

m =

-f-)

log(n)-2--+l.

on the size of elements

coding

of A n and the usual

I-~ f _< (21og(n) e) log(n)-2---
Now assume

the result

for k.

Then I-~ (k+l) (l°g(n) T ) + I) ~=>

m = min(IAn n [i, j)l,

i-~ <=> {@h

: (log(n)

such that

[

I-~ +i) -~ [i,j] @g

m-< (log(n)

~

:(log(n)

2 +i) + (m+l),

(k+l) , g, h increasing

I-~ h(0) = i, h(log(n) T )

I-~ = j, g(0)

I-~ and Vs < log(n) ~ , g(s+l)
h(s+l),

n, g(s+l)

i-~ - g(s) _< (log(n) T ) - g(s)>

or {no such h, g exist and This immediately THEOREM 6.

= 0, g(log(n)

~ B k}

kcl',1,

0 < ~ < I,

A n = {ml c A ^ m < n } Then there is a function

F ~A~

A E A~

_c 2 IOg(n)e

and

I-~ (k+l) m = (log(n) ~ ) +I}.D

and suppose

for all

such taht for all

F(n) = min(IAnl , log(n)k).

k

or

gives:Let

~ ) = m

n ~

that

ne~.

323

Having got this amount of counting we can drop the condition A n ! 2 l°g(n)

Precisely:

T H E O R E M 6'.

Let

ke~,

AeA~

and suppose

that

A n = {ml e A ^ m < n } . IN Then there is a function F ¢ n 0

such that for all n elg,

F(n) = min(IAnl, This theorem follows

from T h e o r e m

6

log(n)k).

D

and the following t h e o r e m w h i c h

will be proved in a forthcoming paper by the authors: T H E O R E M 6".

Let

there is a function

A

be as in Theorem 6'

n

F ¢ A~

and let

such that for all

E > 0.

Then

n ~

i+~ F(n)

: An

We remark here that Theorems k, ~, ~

still standard)

It---> IAnl

5 and 6 hold w i t h

A~

Furthermore

M

p r o v i d e d we interpret

~F, F : log(n) k suitable

I] in place of

"]Anl

~--> A n

1,1

(but

_> log(n) k'' by

for a

function F etc. combining

of the previous

this limited amount of counting with the methods

section we can show

THEOREM 7 •

For

k~l~

and

M ~ ~ ~x,

F E A~ F : log(x) k

Ib-> log(x) k - I.

As an aside we mention here that this can be combined with Woods'

proof

of Theorem 4 to give T H E O R E M 8.

For

k clg,

It is not clear at present

M ~ Vx -3 prime p, p > log(x) k. that T h e o r e m 6' can be similarly

D generalized.

However we can show the following w e a k theorem: T H E O R E M 9. < B Then

Let

A ¢ AM

= ~(i + log(~) -k) A (~) e A M . U

O

< B

and suppose (kelq).

that for all

a,B eA,

324

PROOF.

Let

N c M.

Since

(I + p-l)p ~ 2

for

(I + log(n)-k) ~ e n Hence we can define

F : A nn

-->

p > 0,

for some

log(n) k+2

by

F(x) = least B, (I + log(n)-k) B ~ and count

Ann

SECTION

by counting

x < (i + log(n)-k) B+I

Ann.

D

3. In this section we give an illuminating

of counting mod 2. machine based relations

F"

~ ~ log(n)k+2o

This characterization

description

corresponding

by Bel'tyukov to describe

in

to the function

[2].

C2A ~

is a natural

of the classes

sf,(the f)

characterization

variation

Grzegorczyk

and the class

We shall show that these machines

and hence highlight

what

it means

for

of a

class of

A~

given

can be used A~

to be

closed under counting mod 2. We first describe Such a machine

Bel'tyukov's

M

consists

Stack Resister

Initially

the input goes The program

where each L (i) (ii)

x 0 .... ,xm and the other registers

M

is a sequence

of instructions

t i := ti+l & Vj
tj

q+l'

zI + z2 = z3

goto

Li, else goto Lj,

(iv)

if

zI . z2 = z3

goto

Li, else goto Lj,

z, Zl, z2, z 3 are chosen i

and

at most one of

M M

from

0, x, t, r.

L!,L2...L p contains

eventually

halts with

tially with Bel'tyukov's ly]. f, g : ~ ÷

if there is a SRM (i) (ii)

M halts @i, j el~

M

t k = 0.

and

M accepts

[Our definition

in order to make

~

In this program

the instruction

halts just if he is told to go to Lp+ I.

For functions

are

LI,L2...Lp

:= 0 & goto Lq+ I,

if

each

r.

has one of the forms

(iii)

where

if

q

t0,t I ..... t k and a work register

into

for

(SRMSs).

of a finite number of input registers

Xl,X 2 ..... Xm, stack registers

all zero.

Machines

A c ~]m

for

ti:=ti+l

an input

disagrees

x

inessen-

the later work run more smooth-

we say that

A c SRM Space

(f,g)

with the following properties:-

on all inputs

~

and accepts

such that for any input

iff

x, during

the computation

325

of

M

on

x r ~ fi(max(~))

and

to,t I ..... t k ~ gJ(max(x)). Let

~(x) = i, etc ., P2(X)

0(x) = O, THEOREM

I0.

= x 2 + 2 for all

(a) follows

Lin. Space

= ~

Then

(a) [Bel'tyukov,

[2].]

SRM Space

(p2,P2)

(b) [Bel'tyukov,

[2] .]

SRM Space

(@, p2 ) = A~,

(c)

SRM Space

(~, p2 ) =

C~

(~

Sm~

Space

(2 , p2 ) =

C6A~ ,

(e)

SRM

Space

(B , p2 ) =

C~

immediately

= Lin.Space,

,

.

[Actually Bel'tyukov shows that for reasonable functions Result

x~.

f,

since by the result

SRM Space(f,O = s of Ritchie

[9],

.] P2*

PROOF.

Result

(a) is proved

in

line a proof of (c) borrowing The proofs We first

of

(d) and

show that

suitable machines it only remains tion,

(e)

[2] and

heavily

are along similar

C2A ~ ~ SRM Space to decide

if

to show that

SRM

Space

quantification

this last property

suppose

that

M

is a suitable machine

We may assume tk = I.

We amend

M

M' looks

and

Xl.X 2 = x 3

p2 ) is closed under nega-

and counting mod 2.

A _c ~m+l,

so

A ~ SRM Space

(~ ,p2 ) and

(with the above notation)

that at the end of any computation

of

To show

to accept

M, t k = 0

Ao

or

Let c B

Suppose

( ~,

[2].

Clearly we can find

x I = 0, x I + x 2 = x 3,

bounded

We outin

lines°

(~ , p2 ) .

conjunction,

that

(b) is stated there.

on ideas of Bel'styukov

~=~ C2x < y, <x, ~> c A.

to form a suitable

like

M

but has further

that on input

r = 0

if

machine

z = y

then apply

If

z ~ y

(so

z
stack registers

at some stage

C 2 x < z,

If

M' to accept

M'

t_l,

B.

Phisically

tk+ I, tk+ 2, tk+ 3.

has tk+ 2 = z,

tk+ 3 = 0

<x,u> c A, else r = i. tk+ 3 := tk+ 3 + 1 (etc)

in fact)

set

if

r # 0 and halt.

tk+ 2 := tk+ 2 + l(etc).

Then if

326

r # 0
set u>

tk+ I := tk+ I + i (etc) ioe.

with

of this computation if

t k = tk+ I

Clearly class

we have

for

(i.eo by setting

M' is

Bo

It remains

to show that

SRM

and program

r = 0

that

If at the end

t k + tk+l,

on all inputs.

Let

M

p~ (max(x))), Ii

and

r = i

r = t_l)-

program and the accepted (~, p2 ) is closed under are similar.

(I, p2 ) ! C 2 A 0" M

Let

if

SRM Space

Let

LI,L 2 ..... Lp.

tk+ 2.

quantification

Space

M on the input

:= t_l + i, (etc)

[2].

to, t I ..... t k (bounded by

and run by

out as a suitable

and bounded

is due to Bel'tyukov

space and halting

everywhere

t_l

The proofs

conjunction

r = 0

t k = 0 put

this can be written

negation,

proof

x 0 replaced

and

The method

be an SRM running

of

in this

have stack registers input registers

x 0 ..... x m

be the instruction

amongst

LI,L 2 .... ,Lp(if any) which starts t i := t i + I & Vj < i, tj For convenience that

L I = I0

(accept) Then for

or

we make

i (reject)

appears

in

we can find

(i.e. having

&0

about

we have just applied x, r) ~ >

r

rather

~iq(~,

of

instruction

than

t k-

x, r) e A 0

and

M Ii

the stacks

A0

0

functions

contain

~, ~, r

and

then

[the next instruction ed is

M, namely

the answer

graph and value bounded by a polynomial)

such that if in any computation

0iq(t,

assumptions

and that at the end of the computation

0 ~ i,q ~ k

Fiq(~ , ~, r)

some unimportant

:= 0.

of the form

Ip

to be execut-

I ] or [q = k+l and we will halt without q another instruction of the form Ip].

applying Furthermore

if

9iq(~,

when we are just about eiq'S exist for more

follows

than,

of the form I i, So

M

say,

~, r) then to execute

x, r) is the value of

or halt if q = k+l. q from the fact that no computation of M p(m + k + 3)

otherwise

is equivalent

Fiq(~, I

moves without

it would

to a machine

contain K

whose

applying

That such can go on

an instruction

a loop. instructions

r

are:-

327

(i)q (q-
In state

:= 0 f o r

In state ~q = V i

L

j
L

if

if

~q(t,x,r)

and return ~k+l(t,x,r)

to

set

r:= Gq(~,x,r),

tq := tq + i,

L.

set r := Gk+l(~, x, r) and halt, where

(0iq(~, ~, r) A t i + 0 A A t. = 0) V (q = 0 A r = 0 ^ A tj = 0) j
and Gq(~,x, r) = z <=~ ~/ (0 iq(t,x, t. = 0 A Fiq(~,~, r) = z) ~ r) ^ t i ~ 0 A A i j
K'

~q, ' G'q c C2A 0.

=

0Am

equivalent

=

to

0).

K

with one less

Repeating this process until we

are left with just the final "halt" instruction gives the answer. We shall remove the loop corresponding and consider a computation of L.

K

It will continue to execute

arrive at a value knew

s O of t o

to

(i)0, producing

r0, rl, r 2 .... until we

such that

Since

rs

0
we reset

can only take values

t = tl,t2,...,t k

0, t, x, r and in state

~0(s0,

s o and rs0 in the remaining instructions for

Let

starting with

So, rs0 beforehand we could bypass

we apply (i)q

(i) 0.

t, ~,r).

(i) 0 by replacing t o and r by of

K. (Notice that whenever

t o to 0).

0, 1 we can define

r s = 0 ~=~ ~b < s [b is maximal such that

Now if we

rs

by

G0(b, t, x, 0) G0(b, t, x, I)

.x

and either or

(G0(b, t, x, 0) = 0 A C 2 x < s(b < X A G0(X , t, X, 0) = i))

(G0(b, t, x, 0) = I A

or [no such

b

-~ C 2 x < s(b < x A G 0 ( x ,

t, x, 0) = I))]

exists and either

(r = 0 A C 2 x < s, G0(x , t, x, 0) = I) or(r = IA--, C 2 x < s, G0(x , ~, x, 0) = I)]. Hence d J s o = least s
to

such that

-~ C0(s , t, x, r s)

K' with instructions :-

328

(i)~ (! ~q ~k)

In state

r := Gq(S0, t , ~x ,

L

if

~q(S0, ~, x, rs0)

set

rs0 ), tq := tq + I , t.J := 0 for 0 <j
and return to L o (ii)'

D

In state

L

if

~k+l(S0, t, ~, rs~ set

r := Gk+l(s0,t,~,rs0 )

and halto The theorem now follows, The fact that SRM Space (2, p2 ) = SRM Space (3, p2 ) is rather surprising and suggests that perhaps the same result holds for I and 2 , i.e. that if you can count mod 2 then you can count mod 3. Of course a natural conjecture at this stage is Conjecture.

SRM Space ~ ,

p2 ) = C(k+l)~A~

For other investigations along these lines see Hicks coming paper by Handley-Paris-Wilkie.

[5] and a forth-

By directly applying the last theorem we can obtain a further characterization of C 2 A ~ due to Gandy (by a different proof). THEOREM II. [Gandy, unpublished.] Let G be the class of functions formed from the characteristic functions of the graphs of +,~ by substitution and (bounded) primitive recursiono characteristic functions of the sets in C ~

Then

G

is precisely the 0

SECTION 4. In situations such as we have with these counting problems a currently popular reaction is to prove some oracle independence results, thus confirming (what we already knew~) that these problems are not entirely trivial. results.

In this section we give some such

The relativized versions of A~ are formed in the obvious way. Namely add to LA a new unary relation symbol X, define bounded formulae, A0(X),

of

LA(X)

there is some

as before and for

A, B ~

#(x, X) c&0(X ) such that

B = {~c~

Notice that by induction on the length of is m~ ~ such that if n > i and A n {~n

nm~

=

]~(~, A)} =

say that

iff

l~(n, A)}.

~(x,X) we can show that there

m D n n ~ then {~n

B eA~

I ~(~, D)}.

329

Our first theorem is very straightforward. THEOREM PROOF.

12.

@A _c IN

Suppose that

ly unspecified.

such that

An p

If

p

A0 A

is closed under Counting.

has so far been defined and

A-p is complete-

has the form

m


~, X) ~ , ~, k>

~_
and

peA

~=> [{x
Otherwise put Whilst

F~(x0,

~(x0, ~, X)

p e' A.

¢ A0(X ) set

The result

= k.

follows by the above remark.

this tells us that any proof that

ing cannot relativize however

n0

is not closed under count-

it really gives no new insights.

Our next result

seems more promising.

THEOREM

13 •(*) @A -c IN

such that

A A0

is not closed under counting

rood 2. PROOF.

The proof is an easy application of a very beautiful

due to Ajtai

[i] characterizing

theorem

sets of the form

{A ! n I ~(n, A)} for

~(x, X)

the special Let

e A0(X) o [Actually Ajtai's result is much more general case we shall need. ]

First we introduce

be the language of arithmetic but with

relations

and for

domain n.

n¢ IN

let

n

It is straightforward

there is a formula

~(x)

e L (X)

be the obvious

+,. treated as

structure

to show that for any such that w h e n e v e r

than

some notation, 3-ary

for L with

~(x,X)¢ A0(X) 'A ~ n, ne IN,

~(n, A)<-----> ~ ~(A). This result,

for

result for

L (X)

For

let

f£ n2

subset of

n

L, is proved in

[7]°

Xf = {m Im < n ^ f ( m )

into

2}

and for

Ajtai's

• -

T h e o r e m w h i c h we need is

(*) of

n

[Ajtai,

eventually

[i].]

c= 0}.

fe -~n2 n

T H E O R E M 14.

A-n = ~ the

is a simple g e n e r a l i z a t i o n

n o t a t i o n will only be used w h e n

for all

Since we assume

-~n 2

Let let

Let

There is a striking resamblance

e >0

maps a

f = {he n2I h = f}.(This

is implicit.)

~S c T c i n 2

= {f If

and

Then the version of

~(X)¢

L (X) .

Then

such that

between

this theorem and results

[12] which were unknown to the authors at the moment of writing this paper.

330

(i)

{f I f E T}

(ii)

f cT ~

form a partition of

I dom(f) l

= n - n

l-e

n2,

, l-e

(iii)

l{h en21 ~ 6(Xh) }& U f fcs

I -< 2n-n

Here A stands for symmetric difference. {h E n21 ~ 8(Xh) } simple set

U f. f ES

both sides of h En2

(iii)

The result says then that

can be approximated very closely by the

To see just how good this approximation

is divided

by 2n.

that

This shows that the probability

chosen at random is in one of {h en2I

~

0(Xh)}

~

but not in the other is at most

, ~ fcS

I/2nl-e

As a simple corollary to this we have. THEOREM 15. ly we can find

Let cn qe - 2

g et2,

0(X) E L(X).

such that

Then for all

n

eventual-

q ~ g, Idom(q) l ~ n -vr-n+2 t+l and

either _~ {h on21 p 0(Xh)} _c {h E n21 p ~ e (Xh)}.

or

PROOF. Let

Let

n

be large and let

H = {f c r I f n g ~ ~}

Ifn{hcn21



T

etc be as in theorem with ~ = !~ .

and suppose that whenever

~ 8(Xh)}l,

If n{h En21
f EH,

Xh> ~ ~ O ( X h ) } l

>2t.

Then 2 n-vr~ >

I{h En2] F0(Xh)}A U fl > 2t. IHl . fcs

But

U f ng feH

= U (fug) fcH

IHl

=

•

which gives a contradiction. find f E H such that

Hence, since n

2

g

so by counting elements, >-Igl

=

2n-t

So, without loss of generality we can

If n{h En21 ~ e(Xh)}l~ 2 t c n and f ug E-- 2, Idom(f u g) I -< n - v~ + t. cn is large we can find q ~ - 2 such that q ~_ f u g,

Idom(q) l -< n - ¢~ + t + 2 t<_ n - v~ + 2 t+l

and

331

n {h ~n21

~ 0(X h)} =

as required. PROOF OF THEOREM found

A nt

13.

and

We construct

~(x,X) c A0(X)

{nI~(n,A)} Fix

n

large.

~ {n I I A n n l

We arrange

enough

A n [t,n)



Without

Then

so that by earlier remarks

~ -~(Ann)

it will be

n ~ A (2)

~=> cn

q~-

2

such that

q F t ~ t2,1dom(q) I
and q ! {h~ n2 I ~ ~(X h)}

or

~ ~ {h ~ n21 <~, Xh> = ~ ~(Xh)}.

loss of generality h c n2

assume

such that

the former.

h ~ q

and

Then since

IXhl

is odd.

Idom(q) I
A n n = X h.

n c' A (2) whilst

In view of our present elements

~ ~(Ann).

inability

0

to count even sets with fairly

it would be nice if we could arrange

small.

However

similar

open problem

is closed under Unlike

n ~ A (2)

either

we can find

that

to satisfy

By Theorem 15 we can find Xq ~t = A n t

Suppose we have

to arrange

= 0 mod 2} = A (2)

<=~

A n In, n m~) = ~

to find

in stages.

that

-~ @(n,A) We shall set

A

and we wish

Ajtai's

For example

is whether

A~

results

IAI

here to be very

does not seem to give this. or not we can find

counting mod 2 but not,

some oracle

any proof that

Theorem

say,

few

A

Another

such that

closed under

A

counting.

this one gives rather more than just that

is closed under counting mod 2 will not relativize.

an obvious

way to attempt

to prove

that whenever

0(x) e A 0

then

{x I e(x) }(2)cA 0 would be by induction

on the complexity

of

block on doing this in most natural ways. the theorem if

e.

But the Theorem puts a

For example with

A

as in

332

A I = {2n I nElq}, then we do have

A (2)

A 2 = {2hi n c A} u {2n + II n c' A} A~2) ~ A

'

but we do not have

(AInA2)(2)eAA

0"

In [I] Ajtai also uses his Theorem 14 to show that distinguishing between sets of odd or even size is difficult. T H E O R E M 18.[Ajtai, of Peano Arithmetic,

coded in

K



.

[i].]

a e K,

such that in

T H E O R E M 19.[Ajtai, of Peano Arithmetic,

2

Then

~P ! a ,

K

K

be a countable n o n - s t a n d a r d model

a non-standard. K,

[I]

IPI

.]

is even,

Let

K

Then IQI

@P, Q c a, P, Q is odd and

be a countable n o n - s t a n d a r d model k and R c a coded in K.

a ~ K, a n o n - s t a n d a r d

P coded in K

is even and for all in

Let

Precisely:-

such that for all

AI, .. ., A m _c a coded in

i
l{jl ~ P}J

K @R' , Q, AI, ' "''' A'm

coded

such that


i < a,

,Am> ~
I{jJ c Q}I

Let

F

+ ~x, F : x

lit does not matter if we allow quantifiers above

in

of the PHP.

be a new unary function I&0(F)

140 (F)

F

..,A~>

is odd.

We now turn to look at oracle versions PROBLEM.

Q, A'

symbol.

Then is

i~-->x-I consistent?

to appear in the bounds on the

since we can always replace

F

by the identity

"x".]

At present we know of no full solution to this p r o b l e m although there are several partial results. T H E O R E M 20.

[Woods,

Firstly Woods has shown:-

[Ii].]

Let

Def(G,

F) be the a x i o m

G(x,y) + I Yx,y [ G ( 0

,

y) = 0 ^G(x+I,

Then IA0(G , F) + Def(G,

y)

=

G(x, y)

F) ~ ~ ~x, F : x

if

F(x) ~ y otherwise

].

~e-->x-l.

D

Notice that Theorem 2 is a special case of this. Our next result shows that we can give a positive for the fragment T H E O R E M 21. I ~I(F)

answer to our p r o b l e m

I ~l(f). I @I(F) + Ix, F : x

le--> x-i

is the induction schema for formulae of

is consistent, LA(F)

where

of the form

333

~x I ~x 2 ..... @Xn0, 6 quantifier

free.

PROOF.

K

Let

cK

be a countable

and ~ non-standard.

argument.

Forcing conditions F(Xl)

with

~-<~,

formula

non-standard

We produce

~

<e

from

is defined

F I-I. and

= Y2 ..... F(Xn)

For

~ a forcing

~]I- % ~=>

dependent

If- ~ k(b)

n

k(x) = ~ye(x,y)~ ~ ~

or

and

~T Z o,

e I .... ,ek

for

for % not involving

@I(F)

TI~ %(b)

and

a and

IT - o[ _
appearing

oil- -~ l(b)

and is compatibile

is non-standard decides

0(b,a),

and a is finite. T II-O(b,a)

We can now pick a complete that

~

I~I(F)

that every non-empty have a forcing

F, etc.

Then there is a fixed

Fi(ek ) for

are the constants Then either

i-< j

or

~I(F)

with ~.

Then

n clq,

b c M either To see this notice

i-< j,

in 0.

some fixed J,

Suppose

@v ~_ ~,

This

of

there are

911-6(b,a~ ~(~

is possible

r u ~II- 9(b,2)

for ~(el)

since

so since T

and hence T I~ k(b). sequence

+ F : e+l

condition

a

(f(a) = b) e O,

a. Hence we can pick T =_ o such that T specifies ~(b), ~(al)

Fi(ek )

~(x)

6(x, Yl ..... ym ) we only need to know the values

such values.

some

~=~

M ~ k

F i (x) . Fi(yl . . .) . . .Fi(ym . . ) . Fi(el . . ) where

condition

only on 0 such that for any condition

that to decide

= Yn'

e.g.

~[~ F(a) = b

that

forcing

~ ~ M,

in the obvious way,

Now suppose

axioms,

by a simple

are finite sets of the form

= YI' F(x2)

and

LA(F)

model of Peano's

F : ~+I i~-->~

of forcing

conditions

to ensure

i~-->~. For it is enough

set has a least element.

o and a If- ~(b)

to force

So suppose we

for k ~ ~I(F).

Then for

n

as above {a_
]~ forcing

is definable T 20.

in

K

condition

T ~ o,IT - o I _
and hence has a least element,

e say.

Let

T I~ %(e).

Then by the above remarks

o I~ -, k(a) for all

a <e

so • forces

that e

334

is the least element satisfying

X(x)

in

.

D

Along similar lines we have T H E O R E M 22. (successor)

[Goad,

and let

[4].]

TO

T O + full induction

Let

be the for

L0

be the language with just < and

L 0 theory of ~.

L0(F) + ~x, F : x'

Then l~->x

is consistent.

D

Goad proves

this by a quantifier

be somewhat

simplified by w o r k i n g with models

elimination

argument.

a countable n o n - s t a n d a r d model of Peano Arithmetic,

let

"integers"

L0

J +

~j,

of

J

and

in that order.

standard.

Define

so

F : d + n since

K

be the structure

Then

F : K ÷ K

F(x)=

not,

let

{

K ~ TO .

dcZj

x+~

if

xcJ,

x + G - i

if

J<x
x - d

if

d~x
x

if

d + n ~ x,

is definable

from parameters.

for

Let ~j

can

be

be the

w i t h domain

and

n c J, n non-

+ D,

Also satisfies in

J,

full induction for,if

J must be definable

in

But we can now construct an isomorphism a of

fixes these parameters

J

by

~--> d + ~ - i.



Let

His argument

as follows.

but does not fix

J,

whic~

hence giving the required

contradiction. SECTION

5.

In this short final section we indicate how our previous

results might be improved by making various IA 0.

It must be admitted however

course is

A~

if

we obtain,

for

f : ~ ÷ ~,

T H E O R E M 23.

about

A~

and

that in all cases except for Theorem 26

the general opinion is that the assumption Recall that we know that Lin.Space

assumptions

is false.

is closed under counting and so of

A~ = Lin.Space.

By directly amending this proof

Assume that A 0 Z Spaee(f(n))

and let

A e A 0 ' A -c ~ 2

and such that for all n c~, A n = {m I cA} ! 2f(l°g(n)) Then { I k =

1Anl}~A~

D

3~

The assumption here concerns the amount of computational in

A~.

space available Assumptions about available time also yield more counting.

Precisely let

f :~ + ~

have

A~

graph,

f strictly increasing and

f(n) e n l°g(n) for all n. Define A~(f) as for &~ but with the new function f added to +, (allowing f to appear in the bounding terms in the quantifier). A~(f) =

A0

In [8] an equivalence between the assumption

and the computational

THEOREM 24. that for all

Assume

A~ = A~(f)

and let

A ~ is proved.

A~A~,

A !

~2

and such

n ~,

IAnl = l{m I m < n Then

time available in

& ~A} I ~log(fJ(n)).

{Ik = IAnl}4 A~.

0

This theorem follows easily from Theorem Turning now to the status of the THEOREM 25.

we mention the following result.

Assume that

IA 0 + Vx, x l°g(x) exists Then

AoPHP

6.

~

Matijasevic-Rovinson-Davis-Putnam

IA 0 + Vx, x l°g(x) exists

+ Con(IA 0) ~ AoPHP.

Thm. 0

A proof of a rather more general result will appear in [i0]. The assumption Con(IA 0) can be weakened but our proof still needs some consistency. It would indeed be surprising however if there really was a formal connection between the consistency of

IA 0 and the

AoPHP,

For our next result we show that the problem of section 4 on the relativized AoPHP viz.

has an affirmative answer assuming the Cook-Reckhov conjecture

Vk @n

such that every proof in the propositional A

V

i~n

j
uses more than

nk

THEOREM

26.

~

V

i<e~n

V

j
(Pij ^ P

ej

calculus of

)

symbols. Assuming the Cook-Reckhov Conjecture, IA0(F) + @x,

F : x

le-->x-I

is consistent. PROOF.

Let

K

be a countable proper elementary extension of

~.

Pick

336

non-standard

c, b e K A i~c

uses at least For

8(x)

symbol, calculus

V j
cb

+

in

V i<e~c

K, every proof of

V j
(Pij ^ Pej )

symbols.

a formula

and

such that,

~ ~c

from

L(R),

where

R

we define a formula

is a new binary relation

8(a)* of the propositional

as follows. e

8-

R(a I, a 2)

Pal,a 2

aI + a2 = a3

s

aI

t

new propositional

al,a2,a 3

1

variables

a2 = a3

al,a2,a 3

aI = a2

e

el A e 2

8*

al,a 2 ^ e*

1

me I

1

~e* 1

~x8 (x)

V

@( a ) *

asc

Vxe(x)

A

e(a)* etc.

asc

Not let

TO

consist of

A i-
V ^ A j
A ~ (PijA Pej)... j
t

together w i t h {e*

le is an atomic or n e g a t i o n of atomic sentence of L and e+l p e}.

Then there is still no proof of an inconsistency from T O using less b than c symbols. Now we can form increasing sets of sentences TO ! T I ! T2 ~ (i) less than (ii) n ¢I~

"--

such that each

T n is coded in

There is no proof of an inconsistency

K

and

(in K) from T n using

cb/2n symbols. For every formula

such that

@(a)* ¢ T n

@(x) or

of

L(R)

-~ 0( a )* c T n.

and

a-< c

there is an

337

(iii) e(d)*^

Whenever

V e(a)* ~ T n a~c

then

@n < m ~

and

d~ c

such that

A ~ 0(a)*~ T . a
Define

R c (c+l) x (c+l) by R(al,a 2)

~=>

Pal,a 2 e U

Tn .

nc~

Then it is easy to see that
for

a ~ c,

the formula

where

c ~

some

i ~}.

TO,

+ {6 I e * e

F : c+l

, F> > IA0(F) denotes

d

prove a rather THEOREM

such that

of

a rather

surprising

special

27. Assume

of the

K

a ~,

a.

that

A0

F c A~,

Then for each

{b < a ~ M ~ ~xl
polynomials

It is shown in

between

hierarchy

{d I d < c i

the A0-PHP

[see[8].]

~ < a, ~ and that appearing

r c~

Vx~ 2 < aV

over

in this

there is a

restricted

to

and

First we

x

F :a ~ I~-->a. A0

A0

definition

formula

such that

to any set of the form

...

Q ~r < av

=

PROOF.

with universe

result which we later generalise.

with all quantifiers

g, h

Then because

(as in [7]),

connection

{b < a ~ I M ~ e(a v, b, a)} is not equal

with

R(a,d).

+ F : c+l~-->c

also that the only parameters

6(x,y,z)

rn}-

~--+c, and

the substructure

that is the collapse

of F are

U n~

D

We now develop

Assume

Induction

F(a) = the least

~ is in


AoH,

R> >

[g(x~ I . . . . .~r .

b, a) =

h(~l ..... ~r' b, a)]}

~.

[7] that any set of this form can be written

as

S

*... where

{b < a ~IM ~ @~0 < a~ V~I < a~ the

z m 's

are from

The idea now is to replace F-l(z2 ) + F-l(z 3) = F-l(z 4)

"'" Q~r < av

A

m=l

z

ml

• z

m2

+ z

m3

= z

m4

{~0' ~I ,... ,x r ,i , a, b, 0} . a~

by

etc.

a

and

z I - z 2 + z 3 = z4

Precisely we can write

by F-l(zl ).

the formula

338

in * as ~Y0 <

ark

VYl < a

e T,

k

vk

°'" QYr < a

x I. x 2 + x 3 = x 4]

l

for suitable

vk

F

~Y0 < ak,

y~k ! Rg(F)

where

Pi = & q ~r,

0

if

Pi =
0>

I

if

Pi =
I>

a

if

Pi =
2>

b

otherwise

as

YYl < ak

y~k c Rg(F)

~ T E a l ,

al

where

and hence

as

=

QYr < ak 'Yr"k _c Rg(F)

i ~ i ~4

yq(n)

if

Pi = & q ~r,

0

if

Pi =
0>

i

if

Pi =
I>

F(a)

if

Pi =
2>

F(b)

otherwise

F(a v, a, b, )

in F are restricted

depends

r.

It follows

where

to

that

a v.

such that

a2' ~3' a4 < a

for

quantifiers on

n
T c ~4.

[such that for all

F(ai)

I ~i~4,

if

this can be written

" a2 + a3 = ~4 ]

for

yq(n)

and finite

So using

[such that for all

k, T e ~ .

n
r is A 0,

Furthermore

and all

the formula

{b < a ~ I M ~ ~ F(a ~, a, b, b)}

F only

is not

of the form given in the theorem. COROLLARY F:

a2

28.

|~-->a. Then

PROOF.

that

av

M some

v >~,

{b c M IN ~ ~x I
V~2
g, h polynomials

the hypotheses written

of

F c A Mo

and

M ~AoH.

By M ~ A0H we mean that there is no fixed M A 0 set (without parameters) is of the form

every

with

Suppose

D

...

over

Theorem

in the above

Qx r
form then,

r ~

g(~l ..... x r,b)

= h(~ I ..... x r,b)}

To show this it is enough hold since if any with the notation

A~

such that

to show that

set can be

of Theorem

27, the set

339

{a ~

a

b>[b < a ~ & M ~ e(a ~, b

this set could be written but fixed,

value of

be the parameter, c
Now define

~ F : M + M

and define

F

a, c but

in that form and hence

r) in the form disallowed

if any,

may assume

: a

a)}

appearing

in the

(for a possibly

by Theorem 27.

A0

definition

larger,

So let

of

F.

c

We

and ~ = 26 .

28

by

F(~ e i. i

,e--> a by

a2i)

F(z) =

= ~6

E F(ei)a i i

(z).

F

c 6can now be removed by replacing

that 0(z, a 2 , a, d) defines 28 e(z, a , a, c) defines Fo

where

e i < a 2,

has parameters

i~6by the least

a i-i map from

=

into

d

a

where

.

Annals

a

28

,

such

REFERENCES i

[ i]

Ajtai,

M.

E1-for~mlae

on finite

structures

Pure and Applied Logic, [ 2]

Bel'tyukov,A.

A computer

description

Grzegorczyk Mat. [ 33

Gaifman, H. & Dimitracopoulos, C.

classes.

Inst.Steklov

of

24(1),(1983). and

a

hierarchy

of initial

Zap.Navcu.Sem. Leningrad

(LOMI)

88,

Otdel

(1979).

Fragments of Peano's Arithmetic and the MRDP theorem, Logic and Algorithmic Monographie N ~ 30 de L'EnseignementMathematique

[ 4]

Goad,

[ 5]

Hicks,

C.

J.

Duplicated

Notes

on the pigeon hole principle.

A machine

characterization

primitive

recursion with applications

complexity

classes.

(Submitted

1983).

Ph.D.

of quantification Thesis,

and

to low level

Oxford University

[ 6]

Nepo=mjascii, V.

Rudimentary predicates and Turing calculations. Soviet Math.Dokl. ii, N ~ 6 (1970).

[ 7]

Paris, J. & Dimitracopoulos, C.

Truth definitions for A0 formulae. Logic and Algorithmic, Monographie N ~ 30 de L'Enseignement Mathematique.

840

[ 83

Paris, J. & Wilkie, A.

A 0 sets and induction. Open Days in Model Theory and Set Theory, Proceedings of the 1981 Jadwisin (Poland) Logic Conference, Ed. Guzicki, Marek, Pelc, Rauszer.

[ 93

Ritchie, R.

Classes of predictably computable functions. TAMS 106 (1963).

[I0]

[ii]

Wilkie,A. & Paris, J.

On the schema of induction for bounded arithmetic

Woods, A.

Some problems in logic and number theory and their connections. Ph.D. Thesis, Manchester University

formulas.

(To appear).

(1981) [123

Furst, M., Saxe, J.B.& Sipser, M.

Parity, circuits and the polynomial-time hierarchy. IEEE Foundations of Computer Science (22nd Symposium). (1981), pp. 260-270.

DEFINABLE ULTRAFILTERS AND ELEMENTARY END EXTENSIONS Ramon Pino*

and

Jean-Pierre Ressayre

Universit~ de Paris VII U.E.R. de Math~matiques et Informatique Tour 45-44

5~me ~tage- 2 Place Jussieu

75251 Paris Cedex

05,

France

The purpose of this paper is to show that if existence of

An-based

Hn-ultrafilters on

of MI,M2,...,M k such that

M ~ n+le

M ~ Ad + V = L.

The

(onM) k implies the existence

M1 ~ n+le

M2 ~ n+le

"'" ~n+le

At the end of this paper we will assert without proof that if tion schema,

then the existence of

An-based

Mk"

M ~ founda-

~n-ultrafilters on

with a pseudonormality property is equivalent to the fact that (n, K) - extendible.

(onM) k

M

To begin let us recall some definitions and set some notations.

is

Ad(non

standard Admissible Set Theory) is defined by the following axioms and schema of axioms

: pair, union, extensionality,

&o -separati°n'

Ao-C°llecti°n and

The classes of

An,~n,En formula for

n>-0, are defined as usual.

O n M denotes the ordinals of the model M. O n M.

O n k is the cartesian product of

are denoted

~

i.e.

~ cM

i c {I .... ,n}.

Often we write

On

k-times.

means that

The notation

instead of

The elements of O n k

a

a I .... ,aI is

f(A) _ ~ L

we write it

(a I ..... a n ) and every means that

f

where the fi s are functions whose domains contain

A

,

*Current address:

On

~ = (~i,~2 ..... ~k ).

If ~ is a formula with parameters The notation

foundation (axiom),

(ZlUN l)-foundation (see [B]).

~(a). a. c M, for

is

(fl ..... fm ) m ,, and g fi(A)c_L~ . i=l

Universidad Simdn Bolivar Departamento de Mater0Nticas y Ciencia de la Computacidn Apartado 80659 Caracas 1080A, Venezuela.

342

Often we m a n i p u l a t e M c e M i means and M

b cM

M ~

and

M i means

A class

M. is an end extension of l a eM and M ~ a ~b.

then

Zn

X

as if it were a single function.

that

~ n Mi means is

~

that

Mi

~eM

then

that

M i is a proper

is

is a

In-elementary

M ~ ~[~]

A n (Z n or

~n ) in such that

will only write

An-class,

is a

if

e x t e n s i o n of

Mi~ a~b

M,

i.e. if

M i ~ ~[~].

Zn-elementary M

formula # w i t h parameters X

~>

M, i.e.

end extension of M.

if there exists X = {a c M

a

A n (Z n or

: M ~[a]}.

then the set

M

~n )

Often we

in w h i c h

X is A n

will be clear from the context. A

~n-ultrafilter

following (i) (ii) (iii) (iv) (v)

over

On

is a subset U of

P(On) which satisfies

the

conditions: A~U

and

A~U,

U

~ is

and

AnBeU

and

A c B ~ B~U

A is

H

n

=> AeU or

On\A~U

A is u n b o u n d e d Hn-Complete

where M is resolvable

~

B~P(On)

AcP(On) AeU

BoO

i.e. if

X m = {~EOn

X ! M x On is

: (m,~)eX}

if there exists a

and

aEM,

A I function

~n and VmEa XmeU, then

n Xm E U m~a

f:On ÷ M

such that

M ~ Vx ~a xcf(~)). M

has

A n skolem functions

A ! ~ +I such that a

An-function

iff for every

M ~ Val,Va 2 .... ,Va k @a

f : M k * M such that

k ~i

and any An-class

(a I ..... ak,a)~A

M ~ Val,Va 2 ..... Va k (al, ....ak,f(~)~A

Now we state a theorem of w h i c h our result is a partial THEOREM

I.

(Kaufmann,

countable model of conditions (i) (ii)

Kranakis)

Ad + V = L.

are equivalent:

M ~ Nn_l-COllection

generalization:

(see [Ka],[Kr]).

For every integer

There exists M' such that

there exists

M ~ne M'

n e2

Let

M

be a

the following

~3

(iii)

There exists

In fact if

M ~ Ad

(a)

(ii)

(b)

(i)

(d)

Hn-i ultrafilter

over

:

(i)

if

M is countable

~ (ii)

if

M is resolvable

~

if

M has An_ I Skolem functions

(i)

Always

(i) ~ (iii)

From now on we will w o r k in a fixed model The following

On M

then

~

(c) (iii)

a

two definitions

extend

M

of

Ad.

the concepts

of unbounded

and

~n-ultrafilter. DEFINITION

2.

Let

{~; {~: (~,B)~A}

3.

~n-ultrafilter i) ii) iii)

~

AEP(On k) and A is AeU ~

v)

U is

~n-COmplete

A

U is and

~

n

i.e.

in

k e I.

P(onk), conditions

U is a

hold:

~

BcU

=> AcU or

onk\AcU

if

X i M×0n k is

~n and Vm~a

Xm~U,

Then

An-based

4.

Let a

if for every

AEU there exists A'eU such

A'cA.

n theorem extends

If there exists M I , M 2 ..... M k

} is unbounded.

Xm = {(~I ..... ~k ) : (m'~l ..... ~k )~X}"

A ~n-ultrafilter

The following

k >2

XmcU

m~a

A' is

if

iff

A is unbounded

where n

More generaly,

iff

AnBcU

AcU, BeP(On k) and A c B

iv)

THEOREM

BcU

A is unbounded

is unbounded

On k iff the following

AcU and

On 2.

A is unbounded

Let U be contained

on

in

} is unbounded.

inductively

: {(~2 ..... ~k):(~l'e2 ..... ~k )eA}

DEFINITION

that

be contained

is unbounded

and A c On k, we define {~i

A

M

and

An-based

such that

M

the implication n

be such that

Hn-ultrafilter e ~n+l

c) of Theorem

I.

M ~ Ad + V = L and on

On k

n e I.

then there exists

e M2 e e MI ~ n+! ~ n + l ..... ~ n+l Mk

PROOF. and

We use the following notation

i~{l,...,k-l}

A ! On k

: (x,xi+ I ..... Xk) E A}

= {(x I ..... x i)

D i = {f :

x = (x ! ..... x i)

then we put

A~ = {(x, x i + I ..... Xk) ~i(A)

: if

: ( @ ( x i + ! ..... Xk))(x I ..... xi,xi+ I ..... Xk)¢A}

f:On k ÷ M,

f is

for every x ~ i ( A ) where

An

and there exists

f is bounded over

f is bounded over

B

means

AeU such that

A~ }

that there is B such that

f"(B) ! LB We say that for every

A

is a witness

x c~i(A)

We put We define

D k = {f

Mi =

:

Di/u

f:On k ÷ M

for

~-> { ~ O n k

M i ~ f~g

<--> { ~ O n

if

i
k

: M~

for

PROOF.

5.

Let

f

m~M,

{$EOn k

m

be such that

: M >

By

An .

Put

~n-Completeness

is bounded° m'cm

ioe.

is

A

n

and

An }

Recall

is

if

f,gcM i

then:

, and

A n to ensure

f~M i

it is

that will prove the theorem.

U,

where

meM.

M ! M I"

By definition of

Suppose that for every m ' e m i.e.

f-l(m')~ U.

onk\f-l(m')EU

B m, = onk\f-l(m'), n Bm'cU m'em

,

This contradiction

M k ~ f = m'.

function

~~eOn k . Then

for every

M k ~ fcm,

}cU

Then, by definiton of is

A is

f~D i.

~ = m Cm(~)

: M ~ f(~) = m' } ~U

f-l(m')

f

m is identified w i t h the constant

such that

and

Mk,{~On

'

M _ce Mko

If

Cm:On k ÷ M

AcU

~ f(~)~g6)}~U

f:On k ÷ M

Now, we set a series of lemmas LEMMA

and

: M ~ f(~) = g(~)}~U

and

enough to find a witness

f~D. if i

A~.

i~{l ..... k}.

Mi ~ f = g

Remark that,

for the fact

f is bounded over

but

for every

m'~m, because

for every m'em.

n Bm' = m'~m

shows that

and

f-l(m')EU

~U

because

for some

345

LEMMA

6.

PROOF.

M ~ MI

M i ~ Mi+ I

We have seen that

to take

feD i

e~i(A) A

A

for

and therefore

Let us define

Pi

Pi~Mi

i~{l .... ,k-l}

To see that

f~Di+ I. feD i.

Let

So,

f

M i ~ Mi+ I it suffices be an element of

f"(A[)

by

Pi (~) = ~i' ie{l,...,k}

Pi is bounded over

(onk)~

ie{l .... ,k-l} there is

over

A~ .

That is

Thus

{xeon

Pi+l ~Mi

x ~i(A)

x ~i(onk).

A c On k and for

such that

Pi+l

is unbounded

because the elements of U are unbounded.

M i ~ Mi+ I.

We show that k

That

It is clear

for each

Moreover it is easy to see that for every unbounded every

Di, then

is bounded for every

f c Di+ I.

: Onk ÷ M

because

for

f" (A~) is bounded for every y c~i+l(A).

is a witness for

that

M ! MIo

and show that

there is a witness

is

and

Pl ~ Mo

=

: aex I}

clear that

Let e be an ordinal in M.

Note that

has bounded complement, thus {xcOn

pl~onMl.

So,

M1 ~e{Pl

for every

k

: ~eXl}~ 0.

~eon

And it is

and therefore

Pl ~ M. LEMMA PROOF°

7. Let

tion of geM i.

f

Mi+l, Then

(AnB)~

M i ce Mi+ 1 and

g

for

ie{l,...,k-l}

be such that g e M i and Mi+ I ~ feg. By definik ~ : M ~ f(e) eg(~)}eU. Let B be a witness for

A = {~eOn

AnB~U

for every

and

AnB witness

x c~i(AnB)

and

f

f~M i because

g is bounded over

is bounded by

g.

LEMMA 8. ~o~ lemma for (ZnU~ n) formulas holds for each Mi,ie{l ..... k} That is for every E or H formula n n M i ~ ~[~]

<=-> { ~ O n

k

:

~ M ~ ~[f(~)]} ~ U.

(*)

Before the proof of lemma 8 we have to remark that the hypothesis of theorem 4 holds; a U'

An-based on

On.

M ~n-COllection

if

this is so because if there exists

Hn-ultrafilter U on

On k

then there exists a

By induction, since

M ~ V = L,

~n-ultrafilter

M has A -Skolem functions. n

Then (theorem ic) there exists M' such that

M ~ e M' n+l "

M ~ V = L, M is resolvable and therefore (theorem Ib)

But

'

as

M ~ ~ -collection. n

346

PROOF OF LEMMA 8.

By induction

the lemma is true by definition Note that if (*) holds holds

for the boolean

on the complexity

for the formulas combinations

The proof of (*) for Ao-formulas En-formulas

assuming

(*) for

of (*) for

A -formula.

(*) for

Hn-l"

Then there exists

Then

{~cOn k

~ ! En, then

(*)

of F.

is essentially

the same as we give for

So, we will

omit the proof

for

Suppose

that

g~M i such that

Hn-I formulas

: M ~(g(~),

~(~))}cU,

M i ~ 3×~(x,~)

M i >~(g,f).

because

where

~ is

But, by the above

(*) holds

for En_l-formulas.

and therefore

: M ~x~(x,~(~))¢U.

Conversely,

suppose

such that

A

is

A

n g: On k ÷ M by

Define

-~ k B = {~On

: M >~×~(x,f(~))}~U.

and

because

A~U,

g(~) = y

<--> E~EA ^ ~(y,f(a))

g is well

define because

about M, see [P]).

As

A

also

Clearly

tive hypothesis, Therefore is

An .

Let

C

{~EOn k

If

i
separation

[P] for a proof). Vzea ~y~(y,z). Therefore

to prove

and

~(~)}~U

we must find a witness

for

that

~cM i.

Put

So, by

g"(D~x) ! L B

and if

g is

geM i, by induc-

f"(D~x)nL a

D

g

geMi. D is a witness

~"(D~x) e e

.

But

for

Put

a = ~" (l~x) ne

Vzca 3ycL B ~(y,z)

is a witness

for

~eM i.

M ~ En-

(this fact is standard,

is a set .

Hn-Collection that is

i = k, g ~ M k because

D = AnC.

M > Ad + Hn-Collection So

this proof),

is

so M i ~3×~(x,~). If

for

fact

Vz< L y -~ ~(z,~(~))

the remark before

: M >~(g(~),

^ y = 03.

(this is a standard

An

there exists e such that

because

( ))3 v [ ~ A

geM i.

be a witness x c~i(D )

are

(c.f.

M i >~(g,~),

it remains

~

There exists A ! B

A -based. n

A VZ<ey ~ ~(Z,

and

M > Hn-COllection

An .

U is

M >An+l-foundation

An, since

So if

of some class

of formulas

En_l-formulas.

In_l-formulas.

(*) holds

{a~On k

If ~ is atomic

O

Assume

remark,

of ~.

of M i,

see Thus

for some B.

gEM i.

347

LEMMA

9.

M ~n Mi and

Mi ~n Mi+l for every

i {I ..... k-l}.

PROOF.

Let ~ be a In-formula. Then, by Lemma 8 -~ k M >~(~) ~> {a~On : M ~-~(a)} = onkc U ~> M i }=~(a) i.e.

M 4n Mi" _l

Let

~

be such that

fcM i. By Lemma 8 (~ is In). k ~ ~ <=> {~eOn : M ~%(f(e))}(U<--> Mi+ I ~+(~).

M i >~(~) Joe.

Mi ~ n Mi+l"

LEMMA I0. PROOF.

M ~n+l Mk

Let ~ be a

In+l-formula.

M ~n Mk (Lemma 9) and To show that

M ~n+l ~

by

V×~y~(x,y,z)

M ~[~],

then

it suffices to show that M k.

So, let ~ b

where ~ is ~n-l"

f(x)=y <--+~(x,y,~)

a

Hn+l-formula,

Suppose

^ Vz
M ~ ~(~).

Note that

But it is clear that for every M >~(g(~),f(g(~)),a) for any ~~eOn k . Thus {~ : M > P(g(~), fog(~),~)}EU M k ~ ~(g,fog,~) for any

LEMMA PROOF.

, and

g~M k.

as

fog is

So, for any

In+l-formulas.

Hn+l-formulas satisfi-

M ~Nn-Collection°

therefore

M k ~-~[~] because

In-elementary extensions preserve

ed in M are satisfied in form

If

f

~ of the

Define f:M ÷ M is

A n because

geMk,

An, we have, by Lemma 8,

geMk, M k > @y P(g,y,~), and

Mk > V ×qy P(x,y,~). ii.

If

M !+l

Mi

for every

M i > ~×~(x,f)

So, by lemma 9,

ie{l .... ,k-l}.

where ~ is

Mk ~(h,f)

and hence

Nn'

then

M i > ~(h,f) for some h o

M k > @ ~(x,f).

So by lemma I0,

M ~ @ x%(x,f). The converse is a consequence of lemma 9. LEMMA PROOF.

12.

Mi ! + l

Let ~ be a

Mi ~n Mk (Lemma 9) extensions°

Mk

for

every

In+ I formula. and

If

ie{l ..... k} M i >~[~]

then M k ~ [ f ]

because

In+l-formulas are preserved by In-elementary

848

Conversely,

assume

M k P~[~].

~xV yP(x,y,v), where P is Note that

M k ~ "V

M ~ "V~L~exists"

and

Since ~ is En+l, ~ is of the form

In_ I. L ~ exists" because

"V~Leexists"

M k > 9xVy(eek ~(x,y,~) because M k >VycLPk~(h,y,~).

Since

is ~2"

n >- 1 '

M ~n +I M~,

So, M k > LPk exists.

Mk > @xVy~(x,yT).

Take

h<M k such that

Vy~LPk $(h,y,~) is ~n' by the Lemma 8,

{~eOn k : M > Vy~L k P(h(~),y,f(~))}e[~.

It follows that

B = {~¢On k : M ~ ~×VycL Let

A

be such that

~(x,y,~(~))}~U. ~k A ~_ B, A is An and

AcU.

Define

g-On k + M by:

_A

_A

g(~) = x <--> (~¢A ^ VycL k~(x,y,~(~))

_~

A VZ
v (~ dA ^ x = 0) g is well defined because The formulas because

VycL k~(x,y,~ )

M ~ Hn-COllection.

facts imply that

g

By definition of g, Then, by Lemma 8, since

and A

VZ
n-> I)

g ¢ Mi, then for every

M i >~(g,y,~)

An

These three

~ ~ {~¢On k : M > V y c L ~(g(~),y,f(~))}¢U. _~ k M k > VycLPk~(g,y,f). Now , remark that M i £ LPk

M i ~ V = L, M k ~ V = L (V = L is

Assume

are

is An.

M ~n+l Mi' M ~n+l Mk and

so

M > An+l-foundation.

for any

and

~2' M ~ V = L and

onMi --~ Pk

(Lemmas 6 and 7).

ycM i _c LPk, M k >~(g,y,~) but

YeMi, that is M i > Vy~(g,y,~)

M i ~n Mk'

and therefore

M i ~ ~xVy~(g,y,f). It only remains to prove the assumption M k > @xVy~(x,y,~).

Take

g e M i.

We have

h' such that M k ~ Vy ~(h',y,~). Then, by

Lemma 8, E = {~¢On C I _c E,

k

~ ~ : M ~ Vy P(h'(~),y,f(~))}¢U..

C I is

An

and

CICU.

Let

Let

C I be such that

C 2 be a witness for

~¢M i

Put

349

D = AnCInC 2. witness Take

D is

for

As

f"(D~x) ! L6

D

Recall is a set

nition of D,

L

for

that D is a

~ c M i, there exists

M > En-separation

Put

" L B. b = m~ " '~D - -xJn

M ~ VZcb ~xVy ~(x,y,Z).

on

T

We are going to prove

is a witness that

•

M ~ ~TV Zeb @xcLTVY ed by

DeU.

g ~ M i-

xc~i(D).

~'(D~..)nLD~~

A n and

~(x,y,Z)•

(see [P])

B such

so

It is easy to see, by defi-

But now, by

Hn-COllection,

This last fact implies

that

g

is bound-

D- . x

Theorem 4 now follows

from lemmas

We don't know if the converse

5,6,7,10

and 12.

of Theorem 4 is true.

has carried us to the equivalence

of two concepts

But this question

that we give immediate-

ly. DEFINITION ke2.

14.

Let U be a

U is said to be

for every X such that f:X + M such that On k-i x f-l(a)eU

or

{(al ..... ak )~O~k

:

DEFINITION said to be

15.

i) ii) iii)

An, then there exists

~k-i < rkof Let

n

conditions

if for any

X ~ On i and

and

k iff

be such that there exist

ic{l ..... k}

Mi ~n Mi+l

for every

ic{l ..... k-l}

1 M i ~ Dn-foundation

for every

and

ic{l ..... k-l},

On k-i × XcU

and for every

a~M such that

ne 1

and

M I , M 2 .... ,Mk

k e2

~iEonMi

such that

@xeaO onM<

with

The proof of the following

e A -formula n

~l• < onMi-l\On

i~{2 ..... k}. theorem is given in [PJ.

M is

such that

ie{l ..... k}

DI = {4 : ~ has the form n

for every

On k

hold:

for every

There exists

on

(~k-i+l ..... ak)~U"

Md e Mi n+l

where iv)

is

Hn'

(n,k)-extendible

the following

~ -ultrafilter n

(n,k)-pseudonormal X is

f

A -based n

350

THEOREM 16.

If

for every n e I

and

M ~ Ad + v = L

a (n,k)-pseudonormal

k e 2,

M is

and

M ~

foundation schema then,

(n,k)-extendible

iff

there exists

ultrafilter.

We end with a proposition

relating extendibility with collection.

The

proof is in [P]. PROPOSITION

17.

If

is (n,k)-extendible

M ~ Ad+ v = L,

then

M ~ foundation

schema and

M

M > ~n+k-collection.

REFERENCES [B]

Barwise,

Jo

Admissible Berlin

[Ka] Kaufmann,M.

On existence of 1979-1980o Verlag,

[Kr] Kranakis,Eo

sets and structures.

Springer-Verlag,

(1975). En end extensions,

The University

in Logic Year

of Connecticut°

Springer-

LNM, vol. 859

Definable ultrafilters and end extensions of constructible sets. Z.Math.Logik Grundlagen Math. 28(1982) p. 4.

[P]

Pino,

R.

~n-COllection, bles. (1983).

indicatrices

Th~se de 3~me cycle.

et ultrafiltres Universit~

d~finissa-

Paris VII.

ON

THE AXIOMATIZATION

Alexander Fakult~t

I.

Introduction

In

[5]

Prestel

field

introduced

and

K

is c a l l e d

pseudo

irreducible

(affine)

simple

point

point.

In case

notion

coincides

field.

In c a s e

the

only

proved

for

case

in

us p o i n t be

out

called we

characteristic

The

curve

Throughout We

say

of

in

the

of

K

of a P R C - f i e l d . every K

, has

of

K

shown

in

definition.

closely

admits

empty,

this

closed

(PAC)

that

if

suffices

Meanwhile,

aim

of

a

a K-rational

is

[5]

A

absolutely

, which

algebraically

(cf.[2]) . T h e

which

if

over

orderings

it w a s

xK

proof

(K,P)

a pseudo

finite, curves

(PRC)

, defined

of

notion

this

follows

that

of

notation

of

[3],

Ershov

paper the

is

to

finite

[5].

convenience

2.

is

V

XK

the

closed

closure

arbitrary

an a l t e r n a t i v e

should

real

that

plane

give

Let

space

XK

to consider this

real

with

West-Germany

investigated

variety

in e a c h

Konstanz

Konstanz,

we

PRC-FIELDS

fHr Mathematik

Universit~t 7750

OF

following

,

following

regularly

assume

for

closed the

case

the

with

respeot

XK = ~

to

that

XK

.

a PRC-field For

K

should

and

results

have

zero.

condition

this

that

that

note

a field

holds:

we

use

the

basic

K

satisfies

the

notations curve

condition

if

in

of

[5].

K

the

K

352

(cc)

For

every

simple such

Clearly, the

absolutely

zero

that

every

irreducible

in e a c h f(x,y)

real

= O

closure

(K,P)

6 K[X,Y]

of

K

which

there

has

are

a

x,y

6 K

.

PRC-field

satisfies

(CC).

field

satisfying

The

following

theorem

states

converse.

THEOREM

Before

Every

we

can

consequences

(I)

Every

This

prove from

sum

follows

polynomial

of

K

this

theorem

The

next

Thus

let

s~uares

immediately (cf.[5],

called

equals

from

(CC)

recall

some

this

is c a l l e d

Orderings

this

are

defines

an ordering. known

from

with

K

S,

the

a sum

3, w e

have

two

squares

of

using

the

space

XK

facts

to d r a w

some

absolutely

about

of

XK

S,

0 6 S of

K

K2 c

S,

, contrary if,

-I

a NS b

if a n d

if

the work

S =~{P

.

subset

of

K

.

S c K

is

only

ordering

on

denote

the

K

which

orderings

is by

convention S U -S = K

of

b - a 6 S

of A r t i n - S c h r e i e r ,

IS c P } ,

.

in a d d i t i o n ,

preorderings

always

. A

to

the maximal

us

K

irreducible

orderings

¢ S

exactly

a partial

in

if

S-S c

includes

an ordering

Let

closed.

_ a

basic

a p r e o r d e r i n ~ of

that

real

1.5)

deals

S + S c

(Note

is p s e u d o

in S e c t i o n

a

Prop.

consequence us

(CC)

(CC) :

X 2 + y2

S

f(X,Y)

K

. If w e

in

[5].)

. set

,

linear

in c a s e

P

it

. As

S

is w e l l -

is

353

and

K

which the

admits means

a

(pre-)ordering

that

-1

a sum

of

only

if

squares

K

is

in

K

formally

. Now

we

real

consider

set XK =

which is

is n o t

if a n d

is c a l l e d

fomltally

{P I P

the

real.

with

boolean XK

on

XK

space

form

H(a)

general,

the

to t h e the

(2)

For

all

This

follows

polynomial

is

K

K}

. It

a totally

respect

Thus

the

to

is n o n - e m p t y

clopen

with

inclusion

a clopen

set.

Indeed,

H(a)

do

However,

consequence

exactly

if

topology

(=closed

and

H(a)

compact

XK

is a

open)

as p a r t i a l XK ~

and

sets

K

= H(-a).

In

form

a boolean

algebra

with

they

form

a boolean

algebra

if a n d

of

applying

(CC)

respect only

if

holds:

exists

(CC)

of

ordering.

not

6 K x there

easily

disconnected

this

algebra

sets

a,b

of

with

(cf.[4]).

(see

of

of

sets

, i.e.

inclusion.

following

space

a subbase

a boolean

Clearly,

ordering

= {P 6 X K l a £ P}

a 6 K x form

topology

order

The

H(a)

an

c 6 K x such

to t h e

that

absolutely

H(a)

D H(b) = H ( c ) .

irreducible

[5],Prop.1.3)

a b X 2 y 2 + aX 2 + b Y 2 - 1

Therefore, are

in a f i e l d

exactly

that

a field

(3)

K

From

(3)

different Theorem

the K

is d e n s e

it

sets with

in

follows

topologies 4.1)

K

holds

satisfying

H(a).

In

(CC)

also

every

real

easily on for

that

[5],

(CC)

the

Prop.1.4

clopen , we

subsets

used

(2)

of

XK

to show

satisfies

closure

the

(K,P) .

orderings

p 6 XK

K

. Thus

the Approximation

any

finite

number

induce

Theorem

of o r d e r i n g s

(see

P1,...,Pm

[6], of K.

354

But

even

more

can

LEMMA

(Block

Assume

that

of

into

X

x 1,...,x m

be

proved:

Approximation) K

satisfies

cl0pen

6 K,

(CC)

subsets

and

Xi

(11 _< i <_ m ) .

C Kx

~1'''''~m

X = X I 0...0

there

To

exists

Xm

each

is

pair

x £ K

a partition . of

such

sequences that

for

all

I _
The in

last the

expression

i-th

orderings

duced fields

'block',

Ershov which

Proof:

By

ai

by

for

all

. In

[I] turned

(2),

we

ai(1+a~) P

6 Xi

2

case

-1

XK

one

out

may

of

to be

. Now

exactly

the

consider

the

Since

h(X)

a. ~ a. l 3

in

for such

K2

+

for

mH ((X-xi) 2 - ai8 i 4 ) i=I

=

irreducible.

absolutely f

mod

= y2

a real

all

closure

i • j

implies

assume

i % j

polynomial

, the

separable.

Hence

(xj, /~-h(xj)

with

( ( x - x i)

again

P 6 X. 3

we

PRC-fields.

la i p

P 6 X. 3 6 P

f(X,Y)

)

. Indeed,

. Now

by

) = 0

.

(CC)

- a~

i=I Considering

of

Replacing K

1

) .

m

+

intro-

class

that

that

y

all

above

was

a i 6 K x.

- aia

-h(xj)

the

. Hence

2

Clearly, (K,P)

some

H ( ( X - x i) i=I

is

for

just

a certain

6 P

polynomial

m

f(X,Y)

is

'maximal'

for

may

- ( x - x i)

Approximation

for

X i = H ( a i) we

this

Block

axioms

~

simultaneously

finite

The

necessary,

and

xi

is

the

assume

if

6 P

approximates

Theorem.

as

~ ~l

6 i +(x-xi)

x

Approximation

by

Ix-x i I P

means

P 6 Xi

mentioned

~

see

(as a b o v e )

that

is

is

a simple

a. 6 P 3 there

and are

zero

of

-a. 6 P i x,y

6 K

355

aj~j 4

Thus we

obtain

-

(x-xj)2

~P aj

(x-xj) 2

6 P

4

_

and hence

.x-Xj,p

<_

2 . q.e.d.

Let

us m a k e

theorem: which

From

states

certain

(c)

one

observation

more

(CC) w e e a s i l y the existence

For

every

f(a,b) there

Indeed,

let

of i n f i n i t e l y

many

f(X,¥)

there

a,b

real

are

~f an__dd ~ x,y

f(ai,r)

fI(X,YI)

n

such

of t h e

'curve condition'

K-rational

6 K [ X , Y I]

f

and

fl

irreducible

(K,P)

K

of

y = Ylh(X)

it f o l l o w s

3. P r o o f

. By

points

on

+ r

that

of the satisfy

absolutely

h(x)

of

has

h

. Now

6 K[X]

and

h(x)

let

,

satisfyin~

h(X)

. Choose

Y

~

{0}

% 0

.

some

YI =

r

6

(Y-r)h(X)-1

function

a zero, f

in

field.

Thus

fl

z e r o in e a c h

real

say

.

K

x,Y1£

. From

K

is

closure

Now

the choice

of

r

# 0

theorem the curve

(K,P)

of

point.

Equivalently,

embeds

into

Ks

= 0

(K,P)

in

+ r)

a simple

is a z e r o of

irreducible

closure

fl

closure

, monic

that

the same

and has

(CC),

zeros

= f(X'YI"h(X)

have

absolutely

f(x,y)

I < i K n

be such

E K[X,Y]

, and every

that

for all

fl (X'YI)

Clearly,

• 0

be all

E ~

• O

in e a c h

(a,b)

E K

al,...,a

and

real

to the p r o o f

following

irreducible

are

that

K

the

come

absolutely

= 0

such

Let

obtain

we

curves.

such that

x,

before

over

condition

K-variety K

K

which

, we have

we may

show

, where

(CC). A s s u m i n g has

to show

that the

that

V

a simple

point

in e a c h

that

admits

V

function

field

is an

a K-rational L = K(V)

S56

(4)

K ~ is a

From

the assumption

(5)

An

L/K

IKL+-saturated

i__{s f i n i t e l y

extension

extends

also

Thus,

(6)

L/K

to some

first order note

on

we know

generated,

is c a l l e d

ordering

language

that

V

elementary

L

of f i e l d s

K ~ is d e n s e

and

real,

. Since

of

K

(see e . g . [ 5 ] ,

regular,

(CC)

.

Preliminaries)

totally

if e a c h

real.

ordering

of

can be expressed

(see T h e o r e m

(I) is u s e d h e r e ) ,

in p a r t i c u l a r

that

totally

of

extension

4.1

and

in the

its p r o o f

we also know that

K

in

[5];

K ~ satisfies

(CC).

we have

in a l l

its r e a l

closures,

and

satisfies

(C) a n d

the

Block Approximation.

From

(4),

over

K

Since

L/K

Theorem L

over

(i)

(5),

(6) w e

the e m b e d d a b i l i t y

is r e g u l a r

3D in K

and finitely

[7],Ch.V,

such

L

into

K~

that

there

are

over

and

over

is i r r e d u c i b l e

prove

algebraically

~ ( v I ..... Vn)

below,

there

independent

:m

over

(K*,P)

an a b s o l u t e l y

irreducible

point

in e a c h

(K~,P).

x',y'

6 K•

satisfying

Thus

of

Y

and , such

the a l g e b r a i c

K

curve by

in

exist

Bv,w(f(~,v,w)

real closure

independent

K(Xl,...,Xn,X) monic

As we will

we conclude

are g e n e r a t o r s

algebraically

f 6 K ( x I .... , X n ) [ X , Y ] , f

generated,

from

Xl,...,Xn,X,y

of

that

is a l g e b r a i c

in e v e r y

of

.

x1,.°.,Xn,X

(ii) y

now deduce

elements

, satisfying

over

there that

. Then

over

(6) (i.e.

f(x~,x',y')

= 0

by

(C) and

f(x,y)

x~,...,x n the

= 0

of K ( X l , .... Xn).

in

K*

following

¢

clearly

K • which

is

closure

~f (~,v,w) = O ^ ~-~

K

K

has

formula

o)

f(x~,X,Y)

defines

a suitable

for K ~) t h e r e h(x')

,

• 0

simple

are

, where

357

h 6 K[x*,X] K s is

~

{O}

is given.

JKi+-saturated,

f ( x * , x * , y *) = O x~i.., 'x n" ~ x~ x 1,...,xn,x,y into

we even

and

are

h ( x *)

onto

Using

in

K

% O

has c a r d i n a l i t y

x*,y * 6 K * satisfying

for all

h 6 K[x*,X]

independent

xi,...,X~n,=V*,y~ It r e m a i n s

IKt and

over

resp.,

to find

the

K

~

{O}.

and,

field

by s e n d i n g

L gets

the d e s i r e d

Thus

embedded

elements

K*.

Elimination

extending

K[x*,X]'

find

algebraically

K ~, as c o n t e n t e d .

x~ ..... x~

Since

of Q u a n t i f i e r s ,

, the

formula

over

~(v)

every

real

is e q u i v a l e n t

closed

to s o m e

field

R

finite

disjunction

(pii~) = o

^

A

i where

the

qij

have

Let

P

Then,

conjunction

~3

is a l s o

coefficients

in

K

be a n y o r d e r i n g

ordering

of

L

depending

on

~

Xl,...,x n

must

be

Thus

we may

Q

K s, i.e.

~

and

the p o l y n o m i a l s

Pi

A 3

P 6 X ~ = XK.

(such an o r d e r i n g

exists

= O ^

i.e.

about

3v

P n K

A 3

Pi

the for

some

i

such

. Let

Q

be an

exists

by

(5)).

that

independent

coefficients a while.

over

are

all

K

, Pi(X)

equal

= O

to zero.

From

qij (v) > 0

we conclude (KS,P)

Since such

~

K s is d e n s e that

Hv

in

and

qij (~) > O) .

algebraically

true,

forget

finite

.

, there

(Pi(X)

are

trivially

(L,Q)

of

extending

(L,Q)

Since

qij (~) > o~

j

qij (v) A 3 (K~,P)

by

> O

(6), w e

.

find

elements

%

of

Ks

358

(~'P)

As

Pi(ap)

= 0 , this

By c o n t i n u i t y

~

~ 3

qij (ap) > O

implies

(K~,P)

we can find some

(K~,P)

~

~ ~(ap)

~p 6 K *, ~p # O, s a t i s f y i n g 2

Vv(Jv-apJ

.

_< ~p

~

qij(v)

> 0).

] Expressing

s,r

this f o r m u l a

6 K[X],...,Xn,Z] (K*,P)

~

in q u a n t i f i e r - f r e e

form we find polynomials

such t h a t

(S(ap,~p)

= O ^ A

r

(ap,~p)

> O)

v

and, w h e n e v e r

(7)

(K~,P ') a l s o s a t i s f i e s

(K~,P ') ~

T h e set of

number

ap

=~

(finite) Aj

of c l o p e n

Yp

cover

sets

of

Yp

X ~ . Since

already

covers

X ~ such t h a t e a c h

X ~.

X

and some

i

such that

replaced

by

the B l o c k A p p r o x i m a t i o n

la - a

(7) h o l d s P'

for a l l

a

P' £ X

a finite

X ~ = X I 0...U X 1

and

in some ~

with

% 0

Yp in

%,~p,i

a ,~ ,i

to find e l e m e n t s

for all

(7) h o l d s

Let

is c o n t a i n e d

w e thus can f i n d s o m e e l e m e n t s

Since

the c l o p e n s e t

X ~ is c o m p a c t ,

X

order

then

qij (v) > O)

(7) thus c o n t a i n s

For each

N o w we a p p l y

conjunction,

H(rv(ap,~p)).

the sets

be a p a r t i t i o n

~P2 ~

i ~

P' £ X * s a t i s f y i n g

Yp

Clearly,

Vv(Iv-

this

6 X

a

of

Ip, S 2

for a l l . Thus

(K~,P)

~

K

X~

A j

in

(K~,P ') #

(a)

P' 6 X

, we o b t a i n

f i n a l l y we g e t

V i

to this s i t u a t i o n

such that

for all

P' £ X

of

qij ([) > 0

A j

qij

. K~

359

for all

P 6 X ~. Since for every

course of the proof,

(K*,P) b

Moreover,

Pi

i

w h i c h was a c t u a l l y used in the

was the zero polynomial,

vi (Pi (~) = O ^

given a n o n - z e r o p o l y n o m i a l

assume that xl,..°,x n

g2

is

among the

qij

> 0

g 6 K[X I ..... X n] for each

holds for all orderings

assumption, we also

find

d e p e n d s o n the choice of s a t u r a t e d n e s s of

g(a) ~ 0

$ O

K , clearly of

L . By this

g 6 K[X]. U s i n g once m o r e the

V (Pi (x~) = 0 ^ A 3

for all n o n - z e r o

are a l g e b r a i c a l l y i n d e p e n d e n t over

since

. Here the choice of the elements

i g(x~)

Q

we may

i . Indeed,

K ~, we can even find elements

(K~, P) b

for all

qij (~) > O).

are a l g e b r a i c a l l y i n d e p e n d e n t over

(L,Q) ~ g2(~)

and

¢

we even get

q

~Kt +-

x~ in K ~ s a t i s f y i n g

(x~) > O) ij

g 6 K[X]. Thus the elements K

and satisfy

x~

(K~,P) ~ ~(x~)

P 6 X ~. This finishes the proof of the theorem.

References

[i]

ERSHOV,Yu.L.: T o t a l l y real field extensions. 25,No.2, 477-480 (1982)

[2]

ERSHOV,Yu.L.: Two theorems on r e g u l a r l y r - c l o s e d fields. J. reine angew. Math. (to appear)

[3]

HEINEMANN,B., PRESTEL,A.: Fields r e g u l a r l y closed with r e s p e c t to finitely m a n y v a l u a t i o n s and o r d e r i n g s (to appear)

[4]

PRESTEL,A.: Matem~tica,

Soviet Math.Dokl.

L e c t u r e s on f o r m a l l y real fields. M o n o g r a f i a s de Vol.22.IMPA, Rio de J a n e i r o 1975

[5]

PRESTEL,A.:

[6]

PRESTEL,A., ZIEGLER,M.: Model t h e o r e t i c m e t h o d s in the theory of t o p o l o g i c a l fields. J.reine a n g e w . M a t h . 2 9 9 / 3 O O , 3 1 8 - 3 4 1 (1978)

[7]

SCHMIDT,W.M.: Equations over finite fields. An e l e m e n t a r y approach. Lecture Notes in Math., Vol. 536, B e r l i n - H e i d e l b e r g New York: S p r i n g e r 1976

P s e u d o real closed fields. theory. L e c t u r e Notes in M a t h . V o l . 8 7 2 , S p r i n g e r 1981

In: Set theory and m o d e l B e r l i n - H e i d e l b e r g - N e w York:

FORMALIZATIONS OF CERTAIN INTERMEDIATE LOGICS Part I Cecylia Rauszer Uniwersytet Warszawaski - Instytyt ~ t e m a t y k i Palac Kultury i Nauki 00-901 Warszawa, Polska

It was Godel who

first observed the existence of a continuum of

logics between the intuitionistic predicate logic LI and the classical predicate logic LK. logics".

These logics were named by Umezawa "intermediate

There are many interesting results connected with intermediate logics. One of them asserts that only seven propositional intermediate logics have the interpolation property. Maksimova [7].

This result has been proved by

She showed that the interpolation property is equivalent

to the amalgamation property and then she proved the existence of seven classes of Heyting variaties with the amalgamation property. For predicate intermediate logics the problem of the interpolation property was first examined by Gabbay [2].

He used the theorem that Craig's

interpolation lemma is equivalent to a weaker version of Robinson's consistency theorem to prove Craig's interpolation lemma for the intermediate logics: LI, LM, LMH, where (LM)

LI + ( ~ u ~

(LMH)

LI

+

Vx ~

~ a) ~

a(x)

=

~

~

Vxa(x).

Eight years later in [3] Gabbay extended his model theoretic methods to the so-called logic of constant domains logic (LD)

LI + V x(a(x)u~) ~

where

x

LD, i.e. for the intermediate

(Vxa(x)uB),

does not appear as a free object variable in 8 and he also prov-

ed Craig's interpolation lemma for In 1981 L6pez Escobar

[4]

proof-theoretic methods, used by Umezawa

[i03

LD.

proved Craig's interpolation lemma for LDusing The same formalization for

LD

was earlier

to show some syntactical properties of LD.

It turned out that both papers [33

and [43

Kripke model for an LD theory constructed in domains and the Gentzen type formalization of

contain gaps. [33 [4]

A certain

has no constant is not complete.

361

It should be emphasized papers.

that both authors know about the errors in their

L6pez-Escobar wrote

two papers

the Gentzen formalization

of

Let me cite the following

two sentences

by J.Barwise,

[5],

that every reasonable

[6] partially

and the interpolation

connected with

property for

"Alan Anderson often argued

formal system has both a Hilbert-style notion of proof.

it is certainly

notion of

While this may overstate

true that a Gentzen-style

approach,

emphasis

on rules, rather than on axioms,

inherent

in any given logic in a way not done by a Hilbert-style

Gentzen-style

formalization

of an intermediate

definition was given by Ldpez-Escobar sense of his definition formalization for LD.

logic

L

formalization

we should know

of L.

respect

to

BS

i.e.

S

consists of two parts

[8].

speaking,

~LI

consists

: the first part called

logic

LI

of the system for is complete with

iff ~ has a normal derivation

in

BS, i.e.

The second part of the defined system

is complete,

As an application

logics

LM

and

LMH

the appropriate

formal system

and cut-free. of

L, where

The cut-free

for the

logic considered.

For the intermediate

lemma for

is different

of a certain set of logical rules which are specific

intermediate

LS

S

in [5] and in [6].

a modification

Intuitionistic

there is a cut-free proof for ~. S

A possible

there is no cut-free and complete Gentzen-style

basic system (BS) is roughly given by Sch~tte

system".

in [5] and he proved that in the

from the ones that were considered by Ldpez-Escobar

LI

with its

of a cut-free

In the present paper the basic concept of a formal system

Every formal system

the

lays bare the laws of thought

But if we want to examine the problem of the existence what is to be a cut-free Gentzen-style

LD.

from the paper Stationary Logic

Mo Kaufmann and M. Makkai:

proof and a Gentzen-style case a bit,

LD

calculus

LS

we show, among others,

L = LM, BS

introduced

a formal system for Dummett's

the Craig interpolation

LMH.

logic

in this paper can be extended LC.

to

This is done in a separate

paper. §i.

BASIC SYSTEM. We define a formal language with connectives

u (disjunction), fier),

n (conjuction),

~(implication),

i (falsum), ~ (existensial

quanti-

V (general quantifier).

Assume we are given countably variables,

sentential

infinite sets of free and bound object

variables

and predicate

symbols of each number of

362

el.

arguments

By an atomic

formula we mean every sentential

is an n-ary predicate variables The set

then FORM

I)

(n e i)

and

is defined

if necessary)

finite

(Bu(A ~

s))

as usual.

(possibly

and

F~

(an ~

(Bu(B I ~

p

formula.

F = ~I ..... an' & = ~l'''''~m ' ~,B ~ FORM, r~

falsum and if

a I ..... a n are free object

p(a I ..... a n ) is an atomic of formulas

(with indeees, If

symbol

variable,

Denote by F,A

empty)

sequences

.... of formulas.

then for brevity we write

for Sl~

(~2----> (... ~

(B 2 ~ ( . . . - -

(Sn~S)

...)))))...)))

and ~i ~

(s2 ~

If F is empty, A empty, A

then

('''(Sn ~ F ~

(Bu(A ~

then F ----> (B u(A ~

are empty,

Denote by

then

s)...)))

s))

~)) is the formula

F ------>(Bu (A ~

~))

F ~

Bu(A ---->~), if (Bus)

and if F and

is the formula of the form (Bus).

rules

F ~ (Bu(~(~ r~ (Su(s ~

(str~)

respectively.

is the formula

R ~ the set of inference

(str~)

,

listed below:

y))) y))

r ~

(Suy) ...

r ~

(Su(s~

y))

(str~) r ~

(~u(A ~

F ~

(ur B)

r ~

(Bu(A~ (Bu(A~

r ~

(nl B)

(~----~ ( ~ y ) ) ) ) Yi))

(Su(~ i ~

y))

F ----->(Bu((~ins 2) ~ (ul s)

r ~

(~u(~ I ~ y ) )

i = 1,2

(YlUY2)))

r ~

i = 1,2 y)) (~u(~ 2 ---->~))

F ----> (Bu((elUS 2) ---->y)) (nr B)

F ~

(Bu(A ~ F ~

yl ))

(Bu(A ~

£ ~

(Bu(A ---->y2))

(YlnY2)))

3~3 r~

(~ iB)

.(.S..u...(A---> ~ ) )

r ~

(Su(A ~ r

( I r B)

r ~

r ~ r ~ ~, B, ¥,

r B c RB (sir 2)

Yi c FORM r

i.e.

and

are any sequence of formulas.

is rule of inference

such that the formula

let us take for example as

r ----> r ~ (~ R

r, A

rB

the rule

B

(str~)

is the following rule of inference:

(str2) Denote by

y))

(Bu(~(a) ~ 7 ) ) .... (Su(Vx~(x)~))

then

does not appear, then

(Su(A~

r ~ . . ( S u ( . A ~ ~(a))) ....... r ~ (Su(: ~ -Tx~(x)))

(V i B)

Let

y)

(Su(A=> i))

....................

( ~r s)

where

~

........... S u . ( r , ~ ((~n)~y)))

-¥)

the set of all rules

such that

r

rB c RB

i.e

~

°

R = {r : r B e RB}. By the basic

system

from

and the following

R u RB

(ax I)

(BS)

we mean the set of all rules of inference formulas

(i ----->~)

(ax 2) ri--~ 6

(CUt)

F ~ (@ i)

~(a) ~

and rules:

y

~, 6, y, ~(a) e FORM

y (Vr)

r ~

~(a)

r

Vx~(x)

----->

and a does not occur as a free object variable

in any formula in the conclusion of the rules The formula

e)

6---->y

~x~(x) where

(~

(~ i) and

(V r).

~ in the cut rule will be called the cut formula and denot-

ed cfl. Let us m e n t i o n

that the rules

(jr B) and

some rules that will be introduced

later.

(Jr) are added only to simplify

364

It is not difficult to verify that: i.i

(ax I)

and

(ax 2) can be restricted to

(I ~

a) and (a ~

a),

where a is an atomic formula. 1.2

0

For each rule of inference in

nistically valid,

BS, if all premises are intuitio-

so is the conclusion.

A formula ~ is said to be provable in the abbreviated as

BS

(henceforth

"BS ~ e") iff ~ is obtained from the axioms by means of

the rules of inference. Notice that the subsystem of the cut rule,

(7 i) and

BS consistin~ of (axl),

(ax2), the set

(Vr) is the formal system for intuitionistic

predicate calculus described by Schutte in [8] and [9]. this system by 1.3

IS.

For any formula iff

(Cut-elimination for

a cut-free derivation in PROOF.

Let us denote

Then in the standard way we obtain

BS ~ ~ 1.4

R,

BS).

IS ~ a. For any formula

0 ~, BS ~ ~ iff

e has

BS.

It follows immediately from the Haptsatz theorem for

IS [8] and

1.3.

[]

A derivation in which the cut rule does not appear is called normal. COROLLARY. i.

For any formula ~ , the following conditions are equivalent

~cLl

2.

e has a normal derivation in

3.

BS ~

4.

~ has a normal derivation in

We say that a formal system holds for

S

iff

S

IS BS.

0

is s e p a r a b l e or the separation theorem

the provable formulas of S have derivations using

only the rules of inference containing those logical connectives appearing in the formula. It is not difficult to show that: 1.5

The formal system

BS

without the cut rule is separable.

1.6

The separation theorem holds for the system

IS without the

cut rule. 1.7 In a normal derivation of e (in BS) occur.

0

0 only subformulas of 0

365

Let

g~FORM.

F~

~, where

Recall that we can consider F = ~I .... '~n

and

will be called the antecedents Y = YI ~

Y2

and

Y2

called antecedent

of ~ and y the succedent of ~.

of ~ and

Y2

all antecedents

of ~.

occurring

A pair of E and

, where

Craig's ~cL.

, then YI

If is also

letters w h i c h occur in

[~,~ *] is said to ~*

is obtained

from

in ~.

<~> be the set of all free object variables,

and predicate

y~

the end part of $.

Let Z be the sequence of antecedents

Let

In the sequel ~i .... 'an

is not of the form y~ ~

be a p a r t i t i o n of ~ if ~ is a subsequence by omitting

$ as a formula of the form

~i,YcFORM.

~¢FORM.

sentential variables

In the same w a y we define

F is a sequence of formulas.

Interpolation

Lemma.

Let

L

be an intermediate

logic and let

[~,~*] be any p a r t i t i o n of ~.

Let

A formula y such that I.

c <~> n <$*>

2.

~ ~

y~L

and

y ~*~L,

is called an interpolant an interpolant a formula

for

[~,~*].

Sometimes we say that [~,~*] has

if there exists a y such that I and 2 hold.

~ has an interpolant

We say that

if every partition of ~ has an interpolant.

It is well known that 1.8

(Craig's

interpolation

lemma for LI).

If

~cLl,

then ~ has an

interpolant. 1.9

D

(Craig's

tion in PROOF.

BS.

interpolation

lemma for BS).

Let ~ have a normal deriva-

Then ~ has an interpolant.

Let us assume

that

BS ~ ~.

By the corollary

~LI.

According

to the 1.8 ~ has an interpolant. §2.

LOGIC

LM.

The intermediate following

logic

LM

is obtained by adding to

(~ ~ u ~ ~ e), ~ e

is ~ ~

i •

It is well known that the logic

LM

with the directed set of "worlds", for

LI the

schema

(M) where

D

LM, then

is characterized by Kripke i.e.

if



is a Kripke

structures structure

366

A t,sVu(t

~u

&

s ~u).

The next two lemmas show some syntactical 2.1

The following

formulas

(i)

(((~->

~)

~(~'~

(2)

( ( ~

~ ~) u (

properties

are equivalent

B)) ~

to

of LM.

(M)

---> ( ~ u ~ ) ~ ,

~)),

~

(3) (4) ( ( ~ (5)

~)

((~

~

~

(~

(6) ( ( ~ ~ ~ S ) (7) ( ( ~

~

(8) ( ~ where

~ u~y))

n)

2.2

If

~

u (~ ~

~ S)

~

n

( ~ u ~

then

(A

The main difficulty

(g ~=> ~), ~ n represent

±), respectively. (~aom

(~ ---->m B), ~ ~ ( ~

~, BcFORM,

~¥))),

~

B)),

and as usual

A,B c {(~nB),

~ B) u ( ~

(~au~ ~)),

(n----->~)), (q ~

(B => ~ ~),

((~----->

~)),

( ~ m u ~ ~) +

~, B, y~FORM,

((~

(~u~)),

B),

(~ ~ ~

~ B),

(m ~ B ~

~ B), ~ ~ (B ~-> m ~), ~ ~ (-.~

in constructing

the formal

system

MS

for

LM

MS

and some new rules characteristic

to the positive

should contain all inferences for nonpositive

tautologies

of

the following set of rules:

(mlB) r ----> ( S u ( A ~ (m2 B )

Some of the m e n t i o n e d Prucnal.

(~u~)))

r ----> ( e v ( ~ - - - - > r ~

*

is

rules inference.

Thus the system

Let us consider

B)} where D

It is known that the positive part of LM is equivalent LI.

u~

~>B)~LM.

the choice of suitable

part of

~ ~),

~)) (Su((~

equivalences

r ~ ~ ~

(eu(6~

y))

~) ----> y ) )

were indicated to me by T.

LI LM.

367

r

(m3B)

(Bu(-Tc~¥))

~

r

~

r

(Bu((c~

(Bu(-76~y))

~ ~)

~¥))

(m4B)

2.3

If the premises

of the rule

(miB) , i = 1,2,3,4 belong to

LM, so

does the conclusion. PROOF.

Notice that

immediately ly valid. PROOF of (69 ( ~

~ 6) ~=> ( ~ u ~

(mlB)

for

(m2B).

y)eLM,

(~

and

(m3B).

It is sufficient

then

((~ ~ ~ 6

y) E L M and

6) ELM.

The rule

Thus the lemma follows

(m4B) is i n t u i t i o n i s t i c a l -

to observe that if ( ~ y )

) ~y

) ELM.

E L M and

To prove this let

(6~y)ELM.

Notice that (((-l~y)

n (-~

----->6) n ( ~ y ) )

~

((-i - ~

y) n (-~a----> y))) ELI.

Hence (((-~ -1~u-~ ~)---->y)-->y)--~((-~ - ~ y )

n (-~)

which together with our assumptions We call system ed from

MS

LM

y))ELI

((~8)

as propositional

r

where

from

(m i) is obtain-

rB(cf. § i).

iff

(Cut elimination

for

D MS).

Every theorem of

has a

inductions,

one

the othter on the rank of the derivation.

Recall that the grade of a formula of logical connectives

in ~.

~, g(~), is the number of occurrences

The left rank

RI

of the derivation

defined as the largest length of any thread of formulas with the left hand side premise of the rule formula occurs

LM

MS.

To prove the theorem we carry out two complete

on the grade,

Now

MS ~ ~.

The standard proof is omitted.

THEOREM 2.4.

D

logic.

For any formula ~,~ELM

normal proof in

gously.

----->y)ELM.

part of the basic

(miB) , (mi), i = 1,2,3,4

(miS) in the same way as the rule

THEOREM 2.3.

PROOF.

that

the system based on the propositional

BS, and the rules

we will treat

PROOF.

proves

n (~

in the succedents.

The rank

R

is

R I + R r.

~ is

in ~ that ends

(cut) and in which the cut

The right rank

Rr

is defined analo-

368

According to

1.4

the only new cases arise when one (or both) premise

(or premises) of the rule cut is (are) of the conclusion of the rule (mi6), or (m i) i = 1,2,3,4• I.

Assume that

R = 2, i.e. R I = R r = i, and the theorem is proved

for the grade n. Case of (m I ). Suppose that the following derivation ~.

'{(ml)

r~ ]'~

g(Bu(A~

(Bu(A~(~ ~-~ 6))) (~,u(S~ (--,o~u--,6)))

(m~um

S~¥

6))) = n+l.

(A~

(6 u ( A ~

Consider

(~u-~6))

(~u--,~)))

~

~¥

(ul)

y

(cut) F ~ Then ~ is reduced to

y

~i as follows [

(ml) ....(A---->( 9 ~ ~6))~ (A~ ( ~ ~ 6))

(A~ (~= ~ 6))~ (A~(~u-,6)) (A~ ( ~ u ~ 6))= y : 1

6~

(cutl) y

(A~

(~

~6))

~

(ul)

r----> (6u(A~

(~

-,6)))

(Bu(A~

(~

~ 6))) ~ (cut 2)

Notice that

g(A~

r a y ( m ~ u m 6)) < n and

g(6u(A~

(~

-~ 6))) = n.

Hence

by the induction hypothesis (CUtl) and (cut 2) are eliminable from ~i' that proves that the proof ~, given above, can be transformed into a proof without the cut rule. The case when the right premise of the cut rule in ~ is the result of the rule (str2) is trivial• Case of

(m I) .

Suppose that

g(cf2) = g(-~ ~ u~ 6) = n + 1

the following derivation

(ul)

It is reduced to

and consider

369

r~

(~=

(m 3)

-, ~)

(~

~ 6)

~

~f

(cut) F ~ By the induction hypothesis is eliminable

on the grade

g(~

~ 6 ) = n, the cut rule

from the above proof•

Case of (m2).

Recall

Notice that according not be n •

y

So

that

R = 2 and theorem holds

for the grade n.

to the assumption on the rank the cut formula can

cfl = ( - ~

~) = q and let g(cfl)

= n + i.

Let the

derivation ~ run as follows

(m2)r ~ (-l-~ r ~ (~6)

]I

n) r ~ (~n) ~ n)

((~ ~

~) ~

n) ~ (cut)

Let

~' be a cut free

interesting rule

(~

MS-proof of

((~ ~

6) ~ q ) ~

cases are when the end-formula of

i) or of the rule

(m3).

y

The only

7' is the result of the

In the latter n must be of the form

nI CASE I.

z' runs as follows

(~ ((~ ~ Then ~ is transofrmed

I)

~)~n)~x

into

(cut I) F~

n

n~y (cut 2)

F ~

y

(cut I) and (cut 2) are eliminable by the induction hypothesis CASE 2.

7' runs as follows:

( n= ~ ~ ) i

on the grade•

370

•

,

(m 3) ((~6) Using the fact that the proof

~

~ nl)~

SI ~ (~ ~ ~

y (~63

~ (~ ~

6)))

~' is reduced to Sl-proof

(cuti) -I ~ e

~-~ (~---->~ (-i e ~

6)) ~n I ~

r ~

( ~

(~e~

~ n I)

~nl)~

(~6~

( ~

Y)

%)

y) (cut 2)

-I n I ~ y

%) r~ (6~ ~n I) (cut 3)

(6~ ~nl)~ (r~y) P ~¥

(curl), (cut2) and (cut3) are eliminable by the induction hypothesis the grade.

on

Now let the right premise of the cut rule be the result of (m 2) and let cfl = ( ~

8) , i.e. the derivation ~ run as follows:

f

•

°

(r) P~

(~ e~

(m 2) (-~ ~ ----->6) ~

~)

y

Fay The premise of the rule (r) can not contain ( ~ is i.

So the last step in

~'

must be one of the following cases:

CASE i.

(=l) F ~

( ~

6) as the left rank

~)

371

CASE 2.

8=61u

and

62

r~

(~e~

6 1)

(ur)

CASE 3o

~ = 6

i

n ~

or

~'

r~

(-~ ~ 6 2 )

r~

(-~ ~ 8 )

•( u

r)

and

3

!

I

CASE

4.

r~

( ~

6 = (~i ~

62) ~ 3

r~

(-~ ~ i F~

CASE 5.

*

61)

6 = -~ 61

(-i e ~

r~

(~ e ~ 2

)

and

62 ~

) ((~i ~

63

( ~ I)

62)~63)

and

r ~

(61 ~

-~-~ ~)

r ~

( ~

-~ ~I )

In the case when instead of (ur) we apply a rule of the form (r ~) we proceed in an analogous way as in the case (1)-(7). Note that the subformula (suB) of the formula ( ~ u ~ ) ~ y is decomposable only by the rule (ul).

372

CASE 6.

~ = -~ (6 1 ~

r ~

rl)

and

(-~a~

(-~

r ~

-i-~ 61 )

--~n )

(m 4) r ~

CASE 7.

(-~

-~ (61 ~

n))

and

6 = ~ 61 u ~ 62

r ~

( ~

(61~

-i 62))

r ~

(~c~

(-n 6 1 u-~ 6 2 )

(m I)

We check only the Case 4. is as follows:

Then the derivation

A

or

((61 ---->62) ~

~vv

( ~

((6 1 ~

62) ~

Thus ~ can be transformed

B

7"

of the right premise

C

or

63 ) ---->y (m 2) 63))3 Y

as follows:

6 1 ~ 61

6 2 ~ 63

(31) 6 1 ~ ((61~ 62)~ 63)

((61~ 62)~ 63)~ Y (cut i)

r~

( ~

6 l)

(-~e~ 61)---->y (cut2) Fay

By induction on the grade all cuts are elinfinable. Case of (m3).

Consider only the case when

cfl = (~ ~

~ 6) =~ n

and

373

let q be the form

~ ql"

Let

~'

such that the end formula of (~

~ 6) ~

More exactly,

y and

~'

~ nI ~

((~6)=~nl)=~

y

y

let the derivation

r --~ ( ~

be the proof of

is obtained by application of (m 3) to

~ n I)

r ~

~ ~) ~

~ n I)

~ runs as follows:

(~6~

~ q I)

-~ ( ~

~ 6)~y

-~ n I ~ y

(m 3) r ~

((~

(m 3 ) F ~ As before, we use that

SI ~ -, -~ e ~

Thus ~ can be transformed

y (-~ ~ 6 ~

-~ ( e ~

~))

into:

Sl - proof ~-,~(~-,6~

~ (c~

-,6)

-~ql~

Y

(cutI) (-,-1 ~

~

r ~f~

~

~ n I)

¥)

( ~ n

I) ~

(7~Y)

(cut) r

r ~

(~6~

~

(-,-,6~

n I)

(76~

"~)

F ~(-~ n l ~ ¥

~ n I) ~

(r ~

y) (CUt3)

(r ~

¥)

F~y Note that all cuts are eliminable by the induction on the grade. We omit the case w h e n the right premise of the cut rule is of the form ( ~

~6)

analogous Case of follows:

=> y

and

case for (m4) o

cfl = (~ ---->~ 5)

as the proof is similar to the

(m2).

For simplicity

assume that

A = $

and

let

~ run as

374

(Bu~ ma)

r~

r ~

(Bum 6)

(ul)

(m4B)

(8u-~ ( ~

(~Bu-~ (~------>~))

r ~

6))~ y

(Cut) F ~

y

It is reduced to 7' :

Sl-proof (cutI)

(u1) r~

OB~--,cO

OBu--,-~) ~

(-~ ~ y ) (cut2)

r ~

(~

B ~

~y)

r ~

Y (s~y)

(ul)

r ~

(IBu-,~) => (r~y)

OBu-~(B)

(cut3)

r ~

(r~y)

r=>y Arguing as usual we can eliminate Analogously

for the remaining rules:

It is not difficult

to conclude

the induction hypothesis norml

all cuts from the above derivation.

that if

RI > i

and

Rr> I

then using

on the rank we transform any derivation

into a

proof•

COROLLARY.

For any formula a the following conditions

i)

~LM,

2)

NS~a,

3)

there exists a cut-free proof for ~.

Problem•

Is it possible

to eliminate

from

are equivalent:

D MS

the rules

(m 3) and

(m4B)?. THEOREM 2.5•

(Interpolation

lemma holds for PROOF.

Let

Craig's interpolation

LM.

~eLM.

for any partition

lemma for LM).

To prove the theorem it is sufficient

to show that

[#,E*] of ~ there exists a formula y such that

375

I)

(~

2)

c <~> n <~*>,

where

y) c LM

and

for any formula

(y ~ * )

~, <~>

E LM

is the set of all sentential

variables

ocurring in ~. The m e t h o d w h i c h we use is due to Maehara the fact that an interpolant constructively

of the formula of the form

obtained from a proof of

By the corollary

MS ~ ~.

and its significance (a ~

lies in

(a ~

~) can be

n).

Now we will construct an interpolant

duction on the number of inference rules,

~ by in-

in a normal derivation

At each stage there are several cases to consider.

According

only new cases arise when ~ is a conclusion of the rule

(miB)

~ of ~.

to 1.9 the or

(mi),

i = 1,2,3,4. We deal only w i t h

(mi) ,

such that

~,

derivation

~ of ~.

Let

~

= k

i = 1,2,3,4

and suppose

the theorem holds, where now

= k + I and the last inference be

Let us consider the following partition

(ml) , i.e.

~ = r',

By the induction assumption we know that for an interpolant ~* 1

=

y.

------>( ~

F*

Hence

-~ 6)

for the partition

(r* ~

Moreover

c n
Thus and by

~ =r~

(~u-~6).

~* = F* ----> ( ~ u - ~

~i = F ~

(~

[~i$i] , where

there exists a ~ such that

MS t- ( Y ~

that for every ~'eLM ~$ is the length of

5).

~ 6 ) there is ~I = r',

MS ~ F' ~

y and

-~ 6 ) ) ) .

(~

(~

~6)>.

(ml) we infer that

MS ~ r' ~ It is obvious

and

¥

that

is an interpolant

MS ~ P* ----> ( ~ u - ~

c n
P ~

Let the last inference be

(-~u-~ 6)

6).

w h i c h proves

that y

(~au-16). (m2) , i.e.~ = F ~

((-~=~

6) ~ q ) .

for the simplicity we assume that n is not of the form

nl ~

Moreover q2"

Then we have the following partitions i)

¢ = r'

;

~* = r* ~

( ( ~ )

2)

~ = r'

;

(-~6)

;

We omit I) as it is trivial. ~i = F ~

(-~ ~---->n),

~n)

~* = F* ~

n.

Let us take the latter and consider

~2 = P ~

(6---->q). By the induction

assumption,

376

since [~2'

~El
~E2
, where

~I = F'

are i n t e r p o l a n t s

MS ~ r' ~

Yl

and

71),

(~ ~

, ~ ~a;

that

for the p a r t i t i o n

~2 = F',~;

~i = E2

[~I,EI ]

= r

~

and

n there

72 , i.e.

MS b F' ~

( 6 ~ 72),

MS ~ 72 ~

(r* ~

MS ~ 71 ~

(r* ~ q )

q)

and c <~i > n <El> Using

the rules

MS br'

~

(ur),

(7-~

It is obvious

,

c <~2 > n <E2>

(ul) we o b t a i n

(¥1uY2)), that

<72>

b r'~YlUY2)),

MS

MS ~- (YlUY2)--->

(r*---->n).

:

c <~> n <~*> Now u s i n g

the rule

(m2) we

infer

that

(Yl u y2 )

is an i n t e r p o l a n t

for

the s e c o n d p a r t i t i o n . N o w let us assume E = F ~

((~

w e assume

that the last

~ 6) ~

q).

that n is not of the form

have only one p o s s i b i l i t y ,

namely

On a c c o u n t

of the i n d u c t i o n

[r', -~ ~;

F* ~

n]

Y2' r e s p e c t i v e l y .

IF',

(~

rule

(m3).

~ ~);

Let the last

inference

be

E2 = F ~

m ~.

~ = F'

If ~ # F',

(~ ~

we obtain n]

6) and

[~,E*] i.

E* = F* ~

n.

that for the p a r t i t i o n s

our p r o o f

E = Y ~

such that

Consider

By the i n d u c t i o n

and

then we

there are the i n t e r p o l a n t s

completes

(m4) , i.e.

~ and E = F* ~

n 2.

~ = r',

F* ~

(m3) , i.e. case w h e n

(Yl u y2 ) is an i n t e r p o l a n t

which

is only one p a r t i t i o n

= F',~ ~ and

Thus

F* ~ n ]

is

qI ~

hypothesis

[F', -~ ~;

and

There

inference

As b e f o r e w e omit

this

for case of the

7 (~ ~ # F', case.

assumption

Y1

for

there

~). namely Let E 1 = F ~ are

Y1

and

-i ~ Y2

such that VS~ and

r' ~

( ~

71),

~ < r ] ~ > Thus by

(ur),

(ul),

m }-T 1 ~

n
~I>, I),

(r*~l) ,

<72>

~

~r'

c

(str 3) and

~

n

(~

y 2)

.

(m 4) w e infer

that

MS ~ 7 2 ~

(r*~l)

377

MS ~ r ~

((~)

~

and

~ ~ (YlUY2))

MS ~- -~ -~ ( Y l U Y 2 ) ~

(F* ~

D

l).

Note that 2.6

The system MS without the cut rule is separable.

0

2.7

In a normal derivation of ~ only subformulas of ~ occur.

D

It is not difficult to check that the following two rules r ~

(~Su(A~

~(a))

r ~

(~u(A~

V x~(x))

r ~

~ Bu(~(a)

r ~

(7 ~ogx~(x)~

(VBr)

~y)

(~B1)

where the

a

does not

conclusion logic.

Let

be the

MSQ

system

and the rules

Conjuncture.

as a free

of the mentioned

predicate

BS

occur

MS

(VBr),

X)

object

rules,

enlarged

variable

are valid

in any formula in

LM

by the predicate

in

considered

part

of the

as

system

(gB1).

The f o l l o w i n g

conditions

are

equivalent

i)

~LM,

2)

MsQ ~ ~,

3)

~ has a cut-free proof in MS Q.

Notice that if the conjuncture is true then it is not difficult to prove Craig's interpolation lemma for predicate logic theorem and the subformula property for §3.

LOGIC In [i]

LM,

the separation

MS.

LMH Gabbay proved that

LMH = LI + (Vx ~ ~ e ( x ) ~

complete for the class of all Kripke structures ordered worlds system

T



~ Vx~(x))

with partially

fulfilling

A t V s(t ~ s & A u ( s ~ u + s = u). It follows from the above characterization of ~cLK

iff

LMH that for any formula

~ ~ ~(LMH.

The above theorem is often called the Glivenko type theorem for Umezawa observed in [i0] that the following schemas of formulas:

LMH.

is

378

(*)

~

(**)

((Vx~(x)

~ Vx(~(x)

u~

Vx~(x)

~

~

~ x~(x),

~

(Vx ~ ( x )

~

(gVx~(x)

~ ~

~Vx~(x)),

~

a x ~(x)),

-~ (-~ V x ~ ( x )

n -~ ~ x - ~ ( x ) ) ,

~

~

~ xB(x))

-~-~ @ x Vy(e(x)

~(y)),

((Vx~(x)

B)

~

~

~

~

~(x)),

~

ax(~(x)

~ x(~(x)

~

~

~(x))),

B))

are e q u i v a l e n t to

(MH)

(v x ~ ( x )

v x~(x)).

To construct the formal system IS

MHS

for

LMH

let us take the system

for i n t u i t i o n i s t i c logic and add the following rules: r ~ ( - ~

(mh l )

-~ ~ n ) r ~

( 6 ~ -,--~n)

r ~

((~)

~

r ~

(~

F. ~ r ~

~ -~ ~ ( a ) ~ v x~(x)

-1 ~ q )

~--,n)

(mh 2)

(mh3)

where

a

does not appear as a free object v a r i a b l e in any formula in

the c o n c l u s i o n of 3.1

Let

of the rule PROOF.

(r) (r)

Suffices

(mh3) and

is y ~

be one of the rules belong(s)

to

LMH,

i •

(mh I)

(mh3).

If the premise(s)

so does the conclusion.

to observe that the rules

t i o n i s t i c a l l y valid. from

-, y

(mh I) and (mh2) are intui-

In the case of the rule

(mh 3) the theorem follows

(MH) .

3.2 the form

Let ~ be a normal proof in the system ~-~ ~

y.

MIIS

of a formula of

Then there exists a subformula of y,

Yl'

and a set of subformulas of y ,

say

of the form

is an a x i o m and it occurs in ~ or the

rI ~

(-~ ~

yl )

rl,

say

such that either the formula

379

formulas of the form

F1 ~

m~

and

i ~

Y1

appear in ~ or

Y1 appears

in 7. PROOF.

Let us restore a normal

-7 ~ ~

y.

form ( ~

MHS

Note that the subformula B ) ~

in the latter y must be of form

~ ~ ~

B)

of the formula of the

¥ is d e c o m p o s e d only by the rules

not play any role. form

proof of a formula of the form (~

~ m ¥I

i)

and

(mh!).

and in our case this rule does

So let ~ be a normal

y

(~

M H S - p r o o f of a formula of the

Then ~ runs as follows:

YI

or

FI ~

-~ e

i ~YI

(str I)

(~ FI~

(-~ -~ ~

1 )

¥I )

(r I) ..... (r k)

,

where

FI

~

is a set of subformulas of y (perhaps empty)

a subformula of of ¥.

, ~

y.

The rules

(rl)...(rn)

Note that either a formula of the form

axiom or it is derived from it is derived from 3.3

Y1 by

F1 ~

~

and

Yl

is

operate only on the subformulas F1 ~

and i ~

(9 ~

yl ) is an

Y1 by the rule

(~

I) or

(strl).

D

(Cut e l i m i n a t i o n for MHS).

Any d r i v a t i o n in MHS reduces

to a

normal derivation. PROOF.

By

premises)

1.4

the only new cases arise w h e n one premise

(or both

of the cut rule is (are) the c o n c l u s i o n of the rule

(mhi),

i = 1,2,3. Assume that

R = 2, i.e.

R 1 = R r = i, and that the theorem is p r o v e d

for the grade n. Case of

(mhl).

Suppose that

cfl = ( ~ 6 ) ~

Consider the following d e r i v a t i o n

r ----~ ( ( ~

~)~

and

g(cfl) = n+l.

~ :

F ~ (~l)

~ ~n

(6~

--1 " ~ n )

-~-.n)

(c~6)---->

-~ -, n ) ~ ¥ (cut)

F ~

y

380

Let

~'

be a normal proof of

The only interesting

( ( e ~ 6) ~

~q)

~

Y

cases are when the end formula of

the result of the rule

(~

I) or of (mhl).

~'

is either

In the latter y must be of

the form ~ ~ Y2" CASE i.

~'

runs as follows:

•

°

(~ ((~)

Then ~ can be transformed

~

r ~

(~

r ~

(-~n~

i)

Y

into:

~ (~

~n)~

r ~ ( ~

I~i

~n)

I)

-~-~n)

~

(r~

-~-~n) (cut I)

-,~

-~)

(r~

~n) (cut 2) (cut 3)

r ~ By the induction hypothesis CASE 2.

~'

y

on the grade

(cut I) - (cut 3) are eliminable.

runs as follows:

(mh I) ((~6)~ Then ~ can be transformed

into:

-~-~n)

~

-~-~y

381

r =>

(--,n~

--~

~ ( ~ ) ~

(~))

~

Y (cut i )

r

~

(-~y~

~-~)

-~-~

-~-~y (eut2)

r ~

On account of the induction hypothesis

(~y=>~

Y)

on the grade cut I and cut 2 are

eliminable. The case of (mh 2) is trivial so let us prove the theorem for case of (mh3).

Suppose that

the following

cfl = ~ ~ V x~(x)

and

g(cfl)

= n + i.

Consider

derivation:

F ~

~ ~a(a)

(mh3) r~

~ ~ Vx~(x)

~ Vxu(x)~ (cut) r ~

Let

~'

be a normal proof of

y

~ ~ Yxe(x)~

then by 3.2 the only interesting

case of

y. ~'

rI ~

(e(b)~ I )

rI ~

(Vxe(x)~i)

rI ~

(7 ~ Vx~(x)-->

If we omit trivial cases, is as follows:

(vl) I=>

YI

( ~ i)

-,~ Vx~(x)~

(r I ~

yl ) yl ) (rl) ..... (rk)

--I V x ~ ( x ) ~ This is transformed

into:

y

382

rl~

~(b)

~(b)~

(r~l) (cut I)

rI ~

(r ~ I )

I ~

Yl (cut 2)

rl~

(F~

~i )

r ~

( r l ~ Y 1)

(rl)

P

Above

the formula

rI ~

tion which previously occurrence

occurred

that if

hypothesis

rI ~

that

rI ~

(Vx~(x)

---->i.

rI ~

y

we write

above

the same part of the deriva-

r ---->~ ~ e(a) a

on the grade

(Vx~(x)=> i)

axiom then a reduction Vx~(x)

(b)

of the free object variable

By the induction ble. Notice

~ ~

~

or

by

replacing

~i)

(cut I) and

~ ~ Vx~(x)

is an axiom then the following

~ V~(x)

(cut2)

~

rI

~ ~ ~(a)

; ~

~

(r I ~

yl ) is an Indeed, suppose

is of the form

Vxe(x) ~

are elimina-

derivation:

r ~

~ Vx~(x)

every

b.

of the cut rule from ~ is trivial.

Let us consider

(r k)

.....

(mh3)

(r~i) (cut I)

rI ~

(r~l)

I ~

Y2 (cut 2)

r~

(r I ~

¥i )

(r I) ..... (r k)

383

As usual,

(cut I) and (cut 2) are eliminable.

So the proof of cut-elimination

for

R = 2 and any grade of cut formula

is over. If the rank is

>2, then using induction hypothesis

lemma 3.2 the c u t - e l i m i n a t i o n As a natural COROLLARY.

consequence

theorem can be proved very easily.

D

of the above theorem we obtain.

For any formula ~ the following conditions

I.

~cLMH,

2.

MHS ~ ~,

3.

~ has a normal proof.

3.4

on the rank and

are equivalent:

0

The system MHS without

the cut rule is separable.

3.5

In a normal derivation of ~ only subformulas

3.6

(Craig's

interpolation

lemma).

0

of ~ occur.

For any formula ~,

if

D ~LMH,

then $ has an interpolant. PROOF. MHS.

Suppose that

~ELMH.

An account of

of the rule

(mhi),

1.9

By the corollary ~ has a normal proof in

the only new cases arise when ~ is a conclusion

i = 1,2,3.

Consider only the case of

(mh3).

There are two different partitions i.

~ = r'

2.

~ = r',

;

of

~* = r* ~

~ Vx~(x)

The standard proof for

;

~, namely:

~ ~ Vx~(x), ~* = r* ~

(i) is omitted.

By the induction hypothesis (a

Let ~ be of the form F ~ V x ~ ( ~ .

Let us take

~i = F' ~

~

(2).

there is an interpolant for ~I = F ~

does not appear as free object variable

~I = r*,

i •

e(a)

in r).

be a partition of

~ ~ ~(a)

Let ~I

and let y be a

formula such that MHS ~ Moreover

F* ~

y

and

c <~i > n <~i >.

ing the rules

(~

i),

(mh 3)

MHS ~ r'---->{~V x ~ ( x ) ~ It is obvious required Problem.

MHS ~ y ~

that

a

Thus and ~ ~)

(str 3) and

(r' ~

~(a)).

does not occur free in y.

Apply-

we infer that: MHS ~

~y~

<~ y> c <~> n <$*> which proves

(F* ~ i ) . that

~y

is the

interpolant. To modify the system

0 }ZIS

in such a way that the rules of

384

inference specific for

MHS

are only

LMH valid.

REFERENCES [ I]

Gabbay, D.

Applications of trees to intermediate logics, J.Symbolic Logic 37(1972), 135-138.

E 2]

Semantic proof of Craig's theorem for intuitionistic logic and its extensions, I and II. Proc. 1969 Logic Colloquium, North-Holland Publ. Co., Amsterdam(1971), 391=410.

E 3]

Craig interpolation theorem for intuitionistic logic and extensions, Part III. J.Symbolic Logic 42(1977), 269-271.

[ 4]

Ldpez-Escobar, E.G.K.

[ 5]

On the interpolation theorem for the logic of constant domains.

J.Symbolic Logic 46(1981),87-88.

A second paper on the interpolation theorem for the logic of constant domains. J.Symbolic Logic 48 (1983), 595-599.

[ 6]

A natural deduction system for some intermediate logics.

[ 7]

Maksimowa,L.

J.of Non-Classical Logic, 1(1982),

Craig's interpolation theorem and amalgamable varieties, Algebra i Logica 16(1977), 643-681.

[ 8]

Sch~tte,K.

Der Interpolationssatz der intuitionistischen Pr~dikatenlogik, Mathematische Annalen 148(1962), 192-200.

[ 9]

[I0]

Proof theory, Springer-Verlag, New York (1977). Umezawa, T.

Berlin-Heidelberg-

On logics intermediate between intuitionistic and classical predicate logic, J.Symbolic Logic 24(1959) 141-153.

TYPES IN CLASS SET THEORIES Rolando Chuaqui

William No Reinhardt Depto of Mathematics, University Boulder,

Universidad

Campus Box 426

Catolica de Chile

Casilla II4-D

of Colorado

Santiago,

CO 80309

Chile

This paper deals with the problem of defining order types for well ordered proper classes. equivalence

This is a special case of the problem of defining

types for an equivalence

relation.

ous problem for set theory with regularity method of introducing

equivalence

We recall that the analog-

is solved completely by Scott's

types as the equivalence

class restrict

ed to its members of minimal rank. It is curious that the situation changes so markedly when we pass to class set theory. somewhat peripheral

to most set theorists,

remarks about our interest

in it.

As this problem may seem

we would like to make some

The problem arises naturally when one

tries to develop the theories of constructible theory

(such as Kelley-Morse-Tarski);

became

involved in the problem.

Reinhardt's

attention

this is how Chuaqui originally

Professor Chuaqui drew the problem to

in 1982, when he was preoccupied with problems

volving the notion of intuitive provability. terested

The problem of developing notations

is analogous

in-

Because of this he was in-

in some aspects of proof theory and in particular

notations° ordinals

classes within a class set

in ordinal

for larger and larger

to the problem of getting types for ordinals.

Since

key ideas often show themselves

more clearly and simply in classical

ings than in constructive

one might hope these problems would illu-

minate ordinal notations; results.

ones,

we were thus thinking largely of positive

We have nothing to say f r o m t h e s e

ordinal notations, the problems

investigations

are related.

The positive results we have obtained are In particular,

the method of defining order

types appears to lead to a more conceptual

development

classes than any we know of in the literature°

Also,

of constructible in conjunction with

ideas of Manuel Corrada it leads to a nice characterization part of the impredicative

theory of classes

of the set

(ioe. Kelley Morse-Tarksi

set

The negative results mentioned here were pointed out to us by

several people at the symposium, Magidor,

concerning

but it does still seem to (at least one of) us that

modest but of interest.

theory).

sett-

and W. Mareko

especially

Steve Simpson,

Menachem

386

Suppose that we are given an e q u i v a l e n c e r e l a t i o n X ~ Y We permit ~ to involve p a r a m e t e r s Z.

on classes.

The q u e s t i o n then is w h e t h e r there

is a term r (which may depend on Z and on other parameters U as well)

so

that I)

VZ~u(T(X)

We can ask for more,

: ~(Y) ÷÷ X ~ Y).

that T be a selector:

w h i c h says that if ~ is an equivalence, or Type

~(X)

~ X.

We call the a x i o m

then i) holds,

"~ is an

~-type"

(~,~) or even "T" for short.

Surprisingly,

it appears to be an open p r o b l e m w h e t h e r there is such a

for the e q u i v a l e n c e w h i c h holds b e t w e e n two o r d e r e d pairs w h e n they c o r r e s p o n d to the same u n o r d e r e d pair.

That is, it appears to be un-

k n o w n w h e t h e r u n o r d e r e d pairs can be defined in class set theory. On the other hand, it can be shown using methods of L~vy d e v e l o p e d for i n v e s t i g a t i n g questions of definability,

that there are models of set

theory in w h i c h no such T can be given for the r e l a t i o n of order isom o r p h i s m for well orderings This happens

(i.e. well order types cannot be defined).

in the model o b t a i n e d by collapsing all cardinals b e l o w

the second i n a c c e s s i b l e to the first inaccessible° sidered i n d e p e n d e n t l y by L~vy and Rowbottom) o

(This model was con-

We are i n d e b t e d to Rich

Laver for the following argument. Let M be a m o d e l of ZF w i t h two inaccessible cardinals

K
Let N be

o b t a i n e d from M by collapsing all cardinals ~
<-

The model of K M w e

consider is just V N <+i °

We

observe that a d e f i n i t i o n of types here w o u l d give rise to a function P: I + V K + I in N w i t h P(~) = P(B) ÷ x~=.

~ = ~, namely p(=) = r(x) w h e r e

Such a function in N involves only an initial segment of the collaps-

ing functions.

We may think of N as M o [G]

the r e l e v a n t initial segment. b e t w e e n objects

, w h e r e M o = M [G o ] contains

We may represent P by a r e l a t i o n 9(~,x)

in the ground model:

o(~,x)

if

xeP(~).

By h o m o g e n e i t y

9 is d e t e r m i n e d by the empty condition over Mo, and hence P is in Mo, giving in MoP For example, question° hierarchy;

:I ÷ VK+ I.

This contradicts

for c o n s t r u c t i b l e classes,

(Here "first" means

the i n a c c e s s i b i l i t y of I in M o-

let T p i c k the first class in

in a canonical o r d e r i n g of the constructive

to see that this makes

sense requires k n o w i n g that the level

at w h i c h a class first occurs is independent of the r e p r e s e n t a t i o n of the construction).

Again,

even in models M w h e r e no ~ exists,

the a d d i t i o n

of the existence of ~ to the theory of M may be conservative,

so that in

387

some sense it is as though there were a We first make the trivial observation comprehension indicated° {YEM

I

interested

If M is any model,

in w h e t h e r

Open Problem:

the axiom we have

the external

classes

Thus we are

r can be added so that it both satisfies

and

other schemas of class theory.

Tarski theory

comprehension

consider

, and let T select one member of each.

type axiom,

Kelly-Morse

that if we do not allow • in the

scheme, we may always add a T satisfying

Proof:

M has Y~X

valence

T after all.

(which we shall call KM),

the equi-

In the case of

this is the class

schema.

Is K M together w i t h ~ for well order types a conservative

extension of KM? We do have THEOREM°

Let DC be the axiom of (transfinitely many)

for classes°

That is, DC says that if

quence Z~ (~eOR) sequence

Z

of classes

dependent

Y~Y 0(X~Y).then

choices

there is a se-

so that for all ~,0(,Z ). (Here the

is of course to be construed as an appropriate

class).

Then

F~M + DC + "~ is an ~-type" is a conservative

extension of KM + DC.

Note°

DC is DCon of M a r e k and Zbierski

PROOF:

The proof is really the same as the proof that using DC one can add

a well ordering of the reals without

[1972].

adding any new reals.

only n e e d to use the simplest possible kind of forcing plicated

terms or corresponding b o o l e a n valued functions

was observed by M o s t o w s k i proof is essentially ty) implies

[1972]

the same°

that for analysis,

(Of course it suffices

ce types are definable

for the case of analysis;

again,

formula G(x,y)

to add a well order,

as "the G-first Y such that Y~X").

Two distinct

conditions

We

for the well

as then the equivalenWe force w i t h

A condition may be thought of a class

can contain the same inform-

A condition p will have the form p = (W,Z);

W,Z are proper class-

es, and so we take the ordered pair in an appropriate

sense

(Wxu{0}

(has more

(Zx{l})).

our

(without infini-

of as a well ordering of OR together with a (I-I) assignment ationo

This

but the reverse does not seem to hold.

conditions which well order OR-many classes° to each point of OR.

we

are needed).

The result for set theory

have in the forcing language one new atomic order°

However,

(that is, no com-

We say that one condition extends

(eog. informa-

388

tion than) mer;

another

in case the latter

is an initial

segment

of the for-

e.g.: q extends

p iff @f,XCWp

:f Wq F X

&

Vc~ Z"C{~}=~ Zq{f(~)})).

(If necessary,

we write

We carry out forcing

q e p for this).

in one of the usual ways now,

in the language with

and G° We shall adopt the point of view that we are giving an interpretation of the language

of KM + G into that of KM.

KM + G we associate

a formula

0 whose

gether with one new free variable. variable

of

free variables

7¢ and the basic

formula

pl~W G(X,Y)

the syntactic

for interpreting

= @x,y

important

G(X,Y),

(X = Z" {x} & Y = Z" {y} & (x,y) P q p,X,Y.

to avoid collisions

that ~p ~p e(p) has the form

which

~e(q)]

is equivalent

=

Vq

V

p

CVq~q' ~ e(q))

to

Vq'ep @qeq'¢(q) We may now define

I~ e by induction

on ®:

Atomic: II- (xcy) I~ G(x,y)

=

xEy =

mp~p

(pl~ W G(x,y))

Wp - ]

In the definition of variables

should not occur in ¢o

~p(Vq~p

To carry out the

operation:

= Vqe p ~ e(p/q)

which of course has free variables

Observe

are those of e to-

We shall use X,Y .... for the free

and p,q .... for the new free variable.

definition we first introduce

is of course

Thus to each formula of

of ~ p ,

with q, so q

it

389

Propositional :

IF (0 & ~) = (IF

e)

& (I~ o)

ID (-I e)

e)

=

-~p

(LF

=

Vx

(IF o)

Quantifier :

I~ (Vxo)

To be definite, we m a y always choose p to be the first n e w v a r i a b l e not o c c u r r i n g in ~o

Thus the n e w free v a r i a b l e of Ik0 will be p.

We there-

fore shall w r i t e pl~ 0. Now we w i s h to show that the a x i o m of K M + ing of the u n i v e r s e of classes",

are forced.

sions for c h a n g i n g the e-relation, true u n d e r this i n t e r p r e t a t i o n KM~).

DC, plus "G is a well orderSince there are no provi-

it is trivial that K M + DC remains

(it is the identity on the language of

Thus we have only to see that the c o m p r e h e n s i o n a x i o m of K M remains

true w h e n we a l l o w G: plF

] Z V x ( x e Z +÷e & x~V)

i.e° Vq_>p @r>_q ~ Z V x ( r l k x c Z ÷÷0 & x~V) Given q->p, it w i l l suffice to e x t e n d q to r so that for some Z

&

I)

rll-x~Z ÷ e & x ~ V

2)

rlF @ & x ~ V ÷ xeZ

i.e. i') x~Z + rlF 0 & xeV &

2') Vr'~r

(x(V

& r'IF 0 ÷ x~Z)

We shall extend q to r so that for all xcV, rll- 8 then choose

Z

=

{xEV I rl~ e}

,

or rll- ~ o.

We may

390

in which

case i'),

Of course,

2') will evidently

for each x we can extend r to r x which

may thus form a sequence each x some pe settles OR, and observe one condition

p~

(~e0R)

each of the p~

k, so that the X"{e}

readily indexing Z's

to work in K M w i t h

it has the unappealing

DC, and also wishes

For example,

This is an extension scheme asserting

feature

oVK+I(x,XnVK).

of K M w h i c h

is

using an

corresponding

but this provides principle,

does not carry over°

to introduce

equivalence specific

the strong set theory B of Bernays. by adding

a reflection

of KM true in the universe

In B, DC is provable.

reflection

: x~V & o(x,X)

no justification

Thus the theorem shows

is re-

+ ]< ~ O R

Thus, by the above

theorem we can

for allowing

only in the comprehension

schema.

that adding

them in the

The argument types

does not

KM very much, namely no more than DC does, but it does not

say much about extensions does everything

~--->KM + DC + T ~

of KMo

Since DC is already present

that can be done using types

=---->KM+ DC

I- ~

from B more directly

BI- 0).

This

and we present

The idea is that equivalence

be added by the use of higher

the argument

(classes

But the reflection

if KM + TI-O, we may in B choose

also so that VK+ I ~ o (VK+I,~,T) I=

iff o. KM + T.

(KM + TI- O can be

because

it is

types over a domain may always

type objects

case of the domain of classes).

in KM.

in B, B

fact, however,

effect of being able to continue with such a process particular,

= Wpa,

that it is surprisingly

may be obtained

that any statement

use types:

suggestive.

This however

carry over in the natural way to exten-

consider

in some V + I [V = {V I ~ E OR})

obtained

p~

from the point of view of someone who

It does not necessarily

sions of KM.

already

the proof it is only

D

is very nice

types,

strengthen

into to

the Xe and are increasing;

To complete

The Zr is formed by assigning

wishes

flected

e(X,Y)

this correspondence.

this result

to KM.

V on

all the p (~
to form Wr, we join all the W~

of 0Rx0R by OR.

, and for

to enumerate

The formula

to do the same for the full sequence

under

While

X.

By DC we

that if X is a relation with domain

= pe settle

k+l and continues

accomplished;

0.

such that if ~<6, P ~ P 6

that for any k¢OR, we can amalgamate

pk which extends

then Y has domain necessary

settles

0 for x: it is only necessary

which we apply DC can say for example an ordinal

be satisfied.

But then there

of classes, principles

in the give the

indefinitely.

~ so that V + I ~

In

KM, and

is T : VK+ I ÷ VK+ I such that

391

Hence VK+ I ~ ~ , if ~ only involves

c.

This, however,

shows that BI- ~.

It does not answer the question w h e t h e r B + T is a conservative sion of B. THEOREM.

We d o h o w e v e r , Let B

be like Bernays

strong set theory B, except that for

each natural number n, we assume the reflection a measureable sion of B

u

exten-

have the following result.

cardinal of degree no

Then B

can always be made to

+ T is a conservative

exten-

.

Note I.

We must explain in what sense r is allowed in the reflection

schema.

The simplest way to formulate

not only T but,

for each K
of the classes;

the strongest

this).

÷ ~K ~ M C n

A measurable

w h i c h assigns

PROOF.

cardinals measure

concentrating

of Hanf and Scott [ 1 9 6 ~ ) .

on measurable

(Given a proof in B + ~

VK+I) o

the ~-formulas

the ultraproduct

concentrates

on degree n.

it will involve only reflec-

It w o u l d be desirable under extensions assumptions,

(W+I,~,Tv)

this from

by a measure which this is our iff it is true in

Since almost all v are of

reflection

to degree n w h e n we

the argument°

to have a result of the above sort w h i c h

of the base theory.

Of course,

types and this is perfectly

is stable

if we make restrictive

such as "all classes are constructible",

ly define equivalence

the proof is

inheriting

is true in V + I

in Vv+ I.

that V + I satisfies

This completes

if n

complexi-

to be arbitrary well orders of Vv+ I,

Now a statement 0(T,x,X)

degree n, this shows adjoin T,T v .

, T,T v)

This has the form ( V + I , c , T ) ; holds

to

to degree

of a fixed complexity;

then in (VK+I,

of the

almost all v that 0 ( T v,x,Xnvv)

reflec-

in the sense

It will suffice

in question are true,

To see this, we take the T

choice of To

the Bernays

and we choose a ~ w h i c h reflects

ty n and is moreover of degree n+l,

and consider

of degree n.

which satisfy the reflection

tion to a certain degree, w i t h e - f o r m u l a s is a b o u n d for each of these,

of degree n+l

3rd order indescribable

Let K be of degree n+l.

show that we can adjoin T, Tv(v
I;

cardinals

then it satisfies

(in fact it is universal

valid and hence

are implied by

is of degree I if it has a normal measure

the set of m e a s u r a b l e

n scheme in V +io

statements

is then:

We note that if ~ is a MC,

tion scheme

We add

~ is a well order

0V~+I(~K'x'XnVK)"

cardinal

if it admits a m e a s u r e

this is the following.

(We can suppose

type existence

The reflection principle 0(~,x,X)

Note 2o

r .

then we can actual-

st~b!e under ~urther

392

extensions

of the theory.

There seems to be no reason why such a pheno-

menon could not occur with suitable n o n r e s t r i c t i v e ever, we only have some results of types is possible

at all,

assumptions.

of the following form:

How-

if a definition

then there is a definition of a specified

kind. We were led to the following

idea by an analysis

of well order types and levels

of Chuaqui's

in the ramified h i e r a r c h y

definition

in [1980]. Well

order types seemed to be distinguished by finding some difference complete

theory of the well order,

of the ramified hierarchy, such differences

or rather of the associated

allowing parameters

ceased to appear,

types that a version of the elementary class comprehension° the same complete

But of course,

theory entails

for all the sets.

one obtained sufficiently substructure

When

large order

argument established

this is not surprising,

comparatibility

in the

segment

as having

under elementary

sub-

structure. DEFINITION. understand

Let ~ be a well ordered (VO=, ~ v , <

,X)xc V.

class.

By the structure A(~) we

Here O is disjoint union.

coding of ~ as a well order of V is unavailable Tn(~)

in A(~).

Note that the We put

= the A I theory of A(e) n °

Note that if ~=~, are isomorphic.

then ~(~) = ~(~)

since the two structures A(~), A(B)

Also note that truth for A I sentences n

over V is definable

in KM. THEOREM. ~n(~)

If there is a definition of types • in KM,

then for some n,

is such a type.

PROOF:

If T is defined by a A I formula, then the statements x ~ ( ~ ) for n theory: to see this, note that ~ W s V ~ W = ~ & xcT(W))

x~V are part of the ~ is expressible

in A(~)~

by the assumption

Hence

if ~n(~) = Tn(B),

T(~) = ~(B),

Here is another result of the sort "if anything works, time, however, THEOREM Define

(of B):

so ~=B

that T is a type.

we need a theory stronger

this will".

This

than KM.

Let F : On ÷ V, and let F be a definable

filter on On.

393

• F(X) = {t~V I ~S

~F{<[t e F (XnRK)}}

= {teVl(Fa.e.K)

(tcF ( X n R ) ) }

Suppose that • is a definition of types.

Then there are F,F such that

• (x) = ~F(X)PROOF:

Suppose that • is given by a a I definition. n

Let F be the filter generated by the S such that for some XsV, S =

{~(RK+I,~,XnR K) ~ (Univ,~,X)}

Let F~(X) = TR<+I(x) for XsRK+ I Now teT(X) ÷ o n an F class, t ~ TRK+I(XnRK), t g r(X) +similarly t ~ F (XnRK).

i.e. teFK(XnR ~) and

Consequently ~(X) = TF(X)

as

claimed.

0

Argument works in KM if definition of a is simple enough (E~).

Note:

In conclusion we would like to indicate how to use the definition of types as complete theories to obtain the following conservative extension theorem.

Note that while no DC is required, only sentences rela-

tive to sets are conserved.

Here oV indicates the result of relativiz-

ing all quantifiers in o to V. THEOREM~

KM + ~I- V

implies KM I- V ,

i.e. adding types is conservati-

ve over the set part of KM. Note: KM.

This follows from the development of constructible classes in That such a development can be carried out was noticed by several

people (including Solovay, Marek, Chuaqui), often in connection with extending Cohen's methods to class set theory.

The matter is discussed

in print in Mostowski and Marek [1974] and Chuaqui [1980].

See also

Chuaqui and Marshall [1978]. PROOF.

For this discussion,

let us write ~o(~) for the first order

theory of (V 0(Vx~) , ~v,E ,X)x~V , where E is the c-relation for L (V) imposed on V x ~o (The argument is relatively intensive to the particular version of the ramified hierarchy used). We can now define "small" well orders° Peq(~)

iff ~ ] ~ , y ( B < y ~

& L~(V) < Ly(V) )

394

and "large" Gd(~)

iff

not Peq(~);

we then have the lemma: LEMMA:

Va,B(Peq(~) -+ (a=B +÷to (e) = to(B))

Thus: either all ~ are small, and to is a type; or else some ~ is large, and we easily get comprehension for some LB(V), 6<~. To prove the lemma, it will suffice to suppose e
The

argument assumes familiarity with constructible sets, but uses nothing which presents special problems for proper classes. that e is minimal, and B also.

Let D be the definable elements of L (V)

(allowing parameters from V).

Then D
of the usual sort D = L ,(V).

Thus L

tical point.

We may suppose

but by a collapsing argument

(V)=< L (V)

Thus LK(V) < LK,(V), with K
with say ~ as cri-

But then to(K) = ~o(~');

by the minimality we must have had L (V) < LB(V) all along° REFERENCES Chuaqui, R.

[1980]

Internal and forcing models for the impredicative theory of classes, Dissertationes Mathematicae 176.

Chuaqui, R and [1978] Marshall, V.

Constructibility in the classes, in Mathematical Conference, ed. Arruda, Marcel Dekker, New York,

Hanf, W. and Scott, D.

[1961]

Classifying inaccessible cardinals, Notices AoM.S. 8, p. 445.

L~vy, A.

[1965]

Definability in axiomatic theory I, Proc. 1964 Internat.Congress Logic, Meth.and Philo of Science, ed. Y. Bar-Hillel, North Holland, pp. 127-151.

[1963]

Independence results in set theory by Cohen's method. IV, Notices AmeroMath.Soc. I0, p. 593.

Marek, W. and Zbierski, P.

[1972]

Axioms of choice in impredicative set theory, Bulletin d L'Acad~mie Polonaise des Sciences, S~rie des sciences math., astro et phys., 20.

Marek, W. and Mostowski,A.

[1974]

On extendability of ZF models to KM models, Keil conference 1974, Springer Lecture Notes in Mathematics, vol. 499.

Mostowski,A.

[1972]

Models of second order arithmetic with definable Skolem functions, Fundamenta Math. 75, pp. 223-234.

impredicative theory of Logic: First Brazilian da Costa and Chuaqui, pp. 179-202o

GENERIC EXTENSIONS WHICH DO NOT ADD RANDOM REALS Jacques Stern Universit~ de Caen D4partement

de ~ t h 4 m a t i q u e s

et de M4canique

14032 Caen Cedex, France

We give a sufficient condition ensuring that a partially ordered set P has the property that no random real appears in the corresponding generic extensions. As an application sistent,

the statement

we show that, provided ZF is con-

: "there is a (light face) E2 non measurable

is consistent with various other statements

set"

like:

i

i)

Every

E2 set has the property of Baire.

ii)

Every

~2 set is Ramsey.

iii)

Every set has the property of Baire (in presence of the restricted axiom of choice DC).

I. NOTATIONS AND RESULTS. I.i

In order to state our main technical result, we will use the

framework of topological ordered sets. the author

[7]

,

This framework was introduced by

in order to give an alternative proof of the

following result of Shelah [5]. THEOREM i.

If the theory

ZF

is consistent,

then the theory ZF+DC+

"every set 0f reals has the pFoperty 0f Baire" is consistent as well. A topological ordered set is a partially ordered set endowed with a topo!ogy (which is not necessarily Hausdorff).

A topological ordered set

is separabl e if it has a countable basis of open sets. DEFINITION.

An open subset

if any finite subset of

V

V

of a topological ordered set is coherent

admits a common lower bound.

A topological

ordered set is locally coherent if it has a basis of coherent open sets. In this paper we shall be mainly interested in partially ordered sets which can be endowed with a topology locally coherent.

which makes them separable and

Actually these partially ordered sets have been con-

sidered in the literature. Recall that a partially ordered set o-centered if it is a countable union of subsets

•

is

(Xn) such that, for

396

every n, finitely many members

from X n admit a common lower bound.

it quickly follows from this definition that

the set

~,

Now,

endowed with

the topology generated by the sets (Xn), is separable and locally coherent. Conversely,

any separable,

locally coherent topological ordered-set is

o-centered and therefore the two notions coincide. From the previous remark,

it is clear that the whole paper could have

been written using the notion of o-centered partially ordered set, with some care (and some work)

in order to handle iterations;

the choice that

was made reflects both the taste of the author and his conviction thatthe alternative approach is useful. 1.2

The following will be our main technical tool:

PROPOSITION.

Let

~

be a separable,

locally coherent,

dered set; then, in a generic extension corresponding to

topological or~, no real

is random over the ground model. It should be said that our condition was inspired by several arguments previously used by A. Miller

[ 2 ].

The proof of the proposition will

appear below in section 2. Using the remarks of section I.i, the above proposition can also be read as follows: PROPOSITION.

If

•

is a o-centered partially ordered set, then, in a

generic extension corresponding to

~, no real is random over the ground

model. 1.3

It is well known that if no real is random over L, then there is

a (light face)

I

Z~ set which is not Lebesgue measurable.

In fact, any

real ~ belongs to some Borel set of zero-measure with a code ~ in

L

and

we can pick the first such code ~(~), with respect to the canonical well ordering of L. {(~,B) is a

The set :

~(~)

~

i

~(~)}

Z 2 non measurable subset of

~2;

to build a Z~ non,measurable subset of

from this set it is not difficult ~.

From the previous remark, build models,

it is clear how proposition 1.2 can be used to i in which a non-measurable E2 set exists. In this way, we

prove: THEOREM.

Assume there is a standard model of

ZF;

standard model of

ZFC with the following properties

i)

Z~ set has the property of Baire

Every

then, there is a

397

l

ii)

Every

iii)

E

set is Ramsey

~2

There is a non m e a s u r a b l e

E~ set.

This theorem should be compared w i t h a result of Raisonnier author, w h i c h states all

that the m e a s u r a b i l i t y

of all

~2 sets implies

~z sets have the property of Baire as well [4]

o

theorem shows - inter alia - that there is no converse is not really new;

and the

1

the author had already observed

that

The above implication.

that,

This

in Shelah's i

model w h e r e all sets have the property of Baire set was not measurable;

[3];

the proof that appears

the present paper is straightforward

1.4

some

~2

but this was through an indirect argument

using some w o r k of Raisonnier effective

[5],

in

and gives a result which is more

in character.

Combining p r o p o s i t i o n

[ 5 ],

1.2 w i t h the construction of Shelah

we show that no hypothesis

concerning the Baire property is i strong enough to imply the m e a s u r a b i l i t y of E~ sets. THEOREM.

Assume there is a standard model of ZF;

standard model of i) ii)

ZF+DC,

then,

there is a

such that:

all sets of reals have the property i there is a n o n - m e a s u r a b l e E 2 set.

of Baire.

2. THE MAIN RESULT. 2.1

We start to give the proof of propositi0n

separable,

locally coherent,

any real w h i c h appears belongs

topological

in some generic

space.

extension

to a Borel set of zero-measure

1.2;

we let

• be a

We want to show that corresponding

to

coded in the ground model M. As

usual, we don't w o r k with actual real numbers but with elements 2m

of

2~;

is endowed w i t h the product measure ~ of the u n i f o r m probability

measure on {0,i}.

Any element of

2~ i n

the generic extension

is denoted

by some term T such that the boolean value [ T is an element of

2~

]

equals

I. We fix such a term T.

LEMMA.

For any coherent open subset

exists an element (i)

Vp~V

PROOF OF THE LEMMA. of 2n;

s

of

~q~p

V

of

•

and any integer n, there

2n, such that the following holds: q l~T extends

s.

We fix an enumeration

s I ..... s i .... s k of the elements

if the conclusion of the lemma does not hold,

i, there is an element Pi of

V

such that

then,

for any index

398

Vq
q ~T

extends

si

w h i c h means Pi

i~

Tdoes

not extend si;

now,any common lower b o u n d to all the Pi'S forces T does not extend any element of 2n; this gives a contradiction 2.2

and finishes

We now use the fact that

open sets;

•

we fix an enumeration

has a countable basis of coherent

(Vn) of such a basis w i t h the property

that any element of the basis appears lemma,

the proof of the lemma.

infinitely many times.

for such interger n, we can find an element (2)

Vp~V n

~q~p

q I~

~ extends

Using the

sn of 2n such that

sn.

We note that all this takes place in the ground model;

still w o r k i n g

in

the ground model, we let

Now,

Un =

{~: ~ extends

B

lim sup (Un) =

=

sn} n m

u n>m

U n-

it is well known that if the series z

n

converges,

~

(un)

then,

of zero-measure there is a

lim sup (Un) is of zero-measure; coded in the ground model.

G~ set of zero-measure

sense that it has the same code). tion of the term T,then B belongs exists a condition p

iF T~

p

n n~

of

•

B

thus, B is a G~

set

Now, in any generic extension

w h z c h is built like

B

(in the

We claim that if ~ is the interpretato

B.

If this is not the case,

and an integer

m

there

such that

Uno

This means (3)

Vn~m

b I~ T does not extend s

Now, we pick an open neighborhood fixed;

V

appears

for some q

q~p

n

in the basis

infinitely many times in the sequence

can pick an integer get,

V of p

n ~ m for w h i c h

V n = V.

that we have (Vn) so that we

From property

(2) above, we

:

I~ T extends

sn

but this contradicts

(3).

appears

extension,

in a generic

So we have shown that any real B, w h i c h belongs

to some G~

set of zero measure

coded in the ground model M, hence is not r a n d o m over M. 3. LOCALLY COHERENT TOPOLOGICAL ORDERED SETS. 3.1

We now study the closure properties

of the class of coherent

topological theorem

ordered

1.3.

PROPOSITION. satisfies PROOF. basis

sets.

This wili be useful

Any separable,

locally

If (Vk) is a sequence •

, then,

Vk, hence,

3.2

of coherent

pairs

(k,f) with

where

fFk denotes

(£,g)

of

~

< (k,f)

is a coherent have proved LEMMA.

used to add a new element

f c~;

if

the ordering

£ exceeds

basis;

of

m~ domi-

UD consists

of

on UD is given by:

k, f~k = g~k and ~n f(n) ~ g ( n ) of f to integers

induced by the usual

:

topology

furthermore,


on

UD is endowed

mx~

. Clearly,

if a condition

(k,f)

this is

g~k = f~k}

open set containing

the given condition

locally

Composition with UD. Let ~ be a separable, as in Stern

[7]

,

locally

we will

and UD w i t h a topology.

coherent,

topological

thus, we

coherent,

definition which that

is enough

p c

ii)

k e

iii)

Actually,

of triples

T is a term of the forcing from the integers

iv)

for any

We identify The ordering

on

~

(p,k,T)

we will use a different

£ ~k

and

and more

topo-

straightforward

(p,k,T) where

language

and

p I> T is a mapping

for some j.

and (p,k,o) whenever

~ UD is defined by

(q,£,~)~(p,k,T)

ordered

to the integers

i
conditions

topological

set.

for our purposes.

• ~ UD consists

i)

ordered

endow the one step iteration

logy but it turns out that this gives a simpler

q~p,

(k,f);

the following:

UD is a separable,

3.3

~f

members

then {(k,g)

Recall

set

is a countable

~I has two different

lying in the ground model.

the restriction

topology has a countable

~

ordered

it is not an antichain.

kc~ and

with the topology

of

open sets which

any set of cardinality

UD is a set of conditions

set;

topological

UD

nating all members

given,

coherent,

the c.c.c.

for

in some

to give the proof of

We first note the following:

p I> T = o.

400

q I~ Vn

o(n) ~ ~(n)

q l~ Vi
where

~

~ UD by

A (V,k,u,v)

=

{(p,k,~)

: pcV

and

p I~ T(u)

r (V)

=

{(p,0,T)

: pcV

and

p I~ Vn T(n) = 0}

V

is an open subset of

we endow

~,

and

• ~ UD w i t h the topology

Actually,

the same topology

topology defined on ~

~

~

~

k,u,v

is obtained

are integers;

if we restrict

~;

from this,

UD is separable.

UD is a separable,

= v}

finally,

generated by all these subsets.

to range over a countable basis of

LEMMA:

:

the open set

V

it follows that the

Actually,

locally coherent,

we have

topological

ordered

set. given a condition

PROOF:

of

•

j(i),

containing p;

(p,k,T), we pick a coherent open subset

now for any integer

i
V

there is an integer

such that: p I~ T(i) = j(i)

we claim that the intersection A (V,k,i,j(i))

U

i
is a coherent open n e i g h b o r o o d (Pl,k, T I) ..... (Pr,k,Tr) bound

q

to

of the sets

of

(p,k,T);

indeed if conditions

are given in U, we can find a common lower

Pl ..... Pr;

if o is a term of the forcing language

such

that q I~ ~ is the supremum of

~l,...,Tr

then,

it is easy to check that condition

tions

(Pl,k,Tl) ..... (Pr,k,Tr).

We now define an embedding

k

from

•

(q,k,o)

into

lies below all condi-

• ~ UD by:

k(p) = (p,O,~) where ~ is a term of the forcing language [Vn

~(n) = 0 ] =

It is not difficult i) ii) iii) iv)

such that

~.

to verify that

k

is a h o m e o m o r p h i s m

P

~

k

is increasing

k

has the following properties:

from

P

onto an open subset of

UD

if p,p'

are incompatible,

patible

in

if ~ i s

a maximal

chain in

then,

k(p)

and k(p')

are incom-

~ ~ UD.

• ~ UD.

antichain

in

P, k ( ~ )

is a maximal

anti-

401

Embeddings

satisfying properties

i) to

iv) are called sweet

in

[.7 ] and we keep this terminology. 3.4

Composition with Mathias As in the previous

topological language

ordered set;

forcing.

section,

•

is a separable,

locally coherent,

we assume that some term U of the forcing

is such that [ U is a Ramsey u l t r a f i l t e r

We then consider (see Mathias triples

[i]),

(p,s,T) i)

on ~

~ =

the one step iteration

of

w h i c h we denote by

I P

and Mathias

~ ~ Mu ; ~ ~ MU

forcing consists

of

such that

p c

ii)

s is a finite sequence of integers

iii)

T is a term of the forcing language and

We identify conditions The ordering on

~

~

(p,s,~)

and (p,s,o) w h e n e v e r

p IF T ~ U. p IF T = o.

M u is defined by

(q,t,o) ~(p,s,T) if q
t extends

s,

q IF t-s

In order to define a topology, A(V,s)

= {(p,s,T)

F(V)

! T

and

q IF o!T.

we consider

the sets

: p~V}

= {(p,0,T)

: p~V

is an open set of

V

endow

• ~ M u w i t h the topology generated by all these subsets.

PROOF. P

~

P

, hence,

~ M U is a separable,

Given a condition

containing p;

of (p,s,T);

s

section, we can restrict

countable basis of LEMMA.

and

T~ T = w}.

where

the previous

•

and p

a sequence of integers; the open sets

the topology

is a coherent

term of the forcing language r q IF o = n Ti i=l

lies below all conditions

= (p,~,~)

it is not difficult

q

to

P I ' .... Pr;

if o is a

(Pl,S,rl),...,(Pr,S,rr).

k : ~ ÷~

s M U by

;

to check that it is a sweet embedding.

4. GENERIC MODELS W I T H O U T RANDOM REALS.

V

of

are given in

such that

We finally define an embedding k(p)

ordered set.

open n e i g h b o r h o o d

(Pl,S,~l) ..... (Pr,S,Tr)

A(V,s) we can find a common lower bound

then (q,s,o)

topological

(p,s,@, we pick a coherent open subset

if conditions

As in

to range over a

is separable.

locally coherent

we claim that A(V,s)

indeed,

V

we

402

4.1

In order to prove theorem 1.3, we start with a model of ZFC+V = L

and, in this model, we define a transfinite sequence of Length ~I of separable, locally coherent topological ordered sets ( ~ ), together with sweet embeddings k ~ : P~ + ~ , ~<~<~"i such that whenever ~<~

~o

is

UD

if ~ is even ~ + i is ~ ~ UD, where ~ ® UD is endowed with the topology defined in section 3.3. and k ~ +I is the embedding defined in the same section; k ~ + l if defined by composition for ~<~.

iii)

if ~ is odd, we note that since ~ is a separable locally coherent space, it is c.c.c and therefore the continuum hypothesis holds in any generic extension of the ground model via a generic subset of ~g ; as is well-known, the existence of a Ramsey ultrafilter follows from CH; using the so-called maximum principle we can find a term U in the forcing language corresponding to ~ such that: U is a Ramsey ultrafilter ] = we then define

~+I

to be

~

MU;

I; the topology on P~+I and

the embedding k ~ + l : ~ + i ÷ ~ + I are defined according to section 3.4; k ~ + l is defined by composition for ~<~. iv)

if ~ is a limit ordinal, ~% is the direct limit of the system given by ( ~ ) ~ < A together with the maps (k~)~<~<%; (k~) are the limit maps. The topology on ~% is the limit topology whose open sets are exactly the sets X ! ~ % such that ~
-I k~% (X) is open.

LEMMAo ~% is a separable, locally coherent topological ordered set; furthermore, the limit maps k~% are sweet embeddings. PROOF.

We note that, for any ¢<%, k ~ % ( ~

) is an open subset of

~%;

this is because k ~ is an embedding from ~ onto an open subset of ~; from this, it £oliows that ~% has a countable basis; a basis for ~% is obtained in the following way • ~; apply k ~ ; take the union o~ local coherence of ~% stems from the open sets k ~ ( ~ ) . The fact that onto an open subset of

~%

: take a countable basis for each all these sets° Similarly, the fact that ~% is the union of the k~ is an homeomorphism of ~

is proved by a similar argument.

There

403

remains

to check conditions

beddings; 4.2

We let

maps

ii) to

iv) of the definition of sweet em-

but these are standard facts on direct limits.

k~

•

be the direct limit of the ordered sets

defined in the previous

corresponding

section.

generic extension M[G],

continuum hypothesis

holds;

~

(~)

cardinals

are preserved

also, any real of

M[G] appears

intermediate

generic extension obtained by forcing with some

Because

is a separable,

P~

and

is c.c.c so that in the

locally coherent

and the in an ~

space, no random real can

be found in such an intermediate exist in

extension; finally, no random real can i so that there is a ~ non-measurable set in this model.

M[G],

We now prove the following. PROPOSITION.

In the model

M[G],

l every E 2

set has the property of Baire.

1

PROOF.

It is well known that

~2 sets have the property of Baire if and

only if, for any real 6, there exists a dense

%

of cohen generic reals

over L[~] (see eg [61). Also, it is known that any model obtained by performing iterations of length H I of c.c.c ordered sets has this property,

as long

as iterations with

proof see E71 section 1.3.2). follows.

UD

appear cofinally

From these observations,

(for a

the proposition

4.3 We now turn to the Ramsey property; PROPOSITION. PROOF.

Let

parameter some

In the model

~

~(x,e)

appears

be a

M[G], every ~2 set is R~msey. i ~2 statement with a real parameter

in an intermediate

generic extension

and we may assume that ~ is odd so that

U is a term denoting a Ramsey ultrafilter. generic extension M~+ 1 is obtained

corresponding

to

~+I

6;

this

corresponding =

to

~ $ ~ MU' where

We let M~ (resp. M~+ I) be the

~(resp.

~+i ) .

We now work in M~;

from this model by forcing with Mathias

forcing

[11;

we let y be a term of this forcing denoting the generic subset of m wich is added; (I)

it is well known that for some condition of type (0,a),

(O,a)

I~ ~(¥,~)

or

(O,a) l ~ ( y , e )

where ~ is a term denoting 6.

Now,

it is known also that, if

generic subset of ~ added via Mathias is generic as well. i)

forcing,

From these observations

either any infinite

subset

g'

or else any such infinite M~ (g') ~ ~

~(g',~).

is a

any infinite subset of g

it follows that of gna is such that

M~(g') ~ ~(g',6). ii)

g

subset is such that

404

By Shoenfield absoluteness following holds i)

in

lemma, we get, in the first case,

that the

M[G]

P (gna) ! {x : ~(x,e)} where P (gna) is the set of infinite

subsets of gna;

in the second case, we get similarly: ii)

P (gna) ! {x : ~ 6(x,~)}; l

thus,

the R~msey property

4.4

for

~2 sets is established.

We now turn to the proof of theorem 1.4.

If we look at Shelah's paper [ 5] we realize

that this model of

ZF+DC where all sets of reals have the property of Baire is built by a transfinite

induction of length

~i,

similar to the one we have per-

formed in section 4.3 but -of course- more elaborated; that the intermediate

partially

step of the interaction

we also realize

ordered set built after each countable

is o-centered;

from these observations,

theorem

1.4 follows. For those who are interested

in the approach via topological

we will sketch an alternative Contrary

to what we have tried to do in the rest of the paper,

not be possible

to make this section self-contained;

assume that the reader

is familiar with

notion of sweet topological DEFINITION. satisfies (*)

ordered sets,

proof based on our paper [ 7 ] . we

[ 7 ] and with the

ordered set, w h i c h appears

An ordered topological

it will

therefore,

space is supersweet

there. if it sweet and

the following property:

given conditions

p,q,q ~p,

q, there exists a neighborhood W elements

of

W

Clearly,

any supersweet

together w i t h a neighborhood

V

of

of p, such tbat any finite number of

have a common lower bound in V. space is locally coherent.

In order to prove theorem 1.4, one has to go through the constructions of

E7]

and check that they can be performed using super-

sweet ordered sets instead of sweet ones. supersweet

direct limits

(under sweet maps).

easily handled so we restrict §3 of

This amounts

sets are closed under amalgamation,

[7]

and recall

Direct

ourselves

limits and amalgamation

to composition.

that the topology on

is defined there and w h i c h we consider now, appears above in section 3.3.

to showing that

composition w i t h UD and

~

is not

We claim that if

•

e

are

We refer to UD, w h i c h

the one w h i c h is supersweet,

the

405

same is true of (proposition (*): (**)

P ~ UD.

We note that by section i.i of

[7]

i), it is enough to verify the following weakened form of

For any condition

a neighborhood W

of p

p

and any neighborbood

V

of p, there exists

such that any finite number of elements of

W

have a common lower bound in V. Following the notations of (p,k,~),

[7] ,

we consider a condition

together with an open neighborhood

the intersection of some

V;

ql,...,qs of p lying inBil'"°Bis

ql i~ ~nl ( ~ ) ' ' ' " q s Bjl .... ,Bjs

of

ID ~n s (T);

is

k( ~);

we first choose

respectively

and such that

then, we pick open neighborhoods

ql .... 'qs respectively

any finite subset of

V

B(i,k) with finitely many open sets

A(il,n I) .... ,A(is,ns), plus possibly the set extensions

we may assume that

such that:

Bjl has a lower bound in

Bil (similarly for

Bj 2 .... ). Next, we pick a neighborhood B

B. such that any l element of B has extensions in BJI,B j .... ,Bis.~ Finally, we choose 2 a neighborhood Bj of p such that any finite subset of Bj has a lower T

of

p

included in

bound in B . i

We let

W ° = B(j,k)nA(Jl,nl)

n...n A(Js,ns).

set of integers such that p L~ T~ k = t;

We let

t

be a finite

if conditions

(Pl,k,Tl) ..... (Pr,k,~r) are given in Wo, with (I)

Pl

IF Tl~k = t,...,pr

IF Tr Ik = t

then, we can build a common lower bound by considering an element q of B

below P'Pl ..... Pr and a term o of the forcing language such that T q IF a = sup (T I ..... Tr).

CLAIM i:

Condition

PROOF OF CLAIM:

(q,k,a) belongs to

this is because

q e B

B(i,k). ! B i-

l

CLAIM 2:

Condition

PROOF OF CLAIM.

to A(il,nl) ..... A(is,ns).

We focus on A(il,n I) and we write

confusion can arise. pectively,

(q,k,o) belongs

n

for

n I as no

v

We pick extensions P'I ..... P r of pl,...,pr res-

lying in Bjl and forcing

~n(Tl) .... ,~n(Tr);

we also pick

406

an extension p' of q, lying in B. : this is possible because q belongs Jl to B . Now, any common extension q to p' 'Pl "" 'Pr lying in B. forces ~n(Tl) ^ . . .^~n(Tr) as ~n(T) is a statement like to see that

T(u) ~
~(u) ~>v, Vm T(m) = 0, it is easy

q IP ~n(O) this shows why

(q,k,a) belongs to

A(il,n).

Now, we show that the statement Pl I~ Tl~k = t can be ensured by taking (Pl,k,~l) in some open set WI; for any integer u
WI

be the intersection of all open sets

any of the corresponding statements (Pl,k,Tl) is in

WI

then

Pl

T(u) ~
A(j,n), when

T(u) ~>v;

can only decide

~n is

clearly, if

TIFk to coincide with t.

Finally, we have shown that any finite subset of WonW I has a lower bound in V, except if k(]P) appears in the definition of V; in this case, we simply replace proof of theorem 1o4.

WonW I

by

WoOWlnk(~).

This completes the

REFERENCES [i] Mathias, A.R.D.

Happy families,

Annals of Math.Logic,12(1977)pp.

59-111. [2] Miller, A.W.

Some properties of measure and category.Transactions of the Amer.Math.Soc.

[3] Raisonnier,J.

266(1981),pp.93-I14.

A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel J.Math.

(to appear).

[4] Raisonnier,J. Stern, J.

The strength of measurability hypotheses. appear).

[5] Shelah, S.

Can you take Solovay's inaccessible away?

(to

(to appear) [6] Stern,J.

Some measure theoretic results in effective descriptive set theory,Israel J.Math.20(1975) pp. 97-110.

407

E7] Stern, J.

Regu%arity properties of definable sets of reals. (to appear).