Patras Logic Symposium: Proceedings

PAWS LOGIC SYMPOSION Proceedings of the Logic Symposion held at Patras, Greece, August 18-22,1980

Edited by

GEORGE METAKIDES Department of Mathematics University of Patras Greece

1982

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK OXFORD

'NORTH-HOLLAND PUBLISHING COMPANY - 1982 All rights reserved. No part of rhis publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without rheprior permission of the copyright owner.

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Library ot c ongrras Cataloging in Publication Data

Logic Symposion (1980 : Patrai, Greece) Eatras Logic Symposion. (Studies in logic and the foundatians of mathema:ics 109) Includes bibliographical references. 1. Logic, Symbolic and mathematical--Congresses. I. Metakides, George, 194511. Title. 111. Series. QA9.AlL65 1980 511.3 82-14107 ISBN 0-444-86476-8 (u.s.)

V.

.

PRINTED IN T H E NETHERLANDS

:

ix

PROLOGOS ‘After over two milennia Logicians are returning to meet in Greece’. Thus the headlines ran in the Patras news media during the Logic Symposium from August 18 to August 22, 1980. From all over the world they met indeed, as 23 countries were represented from five continents. In keeping with such cosmopoly we decided to invite a representative selection of topics from most areas of Logic, rather than to focus on a particular aspect. So, h&e you fmd loosely arranged, a broad spectrum of papers ranging from algorithmic troubles to the morasses of set theory.

This was the first International Logic Meeting to be held in Greece. It was sponsored by the International Council to Scientific Unions and the Association for Symbolic Logic and held under the aegis of the Greek Ministry of Culture and the University of Patras. Having the 02-group mept in Patras just before the conference helped to ensure the participation of many eminent Logicians. The Organizing Committee consisted of J.E. Fenstad, A. Kechris, A. Levy, G. Metakides, S. Negrepontis, and G. Sacks. The University officials, the Town officials and its various industries and news media, they all offered their support and hospitality with a generosity that would have pleased Xenios Zeus. Special thanks are owed to Alexandra Pliakoura and Ioanna Riga of Patras who handled the local organizing with efficiency and charm. To Roberto Colon and Marion Lind of Rochester N.Y. belong the credits for the careful typing of this typing together with a personal debt of gratitude. One notable result of the meeting was a proof of the compatibility between having a good time and getting work done. The lectures were very well attended in spite of the lure of the beaches. Theorems were proved and conjectures refuted in classrooms as well as in tavernas and Mediaeval castles.

As Professor Kleene was widely quoted saying to the Press: ‘When I was helping to found the A.S.L. 46 years ago, little did I dream that some day we would have meetings as fine as this one’. May 1982

George Metakides

PATRAS LOGIC SYMPOSION

G . Metakides led.) 0North-Hollandhblishing Company, 1982

1

P.ECURSIVE FVNCTIONALS AND Q U A N T I F I E R S O F F I N I T E T Y P E S R E V I S I T B I ) I11 S. C . Kleene U n i v e r s i t y o f Wisconsin Madison, Wisconsin 53706

9.

INTRODIICTION ANP REVIEW. RFQFTR I 1978 and RFQFTR I 1

9.1. a r e the published versions o f my

l e c t u r e s o f June 13, 1977 a t t h e Second Symposium on Generalized Recursion Theory a t Oslo and June 19, 1978 a t t h e Kleene SvmDosium i n Madison. I n RFQFT I 1959 and RFQFT I 1 s t i t u t i o n of A-functionals In

1978 I proposed

1963,t h e r e

1

were l i m i t a t i o n s on the sub-

and on the use o f the f i r s t r e c u r s i o n theorem.

t o overcome those l i m i t a t i o n s by a l t e r i n g the l i s t o f

schemata and i n t r o d u c i n g new computation r u l e s .

The new schemata i n c l u d e

ones t h a t p r o v i d e d i r e c t l y f o r s u b s t i t u t i o n o f A-functionals and the f i r s t r e c u r s i o n theorem.

The new computation r u l e s do n o t r e q u i r e the values of

p a r t s of expressions being computed t o be computed unless and u n t i l they are needed.

The s t a r k l y formal character which t h i s gives t o the computations

gave reason t o seek a semantics f o r t h e language used i n them. drew the l i n e s I proposed t o f o l l o w i n t h i s semantics.

In

1980, I

Here I s h a l l con-

t i n u e along those l i n e s . I s h a l l endeavor t o repeat enough m a t e r i a l here from the two e a r l i e r

papers

1978 and 1980 o f

t h i s RFQFTR s e r i e s t o enable persons u n f a m i l i a r w i t h

them t o f o l l o w i n the main the present one, though n o t t o appreciate f u l l y how i t f i t s i n t o the c o n t i n u i n g i n v e s t i g a t i o n . Readers f a m i l i a r w i t h

9.2.

In

1978,t h e

may s k i p t o 510.

(An RFQFTR I V i s projected.)

2

primary o b j e c t s were o f the f o l l o w i n g f i n i t e t,ypes:

S.C. KLEENE

2

type 0

=

{O, 1 , 2,...1

( t h e natural numbers);

place functions from type the natural numbers.

i= 1 , For ;

into

SL, yL,.

the superscript

l o , 1 , 2 , ...) . &, b,

. . range over

'lL"

t.ype i t 1 = the ( t o t a l ) one-

type

91,8,

... a r e each

i (i=

over

fl, 1 , 2 , 3 ) .

may be omitted.

,...> )

Then I considered functions ( i n t o (0, 1 , 2 where

c, ... range

+@),

$(w) ,...

a l i s t of variables each ranging over one of the

These functions a r e " p a r t i a l " . A partial function $(a) is

f i n i t e types.

one which, f o r each choice of values of i t s variables from t h e i r respective ostensible ranges, e i t h e r takes as value a natural number o r i s undefined. 3 A partial function which i s defined f o r each such choice i s t o t a l . I defined when a function + @ I )

e

= (0l,...,el)

-

i s p a r t i a l recursive - i 8, ~where

7, > 0 i s a l i s t of "assumed" partial functions

with

(variable o r constant), each of a given l i s t of variables of our f i n i t e types, or equivalently ( w i t h

e variable) when a functional $(ea) i s

partial recursive. Toward developing a semantics t o clothe the bare bones of formal called f o r representing the types

1978,my 1980 proposal

computations a l a 0, 1 , 2 , 3 w i t h i n

types

. . . .

0, 1 , 2, 3.

(Another semantics was developed by

Kierstead in 1980, 1982.) Here type

b = EO,

1, 2

,..., @ I

where

@

= undefined.

For

0, 1 , 2 , type ( L t l ) . = the "unimonotone" p a r t i a l one-place functions it1 a from type 1. i n t o the natural numbers. I shall proceed t o explain

1=

"unimonotone" f o r successive values of

o3

Ishallwrite

= A A ~

type

i-

Take

o o =0 ,

@ ' = A & O @ ,

0, yo

{@'I,

N~ =

> 0

J+l. @2=li1@,

i - I@> = E O , I , 2 , . . . 1 = type 0, i - {Q2>, 13= type 3 - ~ ( 3 ~ 1 .

= type

type

f o r the moment,

Remember t h a t

d

and

l1=

is, being

par-

t i a l functions, need n o t be defined f o r a l l members of t h e i r ostensible

3

Recursive Functionals and Quantifiers of Finite Types Revisited I11

domain type

(i-1)'.

Let

&i=$

( I d

extends

have i t s usual set-

@I1)

t h e o r e t i c meaning, considering t h e f u n c t i o n s as s e t s o f ordered p a i r s w i t h 0 +1 *J I Whenever &i ( 8 ) i s d e f i n e d f o r a given second members E 1 '+1 .i s h a l l wish t h a t t h e f a c t and the value o f aJ ( B ) t o depend o n l y on de-

.

ii,

.

f i n e d values o f $-, never on the absence o f d e f i n i t i o n o f values. So i f '+I .i '+1 .i kJ (B ) i s d e f i n e d and k i 3 $-, then &J ( a ) s h a l l be defined w i t h T h i s being so, I

the same value.

i=

For putting if

kl(

0

8"

I d e f i n e when

0,

:&O = 80

) = E

E

v

monotone l i k e w i s e , a f t e r

b0 =

i s monotone means t h a t ,

&'

1". then

i s the constant f u n c t i o n

&'

i s monotone.

-0

a

h&'n.

&'

i s unimonotone means t h a t which a- 2 ( -a1 )

8'

for

explain.

&'

i s unimonotone means simply t h a t

&'

i s monotone.

i s monotone and, f o r each

w. r. t o &2

The basis

i'

(with

8'

c

&.'

and i 2 ( 8 ' ) = &'(&')),

represents the i n f o r m a t i o n about and

8'

&'

*1

a

for

o u t s i d e of i t s subfunction

&'

as I s h a l l t h a t i s used

being i n t r i n s i c a l l y determined

8'

means t h a t i t i s determined by working from w i t h i n

9.3

0, i . e .

i s defined, t h e r e i s a E i q u e l n t r i n s i c a l l y determined basis

- 2 '1 i n determining t h e value a ( a ),

&'

type

E

without looking a t

8'.

To formulate t h i s , I s h a l l proceed a t once t o t h e n o t i o n o f an "oracle"

f o r a type-;

function

i n i t i o n o f type

&2.4

T h i s n o t i o n served i n

(bottom p. 15).

1980 f o r

the f i n a l def-

The reader should have no d i f f i c u l t y i n

e x t r a p o l a t i n g from t h e f o l l o w i n g account o f oracles f o r type-2 o b j e c t s t o oracles f o r t y p e - i objects, and indeed a d e s c r i p t i o n o f them i s s u b s t a n t i a l l y included i n i t . The major o b j e c t i v e o f t h i s paper i s t o c h a r a c t e r i z e oracles f o r type-3 o b j e c t s ( i n 11.2). An o r a c l e f o r a t y p e - i o b j e c t

i2, or briefly

02 an a -oracle, i s an

agent ( s h a l l we say an agent o f Apollo, and use t h e feminine gender a f t e r

S.C. KLEENE

4

the oracles of Delphi?) who responds t o questions, as follows. t i t l e d t o ask her "What i s &'(&')?",

i1

a question, we p u t an oracle f o r present the envelope t o her. 5

f o r any

;'

E

type

W e a r e en-

i. To

ask such

i n a closed envelope (or chest) and

= -CASE

i2: The i2-oracle pays no a t t e n t i o n t o our envelope ( s t a n d s mute).

Then

&'

Without opening our envelope, the a- 2 -oracle pronounces t h a t

CASE 'L2:

A2(&') is,

@ = @.

i s the t o t a l l y undefined function ;.?

=

&*

CASE 3':

m.

Since she answers

"c"

*1 without knowing what type-; object a

i s the t o t a l constant function

?.&'m.

The &'-oracle opens our envelope, revealing t h a t she will re-1 indeed some values of a , i f she i s to quiresome information about

&',

ll&'(&l)?ll. (Were she willing t o answer

answer our question

without learning some values of

i1,

%'(k')?''

she would do so under Case

T'.)

AS

her f i r s t s t e p toward obtaining such information, she asks the &'-oracle -1 * O who emerges from our envelope a preliminary question 'la ( a ) ? " using an The a* 1-oracle does not reempty envelope (6' = ) . Subcase -1 ) = spond; she stands mute (Case T'). Then so does the a-1 -oracle; a*' ( a

@

m2:

0.

Subcase 3.2': Without opening the envelope, the &'-oracle declares t h a t *1 -0 a ( a ) = n (Case 7'). Thus the i2-oracle learns everything about &',

P

6'

Xi.o!

Depending i n general on the n, she may then -1 ) = m, Subcase stand mute (&2(&') = o r declare t h a t a- 2 (a The a*1-oracle opens the envelope (Case 3'). The a* 2-oracle, observing t h i s , may stand mute, making a- 2( a-1 ) = (She could have been hoping t o get an namely t h a t

=

0)

3.3':

0.

m'.) Or she may pose a f i r s t nonan r, (passing over the f a c t

answer from the &'-oracle under Subcase preliminary question t h a t the ;'-oracle,

'I&'(r ) ? " w i t h -0

E

finding the envelope f o r the preliminary question empty, -1 d i d not answer t h a t ) . Suppose the l a t t e r . As we know, the a -oracle opens

Recursive Functionals and Quantifiers of Finite Types Revisited I11

5

Opening t h i s one, and f i n d i n g r + , i n s i d e , she may stand - 2 01 O r t h e &'-oracle Then so does the &'-oracle, making a (a ) =

a l l envelopes.

0.

mute.

-1 may declare t h a t a ( r ) = + I., I n the l a t t e r event, the a- 2 - o r a c l e may de4 -1 *1 c i d e t h a t t h e information t h a t ( t h e a - o r a c l e opens envelopes) & a (%) =

- 2 -1 a (a )

i s sufficient t o r u l e out

being defined, and accordingly stand mute.

- 2 *1 O r she may decide t h a t i t j u s t i f i e s her d e c l a r i n g t h a t a (a ) may decide t o seek more information by asking another question Altogether, i n t h i s Subcase

r,

3.3',t h e &'-oracle

m. O r she "a * I (q)?".

questions the ;'-oracle

with

a s e r i e s o f d i s t i n c t i n t e g e r s ( p o s s i b l y extending i n t o the t r a n s f i n i t e ) ,

q),

El,

..*, % a

-.-

-1 - 2 - o r a c l e f i r s t asks a question u n t i l e i t h e r (a) t h e a ( i . e . r+ = a -2 *1 -1 which t h e a - o r a c l e does n o t answer, making a (a ) = @,or,with a l l questions

(q)),

%

for

5 < some

- 2 (a - 1 ) -(a

*2 -1 a (a )

=

@)

E < o1 answered, (b) t h e &'-oracle

o r (c), w i t h 5 > 0,

then stands mute

the i 2 - o r a c l e then declares t h a t

m.

Throughout t h e process described, t h e &.-oracle operates determini s t i c a l l y , always doing t h e same t h i n g under t h e same c o n d i t i o n s as she knows them.

Thus,

i n Subcase

ginning, t h e questions f i r s t one

r,

3.3'

w i t h her n o t standing mute a t the be-

%, t--, ..., -5 r , ...

are determined by her, the

o u t r i g h t ( t h e same o f a l l envelope-opening ;'-oracles),

and

r on t h e b a s i s o f t h e e a r l i e r questions and answers ( t h e same -5 of a l l envelope-opening ;'-oracles w i t h h1 3 <- 5 ) ) . each one

{ln

-2

-1 We question t h e a - o r a c l e w i t h an a - o r a c l e i n an envelope.

&.'

But

has an o r a c l e who i s unique (if n o t i n her person, 6 = Using t h i s f a c t and a t l e a s t i n her performance), except f o r

each type-;

object

&' a'.

S.C. KLEENE

6

the determinateness of the behavior of an A2-oracle, her behavior reveals the values of a p a r t i a l one-place function type-i objects $’

o f a variable ranging over

i2

-1

(not j u s t over oracles f o r type-: objects

a ).

I s t h i s function necessarily monotone? I t i s e a s i l y seen t h a t the ‘1 only way i t could f a i l t o be i s when the i2-oracle declares t h a t a- 2 ( a ) =~IJ under Subcase

n =

3.3’ w i t h

-2 7 declare t h a t a ( a ) =

-5

n

for all

i f she does not then a l s o

5 < 5,

3.2’

z1 = xion-.

I (and - 2 *o Apollo) now mandate t h a t in t h i s circumstance she declare t h a t a (la )I I ~ I J under

Subcase

when

=

m.

T h i s mandate completes our specifications f o r an oracle f o r a type-2 object.

A p a r t i a l one-place function $‘

from type-i into

No

is

u-

monotone (and t h u s of type 2) i f f i t i s the function whose values a r e revealed by a (monotonicity-assuri8) oracle f o r a type-2 object.

7

For any $’-oracle f o r whom an a* 2-oracle declares t h a t a.2 ( -1 a ) =

basis 6’ for i1y. l y the values of

k2(i1)

= ~ I J (so

$’

8’

1. &

i2

i s the subfunction of

m,

the

$’ which has exact-

which the k2-oracle discovered before declaring t h a t c

i1

discovered nothing about

and

i2(8’)

6’; so

= n ~ ) . Thus i n Case

the basis

6’

empty s e t of ordered p a i r s ) , the same f o r a l l

is

F2¶ the

0’= X i o @

;’Is.

In Subcase

k2-aracle (the

3.2’ w i t h

&*(&’) = 111, the A2-orac1e in one f e l l swoop learned the whole of A’ = Xa 2, so the basis 8’ i s $’ i t s e l f . In Subcase 3 .3‘ w i t h a‘ 2 ( a0 1 ) = ~IJ, the

*O

basis is

E’

=

I < q a r y I s<

51 (with 5 > 0 ) .

Now I prove t h a t , although I used oracles j u s t now, the basis f o r -1 -2 =1 a w. r. t o k2 (whena ( a ) is defined) i s , as the name suggests, Eique for

i1

and

i2

as type-; and

-;objects.

That the basis i s j n t r i n s i c a l l y

determined ( c f . end 9.2) follows from the manner i n w h i c h an ;?-oracle operates. The independence of the choice o f the ;’-oracle

i s immediate.

6

To prove the independence of the choice of the k2-orac1eY8 take given

Recursive Functionals and Quantifiers of Finite Types Revisited 111

t y p e - i and

-;o b j e c t s

i1 and

*2 a , suppose t h a t

consider a g i v e n choice of t h e a* 2-oracle. a p p l i e s t o no choice of the a- 2 -oracle. choice, then

a‘(@’)

Then

I f Case

* 2 -1

a (a )

i s defined, and

i2# @, so F2

Subcase

then t h e basis i s

i2

I f Case

i s d e f i n e d and t h e o r a c l e i s unique.

3.2’ a p p l i e s ) ,

Case

a p p l i e s t o the given

a p p l i e s t o t h e g i v e n choice, and thus t o a l l choices, and Case (i.e.

I

i1 irrespective

1’

7 ‘ 4’

to

of t h e

choice. I t remains t o t r e a t Subcase

3.3’ f o r

f i r s t e s t a b l i s h t h e f o l l o w i n g remark:

01 ) f ,w i t h a given i 2 - o r a c l e , a92 (a .1

4

i sdefined -- under Subcase 3.3 w i t h t h e b a s i s 71 El 3 8’. Assume t h e with a* 2 (B ) defined,

0;

8

f i n e d i s under Subcase

3.3‘.

il, ;’(a)

=

so by

tioned w i t h the oracle f o r

il, k(@) Now i f

F1, would

I

two choices of the &oracle.

8

,

hypotheses. =

k$

6

then, f o r each

0; so

By Case

3’

l’

c

for

; 2 ( g ) 1 ~ ’ b e i n g de-

8’, t h e given i 2 - o r a c l e , ques( s i n c e F1 c il) ask o f her the same

01 questions and r e c e i v e t h e same answers as o f and from the a -oracle, u n t i l , as must happen f o r some question from t h e domain o f .2 “1 making a (B ) undefined! Applying t h i s remark w i t h k2-oracles under Subcase

m‘, we

8’ get

and

2

2 2

i’, she

i1 w.r. t o

t h e bases f o r

b’

and

8’

3

f a i l s t o answer,

2

78 1 , i.e.

One more observation w i l l complete t h e p i c t u r e . Suppose t h a t -2 - 1 * 2 rl and a (a ) = 5. Then by the m o n o t o n i c i t y o f i2,a (a ) = m. W i l l have t h e same b a s i s as Case

G1?

C l e a r l y so, i f t h e ;‘-oracle

F1 ( a l l

0’).Likewise,

&’Is have t h e same b a s i s 02 * 1 * 2 ’1 a p p l i e s t o both a (a ) and a (a ); f o r , Likewise i f Subcase tioning

3.3‘ a p p l i e s

z1 (,11) w i l l

2 C? & 3

two =

3 =.

8’. i1

rl

a

operates under ifSubcase

A’

=

3.2’

AAoc+g1 = A’.

t o both; f o r , then t h e A2-o+acle, ques-

ask the same questions as she asks o f

&’,

receiving

t h e same answers, up t o t h e p o i n t a t which she declares the value t o be

m,

But i n t h e circumstance i n which I mandated monotonicity above, the basis

S.C. KLEENE

8

for

01

is

a

<

E
2 = Aion

51 and f o r

o f “ s e c t i o n s ” o f a type-? o b j e c t

4’

object

i2.

(In

1980 I used

i n d e f i n i n g bases w. r. t o a type-;

itself.

i3, I

I n d e f i n i n g bases w. r. t o a type-? o b j e c t

9.4

3

is

s h a l l use t h e idea

sections o f a type-:

,

-2

object

a

b u t I have by-

passed t h a t p a r t o f t h e treatment i n t h e present review.)

L2 be a l l t h e bases 8’ f o r a- 1 i . The sections 3S 1 o f i2 a r e

Let over t y p e

i2f{i11&’

i2

sections

b1

=uil 2sl.

8’1

3

for

E

A2.

w.r.

.2

to

&’

as

a

ranges

t h e constant f u n c t i o n s as t h e union of i t s

Thus we can w r i t e

E A

i2

If

0’). i2 . i2 8’ -9’

(A2 = 0 , i2 =

sections

If

c o n s i s t s o f one s e c t i o n

S

sections

E

-$

n’s

If

)

f o r which

8’

Se1 w i t h

sections ills

(with

S

( f o r those

i2,i s t h e empty union of no

has an o r a c l e under Case

B

=

- 2 -1 f o r which a (a )

has an o r a c l e under Case

T2,

has an o r a c l e under Case

S .o

are u n i t sections

Xa

-0

_?.

-2

a

2, i t s -2

-0

= {I

&2(A&on) i s d e f i n e d under Subcase 3.2‘)and {<%,%>I< < 51 ( f o r t h e bases 8’ f o r those

i s d e f i n e d under Subcase

I n Case

m2)1

T2,

the

sections a r e d i s j o i n t , except t h a t i n t h e circumstance i n 9.3 i n which I

5 .o

mandated monotonicity

S

section

3l

(so

82

c

b1

with

A subunion

Acr 11

b2

=

{11; < 51.

of

8‘

of

*;

0’ 4 E2

(then

8’

=

s2

2

i s any

i2

can be a l t e r e d t o an o r a c l e f o r a given sub-

by muzzling her ( i . e .

O, as f o l l o w s :

=u.,€8241B

i2

&2

2).

A given o r a c l e f o r union .2 * 1 B (a =

i s a u n i t s e c t i o n p r o p e r l y included i n a

I n Case

72

0‘).

B

S

with

c

changing her program t o g i v e i f i t a p p l i e s (so A ~-

= @’I),

4 I f Case T2 applies, i n Subcase 3.2 w i t h a

if

Recursive Functionals and Quantifiers of Finite Types Revisited I11

X i 0 ~E

l2ifn e i t h e r

-2 and i n Subcase 3.3

X&'E

nor any constant f u n c t i o n

w i t h an {lc

c

5)

4'

E

I n c i d e n t a l l y , t h i s proves t h a t an,v subunion

6'

9

5)

$,

-- what

she

[ c r ,n>15 <

-5 if [ < r ,n > l c < 5 ) 4 B ~ . -5-5 o f a type-; o b j e c t i s a

type-2 o b j e c t .

10. ;'-ORACLE

TREES, SUBORACLES.

10.1.

The program f o r an 1 2 - o r a c l e as described i n 6.3 and 9.3

w i l l do under any circumstance she c o u l d encounter i n t h e l i n e o f her duty

--

can be represented as a t r e e . -2 *1 "a (a ) ? " .

Her d u t y i s t o respond t o questions

A v e r t e x i n t h e t r e e , by i t s p o s i t i o n , i n d i c a t e s ( n o t h i n 9 o r ) somet h i n g about an

&'

For a given ;'-oracle,

as embodied i n an oracle.

we

f o l l o w a path through t h e t r e e from t h e i n i t i a l vertex up through some ver-

A t each vertex n o t the l a s t on

tex, n o t n e c e s s a r i l y t h e l a s t on a branch.

t h i s path, t h e choice of t h e branching from i t which i s taken n e x t represents a new piece o f i n f o r m a t i o n about t h e

&'-oracle.

i n i t i a l vertex, we know n o t h i n g about t h e a* 1- o r a c l e -1 any a - o r a c l e a t a l l . To see how t h e c l a s s o f t h e &'-oracles

I f we a r e a t the

--

we are considering

p o t e n t i a l l y represented i s de-

creased, o r t h e information about t h e ;'-oracle

i n hand i s increased, as we

proceed along a path, suppose we have already proceeded by zero o r more *2 The a - o r a c l e then,

steps from t h e i n i t i a l vertex t o a c e r t a i n v e r t e x . depending o n l y on the i n f o r m a t i o n about t h e ;'-oracle

represented by the

","

path thus far, may stand mute ( t o make which v i v i d , I then w r i t e "MUM" a t 92 -1 t h e vertex), o r d e c l a r e t h e value o f a (a ) t o be 5 ( I then w r i t e there), o r ask a new question

"@'" o r "r?", according

"&'(&')?''

o f the

&'-oracle ( I then w r i t e

as t h e question i s asked w i t h an empty envelope

S.C. KLEENE

10

(Lo

=

0)o r w i t h an envelope c o n t a i n i n g an

1 E lo (&' = r)).

I n the

f i r s t two cases ("MUM" o r "m_P),the branch t h e v e r t e x i s on ends w i t h it. I n t h e t h i r d case, t h e p a t h s p l i t s i n t o d i f f e r e n t branches according t o t h e 1 h - o r a c l e s c o u l d make t o t h e question.

responses d i f f e r e n t

a For an .1

-1

o r a c l e who does n o t respond r e v e a l i n g more i n f o r m a t i o n about a representing t h a t

a -oracle 1

, the

path

ends w i t h t h a t v e r t e x (even though i t i s n o t

t h e end o f a branch of t h e t r e e ) and

h2 ( h1 )

=

a.

W i t h i n t h e compass o f t h i s general d e s c r i p t i o n , l e t us see what happens i n t h e cases and subcases cataloged i n 6.3 and 9.3. I n Cases

i2 and T2,

6.2 - o r a c l e ' s

the

so t h e r e a r e no branchings.

-1 a c t i o n does n o t depend on a ,

The t r e e c o n s i s t s o f a s i n g l e v e r t e x , as shown

i n F i g u r e 1.

-

- .J

MUM

Case

i2.

Case

P.

F i g u r e 1. I n Case

T2,

t h e i n i t i a l v e r t e x and t h e second v e r t e x on each branch

appear as i n F i g u r e 2, f o r some c h o i c e between t h e a l t e r n a t i v e s a t each second v e r t e x .

MUM o r

MUM

MUM

@?

MUM Case

T2 t o second v e r t i c e s

(complete i f t h e t o p second v e r t e x has "MUM"). Ci",...,.

9

q,?

11

Recursive Functionals and Quantifiers of Finite Types Revisited 111

To keep t h e n o t a t i o n simple, I have shown "MUM o r mJ1l a t t h e end of each o f the

lower w branches, although the 1 ( i f it applies) will depend in

n', in

general on t h e branch. The lower w branches a r e f o r Subcase 01 which t h e a -oracle, w i t h o u t opening t h e envelope, declares t h a t &'(&')

=

n (n =

0, 1, 2,

branch i s f o r Subcase

... f o r

3.3'.

the v a r i o u s & ' I s under Case

I n Subcase

3.1'

7').

The top

(no response by t h e &'-oracle),

t h e path d e s c r i b i n g her ends w i t h the i n i t i a l v e r t e x .

*2 second vertex, i f" t?h e) a , - o+r a rc l e( asks ' ; "

A t the top o r o-th

( t h e second of t h e a l t e r -

-1 n a t i v e s shown), then, since our being t h e r e means t h a t the a - o r a c l e opened -1 t h e o r i g i n a l (empty) envelope, and t h e r e f o r e opens any envelope, t h e a o r a c l e ( a f t e r opening t h e envelope c o n t a i n i n g Q ) may e i t h e r n o t respond ( s o her path ends a t t h e top second v e r t e x )

(n+

... f o r

= 0, 1, 2,

-1 v e r t e x and a i n Figure 3. (IIMUMII) "&'(cl)?"

-1 various a I s ) .

With

responding, we g e t t o one o f

,

-0

l l ~ ? l l a t the top second w

t h i r d v e r t i c e s , as shown

02 A t each o f the t h i r d v e r t i c e s , t h e a - o r a c l e may stand mute

o r declare t h a t ("I~?").

02 - 1

a (a ) =

m ("mJ)

o r ask another question

Again, t o keep t h e n o t a t i o n simple, I have n o t i n -

d i c a t e d t h e dependence of the i n the tree.

9 o r respond w i t h n

m

o r the

c

( i f i t a p p l i e s ) on t h e p o s i t i o n

(My n o t a t i o n s here are subscripted

S.C. KLEENE

12

c = o

c = 1

MUM o r

Case when

"

~

T2 t o

or

MUM o r

mJ o r q ?

MUM o r

~4 o r q ?

t h i r d vertices

i s" a t t h e t o p second v e r t e x . Figure 3.

f o r successive p o s i t i o n s i n t h e path belonging t o a given a d d i t i o n a l l y f o r d i f f e r e n t paths.)

.1

a

, but

A t any vertex w i t h "I~?",

not

t h e ;'-or-

a c l e w i l l e i t h e r stand mute (ending i t s path), o r respond w i t h

51, g i v i n g

... t o

w

branches t o f o u r t h verti'ces.

L e t us index by o r d i n a l s

5

t h e v e r t i c e s along any p a t h a f t e r t h e

rise for

r~, = 0, 1, 2,

i n i t i a l vertex.

( k n i t t i n g t h e i n i t i a l vertex makes t h e v e r t e x i n d i c e s agree

w i t h the indices 'of

r's

a t v e r t i c e s above t h e lower

w

?

branches and o f

Recursive Functionals and Quantifiers of Finite Types Revisited 111

n's

13

on the segments i s s u i n g from those v e r t i c e s . A f t e r any stage i n the c o n s t r u c t i o n o f the t r e e a t which we have an

"r ? " a t the end o f a branch, the c o n s t r u c t i o n continues by branchings -5 f o r n = 0, 1, 2, l e a d i n g t o o n e x t v e r t i c e s indexed by ~ + 1 , -5 a t each o f which t h e a' 2 -oracle, responding t o the i n f o r m a t i o n t h a t ( t h e

...

'1

'1

a - o r a c l e opens envelopes) & Wnn<5a ( r )=n ''rnJll

o r an

lll;+l

I n pursuing

?"

71

.

IIy

w i l l put

"MUM"

o r an

the t r e e c o n s t r u c t i o n , a f t e r a s e r i e s of stages,with no

l a s t , t h e r e may be branches w i t h v e r t i c e s indexed by a l l t h e o r d i n a l s

<

a l i m i t ordinal

5.

For example, t h i s w i l l happen i f we have pushed

t h e t r e e c o n s t r u c t i o n along a l l branches simultaneously, d e a l i n g w i t h a l l the v e r t i c e s indexed by each

T-I

i n turn, w i t h o u t a l l branches having t e r -

n

minated when we have r u n through a l l

c 5.

For each such branch, the

A2-orac1e, responding t o t h e i n f o r m a t i o n t h a t (&' opens) & *1 V T - I ~ < ~(q)=n+ Q w i l l then p u t "MUM" o r an "_m.J" o r an "r ?" -5

a t the

n e x t ( l i m i t ) p o s i t i o n on the branch ( t h e r e being no s p l i t t i n g i n t o d i f f e r e n t branches). Pursuing the t r e e c o n s t r u c t i o n l o n g enough, each branch w i l l have a l a s t vertex (with some

5 <

"MUM"

o r an

"mJII),

For, any sequence

ole

lowhen

must exhaust

%,

indexed except i n Case

..., $, ... o f

E~,

extended through a l l

5 < 5

r2 o r '7

by

d i s t i n c t numbers

f o r some 5 < ol;

i.e.

5, < 61 = No. Then t h e r e w i l l be no new question -2 'lq?''t h e a - o r a c l e can ask, so she can o n l y stand mute ( I W M " ) o r de02 01 Clare t h a t a (a ) = m (IlmJ") f o r some m. f o r some such

Each branch i n t h e t r e e ending w i t h an

w.r.

to

for a l l

- 2 *1 (so a (a ) = )TII

;'Is

n

and maybe beyond.

o f t h e lower

represents the basis

which r u n w i t h the branch t o i t s f i n a l vertex

0'. I n Subcase n', t h e basis i s that

"mJ"

I n t h e t r i v i a l Case

X i o ~=

i1 itself,

o branches as shown f o r Case

T2,

t h e basis i s

and the one f o r represents t h i s

01

a

.

S.C. KLEENE

14

In Subcase

m',

the basis i s

8'

=

CIc < 51,

2 ' mJ"

appropriate one of the upper branches e n d i n g with

"mJ"

Consequently, each branch ending with an section of

represented by the where a- 2(B- 1 ) --

i s associated w i t h a

(5.4 or 9.4), whose domain consists of a l l the ;'Is

i2

r u n w i t h t h a t branch and maybe beyond, w i t h one exception. under Subcase

{lc

<

3.3'

n on

which has the same n

-c

which

For a branch

each segment

El), the domain of the section includes a l s o G1

represented by the Ktl-st of the lower w

hi"^,

=

=

If instead "MUM"

we s t a r t w i t h a t r e e of the kind described, constructed until only

"mJl'

8'

(i.e.

branches.

I have described the t r e e representing a given i2-oracle.

or an

m.

remains a t the end of each branch, i t will i n any case represent

what a n &'-oracle

could give as described i n 6.3 or 9.3 before I mandated

The monotonicity requirement s t a t e d in terms of the t r e e is

monotonicity.

t h a t , i f any branch through the top second vertex represents a constant function the lower

n>Ic < El and ends w i t h an -5w branches (representing h&Ofi)

C
"mJ",

the one f o r t h a t 2 of

must end w i t h the same

"mJ".

I shall say t h a t an i2-oracle i s a suboracle of an ?-oracle, i f f i t s t r e e i s obtained from the t r e e f o r the -2 A -oracle by changing a t the ends

10.2.

of zero o r more branches a label o f the form missible labels, namely, to '@?"

or

"r?" -

"MUM",

"@?"o r

"m/"

t o one of the other ad-

"r?",

and in the case o f

extending the branch appropriately (with new branchings)

t o an additional vertex o r vertices on each extension, b u t i f an

"I&'"

is

obtained on any extension i t must be w i t h the same 5 as i n the label This must be done preserving monotonicity: i . e . we must not change "mS' - a t the end of a branch labeled " -1 a not opening says fill t o "MUM" i f

changed.

our packet of changes leaves o r places extended branch described by

(XIIII)

Given an

-

" ~ d "a t the end o f an original o r

{lc < 5 )

;'-oracle,

an

A'

with t h a t fi ( c f . end 10.1).

E

type

has an oracle

Recursive Functionals and Quantifiers of Finite Types Revisited I11

z 2 - o r a c l e i f and o n l y i f

w h i c h is a s u b o r a c l e o f t h e g i v e n &2

15

2' a .

PROOF.

The " o n l y i f " i s immediate.

- 2 01 the given 3 - o r a c l e , i f a (a )

i2 maybe

01 ) = a- 2 (a

Thus

m;

i2, c a l l

&22; ,

i.e.

a.2

c

.

r;!

a

occurs and i n

"IIJS' occurs.

beyond t o a v e r t e x where the same

To prove t h e " i f " , suppose t h a t oracle for

"mJ"

t o a v e r t e x where

-1) = o n l y i f 72 a (a

m

a suboracle of

i s defined, i t must be by f o l l o w i n g a branch

i n t h e t r e e f o r t h e given ?-oracle the t r e e f o r

-2 a

For, using f o r

By

k2 E

type

i, t h e r e is Some

I f t h i s oracle i s not a 72 92 suboracle o f t h e given a -oracle, we must c o n s t r u c t another o r a c l e f o r a

which i s .

i t t h e "given i 2 - o r a c l e " .

M o n o t o n i c i t y w i l l be assured, s i n c e

I n Case

T2 f o r

(3=

the 2 - o r a c l e

i2 i s of type i. i2c 3 i2= 2,

a2),

given 3 - o r a c l e (represented by t h e one-vertex t r e e for

i2 and a I n Case F2

-2 -1 a (a )

suboracle o f i t s e l f . f o r the 3 - o r a c l e

(2= X & o ~ ) ,

i s defined, i t has t h e value

m,

with

+

and t h e

'I'

MUM") i s an o r a c l e

a- 2

c a

T2 , whenever

-2 and t h e given a - o r a c l e i s a sub-

; i !

o r a c l e of t h e given a - o r a c l e ( w i t h the one-vertex t r e e

'I*

m_J").

T2 f o r the a -oracle. We must c o n s t r u c t a t r e e f o r - 1) i f il, g i v e s "IIJ~' f o r IIJ t h e value o f a'2 (a

There remains Case

i2 which, -2 * 1 a (a )

f o r each

i s d e f i n e d and

"MUM"

otherwise, and which i s obtained by a l t e r i n g ,

as i n the d e f i n i t i o n o f suboracle, zero o r more branches ending w i t h an

"mJ" -

72 i n t h e t r e e f o r t h e given a -oracle. F i r s t , consider any

for

2, Case 7'

for

&'

&').

o f t h e form

(one o f t h e w

92 ( s i n c e by a

-2 a

3.2'

0,

i2(i1) = or f o r our new r e p r e s e n t i n g i1 i n t h e t r e e f o r the given

According as

i 2 - o r a c l e t r e e we take t h e branch ?-oracle

X i 0 ~ f o r some 11 (Subcase

lower branches) and r e s p e c t i v e l y leave i t unaltered

* 1 ) = !+ , a- 2 (a

2 - 1 ) = mJ, a (a

if i t does n o t already have "MUM".

o r change i t s

"mJl'

to

"YUM"

S.C. KLEENE

16

I t remains f o r us to consider a01 I s w i t h envelope-opening oracles (Subcase 3 .3' f o r Case 3' f o r the ;'-oracles). 10

z2,

r 2

a t the t o p second vertex o f her -1 oracle t r e e , i s undefined f o r a l l a 's w i t h envelope-opening oracles, so likewise a- 2 ( -a1 ) by k2 c 7 ) ;and we simply r e t a i n the "MUM" I f the given a -oracle has

"MUM"

a(;')

there f o r our suboracle. Now suppose the given 7a 2-oracle has

" ~ ? " a t the top second vertex of

her oracle tree. If the given i 2 - o r a c l e comes under Case "MUM"

r2,o r under

Case 3' b u t with

a t the top second vertex of her oracle t r e e , f o r our suboracle we

simply change a l l

"I$"'s

second vertex t o

"MUM".

i n the subtree f o r

2

The given i2-oracle cannot come under Case

&*(@') = g would contradict 2 ( @ ' )

=

issuing from i t s t o p

T 2 , as ( w i t h i2c i2)

0.

Finally, suppose the given a- 2 -oracle comes under Case -23 with

at

%

I'

'I

the top second vertex o f her oracle t r e e . Let me c a l l an

i1 bold,

i f f i t has an envelope-opening oracle who

asked of responds (by giving a value "E") t o each o f the questions ?"I" 7' -2 -1 her by the given a -oracle and a l s o by the given a -oracle. The other ci I s w i t h envelope-opening oracles I c a l l sh-y.

In the t r e e f o r the given r a;!-oracle, a bold i1 runs with a branch T1 - (through the top second vertex) representing B = {\c < cf ( w i t h

-

-

'lmJ'l appears ( i t does not fade out

or an

5 > 0 ) to i t s end where "MUM"

e a r l i e r by not answering a question).

G1 w.r.

the basis f o r

*1

oracle, a bold (with 6 > 0)

a

t o 7a 2

appears there,

" ! / I '

i',

is

Likewise, in the t r e e f o r the given a- 2 -

b1

runs w i t h a branch representing

t o i t s end w i t h

W i t h a bold

.

I f an

"MUM"

or an

i n the case of an

"%/"

=

{\s<

51

"mJl' at

b'

in the t r e e f o r the

17

Recursive Functionals and Quantifiers of Finite Types Revisited 111

92 given a -oracle, we have t h e same

2

at

i n t h e t r e e f o r the given 02 01 - 2 - 1 - r;! .1 - 7 2 7 = a ( B ) = a (a ) - a (a ) - a (B ), using

?-oracle

( s i n c e then

m

i2 c c?),

and moreover

F1 8'.

For, w i t h i 2 ( i 1 )

c

F1 b e i n g t h e

defined and

"mJ"

&'

basis f o r

and t h e r e f o r e z 2 ( @ ' )

w.r.

to

2, 8' k

"MUM"

at

i1 i n the

by the i t a l i -

3

c i z e d remark near the end o f 9.3. With a b o l d given a- 2-oracle, (remember t h a t "MUM"

produce

il, in

t h e case o f

a- 2 (y- 1 )

i s undefined f o r every extension

2 , 8'

c

8'

c

at

-1 a ).

;'

;'

t r e e f o r the o f ;* ;B7 u

For, i n t h i s case, the given i2-orac,le

;'.

when queried w i t h

b'

will

Now I propose, f o r our suboracle, t o c o n s t r u c t extensions (proper o r improper) from each branch through t h e t o p second v e r t e x i n t h e t r e e f o r r;!

the g i v e n a - o r a c l e which represents a

&'

l y , f o r each b o l d

k

w i t h an

then t o continue by using t h e p a i r s if any i n

8' m

2

=

a t i t s eid.

Name-

running w i t h t h i s branch t o i t s end, I program our

new o r a c l e t o f o l l o w t h e branch r e p r e s e n t i n g

are n o t i n

Ilny'I

-

{1c < 5)

2

2

t o i t s end i n the

8'

=

{<;,r+>lc

< 51

7a2-tree, that

i n t h e order i n which they are indexed i n

*2 -1 t o go mute o r declare t h a t a (a ) 1 according t o what t h e given i 2 - o r a c l e does a t 8 by

.

5 <

5 , and f i n a l l y ( a t

u

8')

.

I need t o v e r i f y t h a t our new o r a c l e can r e a l l y be programmed t o do

this. There i s no problem i n a l t e r i n g t h e t r e e f o r the given 2 - o r a c l e a t the ends of some of i t s w lower branches as c a l l e d f o r i n t h e treatment o f -2 r 2 Subcase 3.2, o r i n t a k i n g over t h e p a r t o f t h e given a - t r e e c o n s i s t i n g o f the branches through i t s t o p second v e r t e x sans such e n t r i e s a t the ends o f those branches as a r e o f t h e form

"p"', where now I may want something

different. What do I want now, s t a r t i n g a t t h e end of such a branch? represents say

7 B

.

Consider any ( ! ) b o l d

&'

The branch

running w i t h t h e branch t o

--

S.C. KLEENE

18

-1 i t s end ( i . e . a

7 6 ) , and consider t h e

3

-2 t r e e f o r t h e given a -oracle.

b o l d 4"s

i'

associated w i t h

i n the

t a k i n g d i f f e r e n t such

The problem now i s :

w i t h t h e i r respective

4'

am I c a l l i n g f o r d i f f e r e n t t h i n g s

il's,

t o be done? .1 Remember t h a t each step by t h e g i v e n A2-oracle, questioning any a

o r a c l e under Subcase

3.3',i s determined by

answer p a i r s she has so f a r logged).

t h e " h i s t o r y " ( t h e question-

I t may be t h a t , given any b o l d

8'

the same answers, u n t i l she a r r i v e s a t 01

- 1 fl a 3 B ,

7 B , r e c e i v i n g from a l l such

she w i l l ask o n l y questions from t h e domain o f 4"s

-

(C

';1 f? ),

t h e same f o r a l l

she w i l l ask t h e same questions and r e c e i v e from a l l t h e same answers, u n t i l she asks a question r- n o t from the domain of

such

a '5.

Alternatively,

-5

2.

Then d i f f e r e n t branches w i l l be followed i n her tree, and w i l l be

c a l l e d f o r i s s u i n g from numbers

f

= &'(F-)

2

i n my proposed new i 2 - t r e e , depending on t h e

w i t h which t h a t question i s answered.

-5

Continuing on

a given one o f these branches, she w i l l sense no d i f f e r e n c e between d i f f e r ent bold

3

':

u {<&-> ;,]

up through e i t h e r a r r i v i n g a t

- E f

*1 f o r a l l these a I s , o r asking a new question

k.

And so on.

q+ln o t from

;',

t h e same

t h e domain o f

A f t e r my s e r i e s o f v e r t i c e s w i t h no l a s t , t h e step t o t h e

l i m i t vertex, indexed by say

bold

-

e i n our new tree, w i l l be t h e same f o r a l l

i1 u { < ~ , ~ q>
conclusion, when she a r r i v e s a t

i n her t r e e ( w i t h o u t , o r a f t e r asking questions, n o t from t h e domain o f i t w i l l be w i t h t h e same

i'

f o r a l l bold

c1

3

2

6 . So a given branch if she stepped o u t s i d e o f 7

*2 8' u i1 my new a - o r a c l e o r a c l e does a t

*1

B

,

give

"MUM"

or

"mJ",

o r a l l such

-1

B

-;1

B ),

along

my requirement t h a t a t

-2 t h e same as t h e g i v e n a -

can be met.

The new t r e e c o n s t r u c t i o n i s complete; i . e . my p r e s c r i p t i o n does s u f f i c e f o r t h e u l t i m a t e t e r m i n a t i o n o f every branch t h a t extends p r o p e r l y o r i m -

-

p r o p e r l y a branch o f t h e g i v e n A2-oracle t r e e through i t s t o p second vertex

19

Recursive Functionals and Quantifiers of Finite Types Revisited 111

"mJ"

w i t h an

at

2. For,

a bold

&'

(a t o t a l f u n c t i o n i f need be) does

r u n w i t h every extension pursued i n d e f i n i t e l y . Next I c o n f i r m t h a t t h e r e s u l t i n g o r a c l e i s a suboracle o f t h e given

r2

from the given a - o r a c l e 2: o n l y by changes made a t o r beginning a t v e r t i c e s where t h e given a - o r a c l e

z2-oracle.

But indeed, the new o r a c l e d i f f e r s

"mJ".

had an

I n Subcase

3.2'

"mJ"'s

we o n l y changed some

and some p r e l i m i n a r y cases under Subcase to

"MUM"'s.

I n Subcase

3.3'

m',

w i t h h t h the

given t r e e s having questions a t t h e top second vertex, when t h e new o r a c l e gives

"mJ" -

at

2

u

8'

(where

T2

same as t h e a - o r a c l e gave a t

m=

- 2 *1

m

a ( 8 ) ) , then (as we saw)

i s the

k 8'. c

i2. The

F i n a l l y I c o n f i r m t h a t t h e new o r a c l e i s an o r a c l e f o r

pre-

l i m i n a r y cases are immediate. Suppose Case

T2 a p p l i e s

t o both given oracles, each t r e e having a

question a t t h e t o p second vertex. F i r s t , consider any 7 2 rl with a a t which a (P ) = m. I n t h e case t h a t t h e -2 -1 -2 7 gives a ( a ) = m a t 8' (then as we saw m = a (B ) 71 {wc,%>1c < El = B what we have done i s t o take 8'

k1

bold

given i2-oracle and

2

8'

u

m

correct.

then as we saw i2(;')

c

i'),

i n perhaps a

new order, so indeed t h e extended branch represents p r e c i s e l y i2(i') =

running

8',

making

-1 I n t h e case t h a t t h e given i 2 - o r a c l e went mute a t B , i s undefined f o r every extension

-1 i1 of 8

And what we have done i s t o c o n s t r u c t a branch representing

71 B u

8'.

u

81 . So

l e t t i n g t h e new o r a c l e go mute t h e r e i s l i k e w i s e c o r r e c t f o r the f u n c t i o n

i 2 , g i v i n g t h e same r e s u l t f o r i1 a t F1 u 8' as t h e given .1 running w i t h a branch through the t o p a t 8 . For b o l d ;'Is i n t h e given i 2 - o r a c l e t r e e ending w i t h

"MUM"

at

g1,

oracle d i d second vertex

we have made no

'2 * 1 change f o r o u r suboracle, which again i s c o r r e c t , since a (y ) - m f o r -1 - 2 1 any extension y of would (by 6,' c give ;( ) = m, contra-

F'

2)

d i c t i n g t h e f a c t t h a t t h e given 2 - o r a c l e gives

"MUM"

at

?.

S.C. KLEENE

20

Finally, consider any

F i r s t , suppose she f a i l s t o answer some question asked of her by the given 2 - o r a c l e ; so 7 i (A-1 ) = whence (by

which i s shy.

i2 c ?) i2(z1)=

0.Let

(bold) d i f f e r from

@,

z1

by h a v i n g 0 as value f o r each argument f o r which i s undefined. The new i2-oracle will question the ;'-oracle w i t h a l l the arguments in 7B 1 u B* 1 , f o r and

i1 determined

F1

; 1

as above from & I .

Her questioning of the a -oracle

zl(t-)

g'

will run the same up t o the f i r s t argument r i n f o r which @ and a- 7 (c) = 0; t h e n the $-oracle does not answer, so the new oracle makes :'-oracle's - 2 -1 a (a ) =

* 2 1'

0, as desired.

=

4'-

answers a l l the questions t o her b u t f a i l s on a question by the given a- 2-oracle, a (a ) =

If the:'-oracle

0.

Then the new i 2 - o r a c l e will f a i l t o e l i c i t a n answer from the -a1 -oracle a t the f i r s t argument r in b' f o r which :'(I) = @ and

-

&l(y) = 0 , which again renders

- 2 -1 ) =

a (a

@,

as desired.v

We recall the notion of a subunion ( o f the sections) of an

(XXII.2) i n 5.4 o r from 9.4.

Subfunctions of

i2

i2 from

which a r e subunions a r e

represented by suboracles whose t r e e s a r e obtained from the t r e e f o r by simply changing

"mJ" t o

preserving monotonicity.

"MUM"

i2

a t the ends of zero o r more branches,

Indeed, I used this in the concluding paragraph

of 6.3 and of 9.4. 11.

TYPE 3, ASSUKED FUNCTIONS WITH TYPE-2 VAPIARLES. 11.1

T h i s subsection presupposes f a m i l i a r i t y with

1980 5.4.

The cursory reader

may skip t o 11.2. Type 3 , according to 5.2 o r 9.2, consists of the "unimonotone" partial one-place functions from type 2 into Paralleling 5.4 f o r type

0

.

i, a p a r t i a l

one-place function

i3

from type

into

i s unimonotone i f f i t is monotone ( a f t e r 5.2 o r 9.2) and has the following further property i n two parts. (After CY- 3-oracles have been introduced i n 1 1 . 2 , this can be condensed.)

For each

i2

of type

f o r which

21

Recursive Functionals and Quantifiers of Finite Types Revisited 111

* 3 .2 a (a )

4'

of B2

i s defined, t h e r e i s a minimum subunion such t h a t

A2

(so

i3(i2)

i s defined.

i2 =u,,

That i s :

8

8'

i2 c i2,

S

cL2?l

i)

and by (XXII.2) o r end 9.4 E type 03 - 2 - 93 - 2 i s d e f i n e d ( s o by monotonicit,y a ( 8 ) - a ( a ) ) ; and, f o r each c

u

S 8 1€rj2"B1

(unique)

with

8'

E2

c

A2

-3 and a ( 8 )

t h e basis f o r

( f o r each such

i2) i s

-2

a

y.

i2 c 2.l1 I

defined, -3

r.

a

.

S

o f t h e sections

2'"

with

-3 *2 and a (B ) =

call this

-2

D

Furthermore, t h e basis

" l n t r i n s i c a l l y determined", as w i l l be formulated i n

11.2. Again, w i t h o u t i n v o l v i n g the i n t r i n s i c a l i t y , we can c a t a l o g the p o s s i b l e

-2

bases f o r

a

-3 w. r . t o a given type-3 o b j e c t a , as

A'

v a r i e s over type

2.

- 13:

i3 i s the

CASE

i3(Q2)

CASE

Z3:

empty u n i c n a 2 =

CASE k2

33:

t o t a l l y undefined f u n c t i o n

o f sections o f &2) -3 - 2 Then -a ( a )

Otherwise.

must be d e f i n e d f o r some

-3 . ~ ~ : basis.

i2 =

Subcase

A% '; .

3 -2

(a )

S

1;

9

are some

of N , 11.2

bases f o r

and (using the

i s defined f o r some A 2 ' s , and each such 3.13:

&'

&'

=

o2

(Case 12 ).

m

i s i t s own basis.

say; so

4'

(For,

then

and we must take t h i s , as the empty union

i2

1

s,

03. Wo bases.

i s t h e basis f o r every a. * 2,

would p u t us back i n Case 2 . ) Subcase 3.33: -2 -1 a ( a ) i s defined f o r some i ' k ~ = type -3 -2 a ( a ) i s defined, then t h e basis = 2 (4.)

a .3 = Aa

i s defined (Case 22 ) , =

i s defined,

c o n s i s t s of one s e c t i o n

o2

il. Subcase

i2(@')

If i

-2

i s defined, = 5 say, so

0

=

hi2@

&'(a')

i-

@'I

EB 2dl SB

I now describe how oracles f o r type-;

objects

This w i l l g i v e a s e l f - c o n t a i n e d d e f i n i t i o n o f type-3.

If

(Case 32). where

i2 under Case 32, i2 for a'3 ( B- 2 ) t o be

w. r. t o

t h e minimum such subunion of

but

i s undefined,

(fl 2 )

2

E

thus members defined.

i3 shall

perform.

22

S.C. KLEENE

I conjectured ( i n the summer of 1977) t h a t an i3-oracle could operate i n the principal case (Subcase

m3)as

an i 2 - o r a c l e does in the correspond-

i n g case (Subcase -23.3 ); t h a t i s , by questioning the i2-oracle w i t h a nonempty s e r i e s of functions

%,

rnl

i", i,,, ..., bK, ..., receiving

,. . . , x m , . . . , with

determined by

K

{lh<

answers K],

until e i t h e r

a question i s not answered, or otherwise the a- 3-oracle gives u p o r declares -3 - 2 that a (a ) = 2. A counterexample by David Kierstead (reproduced in 8.2) showed t h a t t h i s s i m p l i s t i c program will not always work.

The following

treatment modifies i t (as i l l u s t r a t e d on Kierstead's example i n 8.3). -2 We ask an i3-oracle "What i s a- 3 (a ) ? " by presenting her with a closed envelope ( o r chest) containing an oracle f o r a.2

CASE T3: The a- 3-oracle

simply stands mute.

vndefirred function

=

-

X i 2 0 0".

.

Then

i s the t o t a l l y

G3

CASE T 3 : Without opening our envelope, the i3-oracle pronounces t h a t -3 -2 a (a ) = 2. Then i3 i s the t o t a l constant function XG25.

93 CASE 33: The a - w a c l e opens our envelope, revealing t h a t she will require some information about our question

i2,indeed

some values of

i2, i f

she i s t o answer

'1&3(i2)?". To obtain such information, she begins by ques-

-2 tioning the a -oracle who emerges from our envelope with a preliminary ques-

t i o n , using an envelope containing an oracle f o r

SUBCASE

m3:The i2-oracle

P

-2 does the a* 3-oracle; a- 3 (a ) =

SUBCASE

@

=

xio@.

stands mute (Case T2 i n 6.3 or 9.3).

Then so

@.

m3:The i 2 - o r a c l e , without

i2(i1) = rn (Case F 2 ) ; so iz = .!A&'

opening the envelope, declares t h a t

The i3-oracle then e i t h e r stands mute

*3 * 2 o r declares t h a t a ( a ) = 2.

m3

SUBCASE (Principal Case): The &*-oracle opens the envelope (Case -2 3 ). The i3-oracle observes t h i s , and e i t h e r stands mute, o r embarks on a

23

Recursive Functionals and Quantifiers of Finite Types Revisited 111

*2 program of f u r t h e r systematic questioning of t h e a -oracle. I n t h e l a t t e r case, t o keep t r a c k o f the i n f o r m a t i o n she hopes t o accu-3 mulate thereby, t h e a - o r a c l e e s t a b l i s h e s a c h a r t , which she w i l l update Stages i n t h e f u r t h e r systematic questioning a r e i n -

from stage t o stage. dexed by o r d i n a l s .

L e t us see what happens t o g e t t o Stage 0.

Before Stage 0 the c h a r t i s

-3 Which o f t h e f o l l o w i n g questions the a - o r a c l e opts f o r i n her ap-

blank.

-2 proach t o Stage 0 i s the same o f a l l a -oracles who open envelopes.

OPTION 1. some

-3 -2 The a - o r a c l e questions t h e a - o r a c l e w i t h

11 E yo (Subcase

3.2').

n

-1 -0 a = la

for

i1

That i s , she puts t h e o r a c l e f o r t h i s

.2 i n an envelope, and presents the envelope t o the a -oracle.

IJe know the

-2

-1

a - o r a c l e w i l l open t h e envelope, and question t h e a f o r e s a i d a - o r a c l e who

emerges from i t w i t h

w i l l declare t h a t

0.

i0 =

-1 -0

a (a ) =

Without opening t h e envelope, the ;'-oracle

11. Observing t h i s , the ;*-oracle

may stand

mute; .then so does t h e i 3 - o r a c l e , and the questioning f a l t e r s w i t h o u t reachi n g Stage 0.

OPTION 2 .

O r t h e ;'-oracle

may declare t h a t

- 2 * 1)

a (a

-- m.

.2 The i 3 - o r a c l e plans t o question the a - o r a c l e w i t h one ( o r more)

non-empty p a r t i a l functions on questions her w i t h

a',

0

(Subcase

3.3').

As a beginning, she again

-1 embodies now s p e c i f i c a l l y i n an a - o r a c l e

l1

who opens envelopes b u t i s always d i s s a t i s f i e d w i t h t h e contents.6 -2 -3 a - o r a c l e watches t h e a - o r a c l e question t h i s s a t u r n i n e ;'-oracle. o r a c l e , a f t e r discoverning (by asking "';'(a)?")

The

-2 The a

t h a t t h e ;'-oracle

-

opens

03 envelopes, may stand mute; then so does the a -oracle, and t h e questioning falters.

*2 *1 O r , t h e a - o r a c l e may question the a - o r a c l e w i t h an

determined by h e r s e l f , t h e same o f a l l ;'-oracles

r,

E

who open envelopes.

-1 t h i s case, as the a - o r a c l e opens the envelope c o n t a i n i n g snoops and discoveres what

0

r, N

r,,

In

03

t h e a -oracle

is.

*3

Under e i t h e r o p t i o n , i f t h e questioning has n o t f a l t e r e d , t h e a -oracle

S.C. KLEENE

24

has a r r i v e d a t Stage 0.

She records on her c h a r t f o r Stage 0 what she has

thus f a r learned about the

-2

c1

-oracle.

The most elegant way t o represent i n f o r m a t i o n about the

-2

c1

-oracle, a t

Stage 0 and l a t e r stages, i s t o draw on t h e c h a r t an i n i t i a l subtree of the

.3

A2-oracle t r e e (which, as the a - o r a c l e already knows, i s under Case

T2 i n

l O . l ) , i . e . a connected subset of i t s v e r t i c e s and segments, i n c l u d i n g t h e i n i t i a l vertex, w i t h t h e i r embellishments. As t h i s i n i t i a l subtree i s begun and extended from stage t o stage, each

o f i t s branches having a l a s t vertex ( s o i n p a r t i c u l a r , each f i n i t e branch), and no others, w i l l be indexed by a n o r d i n a l number from a segment o f t h e ordinals.

I n t r o d u c i n g an index records a commitment by t h e

a value o f

&'

t o be defined.

f o r a function

i1 running

- 3- o r a c l e t o o b t a i n

c1

-3 -2 w i t h the branch, if a (a )

is

Each vertex w i l l belong t o a t l e a s t one indexed branch, i n -

deed t o one indexed by t h e o r d i n a l t h a t indexed the branch which t h a t v e r t e x ended when i t was introduced. For Stage 0, under Option 1 w i t h the ;'-oracle for

-1

c1

the w

-0

= ?,a

n,

declaring t h a t

the &3-oracle draws on her c h a r t the one f o r t h a t

lower branches o f t h e ;'-oracle

thus f a r knows the outcome

" m J I I ) ,

i t s second vertex, bearing

"r ?" -0

(a ) of

and indexes i t w i t h 0.

"r+?", t h e

-3

~1

Under Option 2

- o r a c l e draws on the

t r e e from t h e i n i t i a l vertex j u s t t o (as she now knows), and indexes i t w i t h

0. I n the c h a r t a t any stage u which i s reached, I say t h a t indexed branches n o t through the top second vertex o f t h e f u l l ;''tree

-f i r s t kind

-m

t r e e ( t h e o n l y branch f o r which she

-2 01 w i t h the a - o r a c l e asking the -oracle c h a r t the top branch of t h e ;'-oracle

- 2 -1 -

are o f the

( t h e y arose by using Option 1); t h e o t h e r indexed branches, o f

the second kind. A branch (indexed by) K o f t h e f i r s t k i n d records t h a t .2 *1 -0 the C( -oracle, questioned w i t h a = xa %, answered $; and i s s a i d t o

25

Recursive Functionals and Quantifiers of Finite Types Revisited 111

be completed w i t h t h e value the following.

-mK.

A branch

o f the second k i n d records

K

@

A f t e r t h e p r e l i m i n a r y question

by the &'-oracle,

-1

i n response t o which t h e a - o r a c l e revealed t h a t she opens envelopes, the

*2

a - o r a c l e has asked o f the :'-oracle

t h e questions

J,~, q1,..., -rl' r The :'-oracle

..-

i s t h e c r e a t u r e o f t h e i3;oracle,

*2 o r a c l e ' s questioning o f t h e a -oracle.

< CK).

(rl

*3 being a v e h i c l e o f the a -

-3 -1 The a - o r a c l e has p u t i n t h i s a -

o r a c l e ' s mouth the answers

$,, $1,

...) Icn' n

(rl <

.**

c&

a f t e r a l l of which t h e i 2 - o r a c l e has e i t h e r declared asked a new question the

c's

"&'(r

)?".

"i2(&') = mK" o r

(Here, u n l i k e i n 10.1,

-6,

and ~ ' sw i t h a branch index

("%,J"

iscompleted w i t h t h e value

as w e l l as w i t h an index

K,

( o r 6 ) f o r t h e p o s i t i o n on the branch.)

I am indGxing r)

I n the f i r s t case, I say the- branch i s a t i t s end).

I say the branch i s blocked w i t h t h e question

"r

?"

-CK

f o r , t h e c o n t i n u a t i o n o f t h e e f f o r t t o g e t a value of

I n the second case, ( a t i t s end) pending;

i2 f o r

resented by t h e branch i s blocked by t h e want o f a value o f

an

&'(r

i1 rep).

The

-CK

-3

a -oracle, if i 3 ( i 2 )

i s t o be d e f i n e d w i l l , e v e n t u a l l y b u t n o t n e c e s s a r i l y

next, unblock t h i s branch ( i n one o r several ways) by supplying a value o f

A f t e r reaching any stage has j u s t declared t h a t

-3 -2

a

(a )

u, =

-3 n o t the f i n a l stage a t which the a -oracle -3 2, consider what the a - o r a c l e may do.

Contemplating t h e c h a r t a t Stage u,

i2 g i v e n

(from j u s t t h e information about

thereon) she may t u r n aside i n disgust;

- 3 02 a (a )

i s then undefined

and Stage o+l i s n o t reached. O r she may announce w i t h a smile t h a t -3 * 2 a ( a ) = 2. Then Stage u + l i s t h e f i n a l s%age, and a t the r i g h t of t h e

S.C. KLEENE

26

"SJ".

c h a r t as a whole she enters f o r reaching t h e f i n a l stage.

More w i l l be s a i d below on t h e c o n d i t i o n s

-3

O r t h e a - o r a c l e may ask another question

@(;1)?y This question (Option 1) may be of t h e form

&'

A&'$

=

w i t h an +I

n o t already used, w i t h two p o s s i b l e outcomes as described f o r Option 1 under Stage 0.

If t h e question i s answered, the Stage-at1 c h a r t receives a new

K after

branch ( o f the f i r s t k i n d ) , indexed by t h e n e x t g r e a t e r o r d i n a l those indexing branches i n t h e Stage-o c h a r t .

O r , ifOption 2 was n o t p r e v i o u s l y used, t h e i 3 - o r a c l e may use i t now,

w i t h the two p o s s i b l e outcomes described f o r Option 2 under Stage 0. -1 "a ( % ) ? ' I ,

*2

t h e a - o r a c l e asks

If

t h e Stage-o+l c h a r t receives a new branch

( i t s f i r s t of the second kind, w i t h -

cF

= 0),

indexed by t h e n e x t o r d i n a l

K.

As Option 3, t h e &3-oracle may p i c k an uncompleted branch

of the

K

by

"n xcK

q%'(

'%n

' I .

>In

-3 That i s , t h e a - o r a c l e questions the &'-oracle

cKl

<

u i
XCK

,n

-Kc,

>I.

?"

"r

second k i n d i n t h e Stage-o c h a r t , and answer i t s pending question

-CK

w i t h the f u n c t i o n

-2 We know t h e a - o r a c l e w i l l open t h e

envelope and f a i t h f u l l y repeat a l l t h e questions p r e v i o u s l y asked along t h e branch, r e c e i v i n g t h e same answers, u n t i l she asks

"r

and now receives

?"

-CK

the answer

"n

'I.

o r asks

+l)?ll,

"&1(%,5

t h e questioning f a l t e r s

a),

branch the k3-oracle picked i s extended, keeping i t s index "n -6,

'I

t o a n e x t vertex bearing

*3

'I\?'

or

As ODtion 4 , t h e a - o r a c l e may answer a question pending a t an e a r l i e r stage then

6,

-- c$

II

(i3(i2) = o r the

K

bearing

-2 -1 a (a )

According as she then stands mute, o r says

-cK

by a segment

K,

"1;cK+ ,

? I'

"r ?"

x e

.

t h a t was

b u t has meanwhile been answered i n

one o r more wayssand answers i t d i f f e r e n t l y than before, say by

'r~+"

where

Recursive Functionals and Quantifiers of Finite Types Revisited 111

K

i s the n e x t o r d i n a l .

{<&n,qn>ln '%,e+1*

7",

21

According as the k2-oracle, questioned w i t h

< 0 ) u{),

stands mute, o r responds w i t h

or

"mf'

t h e questioning f a l t e r s , o r the Stage-u c h a r t receives a new d i v e r g i n g from a l l o l d branches through the vertex where t h e

branch question

"d" or -K

"G~?'' was once

pending, bearing

"G,~+~ a t?i t"s

y e

'In-

on i t s new segment and

I'

end.

As Option 5, t h e k 3 - o r a c l e may p i c k an (unindexed) branch wiith no l a s t vertex i n t h e Stage-u c h a r t , question the A2-oracle w i t h the f u n c t i o n {<%,%>IT- < 5 1 represented by i t ( 5 a l i m i t o r d i n a l ) , and if the a* 2 -oracle 5 ) a t i t s next ( l i m i t )

responds add t o the branch a vertex (indexed by p o s i t i o n w i t h the a p p r o p r i a t e one o f r e s u l t i n g branch by t h e n e x t o r d i n a l

"$!TI "

or

"r-

?'I,

and index the

F.

4

-3 Now consider how t h e a - o r a c l e may reach Stage u Form t h e union o f t h e stages o f t h e c h a r t f o r a l l

c u

T

for u = l i m

T
i n the obvious

The a- 3 -oracle, contemplating t h i s union, may decide t o close up shop.

way.

Suppose t h e l a t t e r .

O r she may choose t o go ahead.

I n t h e unionized c h a r t , some branches may have been established a t some stage

T,

being t h e same i n a l l Stage-7 c h a r t s f o r

T

17

< u.

In

the unionized c h a r t , each o f these r e t a i n s the index, o r the unindexed status, which i t had i n a l l such stages 7. There may be some branches which were n o t e s t a b l i s h e d a t any stage T <

u , the i n i t i a l segment of such a branch e x i s t i n g a t each stage

subsequently p r o p e r l y extended a t some stage these i n i t i a l segments may have t h e same index great

(T

T K

<

7 c a).

T

being

However

for a l l sufficiently

When t h i s i s t h e case, I c a l l t h e branch a s p i r i n g .

Other branch-3 es n o t e s t a b l i s h e d a t any stage T < u I c a l l unaspiring. The.a - o r a c l e ' s 03 * 2 -2 commitment r e q u i r e s t h a t , i f a (a ) i s t o be defined, the a -oracle, T.

questioned w i t h the f u n c t i o n

A'

=

{<%,%>In

< 51

represented by an

S.C. KLEENE

28

o r ask a new question "r ?".

-I"I' aspiring branch, either answer with an S

-5

The aspiring branches are t o be thus questioned, in the order of the indexing a l l t h e i r s u f f i c i e n t l y long seqments.

K'S

I f from some aspiring branch

neither response i s obtained, the questioning thereupon f a l t e r s , and the aspiring branches with greater

K'S

a r e not called t o the witness stand.

Otherwise, the Stage-a c h a r t comes from the unionized chart by adding t o each aspiring branch a vertex a t the next ( l i m i t ) position bearing the appropriate one of "mS' or

" r ?" and g i v i n g the r e s u l t the index

K.

--5

If there a r e no aspiring branches in the unionized c h a r t , i t i s already the Stage-a chart (provided the

-3

CL

-oracle chose t o go ahead).

A simple example i l l u s t r a t e s how u n a s p i r i n g branches can a r i s e .

In

Figure 4 , the number written over each segment i s the index of the branch ending with i t a t the stage when i t was f i r s t drawn on the chart.

r ' s a t the

Znd, 3rd, 4 t h ,

... vertices

(The

on each branch can be 0, 1 , 2 ,

and the ~ ' ast each branching can be 0, 1 . )

...,

The a3-oracle i s t o start

o u t by drawing the leftmost segment (how e l s e ? ) , then each of the second segments, then each of the t h i r d , e t c . , in order from the bottom u p i n each of these groups.

... Figure 4. The ;X3 -oracle wants t o know k2 (a1 ) on (some extension o f ) each of the branches which eventually r u n horizontally. i n the u n i o n of the charts f o r the stages

These a r e the aspiring branches T < W.

Inadvertently, the u n -

29

Recursive Functionals and Quantifiers of Finite Types Revisited I11

a s p i r i n g branch s l a n t i n g upward ad i n f i n i t u m a l s o gets i n t o the unionized chart.

-2 -1

-3 Ift h e a - o r a c l e wants t o know t h e value of

on t h i s branch,

a (a )

u > w.

she must apply Option 5 t o i t a t some stage

The f i n a l stage (which i s indeed f i n a l , and n o t j u s t the l a s t stage -3 before the questioning f a l t e r e d o r the a - o r a c l e gave up), i . e . the stage -3 ( i f i t e x i s t s ) a t which t h e a -oracle.announces t h a t be t h e successor stage

o+l

on t h e c h a r t a r e completed. each such branch

iKt h e

for

iK is

iK is

. .

<

u

Then t h e

(p > 0)

Xion

x

I+

"$4"

i n the

i n Stage o

C < l ; ~ ~ , $ ~ > )
.

( a ) = 2, can o n l y

i n which a l l t h e index'ed branches

o

a t t h e end of

i s t h e value

f u n c t i o n represented by t h e branch.

second kind, kind,

K

t o a stage

*3 -2

a

k2(;

For a branch

For a branch

K

K K

) = m

X

of the

o f the f i r s t

The analogy t o the corresponding Subcase

f i 2f o r

type 2 ( c f . t h e second paragraph o f t h i s 11.2) i s weak, i n t h a t the questions po, ply

..., iK, ...

(K

< p)

-3 -2 t h e a - o r a c l e poses t o the a - o r a c l e are n o t

i n general completed and answered i n t h e order i n which they are indexed ( i K i s n o t i n general determined by 0+1

C<~,,E~>IX < KI).The f i n a l stage

i s reached t h e f i r s t time t h e i 3 - o r a c l e f i n d s h e r s e l f contemplating a

Stage-o c h a r t ( w i t h a l l i t s indexed branches completed) on t h e evidence ( s h a l l I say the "basis"?) of which, i . e . -2 ' < ~ =a lopes) & v ~ ~ (p,)

5,

t h a t ( t h e i 2 - o r a c l e opens enve-

-3 -2 she can conclude t h a t a ( a ) = 2.

03 Now I s t i p u l a t e , as p a r t o f the d e f i n i t i o n of a ( l i c e n s e d ) a -oracle:

(1) For various oracles f o r a type-; -2 -2 93 3 f o ; - a 1, if t h e a - o r a c l e gives

object

. 3 02

a (a ) =

&*

under Subcase

5 when questioned w i t h any.one

of them, she does so when questioned w i t h any other. -3 Subcase 3.3 oracle.

, the

n3(&

I n the cases o t h e r than

*2 r e s u l t s a r e c l e a r l y independent o f a choice o f the CL

-

So, w i t h t h e s t i p u l a t i o n (l), I have defined what i s the p a r t i a l

one-place f u n c t i o n

i3 from

type

i into

lowhose

values a r e revealed by

a given o r a c l e f o r a type-3 o b j e c t under our questioning o f her w i t h

S.C. KLEENE

30

;'-oracles. Now I d e f i n e t h e

*3 - 2 which a (a )

basis i2 for

i s defined.

-2 the a - and A3-oracles;

i2

w.

1.to A3%

f o r each

A2

for

I begin by doing so r e l a t i v e t o a given choice o f b u t p r e s e n t l y I s h a l l show t h a t the r e s u l t i s i n -

dependent o f those choices." of

A2

The basis

i2 shall

be the type-;

subfunction

d e f i n e d o n l y t o t h e e x t e n t t h a t t h e i 3 - o r a c l e discovered i n her

- 3 .2 a (a )

-2

questioning o f t h e a - o r a c l e up t o t h e moment of her d e c i s i o n t h a t *3 - 2 SO we s h a l l have i2 E type i2 c i2 and ct (B ) = I.

i,

Consider f i r s t the p r i n c i p a l case (Subcase

-3

n3w i t h

03 - 2 a (a ) =

s.

--5.

The

a - o r a c l e made her d e c i s i o n when she had j u s t f i n i s h e d l e a r n i n g t h e s e t

I'bK,$>l~

<

w. r. t o

i2

6,

of

f o r i t s e l f and ( i n Case

*2

a

l2(PK) =

( c f . 9.4).

pK

i s t h e basis

3' f o r kl) f o r every extension a- 1

w i t h an envelope-opening o r a c l e .

knowledge t h a t of

a- 2 . Each

o f argument-value p a i r s f o r

By the monotonicity o f

-2 a , the

$ gives t h e knowledge of t h e whole s e c t i o n S -

A l t o g e t h e r , t h e i 3 - o r a c l e has learned

=u

-pK

i2K
and t h i s I take as the basis. I n Subcase

m3w i t h

learned e v e r y t h i n g about Case

T3* she

- 3 '2 a (a )

i2 =

=

r, t h e a93- o r a c l e so t h e basis

A?',;

learned nothing, so t h e b a s i s

b2

is

i n one f e l l swoop

i2 i s i2 i t s e l f .

o2

In

= hi1@.

-3 - 2 Now I e s t a b l i s h t h a t , when a (a ) i s defined, the basis f o r a type-2 02 o b j e c t a w. r. t o a type-; o b j e c t i3 i s independent o f t h e choices of the oracles f o r those o b j e c t s . I begin by supposing t h e p r i n c i p a l case (Subcase

-2

3 for

k3,

m3,i . e .

-2 a ) a p p l i e s t o one choice o f t h e oracles.

a p p l i e s t o every o t h e r choice.

For

Cases

T3,

But then t h e same

i3(Q2) i s undefined

-3 -2 and a (a )

*3

defined, so Case T3 a p p l i e s t o every a -oracle; and A2(Q1) i s undefined - 2 -1 a2 so Case a p p l i e s t o every a -oracle. and c1 (a ) i s d e f i n e d f o r some

I claim that:

E, w i t h

;',

T2

a given choice of t h e a- 3 and a- 2 -oracles,

-

31

Recursive Functionals and Quantifiers of Finite Types Revisited 111

-3 -2 a (a )

union

i s defined under Subcase 77

B-

of i2

w m

By (1) w i t h end 10.2,

defined,

i2, then, f o r e v e r y subi2. Say t h a t t h e v a l u e o f i3(&')

the basis F2

r e p r e s e n t e d i n t h e f i n a l stage o f t h e c h a r t .

t o g e t t h e v a l u e of

i3(2)we can use f o r 7

a suboracle

a3 Now i f 3 r2 -oracle, her questioning when we q u e s t i o n t h e g i v e n a - o r a c l e w i t h t h i s

t h e g i v e n k 2 - o r a c l e , and we can of course use t h e g i v e n o r a c l e f o r

??+b 2 , Of

k3(2)

IC;~,$>IK < PI

i s obtained w i t h

Of

m3w i t h

r2 -2 t h e B - o r a c l e w i l l r u n e x a c t l y as t h a t o f t h e a - o r a c l e u n t i l she c o p p l e t e s a

f o r which

-

i2(iK) i s undefined,

p,

whereupon (by t h e c o n s t r u c t i o n end 9.4 o r 10.2 of

t h e s u b o r a c l e f o r a subunion) t h e -2 - o r a c l e w i l l n e i t h e r g i v e a v a l u e n o r ask a

- 3 r;! n e x t q u e s t i o n , making a (B ) Applying t h i s , w i t h

undefined!

a3 2 a (k )

d e f i n e d under Subcase

choices o f t h e p a i r o f o r a c l e s w i t h bases a r e subunions of

I n e i t h e r Subcase

23

i2 and 3.2' w i t h i3(k2)

i2, so

i2 and 6 8'3 F , i . e .

m3,t o

two d i f f e r e n t

.

r e s p e c t i v e l y , t h e bases

2 = b2.

d e f i n e d , o r Case

F',

w i t h one c h o i c e of

t h e o r a c l e s , i t i s immediate t h a t t h e same r e s p e c t i v e case a p p l i e s t o each o t h e r choice and t h a t t h e b a s i s i s independent of t h e c h o i c e .

To complete t h e d e f i n i t i o n o f a ( l i c e n s e d ) a- 3- o r a c l e , i t remains f o r me t o assure t h a t t h e f u n c t i o n

k3

whose v a l u e s a r e r e v e a l e d b y t h e o r a c l e i s monotone.

As I remarked f o r i 2 - o r a c l e s i n 6.3 and 9.3, m o n o t o n i c i t y h o l d s vacuously o r t r i v i a l l y i n s e v e r a l cases, which I s h a l l enumerate p r e s e n t l y ( i n i n f e r r i n g (4) from ( 1 ) - ( 3 ) ) . Suppose t h e i 3 - o r a c l e o p e r a t e s under Case with

22 i2 and

-3 -2 a (a ) = 2, under Case

- 3 s;! F o r m o n o t o n i c i t y , we want t o have a (a ) =

T3,

7' (so

-2 and b o t h t h e a Subcase

-

72 and a - o r a c l e s ,

m3a p p l i e s

t o both).

s.

By ( 1 ) w i t h ( X X I I I ) i n 10.2, i t w i l l s u f f i c e t o have t h i s assured when t h e -2 r2 a - o r a c l e i s a s u b o r a c l e o f t h e g i v e n a - o r a c l e . Then i n t h e f i n a l c h a r t i n t h e

-3 -2 a - o r a c l e ' s q u e s t i o n i n g o f t h e a - o r a c l e , each indexed branch (with

"%/"

w i t h t h e same

K

representing

a t i t s end) extends by z e r o o r more v e r t i c e s a branch ; i !

" % P ) o f t h e t r e e "or t h e a - o r a c l p .

';a

(ending

-3

C n m i r l c r how t h e a - o r a c l e r;!

progresses t h r o u g h t h e stages i n h e r f u r t h e r s y s t e m a t i c q u e s t i o n i n g af t h e a

-

P,

S.C. KLEENE

32

*2

o r a c l e , which she must commence s i n c e she d i d so f o r

a

.

The f i r s t t i m e she

r e c e i v e s a response d i f f e r e n t t h a n f r o m t h e & 2 - o r a c l e , i f e v e r she does, w i l l be when she reaches a

-

. 2 r c l a r e s t h a t a (p-)

question

>

c

K

f

bK

_- -$ ( =

"i'(~)?II.

( f o r one o r more

%

f o r each such

Her c h a r t f o r

o f t h e t r e e on h e r f i n a l c h a r t f o r thus f a r acquired about

2

2 -2

a

<

K'S

K),

u).

;! Then t h e r a - o r a c l e de-

-2 w h i l e t h e a - o r a c l e asks a n o t h e r

a t t h a t s t a g e bears an i n i t i a l s u b t r e e ,

-3 Thus t h e i n f o r m a t i o n t h e a - o r a c l e has

i s i n c l u d e d i n what she has a t t h e same s t a g e about q 7

i2, e x c e p t

f o r b e i n g b e t t e r i n t h a t she a l r e a d y knows t h e v a l u e % o f a (G) .1 7 -3 - 2 -1 (and b y m o n o t o n i c i t y , of a (a ) f o r a l l a 3 Q). A w e l l - o r g a n i z e d a - o r a c l e 3 would c o n t i n u e t o pursue t h e same f u r t h e r i n f o r m a t i o n from t h e a - o r a c l e ( t o t h e -2 e x t e n t i t i s w a n t i n g ) as f r o m t h e a - o r a c l e , and n o t hanker a f t e r i n f o r m a t i o n on

*2 s t i l l o t h e r branches of t h e a - o r a c l e t r e e ( i . e . ones h a v i n g no prolongations).

i K ' s as

Indeed, a l l she needs t o do i s t o c o n t i n u e t h e q u e s t i o n i n g she

*2 would pursue with t h e a - o r a c l e , s i m p l y s k i p p i n g t h e s t e p s a l o n g branches ex-

tending

-

b K ' s toward

;,Is,

which have become unnecessary (and indeed i m p o s s i b l e ) .

T?

C o n t i n u i n g thus, h e r c h a r t f o r t h e a - o r a c l e w i l l always remain p r o p e r l y o r i m properly w i t h i n her f i n a l c h a r t for the i2-oracle.

E v e n t u a l l y , she must r e a c h a

stage a t which t h e r e w i l l remain no uncompleted indexed branches.

2

o r a c l e w i l l have a c q u i r e d i n f o r m a t i o n a b o u t

A*

when she d e c l a r e d t h a t

i3(A2)

=

I.

Then t h e

d-

which i m p l i e s a l l she knew about

a -2 .

So she must d e c l a r e t h e same f o r

Thus I conclude t h a t i t i s reasonable t o demand: (2) Under Subcase 3 .3' f o r b o t h -2 -2 ';2 a' and w i t h the a -oracle a suboracle o f the a -oracle, i f the i3-oracle

2,

3 -2

(a ) =

x, she a l s o

a3 P a (a) =

gives

-3 impose on a - o r a c l e s t o assure m o n o t o n i c i t y .

s.

T h i s i s one o f t h e s t i p u l a t i o n s I

The l a s t s t i p u l a t i o n f o r t h i s purpose is (as i n 6.3 and 9.3 f o r $ - o r a c l e s ) : (3)

Under Case

T3,

A2

ifr

on t h e b a s i s o f knowing t h a t

m T 2

she a l s o says t h a t

under Case

-2

a

3

{IK

z2 the k 3 - o r a c l e < p),

t h e n when

says t h a t

A3(i2)

2 = &'n~

under

i3(9) = 2. .a

T h i s completes my d e f i n i t i o n o f a ( l i c e n s e d ) a " - o r a c l e ,

=

the stipulations

( 1 ) - ( 3 ) b e i n g added t o t h e general d e s c r i p t i o n t o assure t h a t t h e o r a c l e ' s

1.

Recursive Functionals and Quantifiers of Finite Types Revisited 111 b e h a v i o r produces b y h e r answers a p a r t i a l one-place f u n c t i o n

No

into

and one which i s monotone.

i s the set of the uni-

monotone p a r t i a l one-place f u n c t i o n s f r o m t y p e - 2 i n t o

k2 c

1.

-2 must o p e r a t e so t h a t , f o r g i v e n a -

and 7a 2- o r a c l e s

T2 ( b u t t h e i 2 - o r a c l e need n o t be a s u b o r a c l e o f t h e a - o r a c l e ) ,

E2

then

0

(1)-(3) are equivalent t o the following version o f monotonicity:

The a 3 - o r a c l e

(4)

f r o m type-;

Such a f u n c t i o n i s unimonotone, i f f i t i s

t h e f u n c t i o n g i v e n by an k 3 - o r a c l e obeying ( 1 ) - ( 3 ) .

I n fact,

A3

33

- 3 7-2

a (a ) =

i2c z2

if i3(i2) =s,

2.

To i n f e r ( 4 ) f r o m ( 1 ) - ( 3 ) ,

A3(i2)i s

with

undefined.

l e t us t a k e cases on t h e a 3 - o r a c l e .

I n Case

F3,

-3 72

a (a

--

)

-3 -2

a (a )

anyway.

I n Case

I n Case

i3,

T3 w i t h

r2c a n n o t a p p l y t o e i t h e r i2 o r ;ar. With * 2 T2 a a , we cannot have Case T2 a p p l y i n g t o 2 ( s o p ( @ ) = 0) y i t h F2 t o k2. I f 7' a p p l i e s t o b o t h k2 and 2 w i t h &2 c 2 , 2 = i2 w i t h t h e same o r a c l e . I n Case T3, i f T2 a p p l i e s t o k2 and T2 t o z2 w i t h a2 2 and and

&3 (&2 )

d e f i n e d , Case

c

c

-3 -2

a (a ) =

and

2

g, t h e n a- 3 (7a2 ) with

&2

c

3

=

and

I n Case

2 by ( 3 ) .

i 3 ( k 2 ) = s,

T3,

if

32 a p p l i e s

t o both

i2

t h e n u s i n g ( 1 ) we can if necessary sub-

s t i t u t e an A 2 - o r a c l e t h a t i s a s u b o r a c l e o f t h e g i v e n z 2 - o r a c l e , whereupon (2) gives

- 2 r;! a (a )

- 2.

To i n f e r ( 1 ) f r o m (4), observe t h a t , i f we have two o r a c l e s f o r a type-2 o b j e c t under Case fortiori

-2 a

T2,

.

r;! c a

-2 s;! say an a - o r a c l e and an a - o r a c l e where

-

(i.e.

a

From (4), (2) and (3) a r e immediate.

-3 I n my d e f i n i t i o n o f typa-3 v i a a - o r a c l e s , tions.

i2= 3 , t h e n

( 1 ) - ( 3 ) o r (4) a r e s t r o n g assump-

The p a y - o f f we can a n t i c i p a t e i s t h a t , i n d e a l i n g w i t h "3-expressions'' e x p r e s s i o n s f o r type-3 o b j e c t s ) , b y h a v i n g imposed these assumptions on t h e

i n t e r p r e t a t i o n s o f t h e type-3 v a r i a b l e s , we w i l l have them s a t i s f i e d by 3-expressions t h a t a r e n o t j u s t v a r i a b l e s . 11.3.

I n 19 I d i d n o t r e v i e w t h e t r e a t m e n t i n 6.4 o f assumed f u n c t i o n s

with v a r i a b l e s o f types

6

and

i;2b u t

-

i t w i l l be s u b s t a n t i a l l y i n c l u d e d i n t h e

f o l l o w i n g t r e a t m e n t o f assumed f u n c t f o n s

h, 1 and 2.

6,

I s h a l l use t h e i l l u s t r a t i o n

8,

-.

.. .

w i t h one o r more v a r i a b l e s o f types

et(a,a1,a2,a

-

-2

)

i n s t e a d of t h e

S.C. KLEENE

34

it(& il, k2) -

We ask t h e

(i1,i2 a r e type-: v a r i a b l e s ) . "i)t(& i,,i2, i2)?" by p r e s e n t i n g it-oracle of 6 . 4

-

envelopes l a b e l e d #1, #2, #3, # 4 , c o n t a i n i n g r e s p e c t i v e l y and o r a c l e s f o r

. . al,

.2 a2, a

CASE ~ ' o ' o y 1 y 1 y 2 ) : for all

The

i t - o r a c l e stands mute.

Then

-

Y(o:091s1921: Without

. . . .2 5, a l , a2, a .

(empty if

4 = 0)

.

. . . .2 a, a l , a 2 , a .

clares t h a t

4

t o her closed

opening any of t h e envelopes, t h e

. a.2 , a').

al,

&&gT ( o : 0 9 1 9 1 9 2 1 :The 6

it(& & l , k2, i2) = @

w.

=

. . it(i, a l , a2, i2)

So

-

=

w

6 t- o r a c l e -

de-

for all

- o r a c l e performs a non-empty s e r i e s o f steps, each

t

o f which c o n s i s t s o f opening one o f t h e envelopes, and/or ( i n t h e case o f #2, #3 o r #4) questioning the

il-, i2- o r

j u s t t h e n o r p r e v i o u s l y opened. t i o n about

4,i l ,a.2 , a.2

I f , as a step, she

-2 a - o r a c l e found i n s i d e when t h e envelope was

By t h e s e steps, she seeks t o accumulate informa-

toward d e c i d i n g whether and how t o answer o u r q u e s t i o n

opens #1, she stands mute ( e n d i n g h e r s e r i e s o f s t e p s ) if

t h e envelope t u r n s o u t t o be empty, and o t h e r w i s e takes n o t e o f t h e i n s i d e as b e i n g t h e v a l u e o f

(E

0) f o u n d

4.

If,i n a s t e p , she opens #2, t h e n as p a r t o f t h e s t e p she asks t h e i l - o r a c l e who s t e p s o u t of i t

"i,,(&')?" w i t h A'

=

o r a c l e stands mute (Case

T'), so does she.

kl-oracle

&,(a *' ) -

declares t h a t

_?.

(Case

w i l l n o t query t h e & ' - o r a c l e f u r t h e r . she notes t h a t t h e hl-oracle,

@

(an empty envelope).

T1),she

= and 1 opens t h e envelope (Case

notes t h a t

If t h e i l - o r a c l e

being queried w i t h

"Qt(&&,,

-

A,-

I f , w i t h o u t opening t h e envelope, t h e

" ~ l ( ~ o ) ? " asks ,

l e t s t h i s s t a n d as t h e i n p u t o f i n f o r m a t i o n f r o m t h e s t e p . a l l y t o answer o u r q u e s t i o n

I f the

i2, i2)?", she

"&'?"

.

She

But, i f she i s eventu-

w i l l i n some subsequent

s t e p s ( n o t n e c e s s a r i l y a t once, and i n general i n t e r s p e r s e d w i t h o t h e r s t e p s ) answer t h e i l - o r a c l e ' s and n o t e t h e i l - o r a c l e ' s

question

"i'?"

answers ( t o

b y one o r more numbers 'I&

1

-0 (a ) ? " )

%1,

T1),

'oo,Q l , ...,

..., s t a n d i n g

mute

Recursive Functionals and Quantifiersof Finite Types Revisited 111

35

s e r i e s of steps) i f the &l-oracle ever f a i l s to answer.

(ending her

Similarly, with # 3 .

I f , in a step, the Gt-oracle opens # 4 , then as p a r t of the same step she asks 2 -oracle stands mute (Case ?), the a2-oracle 11&2(&')?"with = If the

&' 0'.

.2

so does she.

I f , without opening the envelope, the a -oracle answers t h a t 1 i2(&') = m (Case T'), she notes t h a t &* = A& m, and will n o t question the oracle further.

t o the &2 -oracle

2 -oracle opens the envelope (Case

If the

P ) , she notes

&2 -

relative

t h a t the & 2 -oracle when asked "&2(&1)?" asks *2 H&1?" If she i s eventually t o answer i1,i2, a )?", she will in some -2 subsequent steps (not necessarily a t once or consecutive) respond t o the a -oracle's "&'?", i.e.

"Gt(i,

.

question

G 1 ? l *

-2

by commencing further systematic questioning of the a -oracle in

m3of

the manner described f o r an i3-oracle in Subcase

11.2.

Each step of t h i s

f . s. q . i s determined by her on the basis of a l l the information accumulated in the previous steps, which can be a mixture of information about several or a l l of

.. .

.2

.

(Don't do anything without taking i n t o account the whole picture!) 5, a l , a 2 , a She can go mute a f t e r any s t a t e , or before a l i m i t stage; and she must go mute i f rebuffed in attempting t o carry out a step ( b y = @ i f #1 i s opened, or by -2 the a -, a2- or a - o r a c l e ' s not responding t o a question). 1 I f , in t h i s Case 4 3 f o r the &'-oracle, the ;t-oracle does get t o f . s. q . , she then unfurls a c h a r t , t o be used in the manner described in 11.2. I t will be ad-

4

. .

vantangous here t o divide the chart into four parts, # 1 , # 2 , #3, #4, t o receive

4,A1, a2,

the information accumulated about already accumulated about

i, A1,

&2

-2

a

, respectively.

Information

can be p u t on the chart a t once when i t i s

unfurled, and can be updated when further steps relating t o between stages in the f . s. q . of a- 2

* . 4, a,, a2

occur

.

The ;+-oracle can take the step t o a final stage, declaring t h a t -2 bt(& i1, a2, a ) = w, only when she has before her a chart (or i f she never g o t t o f . s. q . , a record of information a b o u t &,, L2 simply) with no question

4,

Pending, l i k e a t o t a l l y unanswered 01

&l ?I, , or an

"r?"a t

in #2 o r # 3 , o r a t o t a l l y unanswered

the end of some indexed branch of the second k i n d , in #4.

S.C. KLEENE

36

Now 1 make t h e assumptions f o r a ( l i c e n s e d ) -3 o r ( 4 ) f o r an a - o r a c l e . I n p a r t i c u l a r , then

6 t- o r a c l e -

corresponding t o (1)-(3)

The c o n c l u d i n g remarks of 11.2 extend.

6,

i s monotone, and, f o r each

(4,Al, A2,

A2)

with

. . a- 2 ) w, t h-e r e i s a E i q u e i n t r i n s i c a l l y - d e t e r m i n e d b a s i s f o r it(& a.,, .- . . a.2 ) w. c. Q it,which i s o f t h e f o r m (4,i1, (L1, d2, i2) where =

a2,

al,

a2,

4, 8, Al, d, Az 6,(4, il, b,, k 2 ) = w. -

-

-2 A2 (i2 a subunion o f a ) and . . . . Here 6,d,, d2, d2 a r e t h e type-0, -1, -1, -2 o b j e c t s which embody e x a c t l y t h e i n f o r m a t i o n about 4,il, i2,i2 which t h e 6-t - o r a c l e . . .2 accumulated b e f o r e d e c l a r i n g t h a t al, a 2 , a ) = w. -

bc

c

11.4

It

c

and

dz

c

i s my b e l i e f t h a t t h i s g i v e s a semantics s u i t a b l e f o r t h e t h e o r y of

p a r t i a l r e c u r s i v e f u n c t i o n s and f u n c t i o n a l s a l a m w i t h v a r i a b l e s up through t y p e 3 and assumed f u n c t i o n s o f v a r i a b l e s up t h r o u g h t y p e 2. i n v e s t i g a t i o n s w i t h t h e aim of c o n f i r m i n g t h i s .

I am c o n t i n u i n g my

13

The g u i d i n g i d e a i n t h i s semantics i s t h a t , when we seek t o compute we s h a l l ask o f

if

k3(i2)

A*

*3 -2 a (a ),

o n l y q u e s t i o n s t o which she must respond i n s p e c i f i c ways

i s t o be d e f i n e d .

Non-response w i l l make

i3(i2) undefined. To

a v o i d p r o v o k i n g non-response u n n e c e s s a r i l y c a l l s f o r c i r c u m s p e c t i o n i n how quest i o n s a r e asked i n sequence.

I n t h i s way we achieve an economy i n t h e q u e s t i o n i n g ,

and i n t h e c o r r e s p o n d i n g computation process. I n K i e r s t e a d ' s a l t e r n a t i v e semantics

1980,1982, computation

s i m u l t a n e o u s l y , o r i n a d o v e t a i l e d f a s h i o n , a l o n g many l i n e s .

A3(a2)

i s pursued

A v a l u e of

( i n my n o t a t i o n ) i s o b t a i n e d i f and when answers a r e o b t a i n e d a l o n g a

s u f f i c i e n t subset o f t h e l i n e s o f computation, t h e o t h e r l i n e s s t a n d i n g uncompleted.

Recursive Functionds and Quantifiers of Finite Types Revisited 111

37

REFERENCES

Kierstead, David P h i l i p 1980 (1978 l e c t u r e ) . A semantics f o r Kleene's 1-expressions. The Kleene Symposium, J . Barwise, H. J. K e i s l e r and K. Kunen, eds., North-Hol 1and Pub. Co. (Amsterdam, New York, Oxford) , 353-366. 1982

(1979 Wisconsin Ph.D. t h e s i s ) . Syntax and semantics i p higher-type r e c u r s i o n theory. Trans. Amer. Math. S O C . ; forthcoming.

Kleene, Stephen Cole 1938 On n o t a t i o n f o r o r d i n a l numbers. 150-155.

Jour. symbolic l o g i c , 3,

1952

North-Hol land Pub. Co (Amsterdam),

I n t r o d u c t i o n t o metamathematics.

P. Noordhoff (Groningen) and D. Van Nostrand (Princeton,, Yew York and Toronto), x + 550 pp. Eighth r e p r i n t , North-Holland'Pub. Co. (Amsterdam, New York, Oxford) and E l s e v i e r North-Holland (New York), 1980.

1959

Recursive f u n c t i o n a l s and q u a n t i f i e r s o f f i n i t e types I . Amer. Math. SOC., 91, 1-52.

Trans.

1963 Recursive f u n c t i o n a l s and q u a n t i f i e r s o f f i n i t e types 11. I b i d . , 108, 106-142. 1978 Recursive f u n c t i o n a l s and q u a n t i f i e r s o f f i n i t e types r e v i s i t e d I . Generalized r e c u r s i o n theory 11, Proceedings o f t h e 1977 Oslo Symposium, ed. J . E. Fenstad, R. 0. Gandy and G. E. Sacks, NorthHolland Pub. Co. (Amsterdam, New York, Oxford), 185-222. 1980 Recursive f u n c t i o n a l s and q u a n t i f i e r s of f i n i t e types r e v i s i t e d 11. The Kleene Symposium, J . Barwise, H. J. K e i s l e r and K. Kunen, eds., North-Holland Pub. Co. (Amsterdam, New York, Oxford), 1-29. Turing, Alan Mathison 1939 Systems of l o g i c based on o r d i n a l s . Proc. London ?lath. S O C . , s . 2, 45, 161-228.

S.C. KLEENE

38

FOOTNOTES I have w r i t t e n t h i s paper as

'See t h e References a t the end o f t h i s paper.

a member o f FOO, and i t c o u l d have been e n t i t l e d "MOO". 'In

1980 and

the present paper, I am n o t attempting t o go beyond type 3.

Thus t h e c l a i m i n

1978 p.

186 t h a t "Is h a l l l o o k f u r t h e r , i n d i c a t i n g how t o

deal w i t h h i g h e r types" must be q u a l i f i e d . functions

And we should l i m i t our assumed

t o having v a r i a b l e s o n l y o f types

'at

5

2.

(An assumed f u n c t i o n

w i t h j u s t one v a r i a b l e and t h a t o f type 3 would be a type-4 o b j e c t . )

So i n

1978 now, on p. 189 "m I' should be omitted, and a number o f 3 3 s i m i l a r omissions a r e a p p r o p r i a t e on pp. 203, 207.

reading

. .

I n 1%

p. 18 l i n e 14 from below, " w i l l " should be q u a l i f i e d by

"if

i t ( & al, a2) i s t o r e c e i v e a value"; and i n p. 23 l i n e 22, "must" should be-qualified by " i f a step ' 4 t ' h o r i z o n t a l l y r i g h t w a r d i s t o be authorized".

The treatment o f

Gt-oracles i n 6.4,

and o f E7 i n 7.4,

can be sum-

marized by saying t h a t t h e sequences o f t h e steps i n t h e questioning o f each

a2

i,

A1 and by t h e ;;-oracle, and o f t h e subcomputations associated of *2 therewith, a r e o f t h e same s z r t s as i n t h e treatment o f iland a -oracles. The t h r e e sequences d r e combined i n t o one ( o f which each i s a subsequence), w i t h the members depending each i n t u r n on what went before i n the combined sequence.

1978 p.

In

ltTl ,. . .

-

e:

,@-I#

206 t h e f i r s t l i n e a f t e r ( V ) , f o r

In

1980 p.

3 l i n e 7, f o r " ( X X I I I ) " read "xXII1';

a f t e r the f i r s t display, f o r read

"63-I s " ; p.

17 1.6,

f o r "2" and "3" read 3Cf.

1978 f o o t n o t e

for

" i l l

1.

'I

and

" R t " read

"R";

read

"<";

""

"il';p.

1952 pp.

p. 4 1.1

1. 14, f o r "'at's'' p. 17 7 . 16 and p. 29 7. 18,

p. 12

26 7 . 16 f o r "a" read "y".

The " f i r s t r e c u r s i o n theorem" o f my

the " r e c u r s i o n theorem" o f

read

352-353 and

1938 t h e

1952 p.

348, and

l a s t two l i n e s o f

§2, o n l y make sense i n a s e t t i n g o f p a r t i a l f u n c t i o n s . 4The idea o f an " o r a c l e " o r i g i n a t e d i n Turing

5I use barred case designations

1939.

Cf.

1978 footnote

2.

"i-2", e t c . because "12", "22'1, e t c . 11$'1,

were used i n a s l i g h t l y d i f f e r e n t c l a s s i f i c a t i o n o f t h e t y p e - i o b j e c t s i n

1980 5.4

b e f o r e I brought t h e i n t r i n s i c determination o f t h e bases i n t o t h e

p i c t u r e i n 6.3.

39

Recursive Functionals and Quantifiers of Finite Types Revisited I11

6@1

has two oracles:

and an o r a c l e under Case

an o r a c l e under Case

7'

i1 who does n o t open envelopes,

who opens envelopes b u t i s never pleased with

the contents (always then standing mute); each o t h e r type-1 o b j e c t has j u s t be d e f i n e d under Case

F2,

independently of

the o r a c l e f o r 7We may s t a r t w i t h a p r e v i o u s l y known p a r t i a l one-place f u n c t i o n n

A'

from

NU; then an o r a c l e f o r a type-2 o b j e c t i s an o r a c l e f o r t h i s 2i2,o r an "a -oracle", i f f the f u n c t i o n given by her behavior coi9cides w i t h &' as p r e v i o u s l y known. O r we may f i r s t know the f u n c t i o n a- 2 type-1 i n t o

through t h e behavior o f t h e o r a c l e . -2 oracle f o r t h e f u n c t i o n a

.

* 2-oracle" i s an I n e i t h e r case an "a

The phrase " f o r a l l i 2 - o r a c l e s " can mean e i t h e r " f o r a l l the oracles for a given type-; o b j e c t i 2 ' I 8 o r " f o r a l l oracles f o r t.ype-i o b j e c t s 4' ( w i t h a- 2 varying)", according t o t h e content.

3.3',d i f f e -r e n t-a02-oracles- may ask t h e i r questions k,cl, ..., $, ... and q,, r--, ..., q,... i n d i f f e r e n t orders, w i t h - 2 (a - 1 ) = 0, the questioning prolonged t o various lengths i f a b u t (as I am about t o prove) w i t h {<$,%>I< < 63 = {<<,%>I< c f> i f a - 2 (a - 1 ) - m. Also f o r 4' = t h e r e i s an o r a c l e under Case r2,besides various 8Under Subcase

a',

oracles under Case

7'

o r l a t e r i n Subcase 'For

i1 = 0'. i t s

o r a c l e under Case

7'

who open the envelope b u t always stand mute (sooner

3.3'). two oracles

6

-2 a r e represented i n the t r e e f o r any a

-

by the path c o n s i s t i n g o f j u s t the i n i t i a l vertex and

the path ending w i t h t h e t o p second vertex. "This

*1 covers a l l the r e s t o f t h e a -oracles (i.e.

except t h e o r a c l e under Case 7' f o r i1 = @',6 -2 9 new a - o r a c l e t h e treatment she deserves. "In

t h i s case, each s e c t i o n of

k2

those n o t under Case

F'),

who w i l l r e c e i v e from our

i s a section o f

?

o r i s properly

contained i n such a section. For, consider any s e c t i o n S df k2. By +1 02 -2 '1 *1 f o r some b' E S* c r3 c B , y E domain S So, f o r each a E 7 1 .1 .1 .1 -1 .1 a 3 y 3 B ; o!. a E domain ZBl. So S c S.,; domain S.l, i.e. f3 7 1 s-1 = -.S1 ' o r y # -61Y B

?j".

xalCs

S.C. KLEENE

40

Furthermore, under simply the circumstances of (XXII.2) , S with i1 E c A2 i s a section of b2 ( a s well as of i2). For, bps domain , so -28 (81 ) i s defined. So by (XXII.2), 8’ has as basis w. r. to 2 i t s basis w. r. t o i2, namely by (XX.2) i t s e l f (since 8’ E g2 c ) 01 01 1 2 Thus S = 8 * r ( a la B 1 with 8’ E (the bases w. r . t o s ) .

82

A

T1

-3 “The p o s s i b i l i t i e s f o r d i f f e r e n t oracles f o r a given type-; object a are similar t o those f o r a given type-; object i2.8 T h u s , under Subcase -33.3 there may be d i f f e r e n t sequences of Stage-a c h a r t s , e t c . I wrote t h i s paper as a member of Friends of Oracles, 13FO0 and and i t could be e n t i t l e d “More on Oracles“.

PATRAS LOGIC SYMPOSION G.Metakides (ed,) @ North-Holland Publishing Company, 1982

41

RECURSION THEORY ON MATROIDS A. Nerode* Department o f Mathematics Cornel 1 U n i v e r s i t y Ithaca,New York 14853

1.

J. Remnel** Department o f Mathematics University o f California a t San Diego La J o l l a , C a l i f o r n i a 92093

Recursion theory on S t e i n i t z exchange systems ( a l s o c a l l e d t r a n s i t i i e a b s t r a c t

dependence r e l a t i o n s ) was i n i t i a t e d i n [ 7 ] .

The m o t i v a t i n g examples were from

vector spaces and f i e l d s , which a r e s u r e l y t h e most important and comnon examples i n everyday mathematics.

For vector spaces, t h e example a r i s e s from an i n f i n i t e

dimensional r e c u r s i v e l y presented vector space

B = Ibl,b2,

...1

over a r e c u r s i v e f i e l d .

w i t h r e c u r s i v e basis

,V

Here t h e independent sets are t h e l i n e a r l y

independent sets, t h e closed sets a r e t h e subspaces, and a t t e n t i o n centers on L(V,),

t h e l a t t i c e o f r e c u r s i v e l y enumerable subspaces o f . ,V

For f i e l d s , the

example a r i s e s from an i n f i n i t e dimensional r e c u r s i v e l y presented a l g e b r a i c a l l y closed f i e l d

F,

w i t h r e c u r s i v e transcendence base

B = Ibl,b2,

...I .

Here t h e i n -

dependent sets are the a l g e b r a i c a l l y independent sets, the closed sets are t h e a l g e b r a i c a l l y closed s u b f i e l d s , and a t t e n t i o n centers on c u r s i v e l y enumerable a l g e b r a i c a l l y closed s u b f i e l d s . L(V,)

(when t h e f i e l d i s i n f i n i t e ) and

L(F,)

L(F,),

the l a t t i c e o f r e -

I n [7] a common property o f

i s i s o l a t e d which accounts f o r many

o f t h e i r common features, a l g e b r a i c and r e c u r s i o n t h e o r e t i c , namely r e g u l a r i t y ( n o t t o be confused w i t h T u t t e ' s n o t i o n f o r matroids).

A d e f i n i t i o n o f r e c u r s i v e l y pre-

sented S t e i n i t z exchange system

(U,cl)

was given, and f o r r e g u l a r r e c u r s i v e l y

presented systems, t h e theory o f

L(U),

t h e l a t t i c e o f r e c u r s i v e l y enumerable

closed subsets, went smoothly, and most r e s u l t s p r e v i o u s l y proved f o r l i f t e d by new smooth proofs ( and t h e r e f o r e obtained f o r

* **

L(F,)).

Research p a r t i a l l y supported by NSF Grant MCS 80-03016 Research p a r t i a l l y supported by NSF Grant MCS 79-03406

L(V,)

were

42

A. NERODE and J. REMMEL

I n t h e present paper a l l unexplained n o t a t i o n i s from [7],

1111.

However, t h e r e i s another s i d e t o S t e i n i t z exchange systems.

They were i n -

vented by van der Waerden [15] e x a c t l y t o capture the common features o f l i n e a r i n dependence i n vector spaces and a l g e b r a i c independence i n f i e l d s .

But e n t i r e l y

separately and i n a language motivated by a w h o l l y d i f f e r e n t c o l l e c t i o n o f examples from graph theory and combinatorial theory, Whitney [16] gave t h e completely equiva l e n t n o t i o n o f matroid, couched i n t h e language o f independent sets. p o i n t o f view a matroid c o n s i s t s o f a s e t U

( c a l l e d t h e independent s e t s o f

U

and a c o l l e c t i o n

U ) such t h a t :

1)

I

From t h a t

o f subsets o f

t h e n u l l s e t i s independent

2)

a s e t i s independent i f and o n l y i f a l l i t s f i n i t e subsets are independent

3)

i f A,B

a r e independent and the c a r d i n a l i t y o f

then there i s an

x

in

B-A

A

i s l e s s than t h a t o f

A u {XIindependent.

with

Beyond v e c t o r space i n -

dependence, Whitney's m o t i v a t i n g example was obtained where

U

i s t h e s e t of edges

of a graph and a subset i s independent i f i t contains no loop ( c i r c u i t ) . t h e van der Waerden closed sets are obtained by saying t h a t of

A, c l ( A ) ,

if

A u {XI

i s n o t independent.)

B,

x

(Of course

i s i n the closure

The importance o f t h e Whitney

f o r m u l a t i o n f o r us i s t h a t i t r e v e a l s a p l e t h o r a of important non-regular examples, many of which r e t a i n some weakened form of r e g u l a r i t y .

A p a r t i c u l a r construction,

w e l l motivated from the m a t r o i d p o i n t o f view but n o t meeting t h e eye from the van der Waerden p o i n t o f view, i s t h a t o f submatroid. X cU,

and

d e f i n e t h e independent sets o f t h e subrnatroid

dent sets i n closure o f of

A

in

Namely, i f

A U.

U which are subsets o f i n the submatroid

X

X.

Equivalently, i f

(U,cl)

i s given,

t o be those indepen-

X

A 5 X, .define the

t o be the i n t e r s e c t i o n o f

X

with the closure

A l i n e a r m a t r o i d i s one which i s isomorphic t o a submatroid o f a

vector space matroid, and these have been s u b j e c t t o very extensive i n v e s t i g a t i o n i n combinatorial theory [l].The reason these do n o t s p r i n g t o t h e eye from t h e van der Waerden p o i n t o f view i s t h a t t h i s makes any a r b i t r a r y subset o f a v e c t o r space i n t o a submatroid, and hence as above i n t o a S t e i n i t z exchange system; i n o r d i n a r y l i n e a r algebra t h i s i s g e n e r a l l y done o n l y f o r subspaces.

The present

paper i s t h e sequel t o [7] promised i n [7, p.381 f o r t h e non-regular case. present paper i s motivated by two questions.

The

One i s , from t h e p o i n t o f view o f r e -

Recursion Theory on Matroids

43

cursion theory, exactly what irreducible minimum p a r t of regularity i s needed t o assure t h a t a given recursion theoretic object (such a s a supermaximal space) e x i s t s ? Second, from the point of view of combinatorial theory, w h a t weaker axioms t h a n regularity a r e s a t i s f i e d by the S t e i n i t z exchange systems corresponding t o the matroids naturally a r i s i n g in combinatorial theory? In Section 2 we introduce seven axioms.

In Section 3 we prove implications

between these axioms, and give examples ( a l l from l i n e a r matroids of coqbinatorial theory) t o show the seven a r e r e a l l y d i s t i n c t and occur f a i r l y naturally. Sections 4, 5 , and 6 we look a t the recursion theoretic case.

In

I n Sectidn 4 the

case when none of these axioms holds i s discussed, the notion of a nowhere simple closed s e t i s introduced, and some existence theorems about nowhere simple closed

.

s e t s are obtained.

In Section 5 we deduce some recursion theoretic consequences

of axioms 11, 111.

In Section 6 we discuss w h a t the proper notion of automorphism

should be f o r these systems, and begin the study of the l a t t e r .

Theorems not proven

here and f o r which no reference i s given will be proven in a l a t e r paper.

2.

AXIOMS AXIOM I . L e t

(U,clv).

be c l o s e d and l e t

V

Then t h e r e e x i s t s a

z

be an i n f i n i t e b a s i s f o r

I

such t h a t i n

(U,clv),

suppIz

has

a t l e a s t two members.

in

AXIOM 11.

Let

(U,clv).

Then i n

b e c l o s e d and l e t

V

(U,clv)

J

be i n f i n i t e i n d e p e n d e n t

t h e dimension o f

- c l (J) c l ( J u {XI)

is infinite.

AXIOM 111. independent {y} 5 F

x

E

5J

For some

J

in

and

c l I ( F ) , with

~ S U P P F X2~ 2

and

k:

f o r a l l i n f i n i t e independent

(U,clI), all

vl,

IF1 = k , a l l

v1 y

,..., v n k

y

in

... , v n

J, all finite

F 5J

with

B c l , ( 0 ) , t h e r e i s an

c l I ( { x l ) , and i n

E S U P P ~ X .

I, all

( U , c l I ) , we have

A. NERODE and J . REMMEL

44 AXIOM I V .

(Downey's s e m i r e g u l a r i t y [3])

No finite dimensional closed

set is the union of two proper closed subsets. AXIOM V A (Federation over

o f Baldwin 121)

A

such that f o r any finite independent set x

in

(Note:

clA(B)

not in any A

In case

clA(B')

There is a finite set B

in

(U,clA),

for any proper subset

A

there exist B'

of

B.

is empty, we call this simply federation, writing

AXIOM V . ) AXIOM V I (Weak r e g u l a r i t y o f Baldwin [Z]) No

is a union of

k

k

dimensional closed set

(k-1)-dimensional closed subsets.

A X I O M V I ( R e g u l a r i t y o f Metakides-Nerode

[7]) No finite dimensional

closed set is a finite union of proper closed subsets. Based on Metakides-Nerode [7], f i r s t Baldwin [ Z ] and then Downey [3] introduced the notions a t t r i b u t e d t o them i n axioms I V , V, V I above as weaker assumptions f o r some o f t h e c o n s t r u c t i o n s i n [7]. We show below t h a t the f o l l o w i n g i m p l i c a t i o n s hold, and by example t h a t no others n o t l i s t e d hold, VII

-+

VI

-+

v

c-t

IV

-f

I1

-+

I

A l l the above a r e s t a t e d as p u r e l y a l g e b r a i c axioms.

L a t e r theorems can be

somewhat generalized, b u t a t t h e c o s t of.more d e t a i l i n proofs, by r e s t r i c t i n g a t t e n t i o n t o r e c u r s i v e l y presented matroids, and then a s s e r t i n g t h a t there i s a dec i d a b l e i n f i n i t e c o i n f i n i t e dimensional closed s e t holds i n

(U,clA).

such t h a t t h e given axiom

Some o f these theorems, however, simply f o l l o w i n t h i s more

general case from t h e observation t h a t i f c u r s i v e l y presented

A

(U,cl),

then

(U,clA)

A

i s a decidable closed subset of r e i s r e c u r s i v e l y presented too.

45

Recursion Theory on Matroids

STRENGTH.

3.

THEOREM 3.1.

Axiom V I I implies Axiom VI, b u t n o t conversely.

The i m p l i c a t i o n i s t r i v i a l .

PROOF.

For t h e counterexample, l e t

Let

bo,bl

be independent.

c(ibol) u cl(ib,l).

Let

Let

bz

be any element o f

be t h e submatroid o f

X

u cl({bo}) u cl(Ib,})

(U-cl({bo,bll))

be an

over an i n f i n i t e r e c u r s i v e f i e l d .

Vm

i n f i n i t e dimensional r e g u l a r system, such as

(U,cl)

u cl({bz}).

cl{bo,bl}

outside

U w i t h domain This gives the desired example.

We have thrown away a l l of a two dimensional closed s e t except f o r three one dimensional closed subsets, and kept a l l the r e s t o f

U. So

clX({bo,bl))

of three one dimensional closed sets, and axiom V I I f a i l s .

How about axiom V I ?

Note t h a t these are the o n l y one dimensional closed subsets o f the l a t t e r i s n o t t h e union of any two o f them. sional closed s e t o f

X

i s t h e union

clX({bo,bl}),

So f o r t h i s particu1ar;two

t h e axiom V I c o n d i t i o n holds.

and dimen-

I n f a c t , f o r every oth,er

closed s e t whatever, t h e axiom V I I r e g u l a r i t y c o n d i t i o n holds, which a f o r t i o r i w i l l verify V I .

We need o n l y check a

V # clX({bo,bl}).

regularity o f cl(V) V,

-

Suppose

V1,

k

2

t

....Vn

V

of

X

V

in

a r e proper closed subsets o f

such t h a t The

X.

U guarantees t h a t t h e r e i s an x i n

-...- c l ( V n ) - ( c l ( V ) ) V1, ...,Vn as required.

cl(V1)

outside

dimensional closed s e t

smallest p o s s i b l e way.

n (cl({bo,b,})).

Such an

x

is in

X,

in

Thus we have perturbed r e g u l a r i t y i n the

One can e a s i l y generalize t h i s t o p e r t u r b a l a r g e c o l l e c t i o n

o f closed sets t o v i o l a t e r e g u l a r i t y .

THEOREM 3.2.

If B

PROOF. VI,

cl(B)

so any

Axiom VI implies Axiom V, b u t n o t conversely.

x

i s an independent s e t w i t h

i s n o t the union of t h e in

cl(B)

k

k

elements

bl,

...,bk,

then by axiom

k-1 dimensional closed sets

cl(B-{bil),

n o t i n these closed sets s a t i s f i e s axiom V since these a r e a l l

the maximal proper subsets o f

B.

For t h e counterexample, use Vm

over

GF(2),

where axiom V I f a i l s because cl({bl,b2,b3})

= cl({bl,b2))

V holds f o r any f i n i t e

u cl({bl,b3})

u cl({bl+

b 2 + b3, b2+b3}),

B by adding t h e elements o f B together.

w h i l e axiom

46

A. NERODE and J. REMMEL

Axiom V i s e q u i v a l e n t t o axiom IV.

THEOREM 3 . 3 .

This i s a case where one notion, clothed in the matroid independent s e t

PROOF.

language (federated of Baldwin), looks on the surface d i f f e r e n t from the exact same notion (semiregular of Downey) clothed in the language of closed s e t s .

For c l a r i t y

we break both notions u p i n t o l i t t l e parts.

Call a closed s e t semiregular i f i t i s

not the union of two proper closed subsets.

Call a f i n i t e independent s e t federated

c l ( 1 ) n o t in

i f there i s an element o f

a proper subset of

I'

c l ( 1 ' ) for

I.

Of course every zero or one dimensional closed s e t i s both federated and semiregular. The theorem follows from the following lemma.

LEMMA 3.4

(i)

If a l l c l o s e d s e t s of dimension 2 a r e s e m i r e g u l a r , t h e n

a l l f i n i t e independent s e t s a r e f e d e r a t e d . (ii)

If a l l independent s e t s w i t h 2 elements a r e f e d e r a t e d , t h e n a l l

closed s e t s of f i n i t e dimension a r e semiregular. F i r s t we prove ( i ) by induction on the s i z e of the f i n i t e in-

PROOF OF LEMMA.

dependent s e t , say

k z 2.

For the base step k=2,

suppose

al,a2

x in

Then by semiregularity of two dimensional closed s e t s , there i s an

n o t in c l ( { a l } ) or i n c l ( { a 2 } ) . This says {al,a2} i s federated.

cl(Ial,a2})

For the induction s t e p suppose the conclusion known f o r be a

k+l

a r e independent.

element independent s e t .

let

k,

B = {al , . . . , a k + l >

By the inductive hypothesis of federation f o r

k element independent s e t s there i s a y in cl(B-{ak+l}) n o t in any c l ( B ' ) f o r B'

a proper subset of

B-Iak+,>.

x in ~ l ( { a ~ + ~ , ynot } ) in cl({y}) or ~ l ( { a ~ + ~ B }y) exchange .

there i s an

y

E

cl ({ak+l,x>) and

y

E

cl({a l , . . . , a k > ) ,

ak+l

E

cl ({x,y}).

Now

so x

E

cl(B) and

suppBx

i s federated by proving would get

x

E

y

E

B = suppBx.

~ l ( B - C a ~ + ~and l ) ak+l

j o i n t l y imply ak+l thesis.

By semiregularity of two dimensional closed s e t s

E

i s defined.

If n o t , some a i E

cl({x,y})

c l ( B - I a k + l > ) . This makes

On the other hand, i f

cl ({ak+l ,y})

.z E

i < k+l,

c l ( k , a k + l } ) , which j o i n t l y imply y

we get E

4

and

suppBx. If

and y

E

B

We verify t h a t

i = k+l, we

cl(B-{ak+l>), which

B dependent, contrary t o hypo-

x

E

cl(B-{ai})

cl(B-{ai>), so

and

suppBy 5 B-Iail,

and

Recursion Theroy on Matroids

4

ai

suppBy,

c o n t r a r y t o t h e choice o f

41

suppBy = B-Cak+l}.

(U,cl)

So

i s feder-

ated. V , V1,

Now f o r ( i i ) l e t

E V.

V = V1 u V2, V1

i V,

be a basis f o r

V1 n V2,

dS+,

V2

Since

V1,

so

V1.

cl({dt+ll).

dt+l

Since

ds+l,

cl(Iz,dt+ll),

dt+l

z

then e i t h e r dt+l

E

V1

i s dependent on

dS+l

E

pendent on

dl,...,ds,

semiregular, and

EXAMPLE 3 . 5 . A,

z V1,

,...,

{bn+,

, bn+2....l,

(U.cl)

let

A

Let

X

(U,cl)

A

E

clA(B)

-

THEOREM 3 . 6 .

W

be

dt+l

in

z

V.

E

V2

indepen-

dt+l

E

or i n By exchange,

i s the union o f

dS+l

E

V1

and

V2,

~ l ( { d ~ + ~ , z5} )V1,

V1 n V2.

So no such

,...,bn},

{bl

A u W.

If

= cl({bi})

and

V1,V2

ds+,

so

i s de-

e x i s t , and

there i s an

iB l

C,

x

let

W

V

is

holds w i t h a non n u l l

be the closure o f

W,

u cl({bj})

and

and

U

note t h a t i f

B

cl(B) (X, c l ) ,

we g e t i n

B i s a f i n i t e independent X

VA

Axiom V d o e s n o t imply axiom 111.

and i n

W.

By

so

u { c l ( B ' ) I B ' ;Bl. so axiom

X,

f a i l s t o s a t i s f y axiom I V

i s independent i n

-

are t h e union o f

X

B 5 X,

while f o r

n z 2, 0 2 i < j < n,

in in

VA

The independent sets o f

To v e r i f y axiom VA,

uIclA(B'):B'

(i)

...,dt ,. where

dl,

be i n f i n i t e dimensional r e g u l a r , l e t

be

then' B n A =

(X,clA),

,...,d,

I n t h e second case

so

and an independent s e t i n

= (Ibi,bjl)

of semiregularity.

x

V1,

dl

i s semiregular,

c l x ( B ) = (B n A ) u c l ( B n W).

r e g u l a r i t y of

certainly

If V

E

Let

cl({ds+,})

We produce S t e i n i t z systems where axiom

any subset o f

set i n

V,

c o n t r a r y t o choice.

be a basis, l e t

cl(Ibi,bj})

not i n

c o n t r a r y t o choice. dS+l

V1.

By f e d e r a t i o n o f two element inde-

I n t h e f i r s t case

V2.

But

but axiom I V f a i l s .

bl,b2

cl({z,ds+ll).

say

V1,

there i s a

cl(IdS+l,dt+ll) are both i n

E

5- V2.

cl({z,dt+,})

in

E

or

g V,

V1

V1 n V 2

we g e t

i s independent. z

E

Since

dl,...,dt+l

pendent sets, t h e r e i s a

dstl

i V1,

V2

extend i t t o a basis f o r

i s defined and n o t i n

dent o f

be f i n i t e dimensional closed sets w i t h

V2

holds.

(ii)

Axiom I 1 1

d o e s n o t i m p l y axiom V . PROOF.

For ( i ) , use

Vm

over any f i n i t e f i e l d ,

For ( i i ) , l e t

bl,b2,

...,

be a

48

A. NERODE and J. REMMEL

base

B for

V_,

that

suppBv

does n o t have e x a c t l y two members.

= cl(Ibi})

cl(ibi,bj})

t h r e e members. of

(V-,

over an i n f i n i t e f i e l d l e t

clI)

n o t i n any

u cl(Cbj}).

clI(F)

n o t i n any

x

Then axiom

V

v

in

such

V

f a i l s since F

has a t l e a s t

r e q u i r e d may be chosen by r e g u l a r i t y

cly(F')

( O f course t h e s e

clI({vi1),

be t h e s e t o f a l l

However, axiom I11 h o l d s i f

F o r t h e n i n f x i o m I11 t h e in

X

x

for

F'

have as

a p r o p e r subset o f

suppFx

the set

F

F,

of

c a r d i n a l i t y > 2).

A x i o m V I implies a x i o m 111, b u t n o t conversely.

THEOREM 3.7.

PROOF. Assume axiom V I , l e t J

I

be an i n f i n i t e independent s e t i n

i s i t s e l f an i n f i n i t e independent s e t , l e t

of

J-I.

an

x

Put

in

F = {y,xo}

clI({y,xo))

f o r axiom 111.

such t h a t

vi

d

vi

and assume a l l

I

such t h a t a l l o f

k t Kn+l

vi

E

clI(Iy,xo}).

,...,vn

E

Given

cli(I

t h e n t h e r e i s no problem e n s u r i n g t h a t

y

J.

Let

xo

vl,...,vnh

vi f c l ( 1 u { X I ) ,

where

be any o t h e r element clI(0),

Now i f vi

U {x}).

(U,clI),

we must f i n d

dclI(Iy,xol),

so we may d r o p t h e s e

Iil,...,ik-2} from

There i s a f i n i t e s e t

,...,ik-2 1 u I x o , y l ) and i n a d d i t i o n n+2. L e t Ki = c l ( { i l ,..., ik-*, vi}), i = 1 ,..., n, l e t = c l ( I i l ,...,ik-2,xo1), l e t Kn+2 = c l ( l i l ,...,i k m 2 , y } ) . Then K1 ,...,Kn+2

are

n+2

v1

c l o s e d subsets o f dimension

,...,i k - 2 , x o , y } )

cl(Iil i n the

k

are i n

cl({il

k,

o f dimension

,

sk-1

a l l contained i n n+2 s k ,

so i f

dimensional space n o t i n any o f t h e

we g e t r e s p e c t i v e l y i n

(U,clI)

that

of axiom 111.

F i n a l l y we need v e r i f y

vi

E

and

x

in

clI(Ixl)

vi

f ~'$31

Ki.

suppFx,

vi f c l I ( { x } ) .

so by exchange

x

E

pendent o v e r cl(Iil,...,ik-2,xo,Y dent over

I,

cl(Iil

x

and

,..., i k - 2 1 ) , I),

or

hence so a r e

vi

both i n

so t h e n

clI({x,vi}) x,vi,

so

x

cl(Iil

cl,(Ivi3).

x f Kn+2,

which v e r i f i e s p a r t

x.

We c l a i m t h i s g i v e s T h i s i s because t h e

~ l ( { i ~ , . . . . i ~ - ~ x ~ , y }and ) inde-

,..., ik-2,x,

=

clI(Ixo,yl),

C

clI(Ivi1).

To see axiom I I I does n o t i m p l y axiom

and

x

Otherwise we have

~l({i~,. vi}), . . ,cio n~ t r ary ~ to , the choice o f

contrary supposition gives

g i v e s an

x f Kn+l

Since

are i n

xo,y

axiom V I

but

vi}) xo,y

= a r e indepen-

V I observe t h a t t h e counterexample used

f o r theorem 3.6 (ii) works h e r e t o o , e x a c t l y t h e same way.

Recursion Theroy on Matroids THEOREM 3.8.

A x i o m V i m p l i e s a x i o m 11.

A x i o m I1 d o e s not i m p l y a x i o m V. The f i r s t two a r e t r i v i a l .

PROOF.

49

A x i o m I11 i m p l i e s a x i o m 11.

A x i o m I1 d o e s not i m p l y a x i o m 111.

Vm

o v e r a f i n i t e f i e l d does t h e t h i r d .

The

same example as f o r theorem 3.6 (ii) works f o r t h e f o u r t h . A x i o m I1 i m p l i e s a x i o m I, b u t not c o n v e r s e l y .

THEOREM 3.9.

As f o r t h e second, l e t

The f i r s t a s s e r t i o n i s t r i v i a l .

PROOF.

f i n i t e dimensional v e c t o r space w i t h b a s i s consist o f a l l

v

(i) v = bk + bk+l (ii)v

bk +

=

...

bo,...,bn,...

Vm

be t h e i n -

GF ( 2 ) .

over

Let

X

o f one o f t h e two forms below.

...

+

bQ + bL + b, + +

To see axiom I1 f a i l s choose

... + bns

J = {bqi

+ b4i+2:

where

k

5

II.

where

k

5

11 < in

i

5

n

2

2).

I t i s easy t o see t h a t i n

o f t y p e (i) span

X

(elements o f t y p e (ii) are

X, c l ( J ) = J , c l ( J u { b o l ) = J u {bo}. Axiom I .

Now we v e r i f y

x

Note t h a t elements

of

X

the sum o f two o f t y p e (i)), so t h e r e i s a subset i s a basis f o r certainly

X

in

dim[X/V]

(X,clv). and

B

i n g axiom I i f t h e r e i s a

z

in

f o r every

IsuppIz(

(X,clv), in

z

b

V,

C

Summarizing

so

(we a r e o v e r

GF(2)!).

x

5

Without pain 111 2 2.

in

1

in = 1,

X,

(X,clv),

We a r e a l r e a d y f i n i s h e d i n v e r i f y -

IsuppIz(

?

(X,clv).

2

in Since

(X,clv).

for all Since

z

z1,z2

So assume

B i s independent i n

and t h e r e i s a u n i q u e

= clv({z})

= clv{il,i23.

o f type ( i ) ,

f i n i t e dimensional closed. C

4.

B with

clv({i(z)))

(We o n l y used t h a t

i s i n f i n i t e independent i n

i s infinite.

lsuppIzl

# z2 i n B, clV{z1,z2)

they a r e elements o f

axiom I .

m,

B we have

we have

suppIz.

z1,z2,z,

E

=

I

Since

B o f elements o f t y p e ( i ) which

in

B.

i(z)

in

I

So i f

a r e elements o f

B,

and t h e i r sum i s of t y p e (ii) o r t y p e (i) suppI(zl

+ z 2 ) = {il,i2}

as r e q u i r e d f o r

A s i m i l a r proof shows t h a t f o r any c o i n Axiom

h o l d s and Axiom I 1 f a i l s f o r

(X,clc).)

RECURSION THEORETIC C O N S E Q U E N C E S Throughout t h i s s e c t i o n

(U,cl)

i s an

n f i n i t e dimensional r e c u r s i v e l y p r e -

A. NERODE and J. REMMEL

50

sented S t e i n i t z system, sets.

A

W 2 U,

in

V

dim(U/W) < = or

and f o r any W1, dim(U,W2v V )

i s maximal i f

L(U)

either

L(U) i s the l a t t i c e of recursively enumerable closed subdim(W/V) <

o u t loss of generality.

L(U)) with

out of elements of

A

B.)

and

dim(W/A) =

and dim(R/A) = =. sive function

f

An

(We say A such t h a t i f

dim(Ri/A) =

m

iff

E

with

L(U)

dim(U/V) = =

dim(U/Wlv V ) < = or

B1 5 B

and

A

-, there

such t h a t

W2 with-

W1,

W1 v W2 = U

A,B

A1, B1

E

L(U)

are decid-

by building a recursive basis f o r

L(U)

e x i s t s an

i s nowhere simple i f f o r a l l

R c W

in L(U) such t h a t

W

U

in

R A A = cl(0

i s effectively nowhere simple i f there i s a recur-

i s the itJ

Wi

in the standard enumeration, then and

in

i s r-maximal i f

V

This i s because i t i s e a s i l y shown t h a t i f

then there a r e A1 _c A

such t h a t

A

W1" W2 = c l ( 0 ) and

A l B1~ = c l ( 0 ) and A1v B1 = U ,

able, with

L(U)

either

W1v W2 = U ,

are decidable (complemented in

and A v B = U ,

-.

and f o r any W

= =,

( I n the definition of r-maximal, we could i n s i s t t h a t

=.

=

with

W2

dim(U/V)

Wf(i)

=

Ri

recursively enumerable closed s e t has

Ri 5 Wi

and

Ri n A = c l ( 0 )

dim(Wi/A) = =.

With no extra axioms we get the following among others. THEOREM 4.1. s u b s e t of

Let B.

B

Then

be a r e c u r s i v e b a s i s f o r cl(M)

i s maximal i n

U.

Let

M

b e a maximal

(See M e t a k i d e s -

L (U).

Nerode [ 7 ] o r Remmel [ l l ] ) .

THEOREM 4.2. an

Let

B

such t h a t

R 5 B

be a r e c u r s i v e b a s i s f o r cl(R)

i n any maximal e l e m e n t o f THEOREM 4.3. subset of

Let B.

B

(i)

A

If

Let

A 5 B

i s r - m a x i m a l and L(U).

is not contained

(See Remmel [ l l ] ) .

Then t h e r e e x i s t s a n

R1 5 M

U.

Let

such t h a t

M

b e a maximal cl(R1)

is

(See Remmel [ l l ] . )

be a r e c u r s i v e b a s i s f o r

U.

i s nowhere s i m p l e ( r e g a r d i n g

B

B

Then t h e r e e x i s t s

cl(R)

be a r e c u r s i v e b a s i s f o r

r-maximal b u t n o t maximal.

THEOREM 4.4.

U.

as a s e t of integers), then

cl(A)

a s t h e i n t e g e r s and

i s nowhere s i m p l e i n

L(U).

Recursion Theroy on Matroids A 5 B

(ii)I f

i s e f f e c t i v e l y nowhere simple,

Si = { b

F i r s t we s h a l l p r o v e ( i ) . B13x

E

E

Wi(b

E

must be i n f i n i t e s i n c e

Given

Wi

where

suppB(Ix})).

Then

Si

i s an r . e .

dim(Wi/cl(A))

=

t h e r e i s an i n f i n i t e r . e . s e t pendent subset o f

Wi,

fective l i s t i n g o f

-.

Given

,...1

and l e t

cl(A)

i s ef-

dim(Wi/cl(A))

=

subset

B

-,

let

and

Si-A

A,

Thus by t h e nowhere s i m p l i c i t y o f

Ti 5 Si-A.

Di = Ido,dl

Wi

then

L(U).

f e c t i v e l y nowhere simple i n

PROOF.

51

Ti,

we t h e n l i s t an r . e .

as f o l l o w s .

...

bo < bl < b2 <

Let

,...

w0,w1

inde-

be some e f -

be t h e l i s t i n g o f

i n in-

B

creasing order. STAGE 0.

Let

STAGE n + l .

d

Jo

= w.

where

i s the least

j,

Assume we have d e f i n e d

suppB(dk) n Ti # 0 # suppB(dj) max {9.1bQE s u p p B ( d k ) l .

Let

and

n Ti

B J

0 s k < j

such t h a t f o r a l l

n Ti)

max {9.1bkE suppB(dj)

>

5

n,

,

where jn+l i s t h e l e a s t j such t h a t Jn+1 t h e r e e x i s t s bLE suppB(wj) n Ti where 9. > max{plb E ( u suppB(dk))}. Again i t p ksn i s easy t o show t h a t s i n c e dim(Wi/cl(A)) = -, d . i s d e f i n e d f o r a l l j. Moreover, J i t i s easy t o see t h a t o u r d e f i n i t i o n ensure t h a t Di = {do,dl, . . . I i s an i n f i n i t e r . e . independent s e t o v e r Ri

c

Wi,

Ri

A

dn+l

...,dn

do,

supp ( w . ) n Ti # 0.

such t h a t

j

=

cl(A).

cl(A) = cl(0),

and

W.

Thus i f we l e t dim(Ri/cl(A))

-

= cl(Di).

Ri

=

so t h a t

then cl(A)

i s nowhere

simp1 e. note t h a t given F o r (ii), dim(Wi/cl(A))

=

(where we do n o t necessary assume

Wi

we can e f f e c t i v e l y f i n d an i n d e x f o r

m),

Si.

A

If

l y nowhere simple, t h e n we can e f f e c t i v e l y f i n d an i n d e x f o r an r.e.

such t h a t

Ti

i s infinite iff

g i v e n an i n d e x f o r

Ti,

Si

-

there i s a recursive and

dim(Ri/cl(A))

=

i s infinite iff

dim(Wi/cl(A)) Di

we can e f f e c t i v e l y f i n d an i n d e x f o r

where we make t h e c o n v e n t i o n t h a t i f i s n o t defined, t h e n

A

Di = {do,

-

f

...,d n l .

such t h a t iff

do,.

.. ,dn

dim(Wi/cl(A)

-

= Ri =

set =

and

Ri

=

Ti 5. Si-

-.

Then

cl(Di),

n+l, dn+, then

Ri n c l ( A ) = c l ( 0 ) , Ri 5 Wi,

so t h a t

cl(A)

i s e f f e c t i v e l y no-

where simole. COROLLARY 4.5.

G i v e n any d e g r e e

6,

A

d e f i n e d as above,

a r e d e f i n e d b u t a t stage

It f o l l o w s t h a t i f

Wf(i)

i s effective-

t h e r e e x i s t s an e f f e c t i v e l y no-

52

A. NERODE and J. REMMEL

where simple

V

L(U)

E

such that

deg(V)

=

dependence

deg(V)

= 6.

This follows from t h e f a c t t h a t Remmel and N i l l e r have shown t h e r e a r e

PROOF.

e f f e c t i v e l y nowhere simple sets i n every r . e . degree [12]. The next theorem r e q u i r e s an e f f e c t i v e n e s s requirement which i s obviously Vm,

correct f o r

and o t h e r l i n e a r matroids b u t which we do n o t know i s e f f e c t i v e

Fa.

a t the time o f w r i t i n g f o r THEOREM 4.6.

Suppose there is a uniform effective method which, applied

to finite

U,

F1, F2

produces

n

dim(cl(F1)

If

cl(F2)).

i s infinite dimensional, then there exist nowhere simple A.

such that Let U.

v

A1

Wo,Wl...

=

C

and

A.

A

=

A1

AO,A1

L(U)

E E

L(U)

cl(B).

be an effective list of all r.e. closed subsets of

We shall construct r.e. independent sets is a basis f o r

Bo u B1

C

C

and

Ai

=

desired nowhere simple closed sets.

cl(Bi)

Bo

and

for

i

B1 =

such that

0,1,

are the

During the course of our con-

struction, we shall try to meet the following set of requirements.

We say that

> n.

cl(Wz) n cl(B:)

P;,

P;, ,!P

STAGE 0.

Let

STAGE s + l . Bi u Bi

cS

E

Let

. . ,cs}.

= fco,.

B:

We priority rank requirements as follows:

=

...

co,cl,

BY

=

be an e f f e c t i v e l i s t i n g o f

Assume t h a t we have defined independent sets

i s independent and

c1(Cs-’),

let

B;+l

c l ( B i u B);

= Bi

for P,e,ni,

c cl(w:)

n (cl(cs)

-

= cl(CS-’).

i = 0,l.

and l e t

Bi

and

Consider

I f c s f cl(Cs-’),

cs.

BY

such t h a t If

look f o r the

which i s n o t s a t i s f i e d and where . < e , n > 5 s

and x

C

0

highest p r i o r i t y requirement

(*) 3

if the dimension of

n

...

P;,

CQNSTRUCTION.

cs

is satisfied a t stage

PSe,n>

cl(cs-’)).

Recursion Theroy on Matroids

(Note we can effectively decide i f dim(cl(W:)

53

x iff

holds since there i s such an

(*)

n c l ( C s ) ) > dim(cl(Wz) n cl(CS-’)).

If ‘<e,n> e x i s t s , then l e t y be the l e a s t x satisfying (*) and s e t Bqtl = u {y} and B t T ] = B f - i . If there i s no such P , l e t Bs+l u Its} and Bit1 = B i .

Bi

F i r s t we shall verify t h a t

This completes the construction. simple.

The argument f o r

dim(W/C) i s f i n i t e .

CASE 1 .

Then we can find

u B1 u {Z0 9 . . . , ~ n } i s independent and

0

R = Id

E

I ~ Ew W(d

D

supp,(w))} n B1.

E

we shall construct an r.e. subset of

a.

Let

= (UX)(XE

an+l = ( ~ x ) ( xWE and

...,z,

W ccl(D).

E

such t h a t .

W

Let

i s i n f i n i t e since dim(W/Ao) =

R

W, E = {eo,el , . . . } ,

-.

Now

which i s independent

suppD(x) n R # 0)

and

and c l e a r l y

,u

1 sn

cl(E)

and

suppD(ai)). Since c

W , c l ( E ) n A.

R

an+l

i s infinite,

= c1(0),

and

-.

dim(E/Ao) =

dim(W/C)

w> c

W

suppD(x) n R

n

i s defined f o r a l l

Let

zo,

by induction.

over A.

CASE 2.

i s r . e . and

Assume W

-.

dim(W/Ao) =

D = B

i s e n t i r e l y analogous.

A1

i s nowhere

A.

=

-.

= { s l 3 x j t z s( x

E

cl(W:)

-

cl(Cs) and

x

E

cl(Ct)}.

Now i f

WevC

is

i n f i n i t e , then i t i s easy t o see t h a t a t i n f i n i t e l y many stages, there e x i s t s an

x

E

cl(Wz) n [cl(Cs)

-

cl(CS-’)J.

’

quirements t h a t requirement

-. dim(V/Ao) = -.

dim(W

n A1)

=

Now suppose

Thus i f

P,e,n> V

=

I t t h e n follows by our p r i o r i t y ranking of re-

W n A1,

WeUC i s f i n i t e .

n and hence

would be s a t i s f i e d f o r a l l

Let

then

V 5 W , V n A.

= c1(0),

to be l a r g e s t element in W e L C .

shall construct an r . e . independent subset of

W , E = {eo,el,e2,

...)

foll ows. STAGE 0.

Let

so

dim[cl(Wz)/cl ( C s ) J )

-

be l e a s t stage

and

s > to such t h a t

> dim[cl(W~-’)/cl(Cs)]. Let eo be the l e a s t s -1 cl(Wpo ) which i s independent over cl(Cs).

W e

in stages as

54

A. NERODE and J. REMMEL

STAGE n+l.

Assume we have d e f i n e d s > sn

be l e a s t stage eS+l

be the l e a s t

1

= dim [cl(W:n+l-l Since

x

E

-

cl(W:"')

-.

We c l a i m

E

Now a t stage

supp&x). S

dim(cl(We")/cl(C

S

and

{x,e 0,...,en3

cl(E) n C = cl(0)

x

c l ( E ) n C.

E

...,sn.

Let

s ~ + ~

sn, en

E

cl(Wen)

dim[~l(W~"+~)/cl(C~"~)]

i s defined f o r a l l

and hence

-

n

and

cl(E) n cl(Ao) = c l ( 0 ) .

be the maximum i such t h a t

n

5,-1 cl(We

S

)/cl(C n),

dim[cl(Wen)/

sn-1 S dim[cl(We ) u { e n l ) / c l ( C ')I. I t thus f o l l o w s t h a t

that is, 5

S

and

Let

i s independent.

en

Let

S

sn-1 S > dim(cl(We ) / c l ( C '))

'))

so,

such t h a t

i t easily follows t h a t

Suppose there i s a n o n t r i v i a l ei

and stages

cl(W;"'-')

u {X3)/Cl(CS"+1)1

-,

...,en

d i m [ c l ( W ~ ) / c l ( C s ) ] > dim[cl(W;-')/cl(Cs)].

such t h a t

dim(Ne/C) =

dim[cl(E)/Aol =

eo.

(C ')]

S

x f c l ( C ')

=

since otherwise

sn-1 S by exchange and hence d i m [ c l (We ) u {en3)/cl (C ')I = sn-1 S 5,-1 S ) u I x l / c l ( C ')I = dim[cl(We )/cl(C ")I. By t h e same argument dim[cl(We en

x

E

6

c l (eo,.

. .,en-l

s -1 cl(Wen ) .

That i s ,

sn

E

,x)

Thus We\

C

x

S

cl(Wen)

E

S

and a t some stage

')

t > sn, x

sn > to which c o n t r a d i c t s our choice o f

and

cl(E) n cl(C) = c l ( 0 )

- cl(C

and

hence

A.

E

t c l ( C ).

to. Thus

i s nowhere simple.

(Note t h a t i n our argument i n t h e second p a r t o f Case 2, we are n o t c l a i m i n g that C

E

i s independent over

because

We

C

but only t h a t

i s i n f i n i t e dimensional over

dimensional over

A.

cl(E) C.

i s i n f i n i t e dimensional over

Thus

cl(E)

i s only i n f i n i t e

b u t n o t n e c e s s a r i l y completely d i s j o i n t from

Thus there i s a stronger n o t i o n o f nowhere s i m p l i c i t y , namely, s t r o n g l y nowhere simple i f f f o r a l l an i n f i n i t e dimensional

R

cW

W

E

such t h a t

L(U) R

with

dim(W/A) =

m,

Ao. A

E

L(U)

is

there exists

i s completely d i s j o i n t from

A.

Note

t h a t our argument i n Theorem 5.4 does produce elements which s a t i s f y t h i s stronger n o t i o n o f nowhere s i m p l i c i t y ) .

5.

W E NOW GIVE RESULTS USING AXIOM I,11, 111.

THEOREM 5.1.

A s s u m i n g a x i o m I, t h e r e e x i s t s a m a x i m a l e l e m e n t w i t h n o

basis extendible t o an i n f i n i t e l y l a r g e r r.e. THEOREM 5.2.

independent set.

A s s u m i n g a x i o m I,t h e r e e x i s t s a n r - m a x i m a l e l e m e n t w i t h

Recursion Theroy on Matroids

no basis extendible t o a recursive basis f o r

U.

Next, theorems u s i n g axiom 11.

Assuming axiom 11, there exists a supermaximal element

THEOREM 5.3.

(with the set

degree).

Assuming axiom 1 1 , there exists a nowhere simple element.

THEOREM 5.4.

V

having any prespecified non zero r . e .

V

V

with no basis extendible to a recursive basis f o r

V

every r.e. basis of

U, but with

extendible to an infinitely larger r.e. inde-

pendent set (in every non-zero r.e. degree). Assuming axiom 1 1 , there exists a

THEOREM 5.5.

R

finite dimensional decidable such that f o r all decidable REMARK.

Note such a

i s extendible (i.e., b u t t h a t no b a s i s f o r

U

and

-

and

R

3

A

...I

H u Bs u {ai,ay,

recursive basis f o r Let and t h a t

= U.

R,

I u J

then

B n V

i s a basis f o r

V,

I

basis

U

of

V

extends . I ) (i.e.,

t h e n if x

E

if

B

B-V,

Let

H u B u {ao,a

S

uo,ul,

R

where

A

be a d e c i d a b l e c l o s e d s e t such t h a t i s as i n Axiom 11.

BS and an i n f i n i t e r . e . sequence ai,a; i s a recursive basis f o r

We w i l l ensure t h a t f o r a l l

l,...)

i s a basis for

U

H

,...

we w i l l such

i s some f i x e d

V

l i m af = ai S

B = u BS. S

exists

Thus

R.

be an e f f e c t i v e l i s t o f t h e elements o f

be an e f f e c t i v e l i s t o f a l l r . e . To ensure t h a t

where

i that

where

w i l l be c o m p l e t e l y d i s j o i n t f r o m

...

U

s,

A t each stage

R.

Vs = cl(Bs).

V = u Vs = cl(B) Let

D

we have

and an in-

i s a d e c i d a b l e c l o s e d s e t c o n t a i n i n g V .)

s p e c i f y a f i n i t e independent that

completely disjoint,

i s extendible t o a recursive basis f o r

V

PROOF O F THEOREM 5 . 5 . dim(U/R) =

R

and

i s a recursive basis f o r

J

i s a recursive basis f o r cl(B-ix1)

D 2 V,

in

w i l l have t h e p r o p e r t y t h a t e v e r y r.e.

V

if

V

with

L(U)

V

U.

Let

Io,I

l,...

independent s e t s .

i s c o n t a i n e d i n no p r o p e r d e c i d a b l e c l o s e d s e t we s h a l l

s a t i s f y t h e f o l l o w i n g set o f r e q u i r e m e n t s .

A. NERODE and J . REMMEL

56 F. : If ui i c l ( I . ) , 1,J' J z E c l ( I j u Iuil)

and such t h a t

Note t h a t meeting t h e requirements any proper decidable space.

and z

E

E

B

-

dim(cl(I/RvV)) = V-cl(1)

and

a,

lsuppIu

ui

E

(Recall

Let

{u},

D 2 V,

i s decidable and

then we see t h a t

b u t c l e a r l y t h e r e i s no

rul(~)I

t

z

E

then we can B

for

U.

u f c l ( I ) , c l ( 1 ) 'V,

cl(1 u

Iul)

such t h a t

2.

F. . i s s a t i s f i e d a t stage

s

1 sJ

Let

0 B = 0

i f either

... be an e f f e c t i v e l i s t o f N x N. {ai,ay ,... 1 u H be a r e c u r s i v e b a s i s f o r U.

,, and l e t

H i s a recursive basis f o r

STAGE s+l.

D

CI(I?), J

CONSTRUCTION. STAGE 0.

-

I= B

We say t h a t requirement (i)

Fi,j

D, B1, and extend i t t o a r e c u r s i v e b a s i s

and

B1,

E

That i s , i f

take a r e c u r s i v e b a s i s f o r Now if u

z

d i m ( c l ( I . ) / R v V ) = m, then t h e r e i s a J V - c l ( 1 . ) and lsupp I.u i U i } ( z ) I t 2. J 3 w i l l ensure t h a t V i s n o t contained i n

R.)

(Assume t h e usual tower o f windows p i c t u r e . )

Look for the l e a s t and t h e r e i s a

z

E

e

5

s+l

such t h a t requirement

c l ( I s u Cuj } ) Je e

-

cl(1S ) Je

with

z

Fi 5

s

e Je

i s not satisfied

such t h a t

Recursion Theroy on Matroids Bs u {zl u {a:,

(b)

...,as)e

i s such an

e,

let

corresponding t o

a:

E

let

BS+l = Bs

r

e ( s + l ) . Let

a;

=

e and

for a l l

I f there

k.

z ( s + l ) be the l e a s t z

be the largest integer such t h a t

s u p p s ( z ( s + l ) ) denotes the support of

...1.

t o the basis H u Bs u {a:,a:,a;,

Note our choice of

B = u Bs

Let

are r.e. and i t i s easy t o see t h a t

S

B u

z ( s + l ) relative

r > e.

z ( s + l ) ensures

from i t s window, and l e t things drop.

BS+l = Bs u { z ( s + l ) 1 , remove a:

This completes the construction. V

a;”

and

e ( s + l ) be the l e a s t such

supps(z(s+l)) where

Then l e t

R.

i s independent over

If there i s no such e ,

51

and

V = cl(B).

independent so t h a t

H is

B and

Th’us

R- and

V

are completely d i s j o i n t .

LEMMA 5.6.

I i m a;

- ak

e x i s t s for a l l

k.

S

We proceed by induction on

PROOF.

so t h a t a.so ,. . . .ak-l only i f

and we s a t i s f y requirement

Note t h a t once requirement

E

-

(z)I C l ( I ? ) & (supp J IjU{Ui1

..,F’.e * .J e.

LEMMA 5.7.

PROOF.

r

i

For a l l

e,

requirement

We proceed by induction on

e a r e met and there i s a stage

e ( s ) i s defined then

e(s)

2

e.

t

a t stage -s+l

.

2.

t > s unless u i

s t i l l i s s a t i s f i e d a t stage t. Thus a s+l k # a; f o r s L so a t most

Fi o, jo,.

‘ e ( s + l’ J e ( s + l )

z c BS+l such t h a t

NOW (*) will continue t o hold f o r a l l

Fi,j

s t sO.a;+l # a;

Then f o r

Fi

s+1, there i s a

cw; u hi))

F:

i s a stage large eriough so

i s s a t i s f i e d a t stage s + l , i t ree(s+l ) * J e ( s + l ) t > s + l . T h a t i s , l e t i = i e ( s + l ) and j = j e ( s + l ) ,

mains s a t i s f i e d f o r a l l

(*I z

Assume so

have reached t h e i r f i n a l values.

e(s+l) < k

then a t stage

k.

e.

k

E

t

cl(1.) J

in which case

times, i . e . , once f o r each of

F. . ‘ev’e

i s met.

Assume t h a t a l l requirements

to large enough so t h a t for a l l

‘ir,jr s

L

for

to, i f

A. NERODE and J. REMMEL

58 F i r s t note t h a t i f s a t i s f i e d a t stage f o r a t most one

then requirement

met.

Thus

e(s) t e+l.

f cl(Ij ) e

dim(cl(1. )/Vv R) = Je i t f o l l o w s t h a t we can t h i n

m.

is e’je Thus e ( t ) = e

tl z to such t h a t i f

Next suppose r e q u i r e m e n t

and

Fi

s > t.

t t to and hence t h e r e w i l l be a stage

i s defined, then e

e ( s ) = e,

and w i l l remain s a t i s f i e d f o r a l l

t

e(s)

ui

t t to and

Fi

Because

.

f a i l s t o be

e 3Je

d i m I c l ( I j ) / V v R] = =, I. t o a n i n f i n i t e s e t I e Je such t h a t I u I u i 1 i s independent o v e r Vv R. Now by Axiom 11, we know t h a t e d i m [ c l ( I u I u i 1) - c l ( I ) ] = m. Thus t h e r e must be a z E c l ( 1 u I u i 1 ) - c l ( 1 ) e e such t h a t IsuppIuIui ,(z)I 2 2 and {ao, ae,zl i s independent over V v R.

...,

Thus f o r any

e

s,

Bs u { a o,...,ae,z~

l a r g e enough so t h a t forces

a. = a t ’ 1 1

t

2

tl, t z z, &

i s e.

for all

can meet r e q u i r e m e n t already satisfied.

Fi

.

Since

i s s a t i s f i e d by stage

t

z

E

e(t+l) > e

t+l,

e(t+l) = e

and hence r e q u i r e m e n t

be a stage

t

Note o u r c h o i c e o f

tl

z would w i t n e s s t h a t we

unless requirement

by o u r c h o i c e o f

I f Axiom I 1 h o l d s for

COROLLARY 5.8,

Now l e t

t c l (I. u {ui 1 ) . Je e

Thus a t stage

and hence

eYJe

i s independent.

u H

Fi

we know t h a t

tl,

.

is

eJe Fi

e Je

i s met. Fie,je then

U,

has a supermaximal

U

element.

PROOF.

Consider

dim(W/VvR) = =. u

E

V-W.

z

E

t h e requirements

and

as c o n s t r u c t e d i n Theorem 10. be an r . e . b a s i s f o r

ue f c l ( I ) , c l ( 1 ) 2 V,

V-cl(1) Fi

..

sJ

such t h a t

Now i f

and

t

2

W

W # U,

dim(cl(I)/VvR)

IsuppIuIul(z)I W 2 VVR

Thus i f

&

W.

Suppose

E

L(U)

and

then l e t

= and y e t t h e r e i s

=

which would v i o l a t e one o f

dim(W/VvR) =

-,

then

W = U

-

i.e.,

i s supermaximal.

REMARK 1. sure

I

Let

But then

c l e a r l y no

Vv R

Vv R

I t i s easy enough t o m i x i n p e r m i t t i n g and c o d i n g so t h a t we can en-

d(V) = D ( V ) = d ( R v V ) = D(Rv V ) = 6 where

V.

D ( V ) = dependence degree o f

REMARK 2 .

Note t o produce a

t e n d i b l e b u t no b a s i s o f

V

6 i s any nonzero r . e . degree,

V

E

L(U)

such t h a t e v e r y r . e .

basis o f

i s extendible t o a recursive basis f o r

U,

V

i s ex-

it i s

&

Recursion Theroy on Matroids

enough t o s i m p l y s t a r t w i t h a supermaximal sum o f a d e c i d a b l e space

R

and an r . e .

S

L(U)

E

space

V.

R e t z l a f f w i l l show t h a t t h e r e a r e supermaximal

S'

d i r e c t sum t o two d e c i d a b l e subspaces.

59 and decompose i n t o a d i r e c t

T h a t i s , techniques used by

Moreover such

S'

such t h a t

L(V")

E

is a

can be shown t o e x i s t

S'

under t h e assumption t h a t axiom I 1 h o l d s . Now, a theorem u s i n g axiom 111 THEOREM 5.9.

6. = * .

Let

(U,cl)

(U) be i t s l a t t i c e o f r . e . c l o s e d sets, l e t L F ( U )

s u b l a t t i c e o f f i n i t e dimensional c l o s e d s e t s . mean t h e r e a r e E

LF(U)

Let

A = [A]

*

B

[cl{p))]

*

Without d i f f i c u l t y = B v C.

dim[C/A]

If <

C

A,

If dim[C/Al <

cl(B) = F

E

LF(U)

C v G = A v G.

(U,clA),

to

l e t L*(U)

P.,

be t h e c l a s s of equiv-

L*(U)

A s* B, [U]

is a

partially

as i t s l a r g e s t element. V,

=* B

A

namely

i s w e l l d e f i n e d on L * ( U ) ,

implies

and L*(U)

and

A

5 C,

then

A =

*

C

if a n d only i f

m.

PROOF.

mensional i n

L(U)

0,l.

L(U)

E

E

we g e t an equivalence r e l a t i o n on

i s a congruence w i t h r e s p e c t t o

[ A ] V [C] = [A v C]

so

A,B

i s t r a n s i t i v e and r e f l e x i v e , and so

B <* A,

[A] s [B] i f

for

be i t s

o r e q u i v a l e n t l y t h a t &here i s an

as i t s s m a l l e s t element,

i s an upper s e m i l a t t i c e w i t h LEMMA 6.1.

A v F 5 B v G,

A s* B and

if

A s* B

Define

O b v i o u s l y s*

Obviously w i t h

ordered s e t w i t h

*

with

be t h e e q u i v a l e n c e c l a s s o f

alence c l a s s e s .

A v C =

LF(U)

E

A v F 5 B v F.

with

if we d e f i n e

L(U).

F,G

recursive hs a set.

be a r e c u r s i v e l y p r e s e n t e d S t e i n i t z exchange system t o i n -

f i n i t e dimension, l e t L

F

V

There e x i s t s a supermaximal space

and

Were (U,clA).

m,

let

8 be a b a s i s f o r

C = A v F.

dim[C/A] = But

-,

Conversely i f certainly

C v G = A v G

C C =

in

*

A,

(U,clA),

then f o r a

dim[C v G/A] =

says f i n i t e

G

-, C

spans

a contradiction.

When i s t h e upper s e m i - l a t t i c e

L*(U)

then

actually a lattice?

v G

C'v G

G

E

is

in

-

LF(U), di-

A. NERODE and J . REMMEL

60 L*(U)

THEOREM 6.2. [A]

-

[B] = [ A A B]

A

and

i s a l a t t i c e under

dim[A/B]

PROOF.

<

i f and o n l y i f f o r a l l imply

Note

a.

A =

A,B,X

B

E

a.

L ( U ) . A 5 B, dim[A/Bl <

A A X #* B A X

but

t h e n f o r any

X

E

L(U), M A X

=

u l a t i o n by lemna 6.1.

Now, t o p r o v e t h a t

let

A =

as

X

E

L(U).

BvF

and

t o get

Since N

as

(BvF)AX =

BAX.

into

AAX

A

=

*

t h e r e i s an

respects F

E

Conclude

AAX =

*

L(U), F

let

A,

with

L(U)

Take

A,B

mod

B,

<

AAX/BAX

*

L (F,)

as

M

base f o r

Fa.

For

,...I ) ,

A = Fa = cl({bo,bl

+ bob2, b2 + bob3,...}).

A

=*

B n X = c1(0),

B and

A

examine t h e map

m,

i n t o a subspace o f

=* X,

Let

+.

v

+

B.

BAX,

A/B.

But a m.

-

be a transendence

,...I ) , m,

and u s i n g

so dim[AA X/BA X I = d i m [ X / c l ( 0 ) ]

=

a.

(In fact i n

so

be any independent s e t , l e t is

v

<

Fa

f

5

AAA

containing BAX

=

z c1(0), L*(U)

Bn = Ibn,bn+, dimensional.

...1 . Then

let

*

makes

L(F,)

collapse.

o r 1 = 0).

i s a l a t t i c e implies a very Namely l e t

s t r o n g form of t h e s u p p o r t lemna used t h r o u g h o u t [61.

X A cl(B)

B

dim[AA X/BA X I <

...

bo,b,,

B = cl({bl,b2

We n o t e t h a t t h e c o n d i t i o n e n s u r i n g t h a t

such t h a t

as

B 5 A, dim[A/B]

Then

t h e s m a l l e s t l a t t i c e congruence

L(F,)

BvF,N

M

i s not a lattice.

We f a l s i f y t h e c o n d i t i o n o f Theorem 6.2.

Jacobians

A = *B, Take

i s a l i n e a r t r a n s f o r m a t i o n , has k e r n e l

PROOF.

X = c l ({bo,bl

L(U)

A v F = BvF.

subspace of a f i n i t e dimensional space i s f i n i t e dimensional, so

Let

E

BAX.

A 5 B, dim[A/B]

so g i v e s a v e c t o r space isomorphism o f

COROLLARY 6.4.

and

LF(U)

E

T h i s i s a f u l l y e q u i v a l e n t form-

(BvF)AX=*AAX.

i n LfV,),

A,B,X

N A X.

E

For t h e other

i s a modular l a t t i c e .

L (Va)

Given

T h i s maps

*

B,

toget

*

COROLLARY 6.3.

PROOF.

A

*

*

M,N

and

m

by lemma 6.1.

d i r e c t i o n , r e f o r m u l a t e t h e c o n d i t i o n as s a y i n g t h a t i f M = NvF,

R 2 A

L(U),

in

A,B,X

dim[Ah X/BA XI <

F o r one d i r e c t i o n suppose

dim[Ah X/BA X I =

[A] v [Bl = [ A v B l ,

X

Xn cl(B,)

B = IbO,bl,

be any c l o s e d s e t i n

...I L(U)

i s i n f i n i t e dimensional

Recursion Theroy o n Matroids

too, and c o n s i s t s o f dim[cl(B)/cl(B,)]

<

Xhcl(B)

7.

AUTOMORPHISMS. (U,cl)

$(cl(A)).

E

cl(al

If

1-1

(U,cl) onto

This i s because

suppBx 5 Ibn,bn+l,...).

dimensional, hence so i s

m

we mean a

dim[Xh c (B)IXh c l ( B n ) l <

i s a S t e i n i t z exchange system, by an automorphism $:U

-f

U

i f and o n l y i f

$(x)

E

isomorphism w i t h r e s p e c t t o t h e r e l a t i o n

A ‘U,

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

A 5 U, x

cl({$(al) “x

..., an

al,

o r t h a t f o r a l l f i n i t e sequences

,...,an)

m,

Xhcl(Bn).)

Note t h a t t h i s says p r e c i s e l y t h a t f o r a l l

$(x) E cl($(A)); x

with

X

E

i m p l i e s by Theorem 6.2 t h a t

m

but

of

is

x

61

E

i.ff

we have

U,

E

,...,$ ( a n ) } ) ;

depends on

cl A

cl($(A)) =

or that’ $

al,...,an”;

i s an

or that

$

Thus t h e d e f i n i t i o n i s n a t u r a l f r o m

maps independent s e t s t o independent sets.

both t h e Van d e r Waerden p o i n t o f view and f r o m t h e Whitney p o i n t of view. Let Call

(U,cl)

be an i n f i n i t e dimensional r e c u r s i v e l y presented S t e i n i t z system. n o n - t r i v i a l i f i t c o n t a i n s a t l e a s t one i n f i n i t e dimensional c l o s e d

C 5 L(U)

s e t and i s n o t

L(U).

We say t h a t

for a l l non-trivial

C 5 L(U)

which map

L(U),

L(U)

to

to

L(U)

L(U)

THEOREM 7.1.

c l o s e d under a l l r e c u r s i v e automorphism of

and f o r any automorphism

of

$

C* = C mod

=* ,

[14] t h a t t h i s h o l d s f o r

We p r o v e t h a t t h i s a l s o h o l d s f o r

lattice

L(V,).

which maps

L(V,)

to

L(V,)

C

(w,cl)

(Vm,cl)

So t h e hypotheses t h a t which map

L(V,)

(where

cl(X) = X

(Vm,cl).

Note t h a t e v e r y automor-

According t o K a l a n t a r i [5] t h e automorphism o f

may be t a k e n r e c u r s i v e . automorphisms o f

$ induces

induces an automorphism of t h e

induced by an i n v e r t i b l e s e m i l i n e a r t r a n s f o r m a t i o n .

that

we have t h a t

The automorphism b a s e p r o p e r t y h o l d s f o r

(Vm,cl)

which maps

(Vm,cl).

PROOF. We reduce t h i s t o a s l i g h t l y s t r o n g e r Lemma 7.2. phism o f

(U,cl)

,(U,cl)

L*(U).

Shore has shown i n [13], X).

has t h e automorphism base p r o p e r t y if

and induces t h e i d e n t i t y on

the i d e n t i t y on a l l o f

for a l l

(U,cl)

to

C

L(V,)

i s itself

Guichard [4] showed t h e l a t t e r i s c l o s e d under a l l r e c u r s i v e

L(Vm)

i s t h e same as t h e h y p o t h e s i s

i s c l o s e d under a l l r e c u r s i v e i n v e r t i b l e s e m i l i n e a r maps which map L(Vm)

62

A. NERODE and J. REMMEL

to

L(V,).

The conclusion t h a t any automorphism of and induces t h e i d e n t i t y on

L(V,),

,

C

(Vm,cl)

a l s o does so on

which maps

L*(Vm)

maps

L(V,)

to

and induces t h e i d e n t i t y

L(V,)

on

to

Vm

a l s o does so on

C*

to

i s t h e same as

Vm

the conclusion t h a t any r e c u r s i v e i n v e r t i b l e semilinear map o f

L(Vm)

which L*(V,).

So we are proving more i f we prove

LEMMA 7.2.

Let

C

be a n o n - t r i v i a l subclass of

L(V,)

a l l r e c u r s i v e i n v e r t i b l e l i n e a r maps w h i c h map Suppose

then

#* @(N).

if

* C

induces the i d e n t i t y on

Using

or

@

that

@-

( R ) 5 N,

as follows.

Let

STAGE 0 .

Let

dependent since STAGE @

-1

so t h a t

6

R n @-

= ro,rl,r2,

bo = r1 and

1

(R) =

...

= b3i+,

bo,

{a}.

Let

...,b3n+3

dim(R) = dim(@(R)) =

I t i s easy t o see t h a t

hence

B

Bo

and Let

m.1

b2 =

r

B, C

E

a;, i% B2 = Cb3n+21n

Vm. E

Let

B

Note

for

\Im

R. and

@-'(rl)

such t h a t f o r a l l b3n+3 = r J n+l such t h a t r J. and Let

r . can be found since J

b3n+5 =

...)

{$I.

are i n -

(ux)Ix f (b,,b,}*).

a r e independent.

Let

rl

(Note t h a t

-.

dim[@(N)/N] =

R n N =

such t h a t

bo,bl,...,b3n+2

B = {bo,bl,

i s a r e c u r s i v e basis f o r

B, = Ib3n+71n

So, f o r a c o n t r a d i c t i o n ,

B u i l d a r e c u r s i v e basis

) where jn+l i s t h e l e a s t jn+l are independent over Ibo,...,b3n+2}*. (Note 8

induces t h e

@

i s nowhere simple and

N

R 5 @(N)

bl = @ - ' ( r l ) .

R n @(R) = @ } .

and

L*(V,).

and

b3n+4 = @ - ' ( r

cl(0)

Then

be an e f f e c t i v e l i s t i n g o f

n + l . Assume we have l i s t e d

(b3i)

.

w i t h o u t loss o f g e n e r a l i t y assume

I$-',

There i s an i n f i n i t e dimensional decidable 1

L(V,)

induces the i d e n t i t y on nowhere simple spaces i n

@

induces t h e i d e n t i t y on a l l o f

@

we may assume @ N

L(V,).

L (l',).

Be Theorem 4.6,

L*(Vm),

C

*

i d e n t i t y on

PROOF.

to

L(V,)

i s a n y r e c u r s i v e i n v e r t i b l e s e m i l i n e a r map, m a p p i n g

@

L(Irm) a n d i n d u c i n g t h e i d e n t i t y o n

to

closed under

(LIX)(X

i s r.e.

L

5

n,

and @-l(rj)

R n $(R) =

{bg,...,b3n+4i*.)

independent, and spans

Bo = I b 3 n l n

IA?, then

i

E

Vm;

IA),

@ - l ( B 0 ) = B1

by c o n s t r u c t i o n i%

are r e c u r s i v e independent sets. be a n o n t r i v i a l element i n t h e c l a s s

C.

Now t h e r e i s an i n f i n i t e d i -

Recursion Theroy on Matroids mensional decidable space V 5 C so let 0

=

=

u

be a recursive basis for

Let L be the recursive linear =

cl(B1) 5 C'. Now since dim(U/C')

=

cl(@-'(B0))

B2 and L(02)

cl(B1) + @-'(C')

=

@,

hence if

@

the only way

m,

#* C', i.e.,

U is if @-'(C')

dicts our choice of

=

Bo and let C'

=

L(C). Now C' E C by hypothesis and C' = L(C) 2 L(V) 2 B1 u B2. Now consider @ - 1 (C'). Note that C' + cl(Bo) = U so that @ - 1 (C') + @ - 1 (cl Bo)) = U. But @-l(cl(Bo)) = transformation such that L(O1)

B1

O2

O1 u

Vm such that D1 is a recursive basis for V.

63

C #* @(C').

But this contra@

induces the identity on C*,

ihduces the

identity on L * ( U ) . REMARK 1.

N, @(N) $(R)

c

A slavish imitation of Shore's original proof does not work. With

as before, let R be decidable, R N,

we get @(R)

n

R

+

= {OJ.

c

@(N),

R

n

N = cl(fl).

Since

Suppose we take a recursive basis B;= B, uB2

for Vm such that B1 is a recursive basis for R. Again we can find a C' with C' R+C'

=

=

However, the desired conclusion does not follow. We get is compatible with @-'(R)+@-'(CI). But R n @ - 1 (R) =

@-l(R)+C'

=

Vm, a space can have disjoint complements, and C'

C

cl(B2).

3

Vm

E

=

@-'(C')

is

possible. REMARK

2.

If F is finite dimensional and A V F

=

B v F and

ible semilinear transformation of Vm, then @(A) v@(F) dim @ ( F )

=

is any invert-

@(B) v @ ( F ) with

dim F. Hence any recursive invertible semilinear transformation of Vm L*(Vm)

induces an automorphism of

as well as one of L(V,).

applies to a non-trivial class of 0 ' s . automorphisms of Ibo,bl,b2,

=

@

...1

L*(V,)

So Theorem 7.1

But Guichard [4] has noted that there are

induced by no invertible semi-linear map of Vm,

is a basis for Vm, then bi

unlike the case of L*(u),

+

(If

bi+l yields such an example.)

not all automorphisms of L*(V,)

So,

lift, and Theorem 7.1

is only about those that do. We conclude with an example which shows that we can have an infinite dimensional recursively presented regular linear matroid X for which the automorphism base property fails due to lack of homogeneity. This X will be a submatroid o f Vm

A. NERODE and J . REMMEL

64

o v e r an i n f i n i t e f i e l d . a basis f o r

Vm

all

Vm,

aai

ai,

where

A, B be d i s j o i n t r e c u r s i v e i n f i n i t e s e t s w i t h

Vm

in

i s any automorphism o f cl(ai)

I n particular,

(U,cl),

E

cl({ail)

for a l l

C = cl(A),

hence on

i d e n t i t y on

E

ai

5 cl(A),

a})

L"(X).

i, $(b2i)

C* and

So

a # O,l,

with

generates i n

i t must map

so

cl(A)

X

=

ai

AuB

omit from

...,i+l.Keep

a l i n e cl(Iai})

o f cardinality

cl(ai)

i+2

F o r each

i+2.

all

'through If

$

t o a c l o s e d s e t o f t h e same

elements,

@(cl(fai}))

= cl(Iai}).

$ ( c l ( A ) ) = c l $(A) ~ c l ( A ) , and i s an i n f i n i t e , c o i n f i n i t e dimen-

c l o s e d under a l l automorphisms o f

X,

c l o s e d under t h e r e c u r s i v e ones. @(ai) = ai

6)

and

l....}.

IOai,la i,...,(i+l)ail

$(cl(A)) = cl(A).

sional closed s e t i n

c l $(Ib2i+,li

B = fbo,b

b e i n g t h e o n l y one w i t h $(ai)

g e t by symmetry

ai

So each

X.

which i s a f i n i t e s e t

size, so

,,...1,

A = {ao,a

(on t h e l i n e t h r o u g h

other points o f

6,

Let

(X,cl),

i n particular

Now l o o k a t t h e i n v e r t i b l e l i n e a r t r a n s f o r m a t i o n = b2i+l,$(b2i+l)

c l ( A ) mod =

*.

=

But

b2i.

I t induces t h e i d e n t i t y on

$(c11b2ili

~ l ( C b ~E ~u}) ( i a r e n o t =*

, so

E

ul) = $

does n o t induce t h e

Recursion Theroy on Matroids

65

REFERENCES

[ I] M. Aigner, C o m b i n a t o r i a l Theory,

( S p r i n g e r - V e r l a g , B e r l i n 1979).

[ 21

J . Baldwin, Recursion t h e o r y and a b s t r a c t dependence, t h i s volume.

[ 31

R. Downey, Ph.D. d i s s e r t a t i o n , Monash U n i v e r s i t y , C l a y t o n - V i c t o r i a , A u s t r a l i a , 1982.

[ 41

D. Guichard, Ph.D.

[ 51

I. K a l a n t a r i , Automorphisms o f t h e l a t t i c e o f r e c u r s i v e l y enumerable v e c t o r spaces, Z e i t , f. math L o g i k 25 (1979), 385-401.

[ 61

G. Metakides and A. Nerode, R e c u r s i v e l y enumerable v e c t o r spaces, Ann. Math. L o g i c 11(1977), 147-171.

[ 71

G. Metakides and A. Nerode, Recursion t h e o r y on f i e l d s and a b s t r a c t dependence, J . o f Algebra 65 (1980), 36-59.

[ 81

G. Metakides and A. Nerode, E f f e c t i v e c o n t e n t o f f i e l d t h e o r y , Ann. Math. L o g i c 17 (1979), 289-320.

[ 91

J . Remmel, Maximal and c o h e s i v e v e c t o r spaces, J . Symbolic L o g i c 42 (1977), 400-418.

[lo]

J. Remnel, An r-maximal v e c t o r space t h a t i s n o t c o n t a i n e d i n any maximal v e c t o r space, J . Symbolic L o g i c 43 (1978), 430-441.

d i s s e r t a t i o n , U n i v e r s i t y o f Wisconsin, Madison, 1982

[ll] J . Remmel, Recursion t h e o r y on a l g e b r a i c s t r u c t u r e s w i t h an independent set, Ann. Math. L o g i c 18 (1980), 153-191. [I21

J . Remmel and D. M i l l e r , E f f e c t i v e l y nowhere s i m p l e sets, t o appear i n J . Symbolic L o g i c .

[131

R. Shore, D e t e r m i n i n g automorphisms o f t h e r e c u r s i v e l y enumerable sets, Proc. Amer. Math. SOC. 65 (1977), 318-326.

[14]

R. Shore, Nowhere s i m p l e s e t s and t h e l a t t i c e o f r e c u r s i v e l y enumerable s e t s ,

J . Symbolic L o g i c 43 (1978), 322-330.

[151

B. L. van d e r Waarden, Modern Algebra (Ungar, New York, 1949).

[16]

H. Whitney, On t h e a b s t r a c t p r o p e r t i e s o f l i n e a r dependence, Am. J . Math. 57 (1935), 509-533.

PA TRAS LOGIC S YMPOSION G . Metokides (ed.) 0North-Holland Pu blishing Company, 1982

61

RECURSION THEORY AND ABSTRAC

DEPENDENCE

J. T. Baldwin* U n i v e r s i t y o f I l l i n o i s a t Ch cago C i r c l e Box 4348 Chicago, I L 60680

I n [Z] Metakides and Nerode examine the a l g e b r a i c p r o p e r t i e s of an a b s t r a c t dependence r e l a t i o n which account f o r c e r t a i n phenomena i n recurs ve algebra.

We continue t h i s study here by i n t r o d u c i n g two p r o p e r t i e s of an

a b s t r a c t dependence r e l a t i o n . on an algebra

A

I f a r e c u r s i v e l y presented dependence r e l a t on

s a t i s f i e s t h e f i r s t o f these c o n d i t i o n s ( i s autcinomous)

then the l a t t i c e o f r . e . closed subsets o f o f r.e. subsets o f

A

A

i s isomorphic t o the l a t t i c e

I f the r e c u r s i v e l y presented dependence r e l a t i o n on

w.

s a t i s f i e s t h e o t h e r c o n d i t i o n ( i s federated) then t h e l a t t i c e o f r . e .

closed subsets o f

A

behaves as t h e l a t t i c e o f r . e . closed subsets of an

i n f i n i t e dimensional v e c t o r space ( (Vm,cl)) field

((F,,cl)).

and Nerode then

That i s , i f (U,cl)

(U,cl)

o r an a l g e b r a i c a l l y closed

i s r e g u l a r i n the sense o f Metakides

i s federated.

More important, i f

(U,cl)

i s fed-

erated then some o f t h e major r e s u l t s proved f o r r e g u l a r dependence r e l a t i o n s i n [Z] h o l d (and by t h e same p r o o f ) .

Thus we g i v e a u n i f o r m explanation f o r

such p r o p o s i t i o n s as t h e f a c t t h a t both

(Vm,cl)

and

(Fm,cl)

have an r.e.

closed subset w i t h no r . e . basis which can be i n f i n i t e l y extended t o an r.e. independent s e t [ Z ; Theorem 5.11: Metakides and Nerode examine t h r e e classes o f examples o f dependence re1a t i ons a)

(U ,cl ) :

v e c t o r spaces over a f i n i t e f i e l d

*Research p a r t i a l l y supported by NSF g r a n t MCS 77-01667

J.T. BALDWIN

68

b vector spaces over an i n f i n i t e f i e l d c

a l g e b r a i c a l l y closed s u b f i e l d s of an a l g e b r a i c a l l y closed f i e l d .

( I n each case

U

i s i n f i n i t e dimensional and r e c u r s i v e l y p r e s e n t e d . )

They p o i n t

out c e r t a i n phenomena which hold i n a l l t h r e e cases and give a uniform proof (based on t h e i r notion of r e g u l a r i t y ) f o r cases b ) and c ) of these r e s u l t s and a l s o of some o t h e r r e s u l t s which hold only i n cases b ) and c ) .

We show t h a t the weaker

notion, f e d e r a t e d , accounts f o r those r e s u l t s t r u e i n a l l t h r e e cases and then show the a d d i t i o n a l results i n [ Z ] which follow from " r e g u l a r " b u t not from federated a r e , i n f a c t , f a l s e i n case a ) .

In p a r t i c u l a r , we show t h a t a federated recursive-

l y presented dependence r e l a t i o n has r . e . supermaximal subspaces b u t a r e c u r s i v e l y presented vector space over a f i n i t e f i e l d has no r e c u r s i v e supermaximal subspace. We a l s o argue t h a t i n the context of autonomous a r e compl ementary concepts.

(Vm,cl) and

( F m , c l ) , federated and

That i s , i f ( * ) "Every i nfi n i t e i ndependent

s e t i s a set of i n d i s c e r n i b l e s . " holds f o r a dependence r e l a t i o n i s e i t h e r federated o r autonomous.

Although both

(U,cl)

(Vm,cl) and

then

U

(Fm,cl) satisfy

( * ) we give some examples o f r e c u r s i v e dependence r e l a t i o n s f o r which ( * ) f a i l s .

W e work e n t i r e l y i n the context of a S t e i n i t z c l o s u r e system a s exposited in [ Z I .

However, t h e importance of the d i s t i n c t i o n between autonomous and feder-

ated dependence r e l a t i o n s was f i r s t pointed out t o us by Shelah i n t h e more general context of the "dependence r e l a t i o n " of forking introduced i n 131. more general because the t r a n s i t i v i t y axiom does not hold.)

(The notion i s

Many consequences

of the assumption t h a t forking i s a t r a n s i t i v e autonomous dependence r e l a t i o n occur in [ l ] . The author acknowledges useful conversations with G . Cherlin and A . H. Lachlan. A dependence r e l a t i o n i s autonomous i f , roughly speaking, t h e whole is no

more than the sum of i t s p a r t s . .1

Definition.

f i n i t e sets

X

Formally,

The dependence r e l a t i o n and

Y

with

XUY

(U,cl)

independent

i s autonomous i f f o r any two cl(XuY) = c l ( X )

There a r e some r o u t i n e v a r i a n t s on t h i s d e f i n i t i o n . .2

LEMMA.

The following a r e e q u i v a l e n t .

u

cl(Y).

69

Recursion Theory and Abstract Dependence i)

(U,cl)

i s autonomous.

ii)

For a l l f i n i t e

iii)

For any independent s e t

X cl(X) = U{cl(Ixl): x

iv)

F o r any independent s e t

X cl(X) = U{cl(Y): Y c X I .

Proof.

cl(XUY)= c l ( X ) u c l ( Y ) .

X,Y

The p r o o f i s easy.

XI.

E

Note t h a t i v ) i m p l i e s i i i ) by i n d u c t i o n and

thence i) .

Of course,

(w,cl)

( t h e l a t t i c e o f r . e . s e t s ) i s autonomous.

A $lightly

l e s s t r i v i a l example i s any model o f t h e mono-unary v a r i e t y d e f i n e d by t h e e q u a t i o n 3 f ( x ) = x. (Here c l ( X ) i s t h e subalgebra generated by X.) A more p a t h o l o g i c a l example i s any model o f t h e t h e o r y o f an e q u i v a l e n c e r e l a t i o n w i t h i n f i n i t e l y many classes, a l l i n f i n i t e . THEOREM.

3.

If

(Here,

(U,cl)

cl(X) = Iy: 3 x

=

U by

Note t h a t

< wl

Ibi:i

U

x

E

cl(y)

U

is isomorphic to the lattice of

i s an e q u i v a l e n c e r e l a t i o n on

be an r . e . b a s i s f o r

$ ( i ) = cl(bi);

subsets of

x Ey}.)

r.e.

w).

sets (subsets of

B

X

is a recursively presented autonomous dependence relation

then the lattice of closed subsets of

PROOF.

E

U.

we can e x t e n d $

Thus i f we map

N

W3.

E

Since

i# j

Let

i n t o t h e power s e t of

t o an isomorphism f r o m t h e l a t t i c e o f

o n t o t h e l a t t i c e o f r . e . c l o s e d subsets o f

$(W) = U { $ ( i ) : i

U-cl($).

implies

$(i)

n

U

r.e.

by

$(j) i s disjoint this i s

o b v i o u s l y an isomorphism.

.4.

Definition.

The weak c l o s u r e o f an independent s e t

XI.

clw(X) = U{cl(Y):Y .5.

Definition.

f i n i t e set an

a

6.

Fact.

E

A

ClA(x)

(n.b.

Y

is

i s properly contained i n

The dependence r e l a t i o n

such t h a t f o r any f i n i t e

-

X

(U,cl) X

X).

i s federated i f there i s a

which i s independent o v e r

A

there i s

clAw(x).

(Vm,cl)

and

(F,,cl)

a r e f e d e r a t e d dependence r e l a t i o n s .

We r e c a l l t h e key d e f i n i t i o n f r o m [ 2 ] .

Icf.2;p.lll.

Note t h a t t h i s n o t i o n i s t o t a l l y

d i f f e r e n t from t h e n o t i o n o f r e g u l a r i t y i n t r o d u c e d i n 131.

7.

Definition [ Z ] .

A f i n i t e dimensional c l o s e d s e t

C

5U

i s regular i f i t i s

n o t t h e u n i o n o f a f i n i t e number o f i t s p r o p e r c l o s e d . s u b s e t s .

We c a l l

(U,cl)

J.T.BALDWIN

70

regular i f a l l i t s f i n i t e dimensional closed subsets a r e r e g u l a r . Note, i n p a r t i c u l a r , t h a t an i n f i n i t e dimensional v e c t o r space over a f i n i t e f i e l d i s federated b u t not r e g u l a r . LEMMA.

0.

If

(lJ,cl)

However, we e a s i l y have

is regular then

(U,cl)

is federated.

As noted below, the proofs of Metakides and Nerode a c t u a l l y e s t a b l i s h many of t h e major s t r u c t u r a l r e s u l t s f o r r e c u r s i v e l y presented federated dependence r e l a t i o n s a s well a s f o r r e g u l a r dependence r e l a t i o n s . emphasized.

One d i s t i n c t i o n should be

Proposition 3.5 o f 121 e s t a b l i s h e s the equivalence of " r e c u r s i v e

dependence" and "recursive presentation" f o r r e g u l a r dependence r e l a t i o n s .

It

seems l i k e l y t h a t t h i s equivalence f a i l s f o r an a r b i t r a r y f e d e r a t e d dependence re1 a t i on. Now we would l i k e t o e s t a b l i s h t h a t "federated" and "autonomous" form a dichotomy. D e f i n i t i o n . The dependence r e l a t i o n

9.

(U,cl)

i s monodimensional ( n o t uni-

dimensional which was taken by Shelah i n 131) i f (*) Every i n f i n i t e independent s e t i s a set of i n d i s c e r n i b l e s . Monodimensionality can be regarded a s a kind o f homogeneity condition.

We

r e q u i r e one more Lemma before e s t a b l i s h i n g the dichotomy. 10.

LEMMA.

cl(Y)

5

If X

5 Y,

Y

is an independent set and

cl(X)

'$ cl"(X)

then

ClW(Y).

Proof.

Suppose cl(.Y)

a

E

cl(X)-clw(X). Since

a

E

cl(XoU{yl).

y

E

cl ( X )

11. LEMMA.

=

clw(Y) and (by induction)

Y = clw(Y) there i s an

B u t then y

E

Xo

Y-X

X

=

{yl.

Choose

such t h a t

cl(XoUIal) s i n c e a r! c l ( X o ) . B u t t h i s implies

contrary t o t h e independence of

Y.

Every monodimensional dependence relation is either autonomous or

federated.

Proof.

Let B be a b a s i s f o r

Lemna 10 and the i n d i s c e r n i b i l i t y of c l ( A ) = clw(A). Now i f (U,cl) i s federated.

X 5B

-

(U,cl).

B

If

(U,cl)

i s not autonomous, by

t h e r e e x i s t s a maximal

A , clA(X) = cl(AUX)

A 5B

clW(AuX)3 c l l ( X )

such t h a t

so

Recursion Theory and Abstract Dependence

71

Thus we have completely c h a r a c t e r i z e d t h e s i t u a t i o n f o r mono-dimensional dependence r e l a t i o n s .

However, i t i s easy t o r i g examples of dependency r e l a t i o n s

which a r e non mono-dimensional and n e i t h e r autonomous nor f e d e r a t e d .

J u s t take

a d i s j o i n t union of an algebra with a f e d e r a t e d dependence r e l a t i o n and one with an autonomous dependence r e l a t i o n . harder t o come by. scheme.

"Natural" examples a r e , however, somewhat

We now show how t h e s p e c i f i c r e s u l t s i n [ Z ] f i t i n t o our

The following r e s u l t s can be proved by replacing each occurrence of the

word ' r e g u l a r ' by the word ' f e d e r a t e d ' i n the proof of the Theorem 5.1 i n [21. 12.

THEOREM.

Suppose

infinite dimension. basis of

V

(U,cl)

is recursively presented, federated, and of

V

Then there exist maximal

in

L(U) such that no r.e.

can be extended to an infinitely larger r.e. independent set.

Moreover a s i m i l a r change of r e g u l a r t o federated and dropping a l l references

t o t h e requirement .14

If

(U,cl)

R

fl

turns t h e proof of 6.1 of [ Z ] i n t o a proof of the following.

i s an i n f i n i t e dimensional, federated r e c u r s i v e l y presented

dependence r e l a t i o n , then t h e r e i s an r . e . closed s u b s e t of

U which i s super-

maximal. 15.

Definition.

ii)

if

W

An r.e. closed set V

-

i s supermaximal i f i ) dim[U/VI =

V

i s an r . e . closed s u p e r s e t of

and

dim[W/V] =

I n a c e r t a i n sense Theorem 14 i s the b e s t possible.

then

-

and

W = U.

To c l a r i f y t h i s we

introduce one f u r t h e r notion. .16

A dependence r e l a t i o n

set

Xo

y

E

(U,cl)

i s e s s e n t i a l l y binary i f t h e r e e x i s t s a f i n i t e

such t h a t f o r any f i n i t e s e t s

c l ( Y ) and

z

c l ( Z ) such t h a t w

E

Y,Z E

if

w

E

cl(XoUY UZ)

then t h e r e i s a

cl(Xo U { y , z l ) .

Z i l b e r [4] d e a l s with a s l i g h t l y more r e s t r i c t i v e notion

( l X o l = 1 ) and

shows t h a t any dependence r e l a t i o n s a t i s f y i n g h i s condition i s representable a s a Note any vector space i s e s s e n t i a l l y binary ( w i t h

p r o j e c t i v e space.

Xo = $ )

while t h e s u b f i e l d s of an a l g e b r a i c a l l y closed f i e l d a r e not.

.17

Definition.

A dependence r e l a t i o n

there i s a function

f:w

+

w

such t h a t

(U,cl)

1x1

< n

i s uniformly l o c a l l y f i n i t e i f implies

Icl(X)I < f ( n ) .

We r e c a l l from [2] the notation f o r dependence over a closed subspace V. .18 D e f i n i t i o n .

If

V

i s a closed subspace of

(U,Cl)

then

J.T. BALDWIN

12

D(V)k

THEOREM.

19.

xo ,...x

= {([email protected]~-~):

Let

k

Fix

Fix

Z = Izo,...zk-ll.

Since

(U,cl)

i s dependent o v e r Z

V

z ~ E- c l~( X o U { y , z } )

f o r some

Z

2

y

E

Z

E

cl(Y), z E cl(Z').

i s dependent o v e r

iff

D(V)k

Xo.

is

D(V)k

i s dependent 2

I f not,

V. V

If not, b u t

such t h a t

Y # @.

So

B u t t h e n by t h e symmetry

Z

i s dependent o v e r

compute

l c l ( Z U X o ) ) < f ( lZUXo])

cl(ZUXo)

of

Y

.. , z ~ - ~ ) .Moreover,

Thus t o determine i f

r e c u r s i v e process, s i n c e we know Now

Z

i s c e r t a i n l y dependent o v e r

Z ' = {zo,.

cl(XOU~z,z,~,l).

f i r s t ask i f

ness.

Xo 5 V

such that

t h e r e e x i s t s a f i n i t e subset

where

E

U

of

i s r e c u r s i v e l y presented, we can f i r s t check whether

z ~ E- c l~( X g UYUZ')

y

V

We a r e t o determine i f

I f so,

Xo.

i s dependent o v e r

lemma,

from the definition of essentially binary.

V.

Proof. V.

X

and every closed subspace

recursive in

over

V).

be uniformly locally finite, essentially binary

(U,cl)

and recursively presented. For every

~ -a r e~ dependent o v e r

cl(ZUXo).

V,

This i s a

by uniform local f i n i t e -

n V # @.

With t h i s theorem i n mind we can see t h e s t a t u s o f Theorems 6.1 and 7.1 o f [21 i n o u r c o n t e x t . 20.

Corollary.

If

(U,cl)

r e c u r s i v e l y presented t h a n Proof. space o f

i s e s s e n t i a l l y b i n a r y , u n i f o r m l y l o c a l l y f i n i t e and (U,cl)

U. L e t (bi:i

<

O)

the

< w)

-.

i s a r e c u r s i v e supermaximal sub-

be a r e c u r s i v e enumeration o f a b a s i s f o r

I t s u f f i c e s t o enumerate an r . e .

(ci:i

V

Suppose f o r c o n t r a d i c t i o n t h a t

w i l l now d e f i n e an r . e . subspace

dim(U/W) =

has no r e c u r s i v e supermaximal s e t .

W

of

U

basis f o r

such t h a t W

o f the

bi

V

over

B u t t h i s i s easy f r o m t h e theorem. which a r e independent o v e r

dim[W/V] =

m

U.

We

and dim[U/Wl

w h i l e guaranteeing

We j u s t generate a subset V

and p u t t h e

c2i

E

W,

c ~ ~ +E , U-W. Note t h a t t h e c o r o l l a r y a p p l i e s t o a v e c t o r space o v e r a f i n i t e f i e l d .

Of

course, Theorem 19 r u l e s o u t t h e f u l l ' s t r e n g t h ( f o r t h e weaker h y p o t h e s i s of f e d e r a t e d ) o f 7.1 i n [ 2 ] . 21.

THEOREM.

Suppose

B u t t h e f i r s t paragraph o f t h e proof y i e l d s .

(U,cl)

is infinite dimensional, recursively presented

=m.

73

Recursion Theory and Abstract Dependence

Let

and federated. (i) i >

is r.e. uniformly in

Ai 0;

A ,A1,...be

5

iii) d(Ai)

such that

d(Ai)

5

d(Ai+l)

d(D(V)i)

a sequence of sets of integers such that: i, i>

for

0;

i >

ii)

0.

5

d(Ai)

d(Ao)

uniformly in

Then there is a n r.e. i > o

uniformly in

and

5

d(Ao)

subspace

i,

V

d(V).

In an attempt t o f i n d a s t r i c t analogue f o r 6.1 of [21 we introduce here a notion, "weakly regular" and show t h a t every weakly r e g u l a r r e c u r s i v e l y presented dependence r e l a t i o n has r e c u r s i v e supermaximal s e t s .

While no natural example

of a weakly r e g u l a r b u t not r e g u l a r dependence r e l a t i o n springs t o mind, the No l o c a l l y f i n i t e

notion has the following apparent advantage over r e g u l a r i t y . algebra i s r e g u l a r .

A p r i o r i , t h e r e may be uniformly l o c a l l y f i n i t e algebras

which a r e weakly r e g u l a r . 22.

Definition.

The dependence r e l a t i o n

k-dimensional subspace of

U

(U,cl)

i s weakly r e g u l a r i f no

can be w r i t t e n a s the union of

k

k-1-dimensional

subspaces. Clearly any r e g u l a r dependence r e l a t i o n i s weakly r e g u l a r and any weakly regular dependence r e l a t i o n i s f e d e r a t e d . of

k-1 dimensional spaces which can cover a

arbitrary. k-1

The choice o f the bound on the number k

dimensional space i s r a t h e r

The l e a s t value which w i l l e a s i l y s u f f i c e f o r t h e next theorem i s

but this seems t o r e l y on the ordering of t h e natural numbers i n making the

construction i n an a r t i f i c i a l way so we f i x e d the value a t over f i n i t e f i e l d s a r e not weakly r e g u l a r . considering

k

k.

Vector spaces

(This becomes more p l a u s i b l e when

much g r e a t e r than the s i z e o f the f i e l d . )

l o c a l l y f i n i t e algebra t o be weakly r e g u l a r the function

For a uniformly f ( n ) = maximal

c a r d i n a l i t y of an n-generated subalgebra must grow very f a s t . 23.

THEOREM.

If

(U,cl)

is recursively presented, infinite dimensional and

weakly regular then there is a supermaximal set of

V

5U

which is recursive as a sub-

U.

T h i s i s the only r e s u l t from [ Z ] where any n o n - t r i v i a l modification of the proof i n [ Z ] i s needed t o extend t h e theorem from r e g u l a r t o our hypothesis. Even here more than two t h i r d s of t h e argument i s i d e n t i c a l .

Thus r a t h e r than

including a complete proof, we w i l l o u t l i n e t h e proof from [ Z ] and then i n d i c a t e our modifications.

J.T. BALDWIN

74

and

(Wi:

a "standard" r e c u r s i v e enumeration o f t h e r . e . c l o s e d subsets o f

U.

24.

Pr6cis.

Vs,(Ws)

Let

(bi:

i < W)

be a r e c u r s i v e b a s i s f o r

V

be t h e e x p l i c i t f i n i t e dimensional subspaces o f

by stage

s.

ent over

Vs.

A t stage

Then

V

s

we w i l l have a sequence

w i l l be

UsVs

and

i n f i n i t e independent sequence w i t n e s s i n g

V,(W)

(a::i

< w)

{ak = l i m s a:; dim[U/V] =

i <

W)

Let

constructed which i s independ-

k < wl

w i l l be t h e

We must meet t h e f o l l o w i n g

m.

r e q u i rements . R:

V

i s a r e c u r s i v e subset o f Ifdim[cl(We U V ) : V l

P(e,n>:

N ( ~ , ~ ) : l i m S as (e,n> 25.

Definition.

P

= a

=

(e,n>

U then

m

bn

E

cl(We U V)

exists.

requires a t t e n t i o n a t stage

(e,n>

s

i f ( i ) and ( i i ) below

hold. bn f cl,(Wz U V').

(i)

There i s an

(ii)

x 6 Wze such t h a t

x & c l l V s U {a: 26.

...,as(e,n)l

Goal o f t h e C o n s t r u c t i o n .

y Vs+'

= cl(Vs

a t stage

s

{y}).

and

i s the s e t o f a)

u

y

u E

x E

cl

y

E

U with

{x,bnl

u < s.

-

cl

y

E

such t h a t f o r some

i s t h e l e a s t p a i r which r e q u i r e s a t t e n t i o n

Our c o n s t r u c t i o n must guarantee:

Ixl VS

V

c l S{x,bnl - c l V

{uj:j < t l

- c l s{bn)

VS

c)

(e,n)

Vs

i s t h e l e a s t element s a t i s f y i n g Defn. 2 5 ( i i ) and

[x,bn}

cl

We want t o c o n s t r u c t t h e

Suppose

VS

b)

U {bnl!

vs

(u.)

all j < t

J

Now we must show a l l r e q u i r e m e n t s a r e met.

I f a ) , b ) and c ) a r e s a t i s f i e d

and R a r e met e x a c t l y as i n [2: Lemma 6.51, [2: Lemma (2,n) 3 (e ,n> 6.61 and [2: Limma 6.41. (The xi's on l i n e 3 o f t h e p r o o f o f 6.4 s h o u l d then

be

P

ui's.) We must m o d i f y t h e c o n s t r u c t i o n t o guarantee a), b ) , c ) m e r e l y on t h e

hypothesis t h a t

(U,cl)

i s weakly r e g u l a r .

Recursion Theory and Abstract Dependence 27.

Modification o f the construction o f

V.

We w i l l guarantee i n t h e c o n s t r u c t i o n t h a t

Stage 0.

L

s

Stage the l e a s t ’(e,n> least

Let

k

0.

U.

x

Vs.

C

Vs.

Vi

= cl(Vs

as

Now

(U,cl)

subspaces

=

2

s.

bpi.

requires a t t e n t i o n a t stage

(e .n) cl(Vs)

u

Let Vs

and l e t

u

{ui:

i < tI

Ix,bnl

VS+’

s,

choose

I f some

= c l ( V s U {b2k+ll).

where

Vi

{XI).

Let y

Define

=

cl(Vs

u

W

cannot be a u n i o n o f t h e if

{uil)

i < t

mo

by induction s e t t i n g

at:;

=

is

s

such

s+2-dimensional subspack

where

a:+’

{x,bn?

l i s t those numbers l e s s t h a n

and

be t h e l e a s t element o f

Let

a”;

By these c o n d i t i o n s ,

generates an

i s weakly r e g u l a r ,

c1(vs u (y)).

d VS+l.

a;

a:

s a t i s f y i n g Defn. 25 i ) and i i ) .

Since

vS+l

P

d

b2k+l

let

dim[Vs]

r e q u i r e s a t t e n t i o n , choose t h e l e a s t such and f o r t h a t p a i r choose t h e

ui

Vt+l

I f no

such

independent o v e r that

Vo = c l { O I ,

75

W

of

W

t+l 2 s+2

Vt = cl(Vs U {bnI),

-

UVi.

Now d e f i n e

i s l e a s t such t h a t t o be

a;

where

m

is

0

l e a s t such t h a t

a;

Z c l ( V S + l u {a:’

received a t t e n t i o n a t stage The n o t i o n s i n [2],

s

’,...,a;+’]).

(using

x

F i n a l l y , we say

P

(e,n>

and y ) .

[4] and here suggest t h a t t h e r e i s c o n s i d e r a b l e work t o

be done i n i n v e s t i g a t i n g s t r o n g p r o p e r t i e s o f a b s t r a c t dependence r e l a t i o n s . The work i n [l]and (31 shows t h a t r e l a x i n g t h e h y p o t h e s i s o f t r a n s v i t i v i t y a l s o yields a f e r t i l e f i e l d f o r exploration.

Both [21 and t h i s paper suggest t h a t

r e c u r s i o n t h e o r y can b o t h m o t i v a t e and t e s t t h e v a l u e o f such axioms.

For

example, t h e a t t e m p t t o prove 6.1 o f [2] f o r v e c t o r spaces o v e r f i n i t e f i e l d s leads t o t h e d i s c o v e r y t h a t v e c t o r spaces o v e r f i n i t e f i e l d s a r e n o t weakly r e g u l a r b o t h by s p o t l i g h t i n g them as an example t o be c o n s i d e r e d and as a C o r o l l a r y o f Theorem 23 and C o r o l l a r y 20.

I6

[l]

J.T.BALDWIN J . T . Baldwin and S. Shelah ( i n p r e p a r a t i o n ) , Second order q u a n t i f i e r s and the complexity o f t h e o r i e s , Proceedings o f t h e l o g i c y e a r i n Jerusalem, ed. J. A . Makowsky.

[ Z ] G . Metakides and A . Nerode, Recursion Theory on Fields and a b s t r a c t dependence, J. o f Alg. 65, 36-95 (1980). [3]

S . Shelah, C l a s s i f i c a t i o n Theory and t h e Number o f Nonisomorphic Models, North-Holland, Amsterdam, (1978).

[4] 8. I . Z i l b e r , Strongly minimal t o t a l l y c a t e g o r i c a l t h e o r i e s , S i b e r i a n Math. J . , v . 21, (1980) p p . 98-112 (Russian).

PATRAS LOGIC SYMPOSION G. Metakides led.) @North-Holland Publishing Company. 1982

MAJOR SUBSETS IN EFFECTIVE TOPOLOGY

I r a j Kalantari* Western I l l i n o i s University Macomb, IL 61455

One of t h e c h a r a c t e r i s t i c s o f c l a s s i c a l i n t e r e s t o f any branch of mathematics i s the question of constructiveness of i t s content.

I n the l i g h t of development

of recursion theory, the e f f e c t i v e n e s s of these constructions become o f special interest.

In t h i s a r t i c l e we i n v e s t i g a t e e f f e c t i v e l y describable open s e t s i n a

topological space and introduce a measure of t h e

effective addressin3 - a b i l i t y

of t h e i r topological connected components. Work i n t h i s approach t o e f f e c t i v e topology began i n Kalantari & Retzlaff [41 and continued i n Kalantari & Remmel 131. & Leggett t1,21.

For f u r t h e r developments, see Kalantari

Studies in e f f e c t i v e n e s s of r e s u l t s in s t r u c t u r e s o t h e r than

i n t e g e r s began w i t h the work of Specker [ l o ] and Lacombe [51 on e f f e c t i v e a n a l y s i s . The new a c t i v i t y i n study of e f f e c t i v e content of mathematical s t r u c t u r e s has been revived i n Nerode’s program and Metakides & Nerode [7,8] work on vector spaces and f i e l d s .

These s t u d i e s have been extended by Kalantari, Remmel, R e t z l a f f ,

Shore and Smith.

Similar s t u d i e s on e f f e c t i v e content of o t h e r mathematical s t r u c -

tures have been conducted.

These include work on topological vector spaces, boolean

algebras, 1i near o r d e r i ngs e t c . Kalantari & Retzlaff [41 began a study of e f f e c t i v e topological spaces by considering a topological space with a countable b a s i s space X

i s t o be fully e f f e c t i v e ;

A

f o r the topology.

The

t h a t i s , the b a s i s elements a r e coded i n t o w

and the operations of i n t e r s e c t i o n of b a s i s elements and the r e l a t i o n of inclusion among them a r e both computable.

*

A r e c u r s i v e l y enumerable ( r . e . ) open s u b s e t of

X

We wish t o thank the University of P a t r a s , Patras Greece, and Western I l l i n o i s University f o r f i n a n c i a l support f o r making t h e presentation of t h i s paper possible. We wish t o acknowledge valuable conversations w i t h Anne Leggett, George Fletakides, Anil Nerode. J e f f Remel and Ted R e t z l a f f .

I. KALANTARI

78

i s then represented by taking the union of basic open s e t s whose codes l i e i n an r . e . subset of

W.

Similar t o open subsets of

E,

the l a t t i c e of

r.e.

subsets of

X

forms a l a t t i c e

L(X)

under t h e usual operations of union

and i n t e r s e c t i o n .

w,

the c o l l e c t i o n of

For a proof of t h e f a c t t h a t the theory of

ducible t o t h e theory of

r.e.

i s not re-

L(X)

and f o r o t h e r r e l e v a n t f a c t s , we r e f e r t h e reader t o

E

t41.

I n this approach t o topology on

i t y of X

except t h a t i t be i n f i n i t e .

X

we have no r e s t r i c t i o n s on t h e cardinal-

X,

Objects of study a r e ' p i e c e s ' of t h e space

given i n t h e form of a b a s i c open set.

We argue t h a t s i n c e i n most c o n s t r u c t i v e

approaches t o mathematics, i t i s a " s u f f i c i e n t l y small neighborhood" t h a t t h e comput a t i o n h a l t s with, a s opposed t o a s p e c i f i c p o i n t , i t is s u f f i c i e n t t o handle and process neighbrohoods a s primary o b j e c t s of study.

Of course, i t becomes necessary

t o require c e r t a i n p r o p e r t i e s (both topological and recursion t h e o r e t i c ) .

In

Section 1 we l i s t these requirements and o t h e r p r e l i m i n a r i e s i n d e t a i l .

I n Section 2 , we introduce t h e c e n t r a l notion of fragment, discuss o t h e r reasons f o r our approach and define some l a t t i c e t h e o r e t i c p r o p e r t i e s o f open s u b s e t s of mented

r.e.

r.e.

In Section 3 , we use a p r i o r i t y argument t o show noncomple-

X.

open sets have major s u b s e t s and observe some c o r o l l a r i e s .

We con-

clude with some remarks i n Section 4. J1.

PRELIMINARIES W e consider a p a i r

(X,A)

where

X

i s a topological space and

countable basis f o r the topology on

X.

sets, i . e . elements of

A, B, C ,...

A.

We use

ploy standard topological n o t a t i o n ; e.g.

the i n t e r i o r of I. 11. 111,

A

A.

We assume t h a t

A

We use a , B , y,. A

..

is a

A

t o denote b a s i c open

t o denote subsets of

i s t h e c l o s u r e of

A

and

and em-

X

A'

is

s a t i s f i e s t h e following topological axioms:

i s closed under f i n i t e i n t e r s e c t i o n s .

0, X

E

A.

Every b a s i c open s e t 6

i s connected, i . e .

d i s j o i n t union of two open subsets of

X.

d

cannot be w r i t t e n a s a

Major Subsets in Effective Topology

IV.

Every nonempty b a s i c open s e t

6

19

contains two nonempty basic open s e t s

with d i s j o i n t c l o s u r e s and the closures contained i n The f i r s t two axioms a r e n a t u r a l .

6.

Connected components play an important

r o l e i n our r e c u r s i o n - t h e o r e t i c notions, t h e r e f o r e Axiom I11 i s a l s o v i t a l .

I v gives us "room t o work"

i n our constructions s i n c e i t implies t h a t

lower s e m i l a t t i c e (under i n t e r s e c t i o n ) i s atomless. a r a t i o n axioms a r e r e l a t e d a s follows:

A

Axiom as a

Axiom IV and the usual sep-

any r e g u l a r Hausdorff space s a t i s f i e s

while IV does not imply e i t h e r r e g u l a r i t y o r Hausdorffness.

IV,

F i n a l l y , note t h a t

I - IV do not impose a m e t r i c upon X. The following topological spaces of i n t e r e s t s a t i s f y axioms I - IV: R , A = the c o l l e c t i o n of a l l open i n t e r v a l s w i t h

( 1 ) X = the real i n e

r a t i o n a l endpoints. (2)

X = the real plane

R 2 , A = the c o l l e c t i o n of a l l open rectangles with

s i d e s p a r a l l e l t o axes and with r a t i o n a l v e r t i c e s .

(3) X = R n , A defined analogously.

(4) Any separable Banach space. (5)

Any topological vector space w i t h a neighborhood base a t each point con-

s i s t i n g of convex connected s e t s and w i t h a countable dense subset.

W e a l s o r e q u i r e t h a t A s a t i s f y some r e c u r s i o n - t h e o r e t i c p r o p e r t i e s .

First,

set A i n a one-to-one correspondence w i t h the p o s i t i v e integers through a Godel coding.

For

6

E

denotes the Giidel number of

A , '6r

x.

notes t h e Gtidel s e t o f

Thus

Cij'j=

6

and

6;

for

x

E

w.

LXJ

de-

T x ~x.=

We say topology

(X,A)

$(x,y)

such t h a t f o r a l l

x,

has an i n c l u s i o n algorithm i f ( 1 ) There i s a p a r t i a l r e c u r s i v e function

LxJnLyJ

Y

6

E

A)

w, -f

(This means t h a t given computes

(2)

E

converges and $ ( x , Y ) =

($(x.y) E,

6

E

~

X

1

~A L Y ,

1.

A,

t h e r e i s a uniform e f f e c t i v e procedure which

A,

there is a uniform e f f e c t i v e procedure which

n 6).

Given

6,

... ,E,,

E

determines whether o r not

6 5 ~ ~ u . . . u ~ " a whether nd

F 5E,u...uE~.

I. KALANTARl

80

I n t h e presence of c o n d i t i o n s ( 1 ) and ( 2 ) , one can e f f e c t i v e l y t e l l w h e t h e r

6 =

as w e l l as w h e t h e r

E

Furthermore, g i v e n f i n d nonempty

E

,,

such t h a t

A

L

E ~ U . .. U E ~

and w h e t h e r

-

.U 6 m

Flu..

EIU...UEn.

C

we can use t h e i n c l u s i o n a l g o r i t h m and e f f e c t i v e l y

A,

E E E~

61u... udm c

Fl

n

E2 = 0 and El

u

E2

This f a c t

c E.

i s a key s t e p i n o u r c o n s t r u c t i o n s . H e n c e f o r t h we s h a l l assume t h a t inclusion algorithm.

-

s a t i s f i e s axioms I

(X,A)

F o r a v i s u a l a i d , we suggest l o o k i n g a t

I V and has an w i t h the

R2

A

d e f i n e d as above.

and

Let

E

n.

Fix

subsets o f

LWJ

be t h e l a t t i c e o f {W:lerw,

A' 5 A ,

= u { ~ x ~ ~ x E W } . An

r.e.

write

Th;l

K5w

set

is

= {r67j16eA'};

full i f

SEO

This

If

U Us

IJ =

K

e'

K

r.e.

i s an

r.e.

f u l l set,

let

W

for

d i s t i n g u i s h between t h e

r.e.

set

L(X)

r.e.

1

uKs.

K=

r.e.

open s u b s e t o f

open s e t

K

X.

O r d i n a r i l y we w i l l n o t

and i t s e n u m e r a t i o n

K.

Next l e t

L(X).

The

x.

and

Ui

U.

I

fl U . = f~ and

U.

can be

t o g e t h e r w i t h t h e s e o p e r a t i o n s i s r e f e r r e d t o as t h e l a t t i c e of

open s u b s e t s o f

DEFINITION:

if

We

KS = U I L ~ - I I ~ ~ K S } and

The o p e r a t i o n s o f u n i o n and i n t e r s e c t i o n can be w e l l d e f i n e d o v e r

r.e.

w, w r i t e

StW

i s c o n s i d e r e d t o be an

collection

E

the r.e.

= ( X I ~ X ~ ~ ~ Lx W < Zs l~, A

Ue: U:

f u l l superset

{Welecw},

f o r some e f f e c t i v e e n u m e r a t i o n

t h a t any {KSlsrwl, V X [ ( ~ S ) ( ~ X ~ + C xcK]. ~ K ~ ~Observe ) u n i f o r m l y e x t e n d e d t o an

u

s u b s e t s o f t h e n a t u r a l numbers under

an a c c e p t a b l e e n u m e r a t i o n o f

SEW},

For

w.

r.e.

Ui

open s e t s a r e complemented i n

r.e.

u

1

U.

I

i s dense i n

X.

L(X)

I n t h e c o u r s e o f our

c o n s t r u c t i o n s we w i l l n e e d t o f i n d b a s i c o p e n s e t s w i t h c e r t a i n p r o p erty erty

P. P

I f b a s i c open s e t s s a t i s f y i n g

P

exist,

and i f t h e p r o p -

i s e f f e c t i v e l y v e r i f i a b l e f o r each b a s i c open set,

the conventional E

p =

we u s e

o p e r a t o r and w r i t e

ll6P(6).

Here t h e i n t e n t i o n i s that

E

i s t h e f i r s t nonempty b a s i c open s e t

81

Major Subsets in Effective Topology

found to have property P(LOd),

P(LL , P(L2-4)

P

during the dovetailing of computations of

... .

I n class cal topology, any pairwise disjoint collection of basic

open subsets o f collection.

X

can be extended to a maximal pairwise disjoint

The latter is of course dense in A

topology, for an open set

X.

Again in classical

there is a pairwise disjoint collection

, and any such can be extended to

A

of basic open sets dense in

X.

another such which is dense in

We consider

effective versions

of these ideas. K

DEFINITION: Let 15i i and

E

w}

for

is a partition

u 5i

X.

be an open subset of K

if the

is contained and dense in

The collection are pairwise disjo nt

&j's

K.

iew

We can show that every

r.e.

a partition

K

open set

{diliew}

r.e.

partition

I n fact, one can prove that given a

(See Kalantari & Leggett [l]). nonempty

open set has an

r.e.

and any

r.e. Turing degree a,

there is

I

for

K

such that deg(f'5;

liew})

=

a. -

(See

Kalantari 4 Remmel [ 3 ] ) . DEFINITION; an r.e.

Let

K

b e an

partition f o r

r.e.

partition

say that

112

we say that

r.e. K.

open set, and let

Then

is an extension of 111

is extendible if there is an

IT1

112 = { ~ ~ l i e w }f o r

IIl.

111 = Icriliew} be

X

such that If no such

Ill C I 1 2 . IT2

We also

exists for

ill,

is nonextendible.

I n contrast to classical setting we have:

PROPOSITION (1.2):

There is an

r.e.

open set

K

for which no

r.e.

partition is extendible. PROOF:

See Kalantari & Retzlaff 141. If for an r . e . open set K no r.e.

partition is extendible, we say K for K is extendible, we say K

i s nonextendible. If some r.e. partition

extendible.

I. KALANTARI

82 92.

BASIC OPEN SETS A N D INFINITY I n ordinary recursion theory, the objects o f study are the p o s i t i v e integers.

As a s t r u c t u r e ,

w

i s a c o l l e c t i o n o f d i s c r e t e and i n d i v i s i b l e o b j e c t s which can

be enumerated o r addressed t o o n l y by d i r e c t and unique h a n d l i n g o f such.

I n most

s t u d i e s , f o l l o w i n g t h e o r i g i n a l concepts o f Post [ 9 ] , t h e n o t i o n o f a measure o f subsets o f

w

enters.

f i n i t e o r as Post n o t e d two c l a s s e s :

i s an

thin".

thin versus =.

e f f e c t i v e l y enumerable and

of subsets of w

The n a t u r a l measure o f subsets o f

r.e.

w

with

Since subsets o f

not e f f e c t i v e l y

contrasting properties.

i s f i n i t e versus i n -

w

w also f a l l i n t o

enumerable, Post conceived

F o r example a s i m p l e subset of

s e t w i t h a complement which i s " c l a s s i c a l l y t h i c k " b u t " e f f e c t i v e l y

T h i s may be compared t o subsets o f r e a l l i n e which a r e

small and large i n

t h e sense o f measure and category. I n t h e s t u b y of subsets o f

w

t h e two s e t s o f m i x i n g p r o p e r t i e s a r e e f f e c t i v e / Metakides and Nerode [ 7 ] i n s t u d y i n g t h e sub-

n o n e f f e c t i v e and f i n i t e / i n f i n i t e .

s p a c e s o f an i n f i n i t e dimensional v e c t o r space, were o f course m o t i v a t e d b y e f f e c t i v e / n o n e f f e c t i v e as one o f t h e p r o p e r t i e s . needed s p e c i a l a t t e n t i o n .

However, t h e n o t i o n o f f i n i t e / i n f i n i t e

There seemed t o be two f a c t o r s a t work here.

t h e o b j e c t s o f s t u d y were v e c t o r s o v e r an i n f i n i t e f i e l d . o f a d d i t i o n between v e c t o r s was n o t one-to-one.

Firstly,

Secondly, t h e o p e r a t i o n

These f a c t o r s t r a n s l a t e i n t o t h e

f a c t t h a t t h e o b j e c t s o f s t u d y can be p a r t i a l l y addressed

to i n

i n f i n i t e l y many

ways ( t w o v e c t o r s i n t h e same d i r e c t i o n a r e p a r t i a l l y i d e n t i f i e d ) , and t h a t t h e y can be nonuniquely addressed t o ( a v e c t o r a d d i t i o n n o t one-to-one).

Metakides and

Nerode found t h a t t h e c o r r e c t n o t i o n o f f i n i t e / i n f i n i t e was when i t r e f e r r e d t o dimension.

Hence, f o r example, a s i m p l e v e c t o r space was t h a t

which i s c o i n f i n i t e - d i m e n s i o n a l r.e.

r.e.

v e c t o r space

( c l a s s i c a l l y t h i c k ) b u t i t s complement has no

i n f i n i t e dimensional v e c t o r subspace. I n d e f i n i n g t h e n o t i o n o f a maximal (and o t h e r l a t t i c e t h e o r e t i c a l ) subset

of

w,

t h e r o l e o f i n d i v i s i b i l i t y o f o b j e c t s o f s t u d y and t h e i r unique addresses

made t h e t a s k s t r a i g h t f o r w a r d : c o i n f i n i t e and t h e r e a r e no

r.e.

M,

an

r.e.

subsets

W

subset o f

u, i s maximal i f i t i s

w i t h both

W-M

and

w-W

infinite.

83

Major Subsets in Effective Topology Metakides and Nerode c o u l d n o t s i m p l y t a k e t h e s e t d i f f e r e n c e between two v e x t o r spaces i n o r d e r t o examine t h e t h i c k n e s s o f t h e d i f f e r e n c e ; t h e y found t h a t t h e c o r r e c t n o t i o n o f t h i c k n e s s o f t h e d i f f e r e n c e between two v e c t o r spaces i s t h e dimension o f t h e v e c t o r space

A

modulo

A

and

B

B.

The d i s c u s s i o n above suggests t h a t i f we want t o examine analogue of above i n we have t o 1 )

L(X),

examine what a r e t h e atomic o b j e c t s o f s t u d y and 2 )

what i s

thick?

the notion o f

The r e s o l u t i o n comes i n examining t h e o b j e c t o f effectively.

Classically, the objects o f

X

b o t h c l a s s i c a l l y and

a r e p o i n t s and i f

X

t h e r e i s no way t o e f f e c t i v e l y l i s t them a l l .

X

i s uncountable,

T h i s a l s o discourages t h e p o s s i b i l i t y Furthermore, s i n c e most con-

o f d e f i n i n g t h i c k t o be i n f i n i t e i n number o f p o i n t s .

s t r u c t i v e approaches i n mathematics e s s e n t i a l l y employ an i d e a e q u i v a l e n t t o des c r i b i n g neighborhoods, we a r e encouraged t o t a k e neighborhoods as o u r addressing units. The n e x t q u e s t i o n i s what t h e n i s t h e n o t i o n o f if we want

as

A'

i s nonempty, t h e r e i s a 6i

with

C &

t e n d i n g so. jects

X,

6 CAo

A'

infinite)?

t o be nonempty.

Clearly,

B u t as soon

and we can f i n d i n f i n i t e l y many

Ai

s

= 0 f o r i # j. T h i s makes A i n f i n i t e w i t h o u t inJ A t t h i s p o i n t however, i t c o u l d be argued t h a t s i n c e a l l of these ob-

and

(ails)

one o b j e c t of

t o be t h i c k we have t o a l l o w f o r

A

thick ( o r

6i

n6.

came from one o b j e c t

and no more.

(6)

with

6 =A,

t h e y should c o u n t o n l y as

T h i s suggests t h a t an i n d i v i s i b l e o b j e c t of

7.

s h o u l d be a connected component o f

A,

a subset

Since i n o u r c o n s t r u c t i o n s we i g -

nore nowhere dense s e t s , i t f o l l o w s t h a t t h e n o t i o n o f connected components o f i s b e t t e r f o r measuring t h e t h i c k n e s s o f A.

fragments o f otherwise.

If

A

A.

The components o f

A'

has i n f i n i t e l y many fragments, i t i s c a l l e d

Note t h a t fragments o f a s e t a r e always open s e t s .

We g i v e two more d e f i n i t i o n s :

Frag(A) =

I

n

if

A

has e x a c t l y

m

if

A

has i n f i n i t e l y many fragments

n

fragments f o r

n < w

3'

are called

thick; thin

I. KALANTARI

84

I

(read fragments of Frag(A,B) =

n

A)

i f exactly with .'8

n

fragments of

have nonempty intersection

A

i f i n f i n i t e l y many fragments of with Bo. (read fraaments of A met by 8) Formally, we may write

n

Frag(A,B) = sup{n/n=O OR

~ E , . . . E ~ ( ~ $6 .

1

(ifj+Ei

= A 08)

fIEj=O)

A

A

[ ( n-< l ) VWi W j [ ( i # j )

(u 2

have nonempty intersection

A

m

A (U

i s open and connected)

A

k j ) l u 3811.

E i

+

And

Frag(A) = Frag(A,X). Frag(A,B) without proving them;

I n t h i s a r t i c l e we use many properties of

C i s a dense subset of

f o r example i f

B

then

Frag(A,B)

=

Frag(A,C).

Techniques

of elementary point-set topology are the only ones needed t o prove the useful f a c t s on Frag(A,B). Next, using the notion of fragment, we give DEFINITION:

i) and ii)

and ii)

a s u b s e t of

is

r.e.

S

Frag(X-S,U)

DEFINITION:

i)

S,

M

open =

r.e.

is

for or

U 3

M

X

r.e.

for

U

and

Frag(X-M, X-U) <

r.e.

r.e.

Frag (X-M,U)

U

nS

f

8.

o p e n , either

X-M

open, e i t h e r <

m

Frag (X-M,X-U) <

or m

Frag(X - M , U ) <

m

m.

r . e . open superset of

U

+

is maximal if

cannot be ' s p l i t ' ( i n the fragmentation

M. That i s X-M i s cohesive. As f o r E ,

i i ) of the above definition i s equivalent t o ii)'

open

open

This s t a t e s t h a t the fragments of sense) by an

U

for

m

a s u b s e t of

M

is s i m p l e i f

X

Major Subsets in Effective Topology

85

S i m i l a r l y , we d e f i n e r - m a x i r n a l i t y as f o l l o w s : DEFINITION:

M M

i) and ii)

a subset of is

if

r.e.

U.

are complementary

1

Frag(X-M,Ui) <

then either

Frag(X-M,U.) <

r.e.

open sets,

Fn

is finite and

or

m

a.

1

DEFINITION:

is r-maximal if

open

and

Ui

X

is a weak array if each

{Fnlncwl

there exists a recursive function m # n

{Fn/ncw) is disjoint if

f

F r7 Lm-l

-t

Fn = Wf(,.).

such that F

LnA

=

b.

Then we have: DEFINITION:

H

and

a subset of is

X

is hyperhypersimple if

1)

H

2)

there exists no disjoint weak array with

i.e.

open

u

Frag(X-H,

F

nru L

n (X-H)

F n,

=

nJ

) =

f o r every

fl

{Fnlnrwl

for which

m

n.

N o t i c e t h a t i n a l l o f t h e above d e f i n i t i o n s , we d i d n o t assume t h a t t h e complement of t h e s e t i n q u e s t i o n has i n f i n i t e l y many fragments. doing so i s t h a t we examine s e t s f o r t h e i r e f f e c t i v e t h i n n e s s . be c l a s s i c a l l y t h i n t h e y a r e c l e a r l y e f f e c t i v e l y t h i n t o o .

The reason f o r I f t h e y happen t o

However, t h e i n t e r -

e s t i n g cases a r e t h o s e w i t h c l a s s i c a l t h i c k n e s s and e f f e c t i v e t h i n n e s s and we w i l l add t h e e x t r a r e q u i r e m e n t when needed. I t i s e a s i l y seen t h a t f o r

i s maximal

a

it'

an open s e t we have

i s r-maximal

U

U

U

',P

i s simple

i s hyperhypersimple

Some o t h e r r e s u l t s which r e q u i r e s p e c i a l c o n s t r u c t i o n s i n c l u d e : THEOREM: S

C

X

(Kalantari-Leggett): with

Frag(X-S)

=

m

There exists an extendible simple s e t

(see

[l])

I. KALANTARI

86

THEOREM:

(Kalantari-Leggett) :

S =X

set

THEOREM:

with Frag(X-S)

(Kalantari-Leggett) :

mal subset of THEOREM:

= m

X

with

X

(see

with

[l])

There exists

Frag(X-M)

(Kalantari-Leggett) :

maximal subset of

There exists a nonextendible simple

M,

a nonextendible maxi-

(see [ 2 ] )

= m

There exists =

Frag(X-M)

M,

a nonextendible r-

which is not maximal

m

(see [ 2 l ) , and in contrast we have: THEOREM: LKalantari-Leggett): therefore maximal subsets

M

There are no extendible r-maximal and of

X

with

Frag(X-M)

=

-.

(See [ 2 1 ) .

A variety of other results of this type may be found in [ l l and

53

[ZI.

MAJOR SUBSETS

In [41 Lachlan introduced the notion of a major subset of an r.e. subset of integers. This was used to show decidability o f some nontrivial sentences (V3). For A an r.e. subset of A , of

w

subset o f

and A-M

with U u A

= w,

w,

M

if M is an r.e.

is a major subset o f A

is infinite, however, whenever U we have w-.(UuM)

is an r.e. subset

if finite. If we interpret this

in Post's terminology of measure, M is a major subset o f A

if A-M

cally thick but effectively it i s almost as thick as A i n covering

In this section we use the notion

is classiw.

o f fragments to define the analogue of

Lachlan's notion for r.e. open subsets of X . DEFINITION:

M

is a major subset o f

1)

M

2)

Frag(A-M)

3)

Ve[Frag(X-(U,uA))

is an

r.e. =

A

if

open subset of

A,

m,

=

3al,

. . .a

[

U a i ) I7 (UeuM) iin

Frag(X-(UeuMu Notice that Frag(B1-B2) = 0 implies that B1 Frag(X-B)

=

0 iff B

is dense in X .

n

U ail) i< n B2

=

=

b

*

01

is dense in B1. Hence

Therefore, intuitively the above definition

Major Subsets in Effective Topology

M

i s that

Ue u A

ever whenever sets

A

i s a m a j o r subset o f

( a l,...,an)

be dense i n

A-M

if

i s dense i n

has i n f i n i t e l y many fragments, how-

i t o n l y takes f i n i t e l y many b a s i c open

X,

Ue u M

d i s j o i n t from

81

t o add t o

Ue u M

X.

t o g e t UeuMu uai i
The i n t e r p r e t a t i o n o f 3 ) i n even a more general s e t t i n g i s t h a t

A

subset o f

if

M

o b j e c t s b u t whenever

c

A,

M

is

Ue u A

r.e.

M

to

i s a major

M misses i n f i n i t e l y many addressable

and

addresses ( a l m o s t ) a l l o b j e c t s o f

X,

Ue

M

needs

addresses o f o n l y f i n i t e l y many independent o b j e c t s t o address ( a l m o s t ) ill o b j e c t s of

X.

T h i s i s c o m p a t i b l e w i t h t h e case o f subsets o f i n t e g e r s and v e c t o r spaces.

I t f o l l o w s t h a t t h e r e l a t i o n o f " i s a m a j o r subset" i s a congruence r e l a t i o n .

Lachlan showed t h a t e v e r y n o n - r e c u r s i v e subset o f

possesses a m a j o r subset and

w

i n LO], t h e a u t h o r d e f i n e d and e x h i b i t e d e x i s t e n c e o f major subspaces f o r noncomplemented ( i n t h e l a t t i c e o f a l v e c t o r space) v e c t o r spaces.

r.e.

v e c t o r subspaces o f an i n f i n i t ' e dimension-

r.e.

L a c h l a n ' s p r o o f uses a p r i o r i t y argument and ex-

p l o i t s the d i s t r i b u t i v i t y o f the l a t t i c e

€.

F o r v e c t o r space case we used. a

p r i o r i t y argument and some s p e c i a l p r o p e r t i e s o f v e c t o r spaces t o overcome t h e d i f f i c u l t i e s i n u s i n g L a c h l a n ' s method ( l a t t i c e o f

r.e.

v e c t o r spaces i s n o t d i s -

tributive). For

L(X),

a d i s t r i b u t i v e l a t t i c e , L a c h l a n ' s method may be a p p l i e d by i n t r o -

ducing and u s i n g t h e n o t i o n o f

However, t h e o b j e c t s o f s t u d y here, t h e

e-state.

I n f a c t , each b a s i c open s e t c o n t a i n s i n -

b a s i c open s e t s , a r e n o t i n d i v i s i b l e .

T h i s makes t h e p r o o f e a s i e r b u t i t i n t r o d u c e s

f i n i t e l y many b a s i c open subsets.

t h e number o f fragments o f a subset o f

X

as a measure o f t h i c k n e s s of t h e subset.

T h i s i s i n t e r e s t i n g p a r t i c u l a r l y i n comparison w i t h c a t e g o r y and measure as measures o f size.

We have

THEOREM:

E v e r y noncomplemented

PROOF:

Let

A

be an

r.e.

open subset o f

We d e s c r i b e t h e c o n s t r u c t i o n o f M,

r.e.

M,

open s e t has a major subset. X

which has no compliments i n

a m a j o r subset o f

A.

I n order t o construct

we s e t some requirements whose s a t i s f a c t i o n produces t h e d e s i r e d

t h e requirements be

L(X).

M.

Let

I. KALANTARI

88

pe:

o

Frag(X-(UeuA))=

lal . . . a n [ ( U a i ) n

+

(U,UM)

i
As usual

pe

such t h a t each

E~ c

A

i # j,

and f o r

Fi

r.

n

MS where we l e t

.uE ~ ) ' .

t h e n e g a t i v e requirements A

ne's

to

subset o f

-.

E ~ =)

E:

Ui,

E;

c E

procedure o f m o d i f y i n g

E~

i

e'

=

0

E~

-

and

Frag(A-M) = i

5

e.

a r e intended

M

i s described i n

A t t h i s stage a l l o f

pels.

M.

T h i s "modifying"

and so t h a t

is a

E;

s t o p s a f t e r f i n i t e l y many t i m e s and t h a t t h e arrange-

Now we d e s c r i b e t h e c o n s t r u c t i o n s f o r m a l l y :

A.

Let

E ~ ' S be

as d e s c r i b e d i n

t h e i n i t i a l stage and b e g i n a simultaneous and d o v e t a i l e d computation of A.

w

We w i l l p r o v e l a t e r t h a t t h i s

ment d e s c r i b e d above does produce a m a j o r subset o f

and

E

by r e p l a c i n g i t by say

We t h e n p u t more i n t o

is done so we s t i l l have Frag(X-(UiuA))

is a

a r e s a t i s f i e d ; however, we do n o t have a m a j o r

I n t h e subsequent stages, we d e c i d e t c i m o d i f y each

ce

@. These The s e t

for

E~

s i n c e we have done n o t h i n g about t h e p o s i t i v e requirements

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

ne

t h e i n i t i a l s t a t e we l e t

Frag(A-M 0 ) = F r a g ( U iEW

Hence,

while

we s e l e c t

SEW

1E W

subset of

uMs.A t

M =

=

J A-M.

t o be t h e f i r s t a p p r o x i m a t i o n t o fragments o f

Mo = ( A -

M)

M).

A d e s c r i p t i o n of t h e c o n s t r u c t i o n i s as f o l l o w s :

of

= 01

i
i s a p o s i t i v e requirement ( f o r c i n g o b j e c t s i n t o

n e g a t i v e one ( f o r c i n g o b j e c t s t o s t a y o u t o f

EL,

A

Frag(A-M) > e f o r each e T

n :

stages

0

u 9) )

u

Frag(X-(UeuM and

=

ui'S

Most f o r m a l l y speaking, t h e r e m a i n i n g stages o f c o n s t r u c t i o n have t o be

d o v e t a i l e d f o r properness, however i n t h e s t y l e o f modern approach, we do n o t become o v e r l y f o r m a l . shrinking t o have

ti's, l i m af

Since we need some n o t a t i o n t o keep t r a c k o f t h e i n c r e d i b l e

we l e t at

denote t h e

ith fragment of

converge f o r each

i. L e t ai = l i m as. s l' t o be a nonempty b a s i c open s e t . T h i s i s so t h a t ai

A-2.

We

will

we i n t e n d t o have each

S

ai

ment o f

becomes t h e

ithf r a g -

A-M.

I n l i g h t o f t h e r o l e p l a y e d by t h e

ails,

arrange

we m o d i f y o u r requirements:

Major Subsets in Effective Topology

and

Pe:

[Frag(X-(UiuA))

Ne:

(a,)'

= 0) A ( i c-e ) ]

+

89

aecUi)

# 0.

We rank t h e requirements:

... .

Po, No, P1, N1, P2,

The c o n s t r u c t i o n progresses

i n stages: STAGE 0:

0

Let

ai

u

Mo = (A-

Let

iE W

=

ci

i

for

S .

w;

and go on t o t h e n e x t stage.

a:)'

Since we need t o know under what e x a c t c o n d i t i o n s we should m o d i f y

af,

we

f i r s t give the following: DEFINITION: E

Pe

requires attention a t s t a t e

s

i f there exist

i

and

such t h a t

We say t h a t

Pe

requires attention

via

i and

E.

Note t h a t c l a u s e s ( 1 ) and ( 4 ) ensure t h e f i n i t e n e s s o f search w h i l e ( 2 ) , ( 3 ) and ( 5 ) a r e e f f e c t i v e l y d e c i d a b l e . a t state s" i s recursive. have

Frag(X-(UiuA))

= 0,

take unnecessary a c t i o n .

Hence whether

Clause ( 2 ) i s s i g n a l i n g t h a t

"Pe

Ui

requires attention i s such t h a t we m i g h t

o f course t h e r e m i g h t be f a l s e s i g n a l s f o r which we However these unnecessary s t e p s w i l l n o t be harmful t o

the r e a c h i n g o f o u r goal ( i t s h o u l d be p o i n t e d o u t h e r e t h a t i f we e l i m i n a t e c l a u s e (2) a l t o g e t h e r , t h e r e i s no r e a l alarm, o n l y t h e number o f unnecessary a c t i o n s increases by a f i n i t e number). i n g a: STAGE S:

Clause ( 3 ) s t a t e s t h a t t h e r e i s a need f o r modify-

w h i l e c l a u s e ( 5 ) i s o f f e r i n g a replacement f o r I f no

Pe

requires attention, l e t

and go on t o t h e n e x t stage.

Otherwise l e t

as = a:-'

ae.

f o r each .im,

MS = M S - l

I. KALANTARI

90

e = smallest e S some E ,

such t h a t

Pe

i

such t h a t

Pe

and

S

E~

= smallest

i

= smallest ( i n Godel

no.)

s via i S and

stage

requires attention via some i requires attention via

S

such t h a t

E

P

eS

i

and

and some

E,

requires attention a t

E.

Then s e t

Finally l e t MS = MS-l u

-

( E ~

Observe t h a t each af f o r each

i >_ e .

fragments of

a:)".

i s effectively defined and t h a t

af c

E~ c

U.

lS

n A

This modification means t h a t there are s t i l l i n f i n i t e l y many but, they are almost a l l a subset of

A-MS

U. ' S

.

Ui

Hence, we have a l -

S

most a l l fragments of

t o be covered by

u

i EU

A-MS ( i . e . a l l except possibly f o r the f i r s t

.

es

fragments)

This completes the description of the construction.

order t o prove t h a t the above produces a major subset of

In

we s t a t e and prove

A,

the following p a i r of lemmas which clearly imply the theorem. LEMMA A:

set of

LEMMA B:

If t h e requirements a r e s a t i s f i e d , then

M

i s noncomplemented, t h e n f o r e a c h

e

A.

If

A

and

1)

Pe

i s satisfied

2)

Ne

is satisfied.

PROOF OF LEMMA A : all

Assume t h a t the requirements

E

w

w e have:

are s a t i s f i e d for

Pe

and

Pe

i s s a t i s f i e d f o r every

Ne

e. F i r s t suppose

e

i s a major sub-

U. u A 1

i s dense i n

2 i . in p a r t i c u l a r , i t follows t h a t

f a i l s t o include

o0, al,...,a

e- 1

X.

ae

c

Since Ui

a t the most.

for all

e

2

i.

Hence

Ui

u M

This implies clause 3 ) of the

Major Subsets in Effective Topology

91

definition holds.

7 a-

i s s a t i s f i e d and n = 0 f o r i # j , (because a t e J the f i r s t stage of and during the construction we arranged Ef n Es = 0 f o r i # j ) Next, since every

Frag(A-M)

we have

N

Frag(

=

u ai) i

j

=

m.

Therefore M

i s a major subset of

0

A.

EU

We show t h a t ( 1 ) and ( 2 ) h o l d by a n induction argument on e .

PROOF OF LEMMA B :

i s noncomplemented and

Assume A

t h a t Pe

Pi

and .Ni

are s a t i s f i e d f o r

i s s a t i s f i e d notice t h a t there are few cases f o r

i < e.

yi

Pi; either

TO see i s sat-

is Pi I n the f i r s t case, Pi

isfied because i t s antecedent i s f a l s e and i t never requiresattention, o r satisfied because i t requires attention and receives i t . causes no alarm and no action.

I n the second case,

tion via

j

j

and some

with

E

5

i.

requires attention via

the same, i t will never require attention via

Pi

i s satisfied for j

5

i,

must have required atten-

An inspection of the definition of

requires attention" reveals t h a t once P i

nitely many integers via which

Pi

j

again.

j

for al l

j

P i could require a t t e n t i o n , we conclude t h a t once i t will never require attention via j again. Hence,

Is s a t i s f i e d f o r a l l

5 i , and N i

so = smalles stage a f t e r which no requires attention,

Therefore

S

lim af = a i o = a i

and i t receives

j

Since there are only f i -

for the induction argument our assumption i s t h a t f o r a l l via

.

"Pe

(ai)' #

with

Pi

i < e,

i < e.

or N i

0 for all i

with

Pi

i s satisfied

Hence l e t i < e

< e.

S

Note t h a t i f the antecedent of

Frag(X-(U. u A ) ) = 0 and j < e. stage beyond

J so,

so

Pe-l

i t must be t h a t

Next, consider

K = (Ue

that i s

at

e

-

u

j < e,

then since Pe-l

never requires attention via

i s satisfied a t j

a t any stage

c

i s also s a t i s f i e d f o r j < e

i s not s a t i s f i e d via

If

U. f o r any i 2 j and any s 2 so. Hence Pe J and i t will n o t require attention via any j < e.

and therefore af

Therefore suppose j = e ,

i s f a l s e we are done, hence suppose

Pe

so,

i <e

a : ' ) .

Frag(X-(Ue u A)) = 0.

t h a t i s ai5O # U e Clearly K

Further suppose

Pe

(otherwise we would be done). i s an

r.e.

open subset of

X.

I. KALANTARI

92 Observe t h a t

Frag(X-(KuAu

F r a g ( X - ( U e u A ) ) < -). i t and

Since

A

K must i n t e r s e c t A;

AS1

n':

and

F

n

K n A # 0.

that i s

(i<eu

Therefore

so,

(Ue-A)O K

=

0 and s1 2 so.

K

c

and

cannot be a complement f o r

Since

K n

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

q u i r e a t t e n t i o n a t some stage l a t e r t h a n s a t i s f i e d by stage

(because

i s noncomplemented,

# 0.

f o l l o w s t h a t Kn(A-ua:')O i<e E c

ago)) = 0

ice

(Ua;')= i<e

and

E

s1

This implies t h a t

0, i t

such t h a t Pe w i 1 r e -

Since P . ' s f o r i < e a r e a l l 0' 1 w i l l r e c e i v e a t t e n t i o n a t t h a t stage s1 v i a e.

Pe

s

w i l l be s a t i s f i e d permanently and p a r t ( 1 ) o f t h e lemma f o l l o w s .

Pe

To see t h a t p a r t ( 2 ) holds, i t i s s u f f i c i e n t t o n o t i c e t h a t a t t h e f i r s t stage that

Pe

w i l l be d e f i n e d which

r e q u i r e s a t t e n t i o n ( i f i t does a t a l l ) , a new a :

i s a nonempty b a s i c open s e t .

The c o n s t r u c t i o n arranges f o r t h i s

t o never be

a:

m o d i f i e d again.

l i m a: = at e S

Therefore

where

[ so

if

pe never r e q u i r e s a t t e n t i o n

s1

if

pe

t =

requires attention a t

s m a l l e s t stage Thus p a r t ( 2 ) i s a l s o s a t i s f i e d

sl.

0.

T h i s completes t h e proof of t h e theorem 0. Next we have: COROLLARY:

PROOF:

There exists a nonmaximal, r-maximal set.

Let

A

be a maximal s e t (see [ 2 ] ) w i t h

ment i s a b a s i c open s e t . let

Since

M be a m a j o r subset o f

A.

A

Frag(X-A) =

and e v e r y f r a g -

i s noncomplemented, i t has a m a j o r subset,

Claim

of a c o n t r a d i c t i o n assume o t h e r w i s e .

M i s r-maximal.

E L(X) with J u. n U. = 0, Ui u U . dense i n X, and Frag(X-M,Ui) = Frag(X-M,U.) = m. Also, 1 J J J s i n c e M i s a m a j o r subset o f A , we must have e i t h e r Frag(A-M,Ui) = or

Frag(A-M,U.)

J

=

Next, c o n s i d e r

m

( o r both). Frag(X-A,Ui)

Then t h e r e e x i s t

To see t h i s , f o r sake

Ui.U.

W i t h no l o s s o f g e n e r a l i t y , assume and

Frag(X-A,U.); J

since

Frag(A-M,Ui)

Frag(X-A) =

m,

one o r

=

m.

93

Major Subsets in Effective Topology both a r e i n f i n i t e . Frag(X-A,AuUi)

Frag(X-A,Ui)

=

This contradicts maximality o f say t h e fragments a r e

u a,))

Frag(X-(UiuMu

k
M

fact that

as w e l l as

A.

I f , say,

5 Frag(X-M,U.- u a k ) k>n

i s a m a j o r subset o f

A u Ui.

Here we have

Frag(X-A,X-(AuUi)) Frag(X-A,U.)

1

t h e n we have

al,...an,

This completes t h e p r o o f .

14

-

Ifb o t h a r e i n f i n i t e , t h a n c o n s i d e r

=

=

Frag(X-(UiuAu

-.

= Frag(X-A,U.) =-.

J Frag(X-A,U.)<m, J u a , ) ) = 0 while k
m

and

This i s a contradiction t o the

A.

0

REMARKS: If

X

i s a m e t r i c space, one can f u r t h e r a r r a n g e f o r fragments o f

U

to

have c e r t a i n m e t r i c measure p r o p e r t i e s ( f o r example, t o t a l measure fragments of be l e s s than, o r g r e a t e r than, a r e a l number a ) .

I n t h i s l i g h t , one can c o n s t r u c t

any o f t h e p r e v i o u s s e t s ( a l s o see [l]and 121) w i t h fragments b e i n g of c e r t a i n size.

T h e r e f o r e t h e i d e a o f t h e " e f f e c t i v e addressing

-

ability",

i f t a k e n as a

measure, mixes w i t h t h e m e t r i c measure somewhat s i m i l a r t o t h e way measure and category m i x . F i n a l l y , t h e r e a r e many open q u e s t i o n s which a r i s e ; f o r example: 1)

I s the l a t t i c e

2)

What o p e r a t i o n , i f any, on automorphisms o f

3)

L(X)

decidable? X

or

A

generates a l l

L(X)?

How many automorphisms does

L(X)

U

have?

I. KALANTARI

94

REFERENCES

I . K a l a n t a r i , Major subs aces o f r e c u r s i v e l J o. u 3r n 0a l 3o -f Symbolic 3 9 ; l & 11 1 I21

I . K a l a n t a r i and A. L e g g e t t S i m l i c i t The J o u r n a l o f Symbolic Logic-

, Maximality

~

~

enumerable v e c t o r spaces. e f f e c t i v e t o p o l o g y ( t o appear i n

e f f e c t i v e t o p o l o g y ( t o appear i n JSL).

& effective

[31

I . K a l a n t a r i and J. Remnel, Degrees o f u n s o l v a b i l i t y ( t o appear i n JSL).

141

I . K a l a n t a r i and A . R e t z l a f f , R e c u r s i v e c o n s t r u c t i o n s & t o p o l o g i c a l spaces, The J o u r n a l o f Symbolic L o g i c , v o l . 44 (1979), pp. 609-625.

topology

A. H. Lachlan, On t h e l a t t i c e of r e c u r s i v e l enumerable sets, T r a n s a c t i o n s o f t h e American M a t h e m a t i G l T i F i e t y d ( l 9 6 8 ) , pp. 1-37.

Les

D. Lacombe, ensembles r 6 c u r s i v e m e n t o u v e r t s ou f e r n & , e t l e u r s a p p l i c a t i o n s a l ’ a n a l se r e c u r s i v e ,Comptes Rendus Hebdomadaires des Seances de ~ d ~ m d i G E F ( - l % r i s ) v, o l . 245 (1957), pp. 1040-1043; i b i d . V O l . 246 (1958) 9 pp. 28-31. v e c t o r spaces, &enumerable 147-171.

G. Metakides and A. Nerode, Recursive1 Mathematical L o g i c , v o l . 11

Annals o f

E f f e c t i v e c o n t e n t o f f i e l d t h e o r y , Annals o f Mathematical L o g i c , -v o l . 17 (T479),8289-320. E. L. Post, R e c u r s i v e l y enumerable s e t s o f o s i t i v e i n t e g e r s and t h e i r d e c i s i o n roblems, B u l l e t i n o f t h e American’MathematicaI S o c i e t y , v o l . 50 m p ; . 284-316. 1101

E. Specker, Nicht k o n s t r a k t i v beweisbare Satze d e r A n a l y s i s . Symbolic Logic, v o l . 14 (1949), pp. 145-148.

The J o u r n a l o f

PATRAS LOGIC S YMPOSION G.Metakides (ed.) 0North-Holland Publishing Company, 1982

95

EFFECTIVE ISOKORPHISMS OF ALGEBRAIC STRUCTURES Edward F. Eisenberg1 Department o f Mathematics

2 Jeffrey B. Remmel Department o f Mathematics University o f California, San Dieao La J o l l a , C a l i f o r n i a 92993

Vassar College Poughkeepsie,N.Y.

14607

INTRODUCTION Algebra and model theory a r e concerned w i t h p r o p e r t i e s which are preserved under isomorphisms.

I t i s n a t u r a l f o r a r e c u r s i o n t h e o r i s t t o con-

s i d e r p r o p e r t i e s preserved under e f f e c t i v e isomorphisms.

This has been done

f o r sets by Dekker, M y h i l l , Nerode, E l l e n t u c k , and others; f o r l i n e a r orderings by Crossley and Manaster; f o r v e c t o r spaces by Qekker and Hamilton; and f o r groups by Dekker, Hasset, Applebaum, and E l l e n t u c k .

A general model

t h e o r e t i c treatment i s found i n Crossley and Nerode's monograph [31. However, t h e r e have been several notions o f " e f f e c t i v e isomorphism" t h a t have appeared i n t h e l i t e r a t u r e .

For example, t h e b a s i c approach i n Crossley-

Nerode [31 i s t o s p e c i f y a c e r t a i n f i x e d r e c u r s i v e l y presented model and study t h e p r o p e r t i e s o f t h e substructures o f under e f f e c t i v e isomorphisms.

U

U which a r e preserved

U can be t h e n a t u r a l numbers N under

E.g.,

e q u a l i t y , t h e r a t i o n a l s Q under t h e usual ordering, an i n f i n i t e dimensional r e c u r s i v e l y presented v e c t o r space group

P(o)

+

over a r e c u r s i v e f i e l d

o f a l l f i n i t e permutations o f

Boolean algebra f: A

Vm

-

Q.

B, we say f

Given s t r u c t u r e s i s eddeuXue i f

N,

F,

the

o r t h e r e c u r s i v e atomless

A, B 5 U and an isomorphism

f

i s e x t e n d i b l e t o a one-one p a r t i a l

'Some o f t h e r e s u l t s o f t h i s paper were based on m a t e r i a l presented i n the f i r s t a u t h o r ' s d o c t o r a l d i s s e r t a t i o n w r i t t e n a t C o r n e l l U n i v e r s i t y under the d i r e c t i o n o f Professor A n i l Nerode. * P a r t i a l l y supported by NSF Grant

E. EISENBERG and J. REMMEL

96 recursive function.

T h i s i s t h e d e f i n i t i o n o f " e f f e c t i v e isomorphism" t h a t was

used by Hasset f o r groups and C r o s s l e y and Nerode [3] i n t h e general model theor e t i c setting.

A second n o t i o n o f " e f f e c t i v e isomorphism" was used by C r o s s l e y

121 f o r o r d e r i n g s and Dekker [4] f o r v e c t o r spaces which we s h a l l c a l l i n t h i s T h a t i s , we say an isomorpnism

paper f u l l y e f f e c t i v e .

i f t h e r e a r e r e c u r s i v e l y enumerable s u b s t r u c t u r e s a p a r t i a l r e c u r s i v e isomorphism

f ' : A'

+

f: A

A' 5 A

8'1 B

and

which extends

€3'

8 i s 6LLeey e 6 6 e d v e

-+

of

and

U

f.

The main purpose o f t h i s paper i s t o show t h a t t h e s e two n o t i o n s o f " e f f e c t i v e isomorphism" a r e d i f f e r e n t i n answer t o a q u e s t i o n r a i s e d by C r o s s l e y and Nerode i n [3].

I n s e c t i o n 1, we s h a l l g i v e some general c o n d i t i o n s on a r e c u r s i v e -

l y p r e s e n t e d model

Bc U and an isomorphism f : A effective.

+

B such t h a t f

i s an e f f e c t i v e b u t n o t f u l l y

I n f a c t , we show t h e r e a r e c o n t i n u u m l y many such

o f t h e examples o f numbers

N

U

under e q u a l i t y , s a t i s f y these g e n e r a l c o n d i t i o n s .

as above so t h a t

U

s p e c i f i e d above, w i t h t h e e x c e p t i o n o f

F

i s f i n i t e , s a t i s f y the stronger conditions. F

B, and

All

f.

and

I n s e c t i o n 2, we

A, €3, and

U which a l l o w s us t o c o n s t r u c t

f

A

A,

s p e c i f i e d above, w i t h o f course t h e e x c e p t i o n o f t h e n a t u r a l

s h a l l g i v e some s t r o n g e r c o n d i t i o n s on

the f i e l d

A,

U which a l l o w s us t o c o n s t r u c t i s o l a t e d s u b s t r u c t u r e s

B a r e cosimple.

We show t h a t a l l o f t h e examples of

N and

,V

where t h e u n d e r l y i n g f i e l d

We n o t e whether o r n o t

V,

where

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

s e c t i o n 2 remains open.

A l s o , f o r many examples f o r which t h e r e s u l t s o f s e c t i o n s

1 and 2 apply, we s h a l l show t h a t we can c o n t r o l t h e isomorphism t y p e s o f

A

and

B as w e l l . There i s y e t a t h i r d n o t i o n o f " e f f e c t i v e isomorphism" which has appeared i n t h e l i t e r a t u r e w h i c h r e l a t e s t o o u r work here.

T h i s t h i r d n o t i o n a r i s e s when we

do n o t r e s t r i c t o u r s e l v e s t o s u b s t r u c t u r e s o f a f i x e d r e c u r s i v e l y p r e s e n t e d model. We

i l l u s t r a t e t h i s p o s s i b i l i t y w i t h a d e f i n i t i o n suggested by Dekker f o r groups

An

which f i r s t appeared i n p r i n t i n t h e work o f h i s s t u d e n t Applebaum 111.

w-gmup

G

as d e f i n e d i n [7] i s a t r i p l e

(A.p,i)

where

A

5N

and

P

and

i

a r e r e s p e c t i v e l y t h e p r o d u c t and i n v e r s e o p e r a t i o n s o f t h e group and a r e e x t e n d i b l e

Effective Isomorphisms o f Algebraic Structures t o p a r t i a l recursive functions. and

p

and

i s s a i d t o be an h . e . ghoup i f

G

i are p a r t i a l recursive.

an isomorphism f : G1

+

Now, g i v e n two w-groups

t h e n as above we say

G2,

91

f

morphism 7:

cl

-+

F,

G2 e x t e n d i n g

and G1 f . Note i f 3

f i x e d r e c u r s i v e l y presented groups l i k e

c22 G2 G1

f

if

and

G2

groups

t o be

cl

and

our c o n s t r u c t i o n s o f s e c t i o n s 1 and 2 w i l l produce subgroups

and

G

I n fact, G2

of

P(w)

Now i n [13], t h e second a u t h o r produces an w-group

G which i s n o t c o n t a i n e d i n any r . e . +

G1

c2

which i s s t r o n g l y e f f e c t i v e and hence e f f e c t i v e b u t

G2

which i s n o t f u l l y e f f e c t i v e .

I:G

i s ex-

f

P(w).

-+

f

and

were subgroups of some

i n t h e d e f i n i t i o n o f s t r o n g l y e f f e c t i v e need n o t be subgrouPs o f

f:G

G2

and a p a r t i a l r e c u r s i v e i s o -

s t r o n g l y e f f e c t i v e b u t n o t f u l l y e f f e c t i v e because t h e r.e.

and an isomorphism

r.e.

i s GtongLy e66ec-

i t i s possible f o r

P(w),

is and

G1

i s e,46ectiue

t e n d i b l e t o a one-one p a r t i a l r e c u r s i v e f u n c t i o n , b u t we say

Live i f t h e r e a r e r . e . groups

A

group.

I t follows that the i d e n t i t y

i s an example o f an e f f e c t i v e isomorphism which i s n o t s t r o n g l y e f f e c t i v e .

Thus f o r w-groups t h e r e l a t i o n s h i p between these t h r e e n o t i o n s of " e f f e c t i v e

iso-

morphism" may be summarized by t h e diagram below. f u l l y effective -strongly

effective -effective

+

+

F i n a l l y , we s h a l l see t h a t an analogous s i t u a t i o n h o l d s f o r l i n e a r o r d e r i n g s , v e c t o r spaces, and Boolean a l g e b r a s .

§I. EFFECTIVE BUT NOT FULLY EFFECTIVE ISOMORPHISMS. The purpose o f t h i s s e c t i o n i s t o g i v e a general s e t t i n g i n which we can show t h a t t h e r e e x i s t isomorphisms

f

which a r e e f f e c t i v e b u t n o t f u l l y e f f e c t i v e f o r

t h e models mentioned i n t h e i n t r o d u c t i o n .

We must s t a r t w i t h some n o t a t i o n .

The

r e c u r s i o n t h e o r y we need i s based on [14] and we r e f e r t o [14], f o r any u n e x p l a i n e d definitions. t i o n s and tions.

Let

$o,$,,...

We w r i t e

xo,xl

,...

be an e f f e c t i v e l i s t o f a l l p a r t i a l r e c u r s i v e f u n c -

be an e f f e c t i v e l i s t of a l l one-one p a r t i a l r e c u r s i v e func$(n)+($;(n)+)

i f after following

f e c t i v e procedure t o c a l c u l a t e xi(n)($i(n)), xi(n)+($i(n)+)

if

s

or l e s s steps o f t h e e f -

we g e t an o u t p u t .

We w r i t e

E. EISENBERG and J. REMMEL

98 3s(xt(n)c)(~s($f(n)J)). Thus

l,...

Wo,W

f u n c t i o n and A = {x

2x1 + 2"

A'

c

A,

Let

Tn

Tn

for

2

n

u

u

if

gr

T:Tn

If

and t h e o r d e r i n g

ij E { l , . , . , n l

fl w i l l a l s o

T.

denotes t h e f u l l w-branching t r e e c o n s i s t i n g of

t. If u = (il,...,ik) E T (T ) n w

k n ( u ) = k.

If

is

i s an i n i t i a l subsequence o f

...,ik)w i t h

and w r i t e

is a

denote t h e f u l l n - a r y b r a n c h i n g t r e e which c o n s i s t s of

2

To a = (il,

a l l f i n i t e sequences

B

+

A'. A

& ~f(x)+)l.

fl i s 0.

u = (i l,...,ik) where each

i s g i v e n by

N.

If f : A

restricted t o

f

and t h e c a n o n i c a l i n d e x o f

denote t h e empty sequence.

+

for

W t = {xlxcs

and

i s a f i n i t e s e t , t h e c a n o n i c a l i n d e x of

a l l f i n i t e sequences on

frA'

we w r i t e

... + 2'n

+

xi

= domain of

Wi

i s an e f f e c t i v e l i s t of a l l r . e . s e t s .

< xnl

1

We l e t

we say

u

i.E N-frl] f o r a l l j o r d e r e d under J i s a node a t l e v e l k of Tn(Tw)

By a n-any bfinnckivlg nwnbeh &ee

a = (il,...,ik) and

j

E

N-to?,

T

u*j

then

we mean a map

w i l l denote

,. . .,ik,j),

(i,

Next we need t o s p e c i f y o u r general s e t t i n g .

Our s e t t i n g i s s l i g h t l y more

general t h a n t h a t o f C r o s s l e y and Flerode [3]. M

i s s a i d t o be a hecwldiwc model

o v e r a r e c u r s i v e language

bl,lMI,

of

N

L

i f the universe o f

and t h e a t o m i c s a t i s f a c t i o n r e l a t i o n

s i v e where $ formula of

L.

I=

$(ao

A, 8 5 M

,...,a n-1 1

DEFINITION 1.1

syotem

guage L P(IMI),

1MI

U

(e.a.c.s.) and into

and

M"

in

@ @

i s recur-

i s an atomic

Throughout t h i s paper we assume a l l languages c o n t a i n e q u a l i t y . over

o n t o and f o r a l l atomic formulas A

satisfies

i s a f i n i t e sequence o f elements of

Given s u b s t r u c t u r e s

IAl,

It$

i s a r e c u r s i v e subset

ck

=

iff

(M,ck)

if

M

B

L,

$(vo

I=

f: A

,...,v

$(f(%),

+

B i s an isomorphism i f f i s

~ - ~o )f

L

and a l l

a.

...,f(an-l)l.

in

i s s a i d t o be an eddective atomic &o&te i s a r e c u r s i v e model over a r e c u r s i v e l a n -

i s a map from t h e s e t of a l l s u b s e t s of

P(IM1)

,...,a n-1

such t h a t f o r a l l

A,

B 5 IMI

IMI,

Effective Isomorphisms o f Algebraic Structures

99

(i)A 5 & ( A ) . (ii) (iii) (iv) (v) (vi)

x

&(A)

E

cp.(A)

i f f t h e r e i s some f i n i t e s e t

i s a s u b s t r u c t u r e of

A

If

If

A

,...,c n1)

c1

,...,f ( c n ) l ) .

x

E

c!Z(A').

L.

,...,c n

-+

M,

in

= cp.(F)

B i s an isomorphism, and then

f o r some f i n i t e s e t

F,

then

E.

f o r some f i n i t e s e t

i s e f f e c t i v e on f i n i t e s e t s , t h a t i s , t h e r e i s a r e c u r s i v e f u n c t i o n such t h a t i f

f

f o r some

B a r e c l o s e d s e t s and B

c

A = c!Z(E)

cp.

M over

such t h a t

A' C A

B a r e s u b s t r u c t u r e s o f M, f: A

and

= Cp.(lf(C,)

B

(viii)

5 cll(B).

cR(cL(A)) = cL(A).

A = cp.(lcl

(vii)

&(A)

implies

A 5B

for a l l

Y

E

N,

A 5 1M1.

x

i s a c a n o n i c a l index of a f i n i t e s e t 1 i f y E cL(A) @f(x)(~)= o i f y cp.(~)'

then

.+

EXAMPLES Sets: cll(A)

M be t h e n a t u r a l numbers under e q u a l i t y

Let =

A

for all

(N,=)

and l e t

A 5 N.

Dense unbounded l i n e a r o r d e r i n g s :

Let

M =

(or<>

be t h e

Q

rationals cp.(A) = A

( p r o p e r l y GBdel numbered) under t h e usual o r d e r i n g snd l e t

for all

A LQV e c t o r spaces: cursive f i e l d

Let F.

Vm

be an i n f i n i t e dimensiona

( F may be f i n i t e o r i n f i n i t e . )

v e c t o r space o v e r a r e The language has unary

o p e r a t i o n symbols f o r s c a l e r m u l t i p l i c a t i o n f o r each element o f b i n a r y o p e r a t i o n symbol f o r v e c t o r a d d i t i o n , t h u s t o say

Vm

F

and a

i s a recursive

model means t h e o p e r a t i o n s o f s c a l e r m u l t i p l i c a t i o n and v e c t o r a d d i t i o n a r e p a r t i a l recursive.

Moreover, we assume

l a t i o n , t h a t i s , g i v e n any , , ,a all

...,an-1 A 5 Vm.

a r e dependent. cp.

ao,...,an-l &(A)

Vm E

Vm

has an e f f e c t i v e dependence r e we can e f f e c t i v e l y decide i f

equals t h e subspace generated by

i s e f f e c t i v e on f i n i t e s e t s because

dependence r e l a t i o n .

V

A

for

has an e f f e c t i v e

100

E. EISENBERG and J. REMMEL

4) Permutation groups: Let M be a proper Gddel numbering of of all permutations f: N

+

N such that Ixlf(x) #

usual functional composition. c!.(A)

XI

the set

is finite, under the

equals the subgroup generated by A for

-

all A 5 P(o). 5)

P(u),

Boolean algebras: Let M be a proper GSdel numbering of Q, the Boolean algebra generated by the left-closed right-open intervals o f language has operation symbols for

A

(meet),

v

Q where the

(join), and -,

(complement).

cilfA) equals the subalgebra generated by A. 6) n-dimensional partial orderings:

Let Dn be a recursive subset of Qn which

is dense in the usual product topology and is such that no two elements of Dn have a coordinate in common. Given x = (xl ,..., xn) and y = (y,,...,yn) in i.

Thus <

is the usual product ordering. We are most interested in the model

(on,<>

On define x ci y if xi

<

yi and x < y if x < 1. y for all

which is the model companion of n-dimenstional partial orderings and has the property that every recursive n-dimensional partial ordering is isomorphic to a recursive subset of Dn under <.

(See [9,10]for details.)

However, for

the purposes of Theorem 1.1 to follow it will be simpler to consider the model M = (On,<, <1

If

&(A)

=

U =

,...,cn )

where cR(A)

=

A for all A 5 On.

(M,c!L) is an e.a.c.s. and A 5 1M1, we say A is a cloned if

A, A is &hLte-cloned if A

=

for some finite set A' 5 I M ! ,

cQ(A')

A is indinite-closed if 'A is closed but not finite-closed, and A is Dede.kind if A is closed and A contains no infinite r.e. set. We note that by condition (vi) of our definition of e.a.c.s, the properties of being finite-closed or in-

finite-closed are preserved under isomorphism. By condition (vii), a finite-closed set can not contain an infinite-closed set. We shall assume for the rest of this paper that if

U =

(M,cf.]

is an e.a.s.c.,

then / M I

closed set. Note that by our assumptions on

U,

=

and [ M I

El

is an infinite-

if A 5 M is finite-closed

then A is recursive although in some examples it may be infinite. Nevertheless, it is clear that we can effectively list all triples I

=

($,A,B)

where

$

is a

One-One partial recursive function and A and 8 are finite-closed sets of M. Let

101

Effective Isomorphisms of Algebraic Structures

0 0 0 I. = (+ ,A ,B >, I 1

(01 .A 1 ,B 1 ) ,... be such an effective l i s t . When we say

=

we can "effectively find" f i n i t e closed s e t s

f: A

-+

of

A, B

M and an isomorphism

B we mean we can effectively find n such t h a t I n

be called an i n d e x of

A, B

n will

(f,A,B).

Next we specify some conditions on closed s e t s

.

= (f,A,B)

U which we ensure t h a t we ca'n construct

M and an isomorphism f : A

of

B which i s effective b u t

+

not f u l l y e f f e c t i v e . DEFINITION 1 . 2 (i)

has t h e

U

i f f o r any f i n i t e c l o s e 6 s e t s such t h a t

f: A

+

set

A 5 IMI

B

f

such t h a t

f l : Ai

Bi

+

B 5 b(

A,

A n (A u

=

B)

for

i

=

1,

...,n

for a l l

(b)

A n (U(Ai

(c)

ui

(d)

t h e r e does not e x i s t c l o s e d s e t

B'

i

f'

11

3

we c a n e f f e c t i v e l y

A,

Bi

2

8,

and isomor-

such t h a t

and

fi

f

3

Bi)) = 0

i s a one-one f u n c t i o n , and

fi

3

Ai

6,

A ~ P * * * Y A , YB 1 * - - - * B n

(a)

i,

and isomorphism

i s p a r t i a l r e c u r s i v e and any f i n i t e

find finite-closed sets phism

(n 2 2 )

n - i n c o m p d b e e Oomohpkinm externion pnopehty

ui

extends

3

f ' : A'

and a n isomorphism

Bi

A'

ui Ai +

and such t h a t

8'

fi. 1

(ii)

h a s t h e OLXOng n-incompatible L5ornohpkidm extenhion phopwLQ

U

g i v e n any C

3

(e)

A

A,

B. f ,

there exists for a l l that

i,

gitA

and Ai,Bi,

A

if

a s i n ( i ) and any f i n i t e c l o s e d and

fi

satisfying (a)-(d) plus

t h e r e i s a n isomorphism

gi:

Ai

+

C

such

is the identity.

EXAMPLE S

We note t h a t in the examples below where we prove the strong n-incompatible isomorphism extension property holds, f o r simplicity, we will assume C the form

cX(A u f c ) ) f o r some c

E

i s of

IM(. I t will be c l e a r however t h a t our

102

E. EISENBERG and J. REMMEL

proofs e a s i l y extend t o cover any f i n i t e - c l o s e d s e t 1.1)

Of course i n t h e s e t case,

U = ((N,=),

C

3

A.

does n o t s a t i s f y even t h e n-

ct)

incompatible isomorphism extension orooertv f o r any 1.2)

n.

U = ( ( Q , < ) , Ck) has t h e strona 2 - i n c o m ~ a t i b l e isomorohism extension orooerty. Let A

A = fa,<

...
...
= {bl<

B

be any f i n i t e s e t d i s j o i n t form c < an,

c < al,

or

and

A u B

ai < c < ai+l.

Say

f(ai)

= bi

for

and l e t

c

Q-A.

E

a. < c < ai+l.

i 5 n.

Let

Now e i t h e r

Since the

1

r a t i o n a l s a r e dense, i t i s c l e a r t h a t we can e f f e c t i v e l y f i n d d i s t i n c t not i n A

x1,x2,y1,y2

Let

bi < y1 < y2 < bi+l.

B2

=

Ai,

u {yll,

B Bi,

and

such t h a t

ai < x1 < x 2 < ai+,

fi

i = 1,2

for

A2 = A u 1x23,

A1 = A u {xll,

fl = f u I(xl,y2)),

and

and

B1 =

f 2 = f u {(x2,yl)l.

B

u

cy21,

It i s clear

s a t i s f y c o n d i t i o n s ( a ) - ( c ) and ( e ) .

For

c o n d i t i o n ( d ) , note t h a t t h e r e can be no isomorphism which can extend both fl C

1.3)

and

< al

Vm

f2 since c > an

and

< x2

x

y, = f 2 ( x 2 ) < fl(xl)

but

= y2.

The cases

a r e handled s i m i l a r l y .

has t h e strong 3-incompatible isomorphism extension orooPrt.v. L e t

and

f: A

B be n-dimensional subspaces,

A u B.

be any f i n i t e s e t d i s j o i n t from

-+

A

B be an isomorphism, and A

Since

Vm

i s i n f i n i t e dimensional

and has an e f f e c t i v e dependence r e l a t i o n , i t i s c l e a r t h a t we can effectivel y find

al,a2,bl,b2.b3

which are independent over

A1 = c t ( A u { a l l ) ,

B.1

= c t ( B u {bil)

A2 = cR(A u {a2]),

i = 1,2,3.

for

be the isomorphisms such t h a t and

f3(al+a2)

= b3.

No isomorphism g

1.4)

C where C

= f

fi: Ai and

+

fl(al)

Bi

for

= b,,

fl u f 2 u f 3 s i n c e

Finally, i f

g j : Ai

+

gl(al)

= g2(a21 = g3(al+a3)

E

P(w),

and i = 1,2,3

to

f 2 ( a 2 ) = b2,

c

= cR(A u { c l )

4

A,

fl(al)+f2(a2)#f3(al+a2)

i t i s c l e a r there e x i s t s isomorphisms

required f o r condition

( e ) such t h a t

= c.

P(o) has the 2-incompatible isomorphism extension property. a

B). L e t

I t i s easy t o check t h a t c o n d i t i o n s ( a ) - ( c ) hold.

can extend

so t h a t ( d ) holds.

A3 = cR(A u la,+a21),

Define

firAi

cR(A u A u

l e t ~ u p p o k t0 6

G i s a f i n i t e subgroup,

a,

supp(a),

supp(G) =

u acG

equal supp(a)

In

E

Given

N l o ( n ) # n).

Note if

i s f i n i t e and can be effec-

103

Effective Isomorphismsof Algebraic Structures t i v e l y found.

Now l e t

A 5 P(u)

isomorphism, and in

(ai,a.)

be a f i n i t e s e t d i s j o i n t from

N

and

fi: Ai

f,((a,,a2))

+

Bi

= (al,a2),

for all

i = 1,2

for

and

(a ,a )o(a3,a4) 1 2

(al ,a2)o(al

whose atoms are = bi

a

Let

4

t o be the isomorphism such t h a t

f 2 ( ( a 3 , a 4 ) ) = (al,a4). No isomorphism

g

has order two w h i l e

= f,

fifA

Again i t i s easy t o

fl

can extend both

and

f2

f((al ,a2))~f((a3,a4)) =

has the strong 2-incompatible extension property.

),cR)

q

be a subalgebra o f bl,

i = 1,

for

Suppose c 1 A

whose atoms are

...,bn '

f: A

...,n,

and

A

...,an,

al

are p r e c i s e l y

Note t h a t f o r each x 5 ai,

ai

-

c

and

x

4 Cy,

ai-yil,

It e a s i l y f o l l o w s t h a t f o r any f i n i t e s e t

x 5 ai

e f f e c t i v e l y f i n d a nonzero

A

,...,a.J

t h e r e a r e i n f i n i t e l y many

y 5 ai,

be a subalgebra o f

B

be any f i n i t e s e t d i s j o i n t from A u 8 .

c,al

A

Let

be an isomorphism such t h a t

B

-+

al,

and we have enumerated t h e atoms o f

C = cR(A u {c))

and i f

Find

A u B.

has order three so c o n d i t i o n ( d ) holds.

,a4)

((G,v,A,-,

f(ai)

be an

[ai,a.l. L e t A1 = ca(Aul(al,a2)l), J and B2 = cL(8 u [(al,a4)l). B1 = cR(B u [(al,a2)l),

o(a) = a

check c o n d i t i o n s ( a ) - ( c ) .

A

B

such t h a t

A2 = cR(A u l ( a 3 , a 4 ) l ) ,

1.5) U =

+

J

J o ( a j ) = ai,

since

P(w), f : A

[a ,a ,a ,a4] n supp(cR(A u A u B)) = 0. 1 2 3 o(ai) = a., denote the permutation CI o f P ( w ) such t h a t

al,a2,a3,a4

Define

be f i n i t e subgroups o f

A, 8

c,a

A

x

in

,...,a n'

c ,ajtl

contained i n

ai

B n A =

0,

we can

cR(A u 1x1) n B =

0.

By repeat-

such t h a t

such t h a t

a

-

cR(Au { x l f n cR(Au { y l ) = A.

then B

so t h a t t h e atoms of

ed use o f t h i s argument, i t f o l l o w s t h a t we can e f f e c t i v e l y f i n d nonzero

,...,x.yJ j

xl,yl ui

c

vi

bi

c

and for all

cR(B u {u.,vi\i 1

A2 = cL(A u {y,

B2

2

,...,uj ' vj

ul,vl

i and

,...,y j l ) ,

Let

We can d e f i n e

f : A1

+

B1

and extending i t t o be a homomorphism. fl(ai-xi)

= bi-vi

for

s

i

5

j

and

c

jl)

fl(ai)

v1 .bl-vl,.

yi c ai

and

u

,..., x j l ) ,

,...,v j l )

Note t h a t t h e atoms o f

,..., ~ ~ , a ~ - x ~,., ..,an a ~ + ~ and

respectively.

xi

A1 = cR(A u {xl

B1 = cR(B u {v,

= cR(B u {u l,...,uj}).

xl,al-xl

[cR(A u Cxi,yili

0.

j))] n A =

such t h a t

and A1

and

B1

..,vj,bj-vj

are p r e c i s e l y

,vjt,

,...,b n

by d e f i n i n g i t on t h e atoms of Let

f,(xi) = bi

= vi for

A1

and

jtl s i 2 n.

Similarly

E. EISENBERG and J. REMMEL

104

we c a n d e f i n e f2: A2 + B 2 so t h a t f2(yi) = u.1 f2(ai-yi) = bi-ui f o r i 5 j and f ( a . ) = b for i 2 1

j+l < i 5 n.

n o t d i f f i c u l t t o check t h a t c o n d i t i o n s ( a ) - ( c ) h o l d .

S i m i l a r l y t h e isomor-

phism r e q u i r e d f o r c o n d i t i o n ( e ) a r e d e f i n e d by l e t t i n g aim a.

1.6)

1

for

both

= g (a -y ) = a.-c

g (a.-x.)

and

1

5

j+l

and

fl

1

2

i 5 n.

isomorphism such t h a t

i and

A = la, f(ai)

A u B.

then t h e s e t o f a l l

but

v1 = fl(X1)’

i and

x1 < x2,

b,},

for

i=

Similarly,

(i)

B2 = B u {y,}.

y2 and

f2(x2) = y

M

U = :M,cR)

Let U

ly

E

Qnly ci b j

Since

xk
c

Define

fi: Ai

iff

E

A,

On-

c <. a 1

A2 = A u Bi

j

i s dense i n

On

iff

+

c
x1,x2,y1,y2

A, = A u {xl},

for Qn,

such t h a t j

iff

{x21,

so t h a t

firA

yk ci b j

8, = = f

and

.

can e x t e n d

fl

be a n e . a . c . s .

U

f

f: A

+

and

f2 since

x .c x2 1

but

n 2 2.

and

8 where A and B

are closed sets

i s effective but not f u l l y effective.

has t h e s t r o n g n - i n c o m p a t i b l e isomorphism e x t e n s i o n p r o p e r t y ,

t h e n g i v e n any r e c u r s i v e i n f i n i t e - c l o s e d s e t many isomorphisms (iii)I f

be a f i n i t e

has t h e n - i n c o m p a t i b l e isomorphism e x t e n s i o n p r o p e r t y , t h e n t h e r e

such t h a t

(ii)If

be an

= Y2.

a r e continuumly many isomorphisms in

B

I t i s easy t o check t h a t c o n d i t i o n s ( a ) - ( c ) 1 F i n a l l y , no isomorphism even w i t h r e s p e c t t o <, much l e s s

< fl(xl)

f2(X2)

If

k = 1,2,

Thus we l e t

w i t h r e s p e c t t o <,c1,...,cn,

THEOREM 1.1

A

+

Qn such t h a t

in

and

and ( e ) h o l d .

Y1 =

which extends

f: A

The main f a c t t o n o t e i s t h a t i f

and f o r

j.

i and

=

=

fp(Y1) = U1’

,...,bJ, 1, ...,n, and

j} a l s o c o n t a i n s an open n-box.

y1 4 y2,

for a l l

fl(xl)

g2(ai)

=

g

B = Ibl

i t f o l l o w s t h a t we can e f f e c t i v e l y f i n d d i s t i n c t

8 u

gl(ai)

x ci a . i f f c
j

x

,...,an]%

= bi

any c o o r d i n a t e s i n common. all

and

=

a l s o has t h e s t r o n g 2 - i n c o m p a t i b l e isomorphism ex-

Let

s e t d i s j o i n t from

i< j

for

i

x 1 = Y1

f2 since

tension property.

i

= g2(yi)

gllxi)

F i n a l l y , t h e r e can be no isomorphism

,. . . ,cn))

U = ((Dn,
i

It i s

cL(A)

f: A

+

C

B as i n (i) so t h a t A

of

M, and

i s a f i n i t e s e t f o r a l l f i n i t e sets

there e x i s t s continuumly

B

a r e isomorphic t o

A

in

M,

C.

t h e n (i)

Effective Isomorphisms of Algebraic Structures

and (ii) hold w i t h the additional properties that

B a r e Dedekind.

and

A

105

PROOF Assume U has t h e n - i n c o m p a t i b l e isomorphism p r o p e r t y .

The f i r s t s t e p i n

our proof i s t o c o n s t r u c t what we c a l l a " n - a r y b r a n c h i n g t r e e isomorphisms".

Technically N.

branching t r e e , i n t o (fa.Ao,B>

where

T i s an e f f e c t i v e map from Tn,

However, we t h i n k of T ( u )

(fu,Aa,Ba)

T ( u ) = k.

Ikand

=

closed s e t s and o u r c o n s t r u c t i o n w i l l ensure fa: A

Ak'

where

=

B

k

= c9.(0)

and

be t h e nodes a t l e v e l

n

h ( r )5 n

h ( T )

that

T(0)

T i s d e f i n e d by i n d u c t i o n .

isomorphism.

or

T o f conflicting

u

Thus

A,

= ko

Tn.

Let

= n+l

and

T

T on a l l

as a t r i p l e

Bu

and

I

where A

kg

.

= (f

k fl Let

We d e f i n e

T(o"k1*i) fi: Ai

f i n i t e - c l o s e d s e t s and

c-1~1f o r some B;

= (fi,Ai,Bi) -+

i 5 k.

where

A1

u

-

BT)]

(A

u

such

,...,An,

B1

,...,Bn

are

a r e t h e isomorphisms we g e t b y a p p l y i n g t h e n-

Bi

on k+l

T

Assume by i n -

,B

A = A

=

'k+l TEBF

...,a nP

UnfT i s a ane-one

so t h a t

i n c o m p a t i b l e isomorphjsm e x t e n s i o n p r o p e r t y w i t h and A = [ U ( A ,

kfl',A kg ,Bk g )

u:,ui,

TcB,,n

function.

are f i n i t e -

be t h e s e t o f a l l nodes

B:

T E

Tn

E

i s a partial recursive

Bu

+

fko i s t h e i d e n t i t y on

o f the

d u c t i o n t h a t we have d e f i n e d

for

t h e f u l l n-ary

B

on k+l

).

,f

B 'k+l

I t i s easy t o see t h a t

,

= f

'k+l

T so con-

s t r u c t e d w i l l have t h e f o l l o w i n g p r o p e r t i e s (a)

WU,T

(b)

Wo

('1 (d)

E

Tn(u C_ T

A.

-+

c

A,

& 9,

c

8, &

f,

c

Tn ( t h e r e i s no isomorphism e x t e n d i n g

E

f,). fu*l,...,fu*n).

i s a one-one p a r t i a l r e c u r s i v e f u n c t i o n .

n'T !f

f o r any p a t h

P through

Tn,

Ap =

u

A,

and

u

Bp =

and (e)

fp: Ap

If P1

and

P2

i s an isomorphism where a r e two p a t h s t h r o u g h

and o*j

o*i

E

Bpl

n Bp2 =

P1

Bp

-+

B,,

E

and

T

fp = and

TEP

u 1

" fPp

are closed sets

E

f,. P1 n

P2 where i # j , t h e n Ap n Ap2 fp

8,

T€P

TEP

P2 b u t = A,,

= fu*

To show t h e r e a r e continuumly many e f f e c t i v e b u t n o t f u l l y e f f e c t i v e

E. EISENBERG and J. REMMEL

106

isomorphisms, i t i s enough t o show there i s a 1 :1 correspondence which associates

A

t o each subset

of

N

Let

(gl,D,,E,),

tive) of a l l triples g: D

u

E

P(A)

through

,...

(g2,U2,E2)

(g,D,E)

P(A)

u

where

i s of l e v e l

if n e A

a ' = u*l

-

,...,f, ,*,,

nodes a t l e v e l

2n+l

so l e t

j

into

are r.e.

n

4

A.

do n o t extend

fu.

By c o n s t r u c t i o n ,

gn

k

A

so

extends

fp(A)

such t h a t

and A # B,

A,B C N

that i f

2n

gn

M and

Let can n o t

gn

does

I t i s c l e a r by our choice o f the

Moreover, by our choice o f nodes a t l e v e l

we have ensured t h a t no

P(A)

substructures of

be the l e a s t

P(A).

...

We c o n s t r u c t

Suppose we have s p e c i f i e d t h a t

g1 ,...,gn-l

if

n = O,l,

for

and

a*2

u'*j

and p u t

fa,*k

P(A) # P(B).

2n

and l e t u '

extend each of not extend

such t h a t

be a l i s t ( n o t n e c e s s a r i l y effec-

D and E

where

E i s a p a r t i a l r e c u r s i v e isomorphism.

+

Tn

i s effective but not f u l l y effective.

fP(A): AP(A) -+ 'P(A) by i n d u c t i o n .

a path

f o r any

then

n = O,l,.,.

for

,

i s not f u l l y

fp(A)

effective. Next assume t h a t property and

has the strong n-incompatible isomorphism extension

U

C i s an i n f i n i t e - c l o s e d r e c u r s i v e s e t i n M.

To prove ( i i ) we

need o n l y show we can c o n s t r u c t a n-branching t r e e o f c o n f l i c t i n g isomorphisms T w i t h p r o p e r t i e s ( a ) - ( e ) and t h e a d d i t i o n a l property, (f)

f o r a l l paths We c o n s t r u c t

l i s t of

P T

through

Ap

Tn,

i s isomorphic t o

by i n d u c t i o n as before.

C i n i n c r e a s i n g order.

Let

u I cj ( s ) l ) where

we have defined

T

j ( s ) = pk(ck

CO

1 Cs).

on a l l

T E

B[

so t h a t

o f length

II

in

B;,

cO,cl,

...

C0cC1 c

f i n i t e - c l o s e d sets defined by i n d u c t i o n as CL(C,

Let

C.

...

be an e f f e c t i v e

be an e f f e c t i v e sequence o f

= ck(0)

and

Cs+l

=

Keeping t h e n o t a t i o n above, assume ( A ) U fT i s a one-one f u n c t i o n , TCB!

(b) 9:,

f o r each

Ck

define

-+

AT,

T

and

T(o*l),.

(C)

for a l l

. .,T(u*n)

U,T E

gu: C n

f i n i t e - c l o s e d sets

E

g: 0

+

Cn

and

u r_

B[,

u + cf+,

where

isomorphism p r o p e r t y a p p l i e d t o TI2

we have deFined an isomorphism

3

A,

.

+ ,A ,

T

implies

We want t o

go 5 9.,

F i r s t use t h e s t r o n g n-incompatible Cn+13

Cn,

and isomorphisms

E such t h a t hrCn i s t h e i d e n t i t y and g l Cn = .4,

and

A = 0

h: U

+

Thus

Cn+l

t o produce and

9eh-l: Cn+l+E

107

Effective Isomorphismsof Algebraic Structures a l s o extends

Note t h a t

9,.

so t h a t we can a p p l y t h e e x t e n s i o n p r o p e r t y t o

d e f i n i t i o n of an e.a.c.s. fa: A,

+

But

€ 2

A = [ u ( A T u

and

AUl

must b e f i n i t e - c l o s e d by p r o p e r t y ( v i ) o f o u r

E

BT) - (Aa u B,)]

t o produce f i n i t e

TEB(:

closed s e t s such t h a t

Ai fi

Aa,

3

f,

3

Bi

Ba

3

girA,

i s the i d e n t i t y f o r

extended t o an isomorphism. 9,*:i

Cn+l

+

identity, path

and isomorphisms

We l e t

be d e f i n e d by

Ai

= 9 .,

go*itcn

g*i,

u g,

=

Bi

and

gi:

...,n,

and

ufi

+

T ( o * i ) = (fi, Ai, = gf 1 0gOh-l.

Having c o n s t r u c t e d

P t h r o u g h Tn , gp

* Ai i' i = 1,

f

Bi>

Ai

E

-t

cannot be

and l e t

Note t h a t s i n c e

girAu

i s the

i n t h i s manner n o t e t h a t f o r any

T

w i l l be an isomorphism f r o m

C = uCn

onto

TEP

A

UA,.

=

Thus c o n d i t i o n ( f ) h o l d s f o r

T

and c o n d i t i o n s ( a ) - ( e )

f o l l o w as

rep before.

&(A)

F i n a l l y , suppose t h a t Then c l e a r l y

A,

r.e.

set, then

and

= Ap2

o*J

E

P2

i s a f i n i t e s e t for a l l f i n i t e sets

i s finite for all

Ap

W

3

P1 # P2.

W

Then

through c

A,

Bp

have t h e p r o p e r t y t h a t one o r b o t h o f t h e r e a r e c o n t i n u u m l y many such

P(A)

W

f o r a t most one

o n l y c o u n t a b l y many o f t h e c o n t i n u u m l y many paths A

and P(A) where b o t h

T

n

.

where

by c o n d i t i o n ( e ) and hence

i s contained i n

W

We c l a i m t h a t i f

P

i# j

W

an i n f i n i t e r.e.

Tn.

E

f o r a t most one p a t h

f o r some p a t h s

f o r some

cs

P(A)

B

P(A) Ap(A)

M.

i s an i n f t n i t e

F o r assume

o*i

P1

E

i s finite. P.

ac

W c Apl

and Similarly

It follows t h a t

constructed previously i s n o t Dedekind. and

Bp ( A )

Hence

a r e Oedekind.

REMARKS

1)

I t i s w e l l known t h a t any r e c u r s i v e l i n e a r o r d e r i n g i s i s o m o r p h i c t o a r e c u r -

s i v e subset o f

Q.

I t i s a l s o w e l l known t h a t any r e c u r s i v e Boolean a l g e b r a i s

isomorphic t o a r e c u r s i v e subalgebra o f

(see [12] f o r d e t a i l s ) .

i n [9] t h a t any r e c u r s i v e n-dimensional p a r t i a l o r d e r i n g t o a recursive subset o f

Dn

under <.

(P,<,)

I t i s proved

i s isomorphic

Moreover i t f o l l o w s f r o m o u r proof of

t h e s t r o n g 2 - i n c o m p a t i b l e isomorphism e x t e n s i o n p r o p e r t y f o r example ( 6 ) t h a t t h e isomorphism fpo: Apo

+

Bpo

produced by Theorem 1.1 i n

e f f e c t i v e o v e r t h e language which i n c l u d e s o n l y

<.

(Dn,<,
are

Thus t h e f o l l o w i n g e a s i l y

E. EISENBERG and J. REMMEL

108

follows from Theorem 1 . l .

COROLLARY 1 .2 (i) L

For a n y i n f i n i t e r e c u r s i v e n - d i m e n s i o n a l p a r t i a l o r d e r i n g

(n 1. 1 )

to

there e x i s t suborderings

and a n i s o m o r p h i s m

L

e f f e c t i v e (where (ii)

2)

+

and

A

8

of

8

isomorphic t o

there exists

B

and an i s o m o r n h j s m

B

which i s e f f e c t i v e b u t n o t f u l l v e f f e c t i v e .

B

we can construct T so t h a t

P(w),

BY construction

f i n i t e recursive s e t s .

o n t o BT.

function from AT

fT =

from N-AT

i s a 1:l recursive function from N onto

f i n e an r.e. g r o u p G = (N,+,i).

on N

i

so t h a t

have Godel numbered P ( w ) That i s , we define

Let

p

0

h:P(w)

U

BT a r e

N-BT

are in-

a r e i n f i n i t e , there i s a

N-BT

onto N I.?,

M.

and

AT

and

Let

We can use

h = f u g

N

i s an isomorphism.

so t h a t the s e t of GSdel numbers of

x

so t h a t

P ( w ) t o de-

and

h

be the group operation f o r -+

With a

i s a I : ] p a r t i a l recursive

f

UET~

and

Since N-AT

1 : l partial recursive function g

+ and

in Theorem 1.1.

T h u s i t will also be the case t h a t N-AT

recursive subsets.

N).

which i s e f f e c t i v e b u t n o t f u l l y

8

Let us consider the special case where U = P(w)

define

isomorphic

Dn

D1 = Q ) .

proper GSdel numbering of

h

+

of

8

F o r any i n f i n i t e r e c u r s i v e B o o l e a n a l g e b r a

subalgebras f: A

f: A

and

A

P(w) and

(We assume we E

P(w)

equals

by

a + b = h(h-’(a)Oh-’(b))

and

i ( a ) = h ( x ) where h-’(a) = y and x i s the inverse of y in P ( w ) .

+ and i are recursive functions and G

These definitions ensure t h a t i s an r.e. group.

Observe t h a t since f o r any path

f p i s a n isomorphism from Ap

subgroups of the operation

P(w), 0

of

N.

onto Bp where Ap

through

T2, h r A p = f p and

BP are considered as

and

i t must be the case t h a t the operation + of

P(w) on Bp.

Considered as a subgroup of

on a l l of

P

G.

For example, l e t

Thus (BP,o)

G

T2

so t h a t

= (Bp,+)

We note, however, t h a t

= (N,+.i)

0

G

agrees with BP can be

and + will n o t agree

then i n the construction of Theorem 1.1

109

Effective Isomorphisms of Algebraic Structures using our proof of the 2-incompatible isomorphism property f o r

to u,

N , a l , a 2 , a3, a4,

4 elements of

fT((a3,a4))

= (al,a4).

h((al,a2)

( a 3 , a 4 ) ) . B u t t h i s means

0

(a,,a,)

while h:P(o)

path

+

P

Thus by our definition of

(a,,a,)

0

and ensured

f T ( ( a l , a 2 ) ) = (a ,a ) 1 2

P(o).

has order

in G

2

Nevertheless, we have shown t h a t

i s an isomorphism of r.e. groups which extends

G

and

(a,,a,) + ( a l , a 4 ) =

+,

( a l , a 2 ) + (a,,a,)

has order 3 in

P(o), we assigned

f p : Ap

.+

Bp for any

T2 and hence gp i s a strongly e f f e c t i v e isomorphism, Thus

through

Theorem 1.1 proves the following. COROLLARY 1.3

where

T h e r e a r e c o n t i n u u m l y many g r o u p i s o m o r p h i s m s

and

A

a r e subgroups of

B

such t h a t

P(o)

f: A

8

+

is strongly

f

effective but not f u l l y effective. 3)

Results analogous t o Corollary 1 . 3 hold f o r orderings, vector spaces, and

To s t a t e such r e s u l t s we must f i r s t define the analogues of

Boolean algebras.

w-groups and r . e . groups.

*

Thus, f o r example:

i s said t o be an w-e66ective f i n e m ohdehing i f

( i ) L = (A,<)

i s r.e. and <

i s said t o be an h . e . finem ohdehing i f , in addition,

L

F where

+

{ p f I f E F ) i s a vector space over a recursive f i e l d

i s the operation of vector addition and f o r each

Operation of scaler multiplication by

space i f

A 5N

and

cursive functions.

pf

f o r each

+ and

E

F

f.

f

f

E

F

h.e. vectoh

space i f

i s r . e . and

A

partial recursive functions. and

1

+

a r e p a r t i a l recursive. i s a Boolean algebra where

t o be an w-ed6eotive Boolean dgebnn i f v,

i s the

are extendible t o partial re-

A,

v,

the operations of intersection, union, and complement respectively.

A,

pf

F,

E

i s said t o be an w-e66cotive vecton

V

f o r each

i s said t o be an

V

f

pf

( i i i ) Suppose B = ( B , A , v , - - ~ )

r.e. and

A

i s a partial recursive.

( i i ) Suppose B = (A,+,

and

and i s extendible t o a partial

i s a binary r e l a t i o n which linearly orders A

recursive r e l a t i o n .

and

A 5N

B

B

N

and

A,

v,

and

7

and

B

are

i s said

extendible t o

i s said t o be an n . e . Boolean dgebna i f

a r e p a r t i a l recursive.

7

B

is

With these definitions, we can

easily define the notion of strongly e f f e c t i v e isomorphism f o r l i n e a r orderings,

110

E. EISENBERG and J. REMMEL

vector spaces, and Boolean algebras by analogy with the definition for groups. Now in general suppose the effective closure system A is Godel numbered so that [ M I

1.1

=

=

u

is a 1:l partial recuracTn fa and there is a 1:l partial recursive function g

onto BT

f u g. Then we can use h to make N into an

=

from N-AT onto N-BT. Let h

(U,c9.) of Theorem

N and AT, BT, M-AT, and N-BT are in-

finite recursive sets. Then by construction fT sive function from AT

=

r.e. structure isomorphic to M as we did in the case of groups. That is, for any n-ary operation p on M,

-

,...,xn)

p(xl

=

h(p(h-’(xl)

on N by letting

we define a n-ary operation

,...,h-’(xn)))

for all x1 ,...,x n in N.

Similarly,

-

for any n-ary relation R on M,

we define an n-ary relation R on N by letting K(x, ,..., xn) hold if and only if R(h- 1 (x,) ,...,h-’(xn)) holds in M.

Let h(M) denote this new structure defined on N. each of the cases M

=

numbering and h(M)

will be respectively an r . e . linear ordering, an r.e. vector

(Q,<>,

M =

It is easy to check that in

Vm,

or M

= (~,A,v,-,),

there is such a GGdel

space, or an r.e. Boolean algebra. As in the case of groups, for any path P through T, Bp can be considered as a substructure of either M or h(M) =

hrAp

since

fp and fp is an isomorphism from Ap onto Bp when Ap and Pp are

considered as substructures of M.

Thus h will show that fp is a strongly

effective isomorphism. Thus as an immediate corollary of Theorem 1 .l, we have the fol 1 owi ng

.

COROLLARY 1.4 (i)

There are continuumly many order isomorphisms

A

where

and

B

are suborderings of

(Q,<) such that

f: A f

8

+

is

strongly effective but n o t fully effective. (ii) f: A

+

There are continuumly many vector space isomorphisms

B where

A

and

B

are subspaces of

Vm

such that

f

is

strongly effective but not fully effective. (iii) f: A

-+

8

There are continuumly.many Boolean algebra isomorphisms where

A

and

B

are subalgebras of

strongly effective but not fully effective.

such that

f

is

111

Effective Isomorphismsof Algebraic Structures

L which

I t i s proved i n [13] t h a t t h e r e i s an w - e f f e c t i v e l i n e a r o r d e r i n g l i n e a r o r d e r i n g , an w - e f f e c t i v e v e c t o r space

i s n o t c o n t a i n e d i n any r.e.

V

which i s n o t c o n t a i n e d i n any r . e . v e c t o r space, and an w - e f f e c t i v e Boolean a l g e bra B which i s n o t c o n t a i n e d i n any r . e . Boolean a l g e b r a .

For each of these

s t r u c t u r e s t h e i d e n t i t y w i l l be an isomorphism which i s e f f e c t i v e b u t n o t ' s t r o n g l y effective.

Thus f o r l i n e a r o r d e r i n g , v e c t o r spaces, and Boolean a l g e b r a s , t h e

notions o f e f f e c t i v e , s t r o n g l y e f f e c t i v e , and f u l l y e f f e c t i v e a r e a l l d i s t i n c t .

12.

COSIHPLE EXAMPLES Let

if A

U = (M,ck)

be an e.a.c.s.

1M1

i s i s o l a t e d and

-

A

A 5M

We say a c l o s e d s e t

i s r.e.

i s cosimple

The purpose of t h i s s e c t i o n i s t o show

t h a t t h e techniques used t o p r o v e Theorem 1.1 may be combined w i t h a p r i o r i t y t o c o n s t r u c t an e f f e c t i v e b u t argument, developed by t h e second a u t h o r i n [ll], f: A

n o t f u l l y e f f e c t i v e isomorphism U

sets i n many e.a.c.s.

+

8 where

A

and

8

a r e cosimple c l o s e d I f we

where c l o s u r e o f a f i n i t e s e t i s always f i n i t e .

consider o u r one example where t h e c l o s u r e o f a f i n i t e s e t i s n o t f i n i t e , namely a r e c u r s i v e v e c t o r space

,V

where t h e u n d e r l y i n g r e c u r s i v e f i e l d

F

we see t h a t any one-dimensional subspace i s an i n f i n i t e r e c u r s i v e s e t are no cosimple s t r u c t u r e s i n . ,V cosimple s t r u c t u r e i n t h i s case. if V

U

We c a l l a subspace

we say a c l o s e d s e t

if A

c o n t a i n s no r . e .

if A

i s pseudo-Dedekind and

d i t i o n s on

U

follows t h a t i f

- A

V c , ,V

A

i s r.e.

c

i s r.e.

I n fact,

M i s pseudo-Dededind (p-Dedehind) A

i s pseudo-coshple ( p - c o s h p l e )

I t i s easy t o see t h a t o u r con-

ensure t h a t t h e c l o s u r e o f an r . e . s e t i s a l s o r.e. U

there

pseudo-coshple

subspace and V-V,

i n f i n i t e - c l o s e d s e t and

1MI

SO

However, t h e r e i s a n a t u r a l analogue of a

c o n t a i n s no i n f i n i t e dimensional r.e.

i n general, f o r any

i s infinite,

It then e a s i l y

i s such t h a t t h e c l o s u r e o f e v e r y f i n i t e s e t i s f i n i t e ,

then

the n o t i o n s o f p-Dedekind and Dedekind and t h e n o t i o n s o f p-cosimple and cosimple coincide. To produce an e f f e c t i v e b u t n o t f u l l y e f f e c t i v e isomorphism

A and

f: A

+

8 where

B a r e p-cosimple c l o s e d s e t s , we must be a b l e t o c o n s t r u c t what we c a l l

E. EISENBERG and J. REMMEL

112

an "w-branching t r e e o f n - c o n f l i c t i n g isomorphisms" branching t r e e o f c o n f l i c t i n g isomorphisms c o u l d g i v e general c o n d i t i o n s on tree

p r o p e r t i e s of Technically

which would a l l o w us t o c o n s t r u c t such a

U

I n s t e a d we s h a l l j u s t o u t l i n e t h e

i s an e f f e c t i v e map f r o m

S(a) =
say

where

S(a) = k

T

into

(I)

for all

and

u, A,

b u t we s h a l l w r i t e

N

i s the index o f

i s an w-btancking .thee oh n-cun&&ng

S

Again we

t h a t we need and show how t o b u i l d i t f o r each o f o u r examples.

S

S

used i n Theorem 1.1.

T

b u t t h e s e become r a t h e r awkward.

S

i n c o n t r a s t t o t h e n-

S

(fo,Au,B,).

T h i s g i v e n , we

iAomahphiAmn

if

a r e f i n i t e - c l o s e d s e t s and

8,

f,:

A,

-t

B,

is

a p a r t i a l r e c u r s i v e isomorphism.

WO,T

(11)

Tw(uC

E

(vu)(vil,

(111)

each o f

f,*i

1

(V)

and B p =

F o r any paths

P

a r e c l o s e d s e t s and

'

to

Bp.)

and

through

P2

where

o*i

E

Tw, Ap,

P1

and

fp =

n Ape = ,A,

a*j

E

P2

P

through

u

TE P

and

Bp

Bpl

i # j.

f o r some

isomorphisms

so t h a t f o r any g i v e n i n f i n i t e r e c u r s i v e subset

S

effective l i s t o f

Q

,. ..

C

C

f o r every path through

and l e t

and l e t

and

Cn = I c l ,

...,c n1

To.

Let

f o r each

c1,c2, n '1.

of

C

...

Q,

be an

Let

be an e f f e c t i v e l i s t o f p a i r w i s e d i s j o i n t dense r e c u r s i v e subsets oo,al,

...

be an e f f e c t i v e l i s t of t h e nodes of

i t i v e l y we t h i n k o f a s s i g n i n g Aui*j

C.

We s h a l l b u i l d an w-branching t r e e of 2 - c o n f l i c t i n g

i s isomorphic t o

AO,A1

so

S

a r e isomorphic t o any g i v e n i n f i n i t e - c l o s e d r e c u r s i v e s e t

U = ((Q,<),c!Z).

T,,

i s an

fT

We n o t e t h a t i n s e v e r a l o f t h e examples below we s h a l l b u i l d

EXAMPLES

of

8 T€P

Ap

1 f p n fp2= f, 1

and

Ap

ik # ij+ t h e r e i s no isomorphism e x t e n d i n g

A

k#j n

isomorphism from

Let

fT).

,...,fUki ) .

TEP

2.1)

c

I t f o l l o w s f r o m ( I ) and (11) t h a t f o r any p a t h

VAT

Ap =

Ap

BT & f,

c

fT i s a one-one p a r t i a l r e c u r s i v e f u n c t i o n . T€Tw (Note:

that

AT & B,

c

...,inE N ) (

U

(IV)

A.

T +

bi*j

for all

j.

Ai

to

ui

We d e f i n e

since S

Ai

Tw.

Intu-

w i l l be used t o d e f i n e

by i n d u c t i o n .

Let

S(0)

be

113

Effective Isomorphisms of Algebraic Structures

(0,0,0).

the l e a s t index o f

so t h a t if k n ( o ) = n, f,(ai)

= bi

9:,

Cn

-+

f o r each

Au

il,i2 5 n, cj

1. ci

2

cil

.

g,(ci)

i 5 n, c ~ <+ ci~

~ <+ ci2~

and f o r a l l

To.

Then s i n c e

and

Bu*i

=

Ak

u lei},

B,

n

either

follows t h a t

g*i,

-+

i < j,

di < d .

but

Au*i

and

gu*i

fo*i(di)

= ei

= g,

f,*i:

no isomorphism

,V

can c o n s t r u c t such a t r e e f o r be a r e c u r s i v e b a s i s f o r

g

Au*i

1

or

F

Vm

and l e t

Bi

of

by i n d u c t i o n .

We d e f i n e where

we have d e f i n e d

S

to

ui

Bo,B

and

S. We do n o t know i f one

l,...

fo(d) = d

Since

n.

Let

uo,ul,

and

Let

d

F

S(0)

i s finite.

B.

...

for

t o f i n d an e f f e c t i v e sequence

B

Again i n t u i t i v e l y ,

i s the effective l i s t

be t h e l e a s t i n d e x o f

i s t h e zero v e c t o r of

u = uk and l e t bl,b 2,...

Bk.

Let

be an e f f e c t i v e sequence

so t h a t i f k n ( o ) = n,

l i s t of

ck(Au u {ai})

fUei

i s i n f i n i t e , we can c o n s t r u c t an

where

S ( o ) = (fu, Au, Bu)

have dimension

ai = bi + Xib2

are iso-

+

> e

where t h e f i e l d

we t h i n k of a s s i g n i n g



It

u i(cn+l,diH.

can extend b o t h

of p a i r w i s e d i s j o i n t i n f i n i t e r e c u r s i v e subsets o f

,V

ci

= f,*j(d.). I t now e a s i l y j J so c o n s t r u c t e d w i l l have t h e d e s i r e d p r o p e r t i e s .

J

S

Cn+l

and

o - b r a n c h i n g t r e e o f 3 - c o n f l i c t i n g isomorphisms

€3,

5

...

where t h e u n d e r l y i n g f i e l d

To.

cj

o r f o r some

be t h e kth element i n o u r l i s t

k

= fu u {(diei)>,

fu*i

Also f o r

since

I n ,V

u = u

Let

i 5 n,

c ~ + ~ ,

The o t h e r two cases cao be

...

morphisms.

2.2)

and

i s dense we can e f f e c t i v e l y f i n d r e c u r i s v e se-

a

i s easy t o see t h a t

fu*j

,...,bn},

< dl < d2 < < ai and bi < e2 < el < bi such t h a t il 2 1 2 = Au u idi}, ei a r e i n Ak f o r a l l i. Then we d e f i n e A*i,

quences di

= lbl

B,

for all

5

j

We c o n s i d e r o n l y t h e t h i r d case.

handled i n a s i m i l a r f a s h i o n . of

,...,an},

There a r e 3 p o s s i b i l i t i e s f o r

= ai.

for all

< c

= la,

i. Moreover, assume we have d e f i n e d an isomorphism

such t h a t

c n+l > ci

namely

A,

then

Bu)

S ( u ) = (fc, ,A,

Assume we have d e f i n e d

Vm. then

Assume

Au

and

be an e f f e c t i v e

F i s i n f i n i t e , we can use t h e dependence a l g o r i t h m

f o r some

Xi

n ck(Au u { a , } )

J

E

F = Au.

ao,al,

...

such t h a t f o r a l l Then l e t

o f elements o f t h e f o r m

i # j,

114

E. EISENBERG and J. REMMEL Au*i

= &(Au

u {ai}),

Eui,

morphism such t h a t i, j, and since

3

and

fa and

= b..

fu,i(ai)

u Iai,aj,ak})

Note t h a t f o r d i s t i n c t

1

P(w),

S.

Let

of

N.

EO,E,

,...

Again i t i s easy t o check

be an e f f e c t i v e sequence o f i n f i n i t e r e c u r s i v e subsets

Let

S(0)

be t h e l e a s t i n d e x o f

and

f

=

( E )

u

Let

E .

=

Ek.

CL(Eu u I(el,ezi)1), such t h a t

fu

3

ui.

to

Define where

Au*i

Ek = 0.

Let

i s the identity

E

S(u) = (fu, Au, BJ

e ,e 2,...

be an e f f e c t i v e

= cL(Au u {(e2i-l,e2i)}),

Au*i

and

fa*i:

and

fu,i((e2i-l,e2i))=

+

Eu*i

(el,e2i).

(e2i-l

,e2i)(e2j-l

has o r d e r 2 w h i l e

,e2j)

Again i t i s easy t o check t h a t 2.4)

I n U = (G,cQ), morphisms

(Recall

C.

Let

eo,el,

...

nonzero elements o f

where

C

Cs = c L ( { d o,...,dS}),

c.

so l e t .,,c,

.. ,csS

1-

i s the u n i t o f

Q

w i l l have e x a c t l y

Cs

be t h e atoms o f Q

set o f a l l intervals

= Ix

a k ' j

E

A.. 1

Thus

through

[a ,a,) k J

KO,$,...

To,

Cs.

Let

Ao,A1

Ap

i s iso-

It i s n o t d i f f i -

C.

an e f f e c t i v e sequence

p a i r w i s e d i s j o i n t subsets o f

a

P

t h e n t h e r e i s a u n i q u e atom

It follows that S

has o r d e r 3.

( e1' e 2 i )(el ,eZj)

be an e f f e c t i v e l i s t o f

,...

eo,el

c

f o r some

e..) J i # j since

we can c o n s t r u c t an o-branching t r e e o f 2 - c o n f l i c t i n g i s o -

c u l t t o construct from

dS+l

fa*j

and

S so c o n s t r u c t e d has t h e d e s i r e d p r o p e r t i e s .

we can ensure t h a t f o r any p a t h

morphic t o

and

ei

(ei,e.)J

Moreover, i f we a r e g i v e n any i n f i n i t e r e c u r s i v e subalgebra

S.

of

C

fu*i

=

t o be t h e u n i q u e isomorphism

6u*i

i s t h e n o t a t i o n f o r t h e t r a n s p o s i t i o n which i n t e r c h a n g e s Then no isomorphism can e x t e n d b o t h

by i n d u c t i o n .

S

and assume we have d e f i n e d

uk

Then d e f i n e

Ei

(fO,fE),{E))

[suPP(Au) u supp(Bu)l n

l i s e of

fu*k

we can c o n s t r u c t a n o - b r a n c h i n g t r e e o f 2 - c o n f l i c t i n g isomorphisms

Again we t h i n k o f a s s i g n i n g

0 so t h a t

and

i s an n+2-dimensional space w h i l e

u Ibi,b.,b

In

fu*j.

fu*i,

I ) i s an n+3-dimensional space. J k so c o n s t r u c t e d has t h e d e s i r e d p r o p e r t i e s .

S

be t h e u n i q u e i s o -

fu*i

no isomorphism can e x t e n d each of

c'l(A,

cL(E

2.3)

k

fu*i

= cR(Bu u I b i I ) ,

1~ = do,d

,,...

such t h a t if c

of

s+l

,. ..

Cs

such t h a t

atoms f o r a l l

s

be t h e sequence of

as i n example 2.1 and l e t

xi

denote t h e

Q l a < x < a.} where ak < a and k J j w i l l be a sequence o f d i s j o i n t r e c u r s i v e E

of

Effective Isomorphisms of Algebraic Structures

F.

subsets o f duction.

s(@) be

Let

,..., b 1 )

cL({b,, for

,...,an])

i 5 n.

such t h a t Cn

where

and

g,(c?)

= ai

dn+l

c

for c.: x1

x.,ao-xi

x,.

are i n

yi

Bu*i

",...,an

F} ) )

Q

Let

4

= ci?(B,

5

=

n.

Rk

=

P,

so t h a t

cL(A,

B,)

u

a r e t h e atoms of

i 5 n. Ak

x2

. ..

c

c

Assume

B,,

A,,

8, =

and

fo(ai)

,bl

fu*i: A*i,

i, we d e f i n e isomorphisms

Cn

9:,

c

,a,

and2, ,b

let

=

~ ~ , f , , ~ ( a ~ - x =~ )bn-yi, = xi,

gu*i(dn+l)

g,*i(c:-

i, d e f i n e

,. .. ,b n +

and

dn+l)

A*i,

= cL.(A,

u {xi])

A*i,

and

B*i,

respectively.

B*,i

and

gu*i:

are

Then f i r each Cn+,

+

=, ,a

f,*i(a.)

-

J

xi,

b. J

=

for

by

Au*i

1 5j

Thus l e t

2

n

and

and

and g*,i = a . f o r 1 5 j 5 n. C l e a r l y fo*i 3 f, J J i. Moreover f o r any k < j , t h e r e can be no isomorphism g xk

A,

such t h a t

gu*i(cn)

since

+

...

y1 3 y23

d e f i n i n g them on t h e atoms and e x t e n d i n g them t o be homomorphisms. f,,i(xi)

= bi

i s t h e unique'atom of

c:

so t h a t t h e atoms o f yi b,-,yi

n

i s dense, i t f o l l o w s t h a t we can f i n d

Then f o r each

and

where

u = uk and Ln(o)

a r e t h e atoms o f

Since

u {yi})

,. . . ,an

,al

0

bv i n -

S

Moreover, assume we have d e f i n e d an isomorphism

such t h a t

and

8

cL.({l-]).

b",..., bn

where

an e f f e c t i v e sequence xi

a

We d e f i n e

ui.

(f ,cL({l-}),ci?({l

S(a) = (f ,A B,) u u'

Assume we have d e f i n e d = ci?({a,,

to

t h e l e a s t index o f

fo i s t h e i d e n t i t y r e s t r i c t e d t o

A,

xi

We t h i n k o f a s s i g n i n g

115

3

g,

for a l l

extending

x . w h i l e f,,k(xk) = ,yk3 y . = f * . ( x . ) . J J O J J Again i t i s n o t d i f f i c u l t t o check t h a t S has t h e p r o p e r t i e s claimed. b o t h fu,k

2.5)

In

and

f,*j

,...,<,),

U =

c o n f l i c t i n g isomorphisms

cL.) S

language which o n l y c o n t a i n s set

C

c

On

tion of

S

we can c o n s t r u c t an w-branching t r e e of 2-

such t h a t t h e isomorphism c o n f l i c t o v e r t h e

<.

we can ensure t h a t

P t h r o u g h To.

c

Moreover, g i v e n any r e c u r s i v e i n f i n i t e C

The c o n s t r u c t i o n o f

i s isomorphic t o S

A,,

f o r a l l paths

i s a m o d i f i c a t i o n of t h e c o n s t r u c -

i n example (2.1) which i s s i m i l a r t o t h e m o d i f i c a t i o n o f example

(1.2) used t o v e r i f y example (1.6) so we s h a l l o m i t t h e d e t a i l s . THEDP.EI1 2.1.

Suppose

U = (M,ci?)

i s an e . a . c . s .

and we c a n c o n s t r u c t

a n a-branching t r e e o f n - c o n f l i c t i n g i s o m o r p h i s m s (i)

If

c&(A)

i s f i n i t e f o r a l l f i n i t e sets

A 5 1M1,

S

in

U.

then there i s a

116

E. EISENBERG and J. REMMEL

P

path fp: Ap

through Bp

+

T,

such t h a t

Ap

and

a r e cosimple c l o s e d s e t s and

i s an e f f e c t i v e b u t n o t f u l l y e f f e c t i v e isomorphism.

If the closure o f a f i n i t e s e t i s not necessarily f i n i t e but there i s a re-

(ii)

X such t h a t f o r a l l a

cursive function

w i t h canonical index

A(u),

furDX(u)

= g

fp: Ap

implies P

a path

through Bp

+

f T,

Ap

such t h a t

Thus i f

the f i n i t e set = Aa

and

Bp

g: A,

and

Bu,

+

grDA(u) =

then there i s

a r e p-cosimple and

i s an e f f e c t i v e b u t n o t f u l l y e f f e c t i v e isomorphism.

X(u)

U = Vm b v e r an i n f i n i t e r e c u r s i v e f i e l d

= Aa

t h a t comes from t h e c o n s t r u c t i o n

A ,

gives t h e canonical index o f t h i s basis f o r

for all

i n ,T,

a

Au,

A

will

I f t h e c l o s u r e o f any f i n i t e s e t i s f i n i t e i n

be a f u n c t i o n d e s c r i b e d i n ( i i ) .

U and

D

CP.(D,(~))

f o r any isomorphism

t h e r e i s a n a t u r a l b a s i s f o r each subspace S.

To,

E

i s such t h a t

Note f o r example (2.2) where

PROOF.

of

Bp

X w i l l a l s o s a t i s f y (ii).Thus

then

s i n c e p-cosimple i s e q u i v a l e n t t o cosimple i n t h e case where t h e c l o s u r e of any f i n i t e s e t i s f i n i t e , we see t h a t (i) i s a c t u a l l y a s p e c i a l case o f (ii). Thus we prove o n l y ( i i ) . We s h a l l c o n s t r u c t i n stages t h e p a t h

A t each s t a g e

theorem.

s

through

P

T ,

r e q u i r e d by t h e

we w i l l s p e c i f y a sequence o f nodes

1. 1,

...

as C us S. C a: where f o r each n 5 s, 0; i s a node a t l e v e l n. We s h a l l 1- 2 S ensure t h a t f o r each n 2 1, l i m un = an e x i s t s and t h a t P = l o l ,a2,. . I .

.

A l s o a t each stage that

A U

n AS =

fl

S

s,

we s h a l l s p e c i f y a p a i r o f f i n i t e s e t s

= B

n BS

I0 ,...,s> 5 A S u A

and

US

S

(As,Bs)

and {O

such

,..., s ) c B s

u

S

We s h a l l ensure t h a t f o r a l l

x,

AS

As+1

and

Bs

A = u AS

BS+l.

E

2‘ S

and

S

B = u Bs S

w i l l be t h e r . e .

respectively

l,... i and

where

Wi

Bp

i s t h e ith r.e. set.

i s an e f f e c t i v e l i s t o f a l l c l o s e d r . e . s e t s i n

recursive function all

and

.

For each i 2 0 , l e t Ui = ca(Wi) Uo,U

Ap

s e t s which a r e t h e complements of

s.

h

such t h a t

for all

Ui = Wh(i)

N o t e t h a t t o ensure

Ap

and

Bp

M

i. L e t

Clearly

and t h e r e i s a

~f

= W’

for h(i) a r e p-cosimple, i t i s enough

Effective Isomorphisms of Algebraic Structures t o ensure t h a t contained i n since

fS =

Ap

i s p-cosimple.

Bp.

u

then

F o r if U

g = f,’rU

=

f,’rU

i s an i n f i n i t e - c l o s e d r.e.

set

i s a p a r t i a l r e c u r s i v e isomorphism

i s a one-one p a r t i a l r e c u r s i v e f u n c t i o n .

fo

117

Thus

g(U) 5 A p

U€TW

would be ensure

an i n f i n i t e - c l o s e d r . e .

Ap

and

Bp

s e t so t h a t

Ap

i s n o t p-cosimple.

Thus t o

a r e p-cosimple, i t i s enough t o meet t h e f o l l o w i n g s e t of

requirements.

9:,

If

i s i n f i n i t e - c l o s e d , then

Ue

Qe

We say r e q u i r e m e n t

U n A # 0. e

i s batin6ied a t stage

s

m.

if U g n AS #

use t h e even stages of o u r c o n s t r u c t i o n t o meet t h e requirements TO ensure t h a t

fp

We s h a l l (Ie.

i s n o t f u l l y e f f e c t i v e , we must meet t h e f o l l o w i n g

s e t o f requirements. Re:

If

Rng $e

and

Dmn $e

isomorphism, t h e n

$e

a r e c l o s e d s e t s and

meet a r e q u i r e m e n t and

$e’

foe-]

f o r a t most

T

L ue-’,

Dmn 9., (i) (ii)

Re.

e

n-1

Note t h a t i f

i. Thus if we f i n d

t h e n we know t h a t

$e

5

s

j, <

X C D

f o r any

X(T)

T

CONSTRUCTION Let

A

i s t h e node of l e v e l

0

=

j, <

S,

... <

$e

Re

e-1

in

could extend

jn such t h a t $,rD

i s batih&&2d a t stage

s

0 B = 0.

jn5 k

c

o;-’*jk,

where

P

*i

x (ue-’

Rng $e

*jk ) -

to

i f either

such t h a t

... <

there are

Stage 0.

oe-l

can n o t be an isomorphism f r o m

Thus we say a r e q u i r e m e n t t h e r e i s an x

i s an

and t h e odd stages o f o u r c o n s t r u c t i o n t o

t h e n by o u r c o n s t r u c t i o n o f

Y

e

fp.

does n o t e x t e n d

We s h a l l use t h e nodes a t l e v e l

Oe:Rmg be+Dmn @

oe = oe-’*k such t h a t f o r a l l s e $g(x)S and $ g ( x ) = f T ( x ) .

118

E. EISENBERG and J. REMMEL

Stage 2s.

5

x

2s

Look f o r an

such t h a t

x

e

5

-

’U:

6

such t h a t

2s

i s n o t s a t i s f i e d and there i s an e If there i s no such e, l e t uZs = u .

A 2s-1.

0

1

‘e i 5 2s-1.

for

L e t A”

~ u ~I p x (- x 5~ 2s & x

= A

4A

A2s-1

2s-1

11 and

u2s-1 BZs = 8’’

5 2s

u Ipx(x

$s-1

4 B 2s-1

& x

)I.

F i n a l l y , l e t u;:

=

uZs

2s-1*j

u2s-1 n AZs =

j = pn(A 2s

where

B 2s

=

u2s-1*n S,

t h e r e must be such a

that

x(2s)

4A

x

e,

let

e(2s)

corresponding t o

2s-,.

(Note by c o n s t r u c t i o n of

j.)

I f t h e r e i s such an

be t h e l e a s t such

B2’).

n

%1*” be the l e a s t such

e(2s).

Note by choice o f

Let

e(2s),

k

U.

k

2

e

and l e t

x(2s)

be the g r e a t e s t e(2s).

such

j

Then define

J

u?’ = o?’-l 1

1

and l e t

for

5

i

A2s = A2s-1

Let

k.

u 1 x ( 2 ~ ) 1u fux(x<2s & x

4A

11

u 6;s

B2s-1 )}.

4 8 2s

u {px(x525 & x

BZs = B2’-l

F i n a l l y , d e f i n e ui2s

‘k for

k c i 5 2s

u:’

by i n d u c t i o n as

j = un(A n AZs = 2s ‘Si-l*n be s a t i s f i e d . )

0

Stage 2s+l.

Look f o r an

and f o r a l l

x

D

E

t h e r e i s no such {ux(x52s+l & x

e,

4

= B 2s n B’S). ‘Ji-l*n

5 2s

e

with

i(T)

= u:sl*j

let

T

u?”’

=

1

$Es+l(x)+ for

0:’

and

BzS+’

Q e ( 2 s ) w i l l now

( v o t e requirement

such t h a t requirement

~u::

A 2s u A‘’))

where

i s not satisfied

Re

and $+’(x)

i 5 2s.

= f 2s(x).

L e t A’’+’

= BZs u {px(x(Zs+l

& x

4 B

O2S

2s+l

u2s+l = uZs*j where

Then define

Ift h e r e i s such an oZs+’ = 02’

i

and B2’+’

for

1

i < d.

e,

let

L e t ’A’‘

n A‘’+’ j = pn(A uZs*n 2s

e(2s+l) = d

=

0

=

B uZs*n 2s

be t h e l e a s t such

= AZs u { p x ( x L Z s + l & x

If

e ‘ = AZs u

4

e.

2s u Bzs)1. u2s 2s+l ) . n B Let

A 2s+l u A2’)} ‘d-1

=

BZs u I p x ( x 5 2 s + l & x

4B

2s+l u B2s ) } . ‘d-1

Next i f

u 2 s~ ( ~ =~ u2s*k, + ~ ) d-l

Effective Isomorphisms o f Algebraic Structures then l e t

B

n BZst1).

02s+l*n d-1

=

0

=

‘d-1

”+’

Finally

AZst1

j = p n ( n > k & A 2s+l

where

= c$f’*j

119

ai

for

e(2s) < i

5 2s+l

T h i s completes t h e d e s c r i p t i o n o f t h e stages.

i s d e f i n e d by i n d u c t i o n

I t i s c l e a r t h a t each stage

i s e f f e c t i v e s i n c e a l l t h e searches f o r

e(s),

A = u AS

We s h a l l now p r o v e a sequence of lemmas

and

S

B =

LJ

S

Bs

a r e r.e.

sets.

x(s),

e t c . a r e bounded.

Thus

which w i l l complete t h e p r o o f o f t h e theorem. LEMMA 2.1.1.

2

t & i< e

e(2s) < e

-+

a:

t = ai).

Qi

2s > t,

a ’:

Q

e(25)

i s s a t i s f i e d a t a stage

stages, t h e r e can o n l y be such t h a t

Now i f

=

e

( a ) e(2s+1) = e,

u,

such stages

for all

a:’-’

= az:l*k

2 s t l > to. Hence

l i m as

s

i s a stage such t h a t

t h e n a’:

# a:’-’

only i f

a t stage

2s.

Now s i n c e once a

i t remains s a t i s f i e d a t a l l l a r g e r

2s.

and

f o l l o w s by o u r d e f i n i t i o n o f s a t i s f a c t i o n o f

such stages

t’e

Thus t h e r e i s a s t a g e

to. Vow i f

2s

( b ) azs”

exists.

Assume

e.

and we s a t i s f y r e q u i r e m e n t

requirement

then

lim u s = u s e

e,

We proceed by i n d u c t i o n on

PROOF. Vs(s

For a l l

Re

2s+l > to and

a’:

where

= c(S1*m

t

0 >.t

2s+l 2s ue # ae , m < k,

t h a t t h e r e can be a i r n o s t

and

n

must e x i s t .

e

I t i s a s i m p l e i n d u c t i o n t o show t h a t f o r a l l

s,

A

n AS =

0

=

n Bs

B

as Thus i t f o l l o w s t h a t Let

P

clearly

A,

n

n A =

be t h e p a t h t h r o u g h

T“

Ba

n

n B =

0 for all

n

where

S

an = l i m an. S

c o n s i s t i n g o f t h e nodes

O,ul,a

2,...

.

Then

Ap n A = Bp n B = 0.

LEMMA 2.1.2.

Ap

= N-A

and

Bp = N - B .

PROOF. By o u r remarks above, we need o n l y show t h a t Ap u A = Bp u B = N. proceed by i n d u c t i o n on

n.

Assume t h e r e i s a s t a g e

t > n

such t h a t

We

E. EISENBERG and J. REMMEL

120

{O,

...,n-11 5 A

u

‘k Thus ukt

u S. = ui).

At where k is such that k 5 t and =

uk. Now consider n. First suppose n

is clear from our definitions that either n Next assume n u

5

q

=

u

-

Ut

+

4A

t, then it “t At or n is put into A t+l .

A t, then

for all s 1. t. Clearly if n

q

E

5k

Let q be the greatest integer such that q 5 t and

A t.

E

Vs(s> t & i

n

E

Ap.

Otherwise q < t

uQ

or n

t be the least stage such that u;+~ # u$:.

s

so let

case, n

E

AS

4A

px(x 5 s+l & x

=

Then clearly n

s+l u As) and n is put into As+1. Thus i n any uq Ap u A. The fact that Bp u B = n is proved in a similar fashion. I

E

and Rp

Thus Ap

are co-r.e. so to complete the proof of the theorem we

need only see that all the requirements are met. LEMMA 2.1.3. PROOF.

A l l the requirements

stage 2s

Re

are met.

First we consider the requirements Re. Let t be a stage such that

t& i 2e

W s(s

Qe a n d

>

Now if $e really extends fp, then there is a t such that for all x E 0 with T c_ ue, $e2s (x)S ,and $,(x) -+

U:

=

ui).

X(T)

=

f, (x). Then at stage 2s+l, we know e(2s+l) > e since uES+’ = uZs s = ‘e* e But e(2s+l) > e only if Re is satisfied at stage 2s by our definition of e(2stl).

Now the only way Re can be satisfied is if there are j, <

...

< jn

k = 1 , ...,n. But then we know $e is not an isomorphism so requirement Re is automatically met. Next consider the requirements 0,. Qj is not met.

Suppose n is the least j

Let u be a stage such that u > n, Ws(s,u

& icn

such that -+

u: = ail,

and for all e < n, if requirement Qe is ever satisfied, it is satisfied at stage u.

Now Un must be infinite-closed and Un

our definition of e.a.c.s.,

it cannot be that Un

2s > u and an x 5 2s such that x

E

UEs

-

A ‘n

.

c

c Ap.

By condition (vii) of

so there is a stage ‘n But then at stage 2s, either A

Qe is satisfied at stage 2s-1 or our definitions force e(2s) 5 e . Now by

121

Effective Isomorphismsof Algebraic Structures

cur choice of

u,

f i e l d a t stage Ue 5 Ap.

n 2s.

5 e(2s)

so t h a t

But then

U ’:

Thus t h e r e can be no such

P

n = e(2s)

A*’ n.

and r e q u i r e m e n t

Q

n

i s satis-

# 0 which c o n t r a d i c t s t h e f a c t t h a t I

We should n o t e t h a t by a p p l y i n g t h e techniques employed i n [ll],one can modify t h e c o n s t r u c t i o n o f Theorem 2.1 t o ensure t h a t any non-zero r . e . degree. set o f

A

and

8

a r e a l s o of

A l s o i n t h e case where t h e c l o s u r e of e v e r y f i n i t e

M i s f i n i t e , one can e a s i l y m o d i f y t h e c o n s t r u c t i o n o f Theorem 3.1 t o

ensure t h a t

A

and

B a r e co-hypersimple.

F i n a l l y , by arguments s i m i l a r t o

the ones used a t t h e end o f s e c t i o n 1, one can show t h a t i n t h e case o f groups, vector spaces, l i n e a r o r d e r i n g s , and Boolean algebras,the

isomorphism

s t r u c t e d i n Theorem 2.1 i s a s t r o n g l y e f f e c t i v e isomorphism.

fp con-

122

E. EISENBERG and J. REMMEL

REFERENCES Applebaum, C. H., Isomorphisms o f w-groups, (1971 ) , 238-248.

N o t r e Dame J. o f Formal L o g i c

Crossley, J. N., C o n s t r u c t i v e Order Types (monograph), Amsterdam (1969), 255 pp.

12.

North Holland,

Crossley, J . N. and Nerode, A., C o m b i n a t o r i a l F u n c t o r s (monograph), S p r i n g e r V e r l a g , B e r l i n (1974), 144 pp. Dekker, J. C. E., J. Sym. L o g i c

Countable v e c t o r spaces w i t h r e c u r s i v e o p e r a t i o n s P a r t 1,

34 (1969),363-387.

Eisenberg, E. F., E f f e c t i v e isomorphisms o f groups and o t h e r s t r u c t u r e s , Ph.D. Thesis, C o r n e l l U n i v e r s i t y , 1974. E l l e n t u c k , E r i k , Sylow subgroups of a r e g r e s s i v e group, Houston J. of Math. 5 (1979), 46-67. Hasset, M. J., R e c u r s i v e e q u i v a l e n c e types of groups, J . Sym. L o g i c 13-20. Hay, L. S.,

The co-simple i s o l s , Ann. of Math.

Manaster, A. dimension: 789-802.

g

34

(1969),

(1966), 231-256.

B. and Remmel J . B., P a r t i a l o r d e r i n g s o f f i x e d i n f i n i t e Model companions and d e n s i t y , J. Sym. L o g i c

46

(1981),

, Recursion t h e o r e t i c a s p e c t s o f dense two dimensional p a r t i a l o r d e r i n g s , Aspects o f E f f e c t i v e Algebra (ed. J . N. C r o s s l e y ) Upside Down A Book Co., Yarra Glen, V i c t o r i a , A u s t r a l i a (1981), 161-189. Remnel, 3. B., (1978)

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Co-hypersimple s t r u c t u r e s , J. Sym. L o g i c

41 (1976),

611-625.

R e c u r s i v e l y enumerable Boolean a l g e b r a s , Ann. o f Math. L o g i c , 75-1 07.

14

, E f f e c t i v e s t r u c t u r e s n o t c o n t a i n e d i n r e c u r s i v e l y enumerable s t r u c t u r e s , Aspects o f E f f e c t i v e Algebra (ed. J . N. C r o s s l e y ) Upside Down A Book Co., Yarra Glen, V i c t o r i a , A u s t r a l i a (1981), 206-226.

Rogers, H. J. Jr., Theory o f R e c u r s i v e F u n c t i o n s and E f f e c t i v e C o m p u t a b i l i t y , McGraw-Hill, New York (1967).

PA TRAS LOGICS YMPOSION G. Metakidex led.) @ North-Holland Pu blishing Company,1982

123

RECURSIVE LINEAR ORDERINGS Manuel Lerman ( l ) U n i v e r s i t y o f Connecticut Joseph G. Rosenstein Rutgers U n i v e r s i t y

The use o f techniques developed f o r r e c u r s i o n t h e o r y on

N,

the s e t o f nat-

u r a l numbers, f o r t h e s t u d y o f a l g e b r a i c s t r u c t u r e s o t h e r t h a n t h e i n t e g e r s goes back many y e a r s .

I n r e c e n t y e a r s , t h i s s t u d y has been g r e a t l y expanded, p r i m a r i l y

through t h e work o f A. Nerode and h i s s t u d e n t s .

T h i s work has taken two d i r e c t i o n s .

The f i r s t d i r e c t i o n i s E f f e c t i v e Algebra; t h e s t u d y o f t h e e f f e c t i v e c o n t e n t o f theorems about a l g e b r a i c s t r u c t u r e s .

The second d i r e c t i o n i s Generalized Recursion

Theory; t h e s t u d y o f q u e s t i o n s r a i s e d i n C l a s s i c a l Recursion Theory b u t based on algebraic structures other than

N.

The focus o f t h i s paper w i l l be t h e s t u d y o f t h e e f f e c t i v e c o n t e n t o f c e r t a i n theorems about l i n e a r o r d e r i n g s .

The q u e s t i o n s we answer were r a i s e d i n Rosen-

s t e i n ' s book [ R l on l i n e a r o r d e r i n g s .

Chapter 16 o f t h a t book c o n t a i n s an e x c e l -

l e n t summary o f r e s u l t s on r e c u r s i v e l i n e a r o r d e r i n g s , w i t h p r o o f s o f many of t h e results. The f o l l o w i n g examples demonstrate p o s s i b l e r e l a t i o n s h i p s between a c l a s s i c a l theorem and i t s e f f e c t i v e c o u n t e r p a r t .

The o p t i m a l v e r s i o n o f a c l a s s i c a l theorem

i s a v e r s i o n i n which e v e r y t h i n g i s r e c u r s i v e .

The f o l l o w i n g c l a s s i c a l theorems

have o p t i m a l v e r s i o n s : CLASSICAL THEOREM 1: subset of

Q,

Every countable l i n e a r ordering i s isomorphic t o a

t h e s e t of r a t i o n a l numbers.

EFFECTIVE VERSION:

Every recursive l i n e a r ordering i s recursively iso-

morphic t o a r e c u r s i v e subset of CLASSICAL THEOREM 2: (1)

Q.

There i s a n o n - t r i v i a l automorphism of

Q.

Research s u p p o r t e d by N a t i o n a l Science Foundation Grant No. MCS 78-01849.

M. LERMAN and J. ROSENSTEIN

124

Sometimes, the construction given in the proof of a theorem is less effective

than the sets which need to be constructed. Effective versions of such theorems will generally have non-constructive aspects to their statements. For example: CLASSICAL THEOREW 3 :

Let

A

order-type of the integers. of

w*

+

the

w,

Then there is a non-trivial automorphism

A.

EFFECTIVE VEF-SION: Let w* + a.

type of

be any set of order-type

A.

A

be any recursive linear ordering of order-

Then there is a non-trivial

IIY-definable automorphism

However, there need not be a non-trivial recursive automor-

phism of

A.

Sometimes, a classical theorem allows several different effective versions. For example: CLASSICAL THEOREM 4: oforder-type

w

Every infinite linearly ordered set has a subset (the order-type of the natural numbers) or

w* (the

order-type of the negative integers). EFFECTIVE VERSION 4a:

(Tennebaum, Lerman [Ll).

Every infinite recur-

sivelinearly ordered set has a recursive subset of order-type w , w*,

w + w*

or

w

+ (w*

+

w)-n + w*

(n

is the order-type of

Q).

Furthermore, there is a recursive linearly ordered set of order-type w

or

w*;

and there is an infinite recursive linearly ordered set

with no recursive subset of order-type w, w*, EFFECTIVE VERSION 4b: ordered set has

a

(!!anaster)

or

w

+

w*.

Every infinite recursive linearly

IIy-definable subset of order-type o or

w*.

In Section 1 , we will study the effective version of another classical theorem: We will try to see how effectively "choice functions" can be defined for certain recursive linearly ordered sets. And in Section 2, we will study the relationship between certain recursive order-types and the functions which they represent.

Recursive Linear Orderings

CHOICE SETS

1.

be a l i n e a r ordering of order-type C f f ( q ) : q

Let A

q

E

125

Q,

e i t h e r f ( q ) i s f i n i t e or

q

by replacing each

ing of

Q

E

from the s e t representing

i s obtained

and preserving the orderA

has a subset

I n f a c t such a subset can be obtained by

i . e . , a subset of

taking a choice s e t f o r A ,

f(q)

The classical theorem s t a t e s t h a t

( a dense subset).

Q

Thus A

lo, w*, w* + wI.

E

Q with a s e t of order-type

between such s e t s .

of order-type

f(q)

QI where f o r each

E

f ( q ) f o r each

tive content of t h i s classical theorem.

q

E

containing exactly one point

A

Q. We will investigate the effec-

Thus we assume t h a t

and determine how effectively we can find dense subsets of

i s recufsive,

A A

and choice s e t s

for A . Rosenstein [R] notes t h a t i f

i s recursive, then we can always find a I$-

A

definable choice s e t f o r A.

Watnick [W] studied t h i s classical theorem under the

assumption t h a t the range of

f

i s e i t h e r constant o r bounded, and determined

when recursive o r recursively enumerable choice s e t s and dense subsets e x i s t .

We concentrate on the case where f ( q ) =

will consider more general functions. w* + w

f o r every

q

E

Q, a case s p e c i f i c a l l y raised in Rosenstein [Rl, and show

t h a t $-definable dense subsets sometimes f a i l t o e x i s t . however, i f we l e t

We

f(q) = w

f o r each

q

E

Q,

if we impose the constraint t h a t the range of

or f

The r e s u l t i s unchanged,

f ( q ) = w*

f o r each

q

Q , or

E

is finite.

W e will use a p r i o r i t y argument t o construct a recursive l i n e a r ordering of the appropriate order-type, and insure t h a t t h i s ordering has no $-definable dense subset.

P r i o r i t i e s will be established in order t o pick a p a t h t h r o u g h a

tree of s t r i n g s on which the construction takes place.

DEFINITION 1 . 1 :

A s t r i n g i s a f i n i t e s e q u e n c e o f n a t u r a l numbers.

The s e t o f a l l s t r i n g s i s d e n o t e d by length of U,T

u(x)

E

S,

u,

is the least

we s a y t h a t

u 5

i s d e f i n e d , and l e t

S.

such t h a t

x T

if

u

IR T

Given

u(x)

u

E

S, l h ( u ) ,

i s undefined.

u(x) = ~ ( x ) f o r a l l

x

b e t h e s t r i n g d e f i n e d by

the Given

such t h a t

M. LERMAN and J. ROSENSTEIN

126

*

u

=

T(X)

u

Given

E

S

I

if x < l h ( u )

if

uu n( xd)e f i n e d

otherwise.

u # 6,

with

lh(u) =

such t h a t

2

T(x-lh(u))

lh(T)

+

lh(u)

u-

we l e t

x < lh(u)

u

THEOREM 1.2: +

(a*

i f either

T

B,

N

C;-definable

p,q

B be a C;

that for a l l

x

E

{(pi,qi):i

dense subset.

B

E

2


such t h a t

(N,

Then t h e r e i s a r e c u r s i v e f u n c t i o n

E

B

Is:g(s)

t+

(pi,qi)

E

i s infinite,

+

N2

such

We w i l l go through t h i s l i s t ,

t

a t which

T h i s p a i r remains designated u n t i l g(t)

While a p a i r

x N u Iqi}

{pi)

E

(p,q)

(p,q)

a t which

i s designated (we assume { x : p
designated a t a l l s u f f i c i e n t l y l a r g e stages.

This s t r a t e g y w i l l guarantee t h a t

B

{p,qI 5 B

i s finite.

f x : p
x N,

then since each p a i r appears i n f i n i t e l y o f t e n i n the l i s t ,

there w i l l be a f i x e d p a i r

+

Hence no

g: N

w i t h o u t l o s s of g e n e r a l i t y ) no new elements a r e placed i n

{@i : i

Z;

{XI x NI i s finite.

a t each stage.

time the next p a i r i s designated.

Let

~


set.

(and i f ) there i s a l a t e r stage

N

i s finite.

)

11 be a r e c u r s i v e l i s t i n g o f a l l p a i r s o f n a t u r a l numbers, w i t h

designating a p a i r

and

(w* + a

o f order-type

{x : p
each p a i r appearing i n f i n i t e l y o f t e n i n t h i s l i s t .

If B

and

& T

N x

p
u

We w i l l guarantee t h a t f o r every i n f i n i t e

o f n a t u r a l numbers.

there a r e

Let

or i f

we Say

S.

U ~ T E

u(x) # ~ ( x ) .

such t h a t

s e t can be a dense subset of t h e order

Let

c T

We w i l l b u i l d a r e c u r s i v e l i n e a r o r d e r i n g

on the s e t

Z;

u

For

S.

There i s a r e c u r s i v e l i n e a r o r d e r i n g o f order-type

a ) * n w i t h no

PROOF:

set

x

5u

T

1.

has higher p r i o r i t y than

u ( x ) < T ( X ) f o r the l e a s t

1h(T)

denote t h e unique

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

+

N}

i s n o t a dense subset o f

(N, cA)

since

be a r e c u r s i v e l i s t i n g o f a l l p a r t i a l r e c u r s i v e functions

N2, w i t h the usual r e c u r s i v e approximation

i

: i,s

E

N}.

Given any

Zo

2

Recursive Linear Orderings set

t h e r e i s an

C,

i s finite. stage

i ' E

N

such t h a t

i f and o n l y i f

C

E

l y : ~ $ ~ ( yE ) { x } x N)

Hence we a t t e m p t t o r e p e a t t h e above 'procedure f o r each

s, we s e l e c t

@j(i),qj(i))

y

S

$i

r

A f i n a l path

no f i n a l p a i r e x i s t s .

i s the s t r i n g o f length

= u{y::

s

N}

E

The e x i s t e n c e o f

r

corres-

0

guesses t h a t

*

must t h e n e x i s t , where

o f h i g h e s t p r i o r i t y such t h a t

s

C

Hence as we pass from

a t an i n t e r m e d i a t e stage, where

S

t.

I f , however, t h e s e t

$i.

i s f i n i t e , t h e r e may be no such f i n a l p a i r .

p a i r t o p a i r , we s e t y (i) = 0

A t each

$i.

guesses a t t h e f i n a l p a i r ( i f any)

S; ys(i)

E

which w i l l be d e s i g n a t e d f o r

ponding t o

l y many

x

127

y:

c yt

ys

for infinite-

f o l l o w s f r o m ( 1 ) - ( 3 ) below, which w i l l , i n t u r n ,

f o l l o w from t h e c o n s t r u c t i o n . (1)

s f t

(2)

Ys f $

(3)

s

+

Ys f Yt.

+

3 t < S(Yt = Y;). = yt

< t & y;

Thus if

r(i) =

0,

then

set o f

1

t &

yuz y; *

0).

w i l l correspond t o a f i n i t e Otherwise,

r(i)

+i

witnessing the fact t h a t

s e t which cannot

w i l l designate a p a i r

does n o t correspond t o a dense sub-

(N, cA).

Once ys

If

$.

(N,
be a dense subset o f (pj(i),qj(i))

5u5

3u(s

-+

i s chosen, f i n i t e s e t s

w i l l be d e f i n e d f o r c e r t a i n x.

I(ys,x)

i s d e f i n e d , i t w i l l be a f i n i t e i n t e r v a l i n t h e o r d e r i n g c o n s t r u c t e d

I(ys,x)

by t h e end o f stage p r i o r i t y ys.

s.

This interval w i l l contain

We w i l l t h e n d e f i n e

Note t h a t by (2),

for a l l

s

I(r,x)

there i s a

x

: s

= u{I(y:.x) t

and w i l l be p r o t e c t e d w i t h

such t h a t

y:

E

NI

x c N.

f o r each

Properties (4)-

= yt.

(10) w i l l be s a t i s f i e d . We w i l l want

{I(r,x)

: x

E

to partition

N}

N

(N,

i n t o i n t e r v a l s of

<*).

The f o l l o w i n g p r o p e r t i e s a l l o w us t o conclude t h a t we have a p a r t t i o n !

(4)

I(ys,x)

i s defined f o r a l l

x < s.

(5)

If

I(ys,x)

i s defined,

x

E

(6)

If

I(ys,x)

i s d e f i n e d and y

E

I(ys,x)

(7)

If

then

I(ys,x). then

I(ys,y)

i s defined and

I ( Y s d ) = I(Ys,Y). I ( y ,x) S

i s d e f i n e d and f o r a l l

p r i o r i t y t h a n y,

then

I(yt,x)

u

such t h a t

- .i s d e f i n e d

and

s < u

5

t

ys

has h i g h e r

I(ys,x) 5 I(yt,x).

128

M. LERMAN and J. ROSENSTEIN

For t h e remaining p r o p e r t i e s , we assume t h a t I(y,,x)

i s defined and f o r a l l

than

u

s,t

x

5

t,

s < u

such t h a t

a r e g i v e n such t h a t has higher p r i o r i t y

ys

The f o l l o w i n g a d d i t i o n a l p r o p e r t y w i l l guarantee t h a t

yu.

interval o f

(N,

(8)

E

If y,z stage

I ( y s . x ) , y
s,

I(I’,x)

i s an

i s placed i n the o r d e r i n g

by


then

v

and

v

I(ys,x).

E

I t w i l l f o l l o w from ( 8 ) and ( 9 ) t h a t f o r a l l

w*

and

x

I(r,x)

N,

E

has order-type

+ a.

(9)

If ys c yt v

then there a r e y,z

I(yt,x)

E

y
such t h a t

1(Ys,X).

E

F i n a l l y , i t w i l l f o l l o w from (10) t h a t t h e o r d e r i n g induced by

: x

{I(r,x) (10)

for all

E

has order-type

N)

If Y, = yt, there a r e and

I(yt.w)

T-,.

I ( Y ~ , Y ) i s defined,

u,v,z

such t h a t

N

E

# I(yt,x)

on


and

u <

I(yt,w)

x
I ( y S , x ) # (yS,y)

and

x
# 1(yt,y)

for a l l

w

E

then

i s defined

fu,v,zl.

Property ( 8 ) w i l l a l s o be used t o preserve designated i n t e r v a l s i n order t o s a t i s f y the non-density requirements.

The c o n s t r u c t i o n i s now given. Stage 0: Stage s + l : 0 : ”

= y,.

yo = $,

I(yo,x)

Step A :

i s undefined f o r a l l

D e f i n i t i o n of

ys+l.

x.

We proceed by i n d u c t i o n on

setting

j,

There are three cases which can be f o l l o w e d a t i n d u c t i o n step

Case I :

(j) i s n o t defined.

L e t ys+, = as+’ j

*

0

j.

6.

and go t o Step

.

s+l us+’ ( j ) = k # 0. F i x t h e stage t < s+l such t h a t yt = uj If j t h e r e i s no u such t h a t t < u 5 s+l and @s+’(u) E { P k l x N u { q k l x N, l e t

Case 11:

us+l .

s+1 = Otherwise, l e t ‘j+l J F i x t h e g r e a t e s t stage r 5 s i n d u c t i o n step

j+l.

Case 111.

( j ) = 0.

u”;

Search f o r the l e a s t

k

5

J

‘I

be t h e s t r i n g o f l e n g t h

such t h a t

Let

‘I

s+l

such t h a t :

yr Z T

*

0,

such t h a t

j

s+l

and l e t yj+l

be t h e s t r i n g of l e n g t h

j

= yr.

such t h a t

.

s+l J GO t o

T c a.

s+l

T c aj

.

Recursive Linear Orderings (11)

For a l l

T

*

k.

(12) (13)

Pk’ qk < s 9 pk
E

I ( a s+l j

and Y

found, s e t

z

then

, p,)

u Ipk}

and

*

k

and go t o i n d u c t i o n step

ordering such t h a t

I f no such

B.

k

is

j+l.

tzs

F i x the g r e a t e s t

I(yS+,,x)

I ( o ’ + ~ , qk) u { p k l j

E

and go t o Step

such t h a t

We proceed by i n d u c t i o n on those

ys+l.

z

i s n o t defined.

ys+l = ‘I

I n t e r v a l Expansion:

B:

x

yt

has higher

already placed i n t o the

has n o t y e t been defined.

There are two cases.

B was reached through Case I 1 1 o f Step A where pk and qk were

designated, and e i t h e r z

I(T,x)

set

= a;+’

a$;

Case I: Step

and

#

S , yt

i s found,

p r i o r i t y than
5

%x$ k

Ifsuch a

Step

t

129

x

E

I(yt,pt)

or

be the l e a s t two elements o f

x

E

(yt,q,)

or

3 x 3 qk.

pk

Let y

N n o t y e t placed i n the
Def ine u Iu : u

I ( y s + l , x ) = fY,zl

w

and f o r a l l i n t o
E

I(YtSPk)

I ( Y ~ + ~ , X ) d, e f i n e

E

or

u

having the same order r e l a t i o n t o a l l as does

Otherwise.

Case 11:

placed i n the


w

!,

Pk $

Or

I ( Y ~ + ~ , W= ) I(yS+,,x).

as the l e a s t and g r e a t e s t elements o f

placed i n
I(Yt,qk)

6

Place y

qkl and

z

I ( Y ~ + ~ , X r) e s p e c t i v e l y , and

I(yS+,,x)

which have already been

x. Let y

ordering.

I(yS+,,w) = I ( Y ~ + ~ , X )f o r a l l

z

and

Define

w

I(yS+,,x)

I(yS+,,x).

E

N not y e t

be t h e l e a s t two elements o f = Iy,zl u I(yt,x)

Place y

and

z

u

{XI and
i n t o the

ordering as i n Case I. A f t e r the i n d u c t i o n i s completed, go t o Step C. Step C:

Interval densification:

and end o f


Place new elements i n t o

a t the beginning

and between any two consecutive i n t e r v a l s o f t h e form

defined i n Step 6.

For each new

x

This completes the construction. ordering.


placed i n t o Clearly


define

I(y,+l,x)

I(yS+,,x) =

(XI.

(N, < A ) i s a r e c u r s i v e l i n e a r

We leave t h e v e r i f i c a t i o n o f ( 1 ) - ( 1 0 ) t o the reader.

These v e r i f i c a -

t i o n s a r e r o u t i n e w i t h t h e exception o f (10) which f o l l o w s from (13).

We thus

M. LERMANand J. ROSENSTEIN

130

r

conclude t h a t

(N,

i s d e f i n e d and

t o be shown t h a t t h i s o r d e r i n g has no

i

Fix

E

s

and the stage

N

such t h a t

F i r s t assume t h a t

lower p r i o r i t y than

y,

nated a t stage

s,

pk

I(ys,qk)

E

I x : @i(x)

{qkl x N I

E

contains both

pk

and

Oi

fined.

T E

r ( i ) = 0.

that

can be i n

I(r,x)

order-type

(w*

[I(r,p)

B

k

Assume t h a t t h e

Hence since

and

u

i s desigi s finite.

x N I and

(N,

p,q

E

Ci set

x

B corresponding

such t h a t

(N,
Note

I(T,x)

i s de-

a t most one element of

B

i s i n f i n i t e and

B

such t h a t

such t h a t f o r a l l

Oi

cA).

i n order t o o b t a i n a c o n t r a d i c t i o n .

k # 0 because o f Step A, Case 111.

sired contradiction.

(pk,qk>

has

'

s e t corresponding t o

(N,
p
t

2

u

such t h a t

Hence r ( i ) # 0,

B

has

and

i s defined] =

u I x : p
But then there w i l l be yt(i) =

x.

t h e r e must be

+w)-Q,

Ip,)

E

Hence the

i s a dense subset o f

f o r any

u I(r,q)


(N,

t > s , yt

{ x : pk
t h e r e a r e o n l y f i n i t e l y many

S,

Also, since

Thus a p a i r

and so i s n o t a dense subset of

qk

i s a dense subset o f

t h a t f o r each

Then f o r a l l

I x : Ipi(x)

t,

a r e both f i n i t e .

Suppose f i n a l l y , to

r ( i ) # 0.

and by (8)

Since y t ~ y , f o r i n f i n i t e l y many

Note t h a t by ( 2 ) , such

ys = Y:+~.

r ( i ) # ys(i)).

(else

I t remains

w).Q.

dense subset.

Z;

a stage must e x i s t .

(a* +

has order-type

cA)

0. yt

3

yq,

y i e l d i n g the de-

This completes the proof o f t h e theorem.

We thus note from Theorem 1.2 t h a t the f o l l o w i n g c l a s s i c a l theorem has the f o l l o w i n g e f f e c t i v e version. CLASSICAL THEOREM 5:

E v e r y l i n e a r o r d e r i n g of o r d e r - t y p e

has a s u b s e t o f o r d e r - t y p e

n.

In f a c t ,

(w"

+o)-n

a c h o i c e s e t for t h e l i n e a r

ordering i s such a subset. EFFECTIVE VERSION: (w*

+w)*Q.

Then

A

has a

s u b s e t of o r d e r - t y p e M o d i f i c a t i o n s o f Step otheer order-types.

b e a r e c u r s i v e l i n e a r o r d e r i n g of o r d e r - t y p e

A

Let

If y

II;

c h o i c e s e t but n e e d n o t h a v e a

Zi

0.

B o f the construction w i l l y i e l d similar results for i s omitted i n Step B, then

(N,


w i l l have order-

131

Recursive Linear Orderings type

if z

w * q;

w*

and if both y

q;

order-type

a, cA>

i s o m i t t e d i n Step B, then

Z{f(q): q

E

and

QI

z

w i l l have order-type (N, cA> w i l l have

are omitted i n Step B, then

f o r some

recursive l i n e a r orderings w i t h no

f

whose range i s f i n i t e .

Thus there are

Zi dense subsets f o r each of these order-types.

REPRESENTATIONS OF LINEAR ORDERINGS

2.

Many l i n e a r orderings a r e constructed from sets o f i n t e g e r s o r f u n c t i o n s N

-+

N

and thus represent those sets o r f u n c t i o n s .

I t i s i n t e r e s t i n g t o study the

r e l a t i o n s h i p between a l i n e a r o r d e r i n g and the s e t o r f u n c t i o n which i t represents. Sometimes, representations can be used t o prove more general theorems about l i n e a r orderings.

For example, consider the statement:

orderings has a r e c u r s i v e model" where

X

X

"Every

i s a c l a s s i n t h e a r i t h m e t i c a l hierarchy. X = Cy

P e r e t y a t ' k i n [ P I showed t h i s statement t o be t r u e f o r

X = A;

The proof t h a t the statement i s f a l s e f o r

and Lerman and

X = Ci

Schmerl [LS] l a t e r showed t h i s statement t o be t r u e f o r

X = A;.

theory o f l i n e a r

but false f o r

r e l i e s on a represen-

t a t i o n theorem f o r f u n c t i o n s . The simplest types o f representations t o consider are a-representations of sets, where of

A

lists

a

L e t A = {ai

i s an order-type.

i s a l i n e a r o r d e r i n g o f order-type

A

: i

C{a+ai

E

: i

NI. E

An a-representation

If

NI.

: i

{ai

E

NI

i n order o f magnitude, then t h e corresponding a-representation i s s a i d

t o be i n order o f magnitude. a-representations have been s t u d i e d f o r [R],

Fellner

THEOREM 2.1:

Ci

and

a = q

by Rosenstein

[F], and Lerman It]. We l i s t some r e s u l t s which have been obtained. If

s e t has an

THEOREM 2.2:

a = w* + o

If

A

h a s an

w*

+

A

o* + w - r e p r e s e n t a t i o n

A

E

Every

C;.

w - r e p r e s e n t a t i o n i n o r d e r of magnitude. has an q-representation,

A

then

Zi.

E

If t h i s

r e p r e s e n t a t i o n i s i n o r d e r of magnitude, t h e n

A

s e t h a s an q - r e p r e s e n t a t i o n but e v e r y

s e t has an v-represen-

t a t i o n i n o r d e r of m a g n i t u d e .

an v - r e p r e s e n t a t i o n .

Ci(IIi)

There i s a s e t i n

E

A;.

C;

Not every

= A;

A;

w h i c h has

132

M. LERMAN and J. ROSENSTEIN Ne v i l l now i n v e s t i g a t e t h s r e l a t i o n s h i p between the r e c u r s i v e order-type

Clf(q) : q

E

QI

and functions

which are represented by t h i s order-type.

f

Rosenstein [Rl notes t h a t given a r e c u r s i v e l i n e a r o r d e r i n g Zlf(q) : q

E

QI, t h e r e i s a A:

zCg(q) : q

E

F e l l n e r [ F l shows t h a t if f QI.

a r e c u r s i v e order-type. Question 2.3:

If

If

g

such t h a t is

L

has order-typ.

I$ then

Zlf(q) : q

E

QI is

Two questions had been l e f t open.

L i s a r e c u r s i v e l i n e a r o r d e r i n g o f order-type C l f ( q ) : q

i s there a I$ f u n c t i o n Question 2.4:

function

L o f order-type

g

is a

f

such t h a t

L

has order-type

function,.is

A;

Z{f(q) : q

Clg(q) : q

QI

E

E

E

QI,

QI?

a r e c u r s i v e order-

type? We have been unable t o answer Question 2.3.

The next theorem answers

Question 2.4.

THEOREM 2 . 5 :

T h e r e i s a A;

function

f

such t h a t

Clf(q) : q

QI

E

i s n o t a recursive order-type. Let

PROOF:

{We : e

be a r e c u r s i v e l i s t o f a l l r e c u r s i v e l y enumerable

We w i l l d e f i n e a one-one f u n c t i o n

p a r t i a l orderings. oracle.

N3

E

To show t h a t

We

a l l maximal f i n i t e i n t e r v a l s o f

W

are maximal f i n i t e i n t e r v a l s o f

W

enumerated by

f

We note t h a t a

A;

N

using a A'

3

have u n i f o r m l y bounded length, o r t h a t there

e

o f lengths

e

m and

i n t h e reverse o f t h e i r o r d e r i n g i n We.

a recursive l i s t i n g o f

+

we w i l l guarantee t h a t e i t h e r

f,

does n o t represent

f :Q

n

r e s p e c t i v e l y which are Let

{qi

: i E NI

be

Q. o r a c l e can be used t o answer the f o l l o w i n g questions:

(1)

Is We

a t o t a l l i n e a r ordering?

(2)

Does

(3)

I s an i n t e r v a l o f

(4)

Does a given i n t e r v a l o f

We

have an i n t e r v a l o f c a r d i n a l i t y We

of cardinality We

n

n? a maximal f i n i t e i n t e r v a l ?

have c a r d i n a l i t y

A t each stage of the construction,

We

n?

w i l l e i t h e r be undischarged, P a r t i a l l y

discharged, o r discharged and w i l l have associated w i t h i t a f i x e d ,set c a r d i n a l i t y 52

and a moving s e t

M Se o f c a r d i n a l i t y 51.

numbers which we would l i k e t o place i n t h e range o f

f

FE of

These sets w i l l c o n t a i n t o show t h a t

we

cannot

133

Recursive Linear Orderings

represent f. In order to preserve the one-oneness of f and to satisfy the requirements for all We, the construction will preserve the following properties: (5)

x

E

F:

(6) e # n (7)

& y -f

ME+ x

E

u MZ) n ( F i u Mi) # 0.

(F:

e< n & x

(8) e < n & y <

(9)

t

+

< y.

u

F:

:F & x

E

M;

& y

M;

<

x

-+

E

Mi

+

y

< x.

E

ME

-+

x

<

y.

Fs c Ft. e- e

(10) s < t & x

E

:M & y

y.

The construction proceeds as follows. Stage 0: FE

=

ME

=

0 and We is undischarged for all e

E

N.

Stage s+1: Proceed by induction on {j:j 5 s+ll. Adopt the appropriate case below at step j.

Wj is undischarged and Ms

Step j 5 s:

Case 1:

whether W

is a total linear ordering. If W j

Place

wj

j

J

0. Use (1) to determine

=

is not a total linear ordering,

in the discharged state, let MS+l = FS+l = 0, and go to the next

induction step. A s s m e that if any, such that n

>

wj

J

j

is a total linear ordering. Fix the least n,

x for all x

E

: e < jl. If no such n exists,

u{M:+l

go to the next induction step leaving W

j

undischarged and setting MS+l = Fs+l = B . J

J

Suppose that n exists. Use ( 2 ) to ask the A:

oracle whether W

j

has a finite

interval of cardinality n. If the answer is no, place W in the discharged j state, let M s+l -- p+1 = 0, and go to the next induction step. If the answer j

of W .

is yes, use (4) to fix an interval :1

J

of cardinality n. Place n

E

MS

j

and go to Case 2. Case 2:

W

j

is undischarged and M . # 0 J

.

Then Ms will contain one element, J

say n, and an interval Is of cardinality n will have been defined. j

to determine whether Is

j

is a maximal finite interval o f W

use (4) to find an interval I:+1

j'

If it is not,

of Wj o f cardinality n+l.

2 :1

case, Wj remains undischarged; set F?+l = 0 and M:+' J

Use (3)

= {n+ll

In this

and go to the.

M.LERMAN and 3. ROSENSTEIN

134

next i n d u c t i o n step.

Suppose t h a t

I?i s a maximal f i n i t e i n t e r v a l o f

J

W

j* Then W . becomes p a r t i a l l y discharged and we s e t F?l = I n ) and r e s e t J J MS+l = 0. For a l l k such t h a t j < k < s , Wk r e t a i n s i t s c u r r e n t c l a s s i f i j = 0. We now proceed t o t h e next stage. = Fi, and cation, F;+l Case 3:

such t h a t

n > x

x

: e < j).

Wj

E

uIM;+l

for all

x

E

I f no such

j e x i s t s , go t o the next i n d u c t i o n step l e a v i n g

n

p a r t i a l l y discharged and s e t t i n g

exists. n.

M? = 0. F i x t h e l e a s t n, ifany, J u(FZ+l : e < jl u Fs and n+l < x f o r a l l

i s p a r t i a l l y discharged and

Wj

FS+l = F? and M;+l = 0. Suppose t h a t n J J Use ( 2 ) t o determine whether W has a f i n i t e i n t e r v a l o f c a r d i n a l i t y

j

I f no such i n t e r v a l e x i s t s , place

and M?+l = J J an i n t e r v a l J?

FS

J

Case 4:

0, of

W . i n the discharged s t a t e , l e t F;+l

J and go t o t h e next i n d u c t i o n step.

W

j

of cardinality

n.

Place

n

=

Otherwise, use ( 4 ) t o f i x E

M?

J

and go t o Case 4.

i s p a r t i a l l y discharged and Mf = 0. Then M; w i l l c o n t a i n one 3 n, and an i n t e r v a l J? o f c a r d i n a l i t y n w i l l have been defined. J Use (3) t o determine whether J s i s a maximal f i n i t e i n t e r v a l o f W If i t i s j j' not, use ( 4 ) t o f i n d an i n t e r v a l 2 J; o f W j o f c a r d i n a l i t y n+l. I n t h i s

W.

J element, say

W . remains p a r t i a l l y discharged; s e t J go t o t h e n e x t i n d u c t i o n step. Suppose t h a t

case,

of

Wj.

For a l l

F?+l = Fs and M;+l = ( n + l l and J j J S i s a maximal f i n i t e i n t e r v a l J Then W becomes discharged, and we s e t FS+l = FS u I n 1 and M;+l = 0. j j J k such t h a t j < k 5 s , Wk r e t a i n s i t s c u r r e n t c l a s s i f i c a t i o n ,

FS+l = Ff, and k L e t Pe and p,

=.

0.

F i x the g r e a t e s t

be t h e two elements o f

-

Q

t < s+l

such t h a t

It was defined.

j o f s m a l l e s t index on which

f

has

FS+l has c a r d i n a l i t y 2; l e t F:+l = Cn,rl. j If I : precedes J? i n t h e o r d e r i n g o f W d e f i n e f ( p e ) = n and f(p,) = r. J j' Otherwise, d e f i n e f ( p e ) = r and f ( p ) = n. Go t o the next stage. m n o t y e t been defined, w i t h

Case 5:

Wj

i s discharged.

pe < pm

W . remains discharged.

J and go t o t h e n e x t i n d u c t i o n step. Step s + l :

Set

FiC1 = Fsj, Wj S+l =

wj

Go t o the next stage.

This completes t h e construction.

f

i s c l e a r l y a A!

function.

We leave

Recursive Linear Orderings the v e r i f i c a t i o n o f (5)-(10) t o t h e reader. one-one.

135

I t f o l l o w s from (5)-(10)

that

is

f

Furthermore, since t h e r e are i n f i n i t e l y many t o t a l l i n e a r orderings

w i t h a r b i t r a r i l y l a r g e maximal f i n i t e i n t e r v a l s b u t no i n t e r v a l s o f order-type or

w

must be t o t a l ( s i n c e each such

f

w*,

W

j

f

causes

t o be defined on

two new elements). Suppose t h a t a contradiction. MStl

J

C{f(q) : q Let

W

can be r e s e t t o

0

- j. k <.

E

Q}

i s a r e c u r s i v e order-type i n order t o o b t a i n

be a r e c u r s i v e l i n e a r o r d e r i n g having t h i s order-type.

j

o n l y f i n i t e l y o f t e n through t h e a c t i o n o f

for

Wk

The character o f t h i s l i n e a r o r d e r i n g causes i t t o proceed through a l l

cases and d e f i n e C{f(q) : q

E

Q}

f

on two elements,

pe

and

reverses the order i n which

hence W

cannot have order-type j the theorem.

C{f(q) : q

.p,

But

f

f(pe)

and

f(pm)

E

i s one-one and appear i n Wj;

Ql. This completes the proof of

Other i n t e r e s t i n g questions can be r a i s e d about r e c u r s i v e l i n e a r orderings o f order-type

Clf(q) : q

E

Q}. One such question r a i s e d by Rosenstein [R] deals

w i t h automorphisms, and we would l i k e t o repeat i t here. Question 2.6:

Let A

be a r e c u r s i v e l i n e a r o r d e r i n g o f order-type

e f f e c t i v e l y can we f i n d a n o n - t r i v i a l automorphism o f Rosenstein notes t h a t a

A;-definable

-

q.

How

A?

automorphism can always be found, b u t

t h a t a r e c u r s i v e automorphism does n o t always e x i s t . 1;-definable

2

automorphism can always be found.

I t i s unknown whether a

136

M. LERMAN and J. ROSENSTEIN

REFERENCES

[F]

S. Fellner, Recursiveness and finite axiomatizability of linear orderings, Ph.D. Thesis, Rutgers Univ., 1976.

[L]

M. Lerman, On recursive linear orderings, In: Logic Year 1979-80; at the University of Connecticut, Lerman, Schmerl , Soare, Eds., Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, to appear.

[LSI

, and J. Schmerl, Theories with recursive.models, J. Symbolic Logic 44 (1979), 59-76.

[PI

M. Peretyat'kin, Every recursively enumerable extension of a theory of linear order has a constructible model, Algebra i Logika 12 (1973), 211-219.

[Rl

J. Rosenstein, Linear Orderings, Academic Press, New York, to appear.

[Wl

R. Watnick, Recursive and constructive linear orderings, Ph.D. Thesis, Rutgers Univ., 1979.

PATRAS LOGIC SYMPOSION G . MetaMdex led.) 0North-Holland Publishing Company,1982

137

EXTERNAL AND INTERNAL ALGORITHMS ON THE CONTINUOUS FUNCTIONALS Dag Normann U n i v e r s i t y o f Oslo Oslo, Norway

1.

INTRODUCTION

Recursion t h e o r y i s t h e t h e o r y o f a l g o r i t h m s .

Over t h e n a t u r a l numbers o r

t h e h e r e d i t a r i l y f i n i t e s e t s t h e a l g o r i t h m s a r e w e l l understood.

There a r e

m a t h e m a t i c a l l y p r e c i s e d e f i n i t i o n s o f t h e p a r t i a l r e c u r s i v e f u n c t i o n s which a l l c o i n c i d e and which a r e g e n e r a l l y accepted as a good mathematical model f o r e f f e c t i v e computa ti ons. I n g e n e r a l i z e d r e c u r s i o n t h e o r y a l g o r i t h m s a r e extended o r g e n e r a l i z e d t o o t h e r domains such as a l g e b r a i c s t r u c t u r e s , a d m i s s i b l e s e t s , h i g h e r t y p e f u n c t i o n als etc.

There i s a l a r g e v a r i e t y o f g e n e r a l i z a t i o n s r a n g i n g f r o m e f f e c t i v e group

t h e o r y t o r e c u r s i o n o v e r i n a d m i s s i b l e , uncountable l i m i t o r d i n a l s .

Moreover t h e r e

i s sometimes debate on t h e c h o i c e o f a g e n e r a l i z a t i o n o f a l g o r i t h m s t o a c e r t a i n structure. I n modern g e n e r a l i z e d r e c u r s i o n t h e o r y s i m i l a r i t i e s between t h e v a r i o u s g e n e r a l i z a t i o n s a r e s t u d i e d and d i f f e r e n c e s a r e c l a s s i f i e d .

This also involves

an a x i o m a t i c approach t o r e c u r s i o n t h e o r y , and v a r i o u s axiom systems f o r precomputation t h e o r i e s and computation t h e o r i e s have been g i v e n .

See Fenstad [41

f o r a general survey. I n t h i s paper we w i l l d i s c u s s a d i s t i n c . t i o n t h a t has n o t been c l e a r i n t h e r e c e n t d i s c u s s i o n s o f g e n e r a l i z e d r e c u r s i o n t h e o r y , t h e d i s t i n c t i o n between i n t e r n a l and e x t e r n a l a l g o r i t h m s o v e r a s t r u c t u r e . A g e n e r a l i z e d s e t o f a l g o r i t h m s o r e f f e c t i v e o p e r a t o r s i s i n t e r n a l if i t i s generated f r o m t h e o p e r a t o r s o f t h e s t r u c t u r e and some (imagined) computation power.

As examples of t h i s k i n d o f g e n e r a l i z a t i o n s we w i l l mention K l e e n e ' s [81

computations i n h i g h e r t y p e f u n c t i o n a l s ,

a- and @ - r e c u r s i o n t h e o r y , s e t - r e c u r s i o n

D.NORMA"

138

[15] and f i n i t e a l g o r i t h m i c procedures o v e r a l g e b r a i c s t r u c t u r e s (see Tucker [161 f o r a recent survey). Moreover t h e computation t h e o r i e s o f Moschovakis [ll]and Fenstad [4] o r t h e i n d u c t i o n - a l g e b r a s o f Moschovakis [ I 2 1 c l e a r l y a r e a x i o m a t i z a t i o n s o f i n t e r n a l generalizations o f recursion theory. The e x t e r n a l n o t i o n o f e f f e c t i v e w i l l be used when t h e s t r u c t u r e has a n a t u r a l coding as a subset

A

N o r as a f u n c t i o n

of

f 6 NN,

o r when t h e o b j e c t s of

An o p e r a t o r on t h e s t r u c t u r e w i l l be c a l l e d

t h e s t r u c t u r e have such n a t u r a l codes.

e f f e c t i v e i f t h e r e i s a n a t u r a l code f o r t h e o p e r a t o r u n i f o r m l y r e c u r s i v e i n a code Thereby we reduce t h e problems

f o r t h e s t r u c t u r e o r t h e codings o f t h e o b j e c t s .

o f e f f e c t i v e n e s s t o a concept we a r e f a m i l i a r w i t h .

Moreover, t h e a l g o r i t h m s w i l l

work o n c o n c r e t e o b j e c t s l i k e n a t u r a l numbers and n o t on a b s t r a c t elements o f , say, some a l g e b r a i c s t r u c t u r e .

The t r a d i t i o n s o f e f f e c t i v e a l g e b r a and t h e t h e o r y of

If a

enumerations belong t o t h e e x t e r n a l g e n e r a l i z a t i o n s of r e c u r s i o n t h e o r y .

s t r u c t u r e p e r m i t s b o t h an i n t e r n a l and an e x t e r n a l g e n e r a l i z a t i o n t h e e x t e r n a l g e n e r a l i z a t i o n w i l l be t h e s t r o n g e r . I n t h i s paper we w i l l survey some r e s u l t s c o n c e r n i n g t h e d i f f e r e n c e and s i m i l a r i t y between t h e i n t e r n a l and e x t e r n a l a l g o r i t h m s o v e r t h e c o n t i n u o u s f u n c t i o n a l s . I n t h e f i n a l paragraph we w i l l d i s c u s s a q u e s t i o n o f Feferman [31 i n view o f t h e r e s u l t s and t h e d i s t i n c t i o n we m e n t i o n here. 2.

NOTATION AND BACKGROUND

H i g h e r t y p e o b j e c t s a r e e s s e n t i a l l y uncountable and may i n general n o t b e coded b y f u n c t i o n s

f:N

+

N.

A continuous map

F: N

+

N w i l l however be

t o t a l l y decided by C(B,n);

B

i s a b a s i c neighborhood on which

T h i s s e t may be coded b y a f u n c t i o n

F

[lo]

defined a hierarchy

o f c o u n t a b l e o r continuous f u n c t i o n a l s , where each

and each

$

+

nl.

f.

I n d e p e n d e n t l y Kleene 191 and K r e i s e l

$: Ct(k)

has c o n s t a n t v a l u e

$

E Ct(k+l)

(Ct(k)& i s a map

IN

has codes, c a l l e d a s s o c i a t e s , w i t h t h e f o l l o w i n g e s s e n t i a l p r o p e r t y :

External and Internal Algorithms on the Continous Functionals I f 11, E C t ( k + l ) resp. t h e n

and

0f

a

Ct(k),

6

and

i s u n i f o r m l y computable i n a

$(Q)

139

are associates f o r

$

and

$

6.

and

For a d e t a i l e d d e f i n i t i o n see t h e o r i g i n a l papers o r Hyland [6] o r Normann U31. The i n t e r n a l n o t i o n o f computation i s g i v e n b y K l e e n e ' s Sl-S9 f r o m Kleene [El, t h e e x t e r n a l n o t i o n o f r e c u r s i o n i s g i v e n by u n i f o r m T u r i n g - r e c u r s i o n on t h e associates. The purpose o f t h i s paper i s t o d i s c u s s t h e d i s t i n c t i o n s and connections between these two n o t i o n s . and j u s t g i v e

As t h e paper i s a survey we w i l l l e a v e o u t t h e p r o o f s

references.

We denote Kleene-computations by

{el(@,

Qn) = t

,-.,

where

e

i s t h e index

f o r t h e a l g o r i t h m used. The r e c u r s i o n s a r e denoted [el($l

,..,

which means t h a t $1,.

. ,@n

Iel(al

= t

,.., an)

= t

whenever

a

a

l,.., n

are associates f o r

resp.

Kleene [91 showed t h a t t h e r e i s a r e c u r s i v e f u n c t i o n

p

such t h a t f o r a l l

$1,. . ,@nst {el(&,

, - - 9

= t

,.

[P(~)I(Q~

= t

The p r o o f i s by c o n s t r u c t i n g a u n i v e r s a l a s s o c i a t e f o r a l l computations.

The

consequences were THEOREM 1

(Kleene [91)

a If $

is computable, then

b

is computable in

If

Q

$

is recursive, i.e. has a recursive associate.

11, then Q

cursive map sending any associate for

is recursive in $

9, i.e., there is a re-

to an associate for

4.

T h i s theorem f o l l o w s t h e general p a t t e r n , t h e e x t e r n a l n o t i o n i s s t r o n g e r than t h e i n t e r n a l n o t i o n . T a i t showed i n an u n p u b l i s h e d n o t e t h a t i t i s s t r i c t l y s t r o n g e r . ?HEOREM 2

(Tait)

The fan-functional

8

is recursive but not computable from any

f E NN.

D.NORMA"

140

The f a n - f u n c t i o n a l

i s a type-3 o b j e c t t e s t i n g t h e modulus o f u n i f o r m c o n t i n -

u i t y o f a type-2 f u n c t i o n a l o v e r a compact s e t .

As any element o f

Ct(2)

is

u n i f o r m l y computable from any o f i t s a s s o c i a t e s t h i s i s t h e f i r s t l e v e l where a counterexample may be found. Normann [13].

F o r a d e t a i l e d p r o o f see Gandy-Hyland

[51 o r

The u n d e r l a y i n g i d e a o f t h e p r o o f i s t h a t a computation i n

w i l l only involve evaluations o f f u n c t i o n a l wants t o know

F

F

at

F,g-computable

F,g

points, w h i l e the fan-

o v e r a compact s e t .

F o r a w h i l e t h e f a n - f u n c t i o n a l seemed t o be e s s e n t i a l l y t h e o n l y noncomputable r e c u r s i v e f u n c t i o n a l and t h e c o n j e c t u r e t h a t e v e r y r e c u r s i v e f u n c t i o n a l i s computable i n # the functional and any

r

f

r

B u t t h e n R.Q.

e n j o y e d l i f e f o r some years. and

r

J.M.E. Hyland showed t h a t

Gandy d i s c o v e r e d

i s n o t computable f r o m

@

t e l l s us t h a t t h e r e s h o u l d be an i n d e x f o r

r

E NN.

i s bar-recursively defined by

* ** where

r(F) = n

if

F

i s constant

r ( F ) = FO(Xnr(Fn+,))

Fn(f) = F(n*f),

*

n

otherwise

i s concatenation.

The f u n c t i o n a l - e q u a t i o n

**

b y t h e r e c u r s i o n theorem. The i n d e x e x i s t s , b u t d e f i n e s t h e everywhere u n d e f i n e d o b j e c t .

However, if we

use K l e e n e ' s c o n s t r u c t i o n f r o m Theorem 1 on t h i s i n d e x we w i l l g e t a r e c u r s i v e associate f o r

I'.

T h i s shows t h a t K l e e n e ' s r e d u c t i o n i s n o t a r e d u c t i o n i n t h e

s t a n d a r d sense, t h e i m p l i c a t i o n l e a d i n g t o Theorem 1 i s n o t an e q u i v a l e n c e . F u r t h e r s t e p s i n e s t a b l i s h i n g t h e gap between c o m p u t a b i l i t y and r e c u r s i v e n e s s were t a k e n i n i n v e s t i g a t i n g t h e c o m p l e x i t y o f s e m i - c o m p u t a b i l i t y and t h e semirecursive relations.

A r e l a t i o n i s s e m i - a l g o r i t h m i c i f i t i s t h e domain of some

p a r t i a l algorithm. It follows d i r e c t l y from the d e f i n i t i o n t h a t f o r

o f associates f o r functionals o f type i s t h e same as o r d i n a r y r e c u r s i o n . plexity

1

k+l

is

1

k 2 1

IIk.F o r

k=l

Kleene-computability o f

I$.B e r g s t r a [l]showed t h a t f o r

k>2

the s e t

As(k+l)

countable recursion

Ct(2)

i s of.com-

t h e Kleene-computations o v e r

141

External and Internal Algorithms on the Continous Functionals Ct(k)

i s o f complexity

1

IIkk-l.

These e s t i m a t e s o f c o m p l e x i t y cannot be improved by t h e f o l l o w i n g r e s u l t s . THEOREM 3

a (Kleene) All

Il1 -subsets of N N 1

are uniformly semi-computable in any typel2

functional b

(Bergstra [2]) All

II12-subsets of N N are uniformly semi-computable in any type 4 func-

tional

c (Normann [131) 1 all ll -subsets of k

For k > 2,

continuous type k + 2 THEOREM 4 For

NN

are uniformly semi-computable in any

NH

are uniformly semi-recursive in any

functional.

(Hyland [7]) 1 all ll -subsets of k

k l 2,

continuous functional type k.

These r e s u l t s show t h a t t h e r e i s a c l e a r gap i n c o m p l e x i t y between t h e c l a s s e s o f computations and r e c u r s i o n s . this

The n e x t r e s u l t i s another example of

.

THEOREM 5

(Normann)

For each

kl3

for all @ECt(k)

there is a recursive operator

J (@) is not computable in @ k

Jk:Ct(k)

and any

+

Ct(k)

such that

$ECt(k-l)

REMARK

The p r o o f i s by t h e same method as t h e p r o o f o f Theorem 12 i n Normann [141. That theorem i s a s l i g h t 1 . y weaker r e s u l t . 3.

CLOSING THE GAP

I n t h i s s e c t i o n we w i l l see t h a t c o u n t a b l e r e c u r s i o n i n a c e r t a i n sense can be c a p t u r e d by K l e e n e ' s Sl-S9 when r e l a t i v i z e d t o a s u i t a b l e i n f i n i t e l i s t of functionals.

F i r s t we w i l l see t h a t Theorem 5 seems t o be t h e b e s t p o s s i b l e r e -

sult i n i t s line. THEOREM 6

(Normann [141)

D.NORMA"

142

k20.

Let $€Ct(k)

There is a recursive functional

is uniformly computable in

Pk

PkECt(k+l)

such that every $.

and any associate for

From theorems 3 and 4 we see t h a t i f we want t o i m i t a t e countable r e c u r s i o n by computations i n a f u n c t i o n a l we must a t l e a s t move two types up i n the hierarchy. But t h i s i s a l s o s u f f i c i e n t . THEOREM 7

(Normann [ 1 4 ] )

k12

For each

there is a recursive functional

recursive function

such that for all

pk

.

[el (@l,. ,tJn) (c

LI

,..,On

QkECt(k+2)

and a primitive

E Ct(2k)

$1

I P p } ( @ k , $ l , . .,$J

means both sides undefined or both sides defined and equal).

If

The proofs o f the two theorems are by the same method.

takes a value we have an approximation t o t h i s value u n i f o r m l y p r i m i t i v e r e c u r s i v e

in $

The f u n c t i o n a l

%,..,@n.

E Ct(k+l).

ak takes as arguments e +

There w i l l be a computable r e l a t i o n

R

[el ($1

,..,$n)

takes no value => V @ c C t ( k + l htlR(e,$l

[el($?

,..,)$,,

takes a value => apXt(k+l)YtR(e,$,

where

and a

such t h a t

,. . $,@ , )t,

,.-,$,,$.t)

w i l l be u n i f o r m l y computable i n e,$ ,. . ,@n.

$

We l e t

,. . ,Qn

Qk(e,@l

,.., $,$)

,.., $),

be the l i m i t o f approximation t o

if

w h i l e i t w i l l otherwise be t h e t ' t h element o f t h e approxi-

VtR(e,$l,..,$n$,t) mation f o r the f i r s t 1R(edl

t

such t h a t

,..,$n,$,t)

This i s a rough sketch of the c o n s t r u c t i o n o f

A d e t a i l e d proof i s t ~ ~ .

given i n Normann [14]. COROLLARY

Uniformly i n

k

there i s a recursive functional

countable r e c u r s i o n and Kleene-computability i n bk REMARKS

#k€Ct(k+E)

such t h a t

are e q u i v a l e n t over

Ct(k).

Theorem 4 i s a l s o used t o prove t h i s c o r o l l a r y .

By e q u i v a l e n t we mean t h a t t h e r e a r e r e c u r s i v e f u n c t i o n s

that f o r a l l

e,@l

I . . ,

@,,

E Ct(k)

p1

and

p2

such

External and Internal Algorithms on the Continous Functionals

143

[el ($1 ,..,0,) .- b1W Q k , t J 1 , .. ,@J I e } ( @ k , @ l ., .,t~,,) 4.

= [ P ~ ( ~( $) lI , . .,@,,I

I S COUNTABLE RECURSION INDUCTIVE?

Feferman [31 asked i f countable recursion i s inductive, and the question was repeated in Hyland 171.

In a sense Theorem 7 and i t s corollary gives a

positive answer t o t h i s problem, countable recursion i s equivalent to a computation- theory defined from schemes.

B u t t h i s i s equivalence i n the sense of the final remark of the previous section, which i s an equivalence between precomputation theories, see Moschovakis [ l l ] and Fenstad (41.

I n a precomputation theory we do n o t have sub-computations

or lengths of computations l i k e i n a computation theory, we j u s t have the most elementary computation s t r u c t u r e . Countable recursion i s a precomputation theory and we have shown t h a t i t i s equivalent, as such, t o a computation theory.

B u t t h i s will n o t turn countable

recursion into a computation theory. The background f o r Feferman's Question was a feeling t h a t countable recursion i s not captured by the recent attempts t o axiomatize general recursion theory, see Moschovakis [ l l ] , Fenstad [4], Moschovakis [12] and Feferman [3].

As the reduc-

tion of recursions t o computations adds no computation-structure t o the s e t of recursions, we must agree t h a t Theorem 7 i s not a positive answer t o the problem.

B u t the motivation behind computation theories or the induction algebras of Moschovakis [121 i s c l e a r l y t o axiomatize internal generalizations of recursion theory so there i s no reason t o expect t h a t they will capture the external variants like countable recursion. Based on these considerations we feel t h a t Theorem 7 i s the best positive answer t h a t may be given t o Feferman's question.

This does n o t indicate a f a i l u r e

of the abstract approach since i t was never intended t o cover the situation of countable recursion.

D.NORMA"

144

B e r g s t r a , J.A.: C o m p u t a b i l i t y and c o n t i n u i t y i n f i n i t e types, Thesis, U n i v e r s i t y o f U t r e c h t , 1976. B e r g s t r a , J.A.: 1-envelopes o f Continuous F u n c t i o n a l s , R e c u r s i v e F u n c t i o n Theory: N e w s l e t t e r No. 16 (1977), I t e m 196. Feferman, S: I n d u c t i v e schemata and r e c u r s i v e l y c o n t i n u o u s f u n c t i o n a l s , i n R.O. Gandy and J.M.E. Hyland ( e d s . ) , L o g i c Colloquium ' 7 6 , NorthHol 1and (1977), 373-392. 4.

Fenstad, J.E.:

General R e c u r s i o n Theory, S p r i n g e r Verlag, 1980.

5.

Gandy, R.O. and Hyland, J.M.E.: Computable and r e c u r s i v e l y c o u n t a b l e f u n c t i o n s o f h i g h e r t y p e , i n R.O. Gandy and J.M.E. Hyland ( e d s . ) L o g i c Colloquium '76, N o r t h - H o l l a n d (1977), 407-438.

6.

Hyland, J.M.E.: F i l t e r s p a c e s and c o n t i n u o u s f u n c t i o n a l s , Ann. Math. Log. 16 (1979) 101-143.

7.

Hyland, J.M.E.: The i n t r i n s i c r e c u r s i o n t h e o r y on t h e c o u n t a b l e o r continuous Generalf u n c t i o n s , i n J.E. Fenstad, R.O. Gandy and G.E. Sacks (eds.): i z e d R e c u r s i o n Theory 11, N o r t h - H o l l a n d 1978, 135-145.

8.

Kleene, S.C.: T.A.M.S.

9.

Kleene, S.C.: Countable f u n c t i o n a l s , i n A. H e y t i n g ( e d . ) C o n s t r u c t i v i t y i n Mathematics, N o r t h - H o l l a n d (1959) 81-100.

10.

K r e i s e l , G.: I n t e r p r e t a t i o n o f a n a l y s i s by means o f f u n c t i o n a l s o f f i n i t e t y p e , i n A . H e y t i n g ( e d . ) , C o n s t r u c t i v i t y i n Mathematics, N o r t h - H o l l a n d (1959) 101-128.

11.

Axioms f o r comoutation t h e o r i e s - f i r s t d r a f t . i n R.O. Moschovakis. Y.N.: Gandy and C.E.M. Yates, ( e d s . j : L o g i c Colloquium ' 6 9 , N o r t h l H o l l a n d ( 1 971 ) 199-255.

12.

On t h e b a s i c n o t i o n s i n t h e t h e o r y o f i n d u c t i o n , i n Moschovakis, Y.N.: R.E. B u t t s and J . H i n t i k k a , L o g i c , Foundations o f Mathematics and C o m p u t a b i l i t y Theory, 0. R e i d e l P u b l i s h i n g Company 1977, 207-236.

13.

Normann, D.: Recursion on t h e Countable F u n c t i o n a l s , S p r i n g e r L e c t u r e Notes i n Mathematics No. 811, S p r i n g e r V e r l a g 1980.

14.

Normann, D.: The Continuous F u n c t i o n a l s ; Computations, r e c u r s i o n s and degrees, P r e p r i n t S e r i e s i n Mathematics No. 2, 1980, U n i v e r s i t y o f Oslo.

15.

S e t Recursion, i n J.E. Fenstad, R.O. Gandy and G.E. Sacks ( e d s . ) Normann, D.: G e n e r a l i z e d R e c u r s i o n Theory 11, N o r t h - H o l l a n d 1978, 303-320.

16.

Tucker, J.V.: Computing i n A l g e b r a i c Systems, i n F.R. Drake and S.S. Wainer ( e d s . ) : Recursion Theory: i t s G e n e r a l i z a t i o n s and A p p l i c a t i o n s , Cambridge U n i v e r s i t y Press 1980.

R e c u r s i v e f u n c t i o n a l s and Q u a n t i f i e r s o f f i n i t e types I , 91 (1959), 1-52; and I1 (1963), 106-142.

r n i - ~ ~~ W V I C. ) I I I I ~ V J I V I V

C . Metakkia (ed.) 0North-Holland Publiahing Company. 1982

145

THE TURING DEGREES AND THE METADEGREES HAVE ISOMORPHIC CONES* Sy D. Friedman

MIT Cambridge, MA 02139

I n t h i s paper we study t h e g l o b a l s t r u c t u r e o f the of

8-degrees i s a s e t o f

@-degrees o f the form

A

8-degrees.

(cI(d2do)

-

2

f o r some f i x e d

The base o f t h i s cone i s t h e 8do, p a r t i a l l y ordered by < -8' degree do. Our main r e s u l t i s t h a t i f B i s countable then the Turing degrees and t h e 8-degrees have isomorphic cones. I f 8 = u i k thenXhe 8B-degree

degrees are t h e metadegrees and i n t h i s case the cone o f metadegrees w i t h base 0 ' = complete meta-RE degree i s isomorphic t o the cone of T u r i n g degrees w i t h base the Turing degree o f Kleene's If

V = L we a l s o c o n s t r u c t some isomorphisms i n t h e uncountable case

(sections two and t h r e e ) . < K-closed then the

If 8

6:

If

8 has r e g u l a r c a r d i n a l i t y

8-degrees and the

has s i n g u l a r c a r d i n a l i t y

K

i s t h a t t h e r e e x i s t a tame i n j e c t i o n the p r o p e r t y t h a t

g-l

P, F

S8

and

Sg

is

K-degrees have isomorphic cones.

then some cone o f

isomorphic t o a cone i n t h e r e g u l a r

K

8-degrees. S ->

B

f o r each y <

K;

K-degrees i s o f t e n

The hypothesis i n t h i s case i.e.,

an i n j e c t i o n g w i t h

K.

Much o f t h i s work ' i s based on techniques o f Maass from Maass [19781 and Maass 119791. a.b,cl

The uncountable case draws h e a v i l y on Friedman [1981

which p r o v i d e " f i n e s t r u c t u r e " c h a r a c t e r i z a t i o n s of t h e hypotheses

used. These r e s u l t s are i n f a c t an a p p l i c a t i o n o f ideas from inadmissible recursion theory t o degree theory on admissible (and i n a d m i s s i b l e ) o r d i n a l s . *The p r e p a r a t i o n of t h i s paper was supported by NSF Grant #MCS 7906084

S.D. FRIEDMAN

146

The isomorphism mentioned between a cone o f T u r i n g degrees and a cone o f metadegrees c o n v e r t s t h e T u r i n g jump i n t o t h e

weak metajump,

a notion which f i r s t

arose o u t o f t h e s t u d y o f i n a d m i s s i b l e o r d i n a l s . SECTION ONE

COUNTABLE CARDINALITY

B = (S ,B) B

Suppose

i s amenable; i . e . ,

8

n

y < B.

f o r each

Sy 6 SB

Then

we d e f i n e : = least

llcf(B)

s.t.

y

11proj(8) = l e a s t y If

Ilcf(B)

= llproj(B)

there i s a

s.t.

then there i s a

= w

collapse o f THEOREM 1.

A

=

ll(Sw,A).

is

C

11(8)

A 5w

f:B

bijection

s.t.

The s t r u c t u r e

A

f:y

=

C

for (sw,A)

is amenable and

B

8.

be the Maass collapse of

the Turing degrees

+

y.

g:B

c_ w ,

-+

w

C

i s I1(B)

i s c a l l e d t h e Maass

2

Turing degree (A)

Ilcf(B) =

l1 proj(8)

For

C,D 5 SB

IDlC

2

write

2 Cl.

C

5

iff

D

= w.

Then there is an isomorphism and the

B-degrees

2

C _I

D v

between

(B).

B-degree:

B. Then 8-degree (C)

The A-degrees a r e d e f i n e d s i m i l a r l y .

Let j

B-degree

I n t h e p r o o f i t w i l l be c o n v e n i e n t t o r e f e r t o t h e n o t i o n o f

D,D

B

+

B. Suppose = (S ,B)

(S ,A)

unbounded

there i s a 11(8)injection

and i n t h i s case Maass [1978] c o n s t r u c t s i f and o n l y i f

I1(B)

=

I n t h i s terminology

Theorem 1 a s s e r t s t h a t t h e A-degrees and t h e 8-degrees a r e i s o m o r p h i c . Define

C c_w

n i t e B - f i n i t e set. LEMMA 2 .

Each

t o be B-bi-immune i f n e i t h e r

C

n o r w-C

c o n t a i n s an i n f i -

The f o l l o w i n g lemma i s proved on pages 154-5 o f Maass [19781.

A-degree contains a 6-bi-immune representative.

The s i g n i f i c a n c e o f 6 - b i - i m m u n i t y i s t h i s : on B-bi-immune s e t s .

The r e d u c i b i l i t i e s

5 A,(

agree

For, by t h e key p r o p e r t y o f t h e Maass c o l l a p s e t h e r e d u c t i o n

procedures c o i n c i d e f o r these r e d u c i b i l i t i e s and B-bi-immunity i n s u r e s t h a t t h e neighborhood c o n d i t i o n s a r e a l s o t h e same. P r o o f o f Theorem 1.

Define

f o r any B-bi-imnune

D E

ism.

We show t h a t

j

4.

j:

A-degrees ->

8-degrees by

j ( 4 ) = 8-degree (D)

By t h e above remarks t h i s i s a w e l l - d e f i n e d monomorph-

i s onto.

To do so we w i l l make use o f t h e r e d u c i b i l i t i e s :

147

The Turing Degrees and the Metadegrees have Isomorphic Cones

Choose a B - r e c u r s i v e ( t h a t i s , I 1 ( B ) )

bijection h

E 5S

Given

B-degree.

we c o n s t r u c t a 6-bi-immune

6

E

f i n a l l y choose

E

cw

5

C,D 5 W , C

A D

iff

E

5

E** < fB

f B E**,

fB 0 ) .

C

h

is

2

E*

E** = g[E*l

fE E**

and

and

E**. E**

5

h

fB E

Putting t h i s chain of, reducih

5

we see t h a t

E* and E*

h

E

Then l e t

have t h e same

5

E

E.

h

h

Moreover

E,E

t o be any 8-bi-immune s e t o f t h e same A-degree as

b i l i t i e s t o g e t h e r and u s i n g t h e d e f i n i t i o n o f

Since

such t h a t

I z c : S B ( z z E I v { z c S B t z n E E = 0 } Zfs E*,

( s i n c e f o r any

w.

h

E* = { z E S 8 1 z ~ E Iv I z E S g I z n E + O 1 .

Let

Then

g:SB <--a

B-bi-immune we have

E

5

E.

E.

This vields

E

5

fB E.

1-

The a p p l i c a b i l i t y o f Theorem 1 depends on t h e p o s s i b i l i t y , g i v e n a eountable

6, o f choosing B 5 SB such t h a t B

l i m i t ordinal

llc f ( 8 )

=

l1 proj(B)

THEOREM 3.

such that

Proof.

B

=

f: 8 <->

(SB,B)

COROLLARY 4 .

Moreover

B = wlCK

we have:

The Turing degrees

degrees

L 0'

Proof.

There i s a A ( L

choose a Theorem 1. iff A

5

zT 0' s i n c e

= i

f

is

2 Turing

l1 proj(B)

11(S8,B)

and

degree (@

B

LfB f

= w.

il. Then B

for all

SB, B

f.

nS

1-

Y

i s the

are isomorphic to the meta-

wlCK

and

w.

By Theorem 3 we can

CK such t h a t (L CK,B) obeys t h e h y p o t h e s i s o f w1 w1 A2 master code f o r L CK; i . e . , A SI,!i s A ( L CK) w1 ( f o r a p r o o f see Jockusch-Simpson [1976]). I f (Lw,A) i s the

A

(L is

B

CK,

w1

5L

B v C)

A*(L

Theorem 1 t h e T u r i n g degrees degrees

cf(B) =

b i j e c t i o n between

CK)

CK) s e t w1 But Q i s a

Maass c o l l a p s e of A

1,

5

B

There is

= the complete meta-RE degree.

A (L

0

is a bijection.

is amenable and

y < B.

finite for a l l I n case

w

6 = {(6o,...,Bn)\f(Bi)

Define

i s amenable and

B

Our n e x t r e s u l t says t h a t t h i s i s always p o s s i b l e .

= w.

Suppose

= ( S ,B)

2 metadegree

where

CK) and

2 Turing

(B v C ) = 0 ' .

For other o r d i n a l s

B <

U:

C = a r e g u l a r complete meta-RE s e t , t h e n

O'zT A

degree (@)

A (L

CK, C). So by w1 a r e i s o m o r p h i c t o t h e meta-

since

U' i s

-1

we can choose t h e l e a s t p a i r

(y,n)

so t h a t

S.D. FRIEDMAN

148

there i s a degree(A) imnune

A (S ) b i j e c t i o n B <--A w. Then t h e T u r i n g degrees 2 T u r i n g n Y a r e i s o m o r p h i c t o t h e B-degrees B-degree (A) where A i s a B - b i -

master code f o r

An

Y'

2

isomorphic t o t h e T u r i n g degrees

O(B)

O(B)

Ha

i s j u s t t h e T u r i n g degree o f

greater than

then the

w

A-degrees

> @(n) = nth t o t h e T u r i n g degrees -

where

1. A-degree

6-degree

Felative t o

if C

i s tamely

of

on t h e

0.

hyperjump o f

A-RE

If

j

D E

I f 8 = (S ,B)

D.

8-RE

If

then the

A = U wnCK n

where

1. T u r i n g

&w) =

o f Theorem 1 has on r e l a t i v e

j

i s amenable

B

relative to

B.

C

then

i s tamely

8-RE

The f o l l o w i n g r e s u l t d e s c r i b e s t h e range

degrees. is the isomorphism of Theorem 1 then an A-degree

j(g)

set iff

If

Proof.

admissible

C,D c_ SB t h e n C i s tamely 6-RE r e l a t i v e t o D i f I z E S B t z _ c C }

If

A-RE

a E ~ 9 i 9s a n o t a t i o n f o r

(complete A-RE s e t ) a r e i s o m o r p h i c

We now d i s c u s s what e f f e c t t h e isomorphism

THEOREM 5.

(The

A = w+B.

(e,&2),&(3) ,...>.

recursive j o i n o f

i s B-RE

where

A = unCK= nth

If

A-degrees a r e isomorphic t o t h e T u r i n g degrees

RE-ness.

0,

Bth jump o f

=

B and Ha i s t h e a s s o c i a t e d Kleene H-set.)

an

i s the largest

such t h a t b o t h { z E S I z E_ D l and I z E S I z n D = 01 a r e B B B F o r example, if A i s a r e c u r s i v e l i m i t o r d i n a l t h e n t h e A-degrees a r e

1n~(S 1.

j

A

D c_ S

of a s e t

degree

The 6-degree o f

S

8-RE

contains a tamely

contains

set.

h

d

is

A-RE

t h e n so i s

D, t h e c a n o n i c a l B-bi-immune s e t o f t h e h

same A-degree as

D o b t a i n e d f r o m Maass [1978].

h

since

h

D i s 8-RE

c o n t a i n s t h e tamely Conversely i f

Iz E S I z 5 0 1

and

B

8-RE E

set

5 SB

8-degree as a 5-bi-imnune proof o f Theorem 1.

As

E

Iz E

D i s tamely 8-RE h

S Iz

B

z 501.

i s finite,

So

j(d)

0.

i s tamely A-RE

=

But then

set.

i s tamely

8-RE

t h e n we show t h a t

E

Consider t h e s e t s

E*,E**,

8-RE

E*

we see t h a t

has t h e same E

is

from the

8-RE.

Hence

E**

h

is

8-RE

and t h e r e f o r e

A-RE.

Once again,

E

i s also

as t h e c a n o n i c a l B-bi-immune s e t o f t h e same A-degree as h

Theorem 1 showed t h a t Note t h a t i f

E,E

8 = ( S ,B)

B

have t h e same 8-degree.

A-RE

as i t i s o b t a i n e d

E**.

The p r o o f o f

1-

s a t i s f i e s t h e h y p o t h e s i s o f Theorem 1 t h e n e v e r y

8-degree i s r e g u l a r ; t h a t i s , e v e r y 8-degree c o n t a i n s a s e t

C

such t h a t

(SB,C)

The Turing Degrees and the Metadegrees have Isomorphic Cones i s amenable. then any

E

B i s an o r d e r - p r e s e r v i n g unbounded I 1 ( B )

For, if f : w ->

h

has t h e same B-degree as

5 SB

149

.,

E

where

f [El,

comes f r o m t h e

proof of Theorem 1.

As a r e s u l t Theorem 5 can be a p p l i e d t o y i e l d :

COROLLARY 6.

is the isomorphism of Theorem 1 then for A-degrees

dl

A-RE

is

Proof. (SB,C) j

j

If

relative to

d

J u s t note t h a t i f

2

<->

is tamely B-RE relative to

j(d ) 1

i s tamely

2 d2).

C

Thus by Theorem 5 j(d )

RE <->

(SB,C)-

€3-degree

j(d2),

B-RE degrees.

i s A-RE

dl

relative to

i s tamely B-RE r e l a t i v e t o

1

The answer comes f r o m work o f Maass [1979].

paper i m p l i e s t h a t i n case

B

(S

=

B

a B-degrec. c o n t a i n s a r e g u l a r B-RE

set (i.e.,

set

a B-RE

E

&

D.

set

C

is

Proof.

B-RE

set.

B

%RE

and l e t

4

iff

SB

A-RE

is

i s B-RE

= g$?*].

D

Then

r e l a t i v e t o t h e A-RE is

A-RE

set

relative to

o r (g-’(w) -1 9 ( x ) €2) relative to

D

Thus t h e

is

B-RE relative

=

(1 ,z),

or

3x

g[Cl

2,

is the isomorphism of Theorem 1 then for any A-degree

relative to some

A-RE

degree

g: S

c5

such t h a t :

We g i v e Maass’ p r o o f s p e c i a l i z e d t o t h e p r e s e n t c o n t e x t .

If, j

Let

(SB,C)

s a t i s f y i n g the conditions o f

<>w be a I 1 ( B ) b i j e c t i o n and f o r B canonical B-bi-immune s e t of t h e same A-degree as D. If

S

such t h a t

t o t h e A-degrees which a r e A-RE

j

Maass [1979], t h i s a l s o works f o r any s t r u c t u r e

THEOREM 7.

o f the

j

As Maass remarks i n s t a t e m e n t 3 ) a f t e r Theorem 1 o f

t o an A-RE predecessor.

our Theorem 1.

-1

j(d2).

Our remark a f t e r Theorem 5

i m p l i e s t h a t each B-RE degree c o n t a i n s a r e g u l a r degrees i n t h i s case correspond under

j(dl)

obeys t h e h y p o t h e s i s o f o u r Theorem 1 t h e n

J)

E f o r some A-RE

relative to

d2 <->

Theorem 1 of t h a t

i s amenable) if and o n l y i f i t c o n t a i n s a R-bi-immune s e t

j(d)

j(d ) . 2

(S ,C)

We end t h i s s e c t i o n by d e t e r m i n i n g t h e i n v e r s e image under

A-RE

dl,d2:

amenable, t h e n B obeys t h e h y p o t h e s i s o f Theorem 1 and t h e r e s u l t i n g isomorphism i s

r (A-degrees

has

function

C =

n

C* =

Iz

D

and

B

g[C]:

g[Cl.

z

4

let

g[C](g-’(w)

sB I z n

C

D 5 w l e t D be t h e

+ 01 v { z

D i s B-bi-immune.

n

E SBlz

But

D

(SB-C)

is

w E g[C*l <>-

+ 01

<->

= (1,z)

[ ( g - l ( w ) = (o,z),

[3x E g[CI(g-’(w)

and

g-l(x) E z)].

+ 0)

A-RE

To see t h i s , i t s u f f i c e s t o show t h a t

But

(SB-C)

E

e Zg.

z

g[C*l

n

C

+ 0)

= ( 0 , ~ )and

So

u s i n g o n l y f i n i t e neighborhood c o n d i t i o n s on

g[C*] g[Cl.

is Thus

B-RE

S.D.FRIEDMAN

150

gK*l gtC1

is

A-RE

relative to

relative to

A-RE

f E S l s E w) f: w

D,E

+

a

E.

,!,(A)

is

zfB g[C*l

co-B-RE

n

as

DS+l

E D'+'

4

Choose a

E.

C

To show t h a t where

n

D

s.t.

E

t

5

n = DS

ns

enough

t

and

i s 6-bi-immune.

s

=

E

x

n

s,t =

t

n

s =

s.t.

D

Lw

0:

zfB C

E

B-jump o f

x

Lw

D and

b.

3s,K,tELw((f(n)s,K,t)EC)

As

i s 8-bi-immune t h i s gives

c_ SB-C.

D

nS

C

n n.

one need o n l y know a f(n) This can be determined from D n n given a l a r g e

n s.

Thus

C <

4

D v E

and we are done since

b

B-RE

relative to

For any

Some Remarks

1)

A-degree

recursive ordinal). theorems h o l d i n in

A,

B-RE j(cJ')

3

D

This B-degree i s c a l l e d the weak

(b)) = b'

relative to

b.

Thus we have:

B-jump(j(9))

= weak

=

and

= j@)'

j@")

Theorem 7 e s t a b l i s h e s the d e n s i t y and s p l i t t i n g theorems f o r

8-RE degrees whenever

RE

b.

E

there i s a l a r g e s t

Theorem 7 i m p l i e s t h a t weak €3-jump (weak B-jump

COROLLARY 8 .

is

t

n c D <->

To determine

B-jump(b) = l a r g e s t B-degree which i s

2)

D

F n sl.

E n s and the c o n d i t i o n on as

1-

B-degree which i s tamely

is

3 D,

increasing, unbounded

A consequence o f C o r o l l a r y 6 i s t h a t f o r any B-degree

the

E

enumeration o f

11(8)

= E

depends on

Then

{ f f n ) } x Lu

D <->

s

E

I1(A,E)

be a

d

-D',K

t

only

(hence co-B-RE).

3 C.

stage

C

and hence

Then d e f i n e :

8.

co-A-RE and

{DSls E

enumeration of

least

D

g[Cl

S W are-.B-bi-imnune and E i s A-RE,

Let

c=~(f(n),s.K,t)ln

C

Also note t h a t

3 gtC*1 3 D. Conversely, suppose

is

gtC1.

I1 cf(B)

= I1proj(B) = o

For i t i s n o t d i f f i c u l t t o show t h a t f o r any

{Turing degree ( B ) 1A

zT B,B

is

RE

B = SB, B

( f o r example

i n some

C

Ac_w

zT B

a

these such t h a t

A).

Shore p o i n t e d out t o me t h a t another c o r o l l a r y o f t h i s work i s :

degrees w i t h jump are

not isomorphic

The Turing

t o the metadegrees w i t h metajump.

t h i s were the case then the Turing degrees between morphic t o the metadegrees between metajump (0)

0'

and

0"

For, if

would be i s o -

and metajump (metajump (0)). By

the above r e s u l t s the l a t t e r i s isomorphic t o the Turing degrees between

@'

and

~9)'. But as i n Feiner [1970] any a r i t h m e t i c a l l y presented l i n e a r o r d e r i n g can be

The Turing Degrees and the Metadegrees have Isomorphic Cones

embedded as an i n i t i a l segment o f t h e T u r i n g degrees between

O(N)

is

presentable.

which a r e n o t SECTION IWO --

O and 0",

N any topped i n i t i a l segment o f t h e degrees between 0 '

b u t f o r some l a r g e

0"

151

and

There a r e a r i t h m e t i c a l l y - p r e s e n t a b l e 1 i n e a r o r d e r i n g 5

O(N)-presentable. UNCOUNTABLE REGULAR CARDINALITY

We now i n v e s t i g a t e how t h e work o f t h e p r e c e d i n g s e c t i o n can be adapted t o

B o f cardinality

ordinals llcf(B)

= llproj(B)

C

is

a B-bi-immune s e t .

i s amenable and

such t h a t f o r

K

C

c_

C

K,

is

Moreover Maass' p r o o f o f L e n a 2' a l s o

ll(LK,A). Thus, i f

covers t h e p r e s e n t s i t u a t i o n :

B = (SB,B)

Assume

As b e f o r e t h e r e i s a A c_

= K.

if and o n l y i f

ll(B)

regular.

K,K

A = (LK,A)

t h e n each A-degree c o n t a i n s

We must now examine t h e p r o o f o f Theorem 1.

The f u n c t i o n

A-degrees ->

j:

B-degrees can be d e f i n e d as b e f o r e .

It i s

c l e a r l y a monomorphism b u t a f u r t h e r h y p o t h e s i s i s now needed t o stlow t h a t i t i s ,. Onto. Given E 5 s d e f i n e E*, E**, E as i n t h e p r o o f o f Theorem 1. Also, f o r

B

define

C1, C2-cS

B-card(zl

U z2) <

5 KB

C1

{ ( z l ,z2) Izl

iff

C2

KI ~ f B I z 1 , z 2 ) I z 1 5 C2,z2 n C2

have

E

,. E.

5 E*

E**

h

The problem i s i n showing

K

and

E z(w

w E z

for all

n

E

0)

+

Sg

THEOREM 9 .

regular. is

y <

then there i s

(similarly for

K

belongs t o

Suppose Let

(K-closed.

8

SB-E

SB.

= (SB,B)

A = (LK,A)

E

2

E we need t o have:

a r e guaranteed i f we assume t h a t

f: y

0,

SB

0,

=

U z2) <

K).

'Then as n

=

01 L ~ ~ EL *~ E * * ' ~ ~ E n

E.

If

We have

whenever

z

E SB

E $B

E**

x E LK(g

but t o i s tame).

has 8 - c a r d i n a l i t y

0 B s . t . z ' 5 E and w n 2 ' O f course b o t h o f these p r o p e r t i e s

z' E S

as w e l l ) . is

n C1

B-card(zl

g - l [XI E SB

we need t h e p r o p e r t y :

More s e r i o u s l y , t o g e t E* $ B <

=

,z2

z, 5 E, z2 n E

before one can show t h a t IJzl ,z2) lzl ,z2 E SB, and hence

5 C,

(K-Closed;

i.e.,

i f we assume t h a t any

Thus we have: i s amenable and

l,cf(B) = 11proj(B)

be the Maass collapse of

Then there i s an isomorphism j

= K,K

8. Further assume that

between the

sB

A-degrees and the

B-degrees.

Theorem 3 goes o v e r i n t h i s s i t u a t i o n i f one assumes t h e

(K-closure o f

S

B'

S.D.FRIEDMAN

152

For, i f

f : 5 <-->

t h e a m e n a b i l i t y of 6

2 y}.

(S5,B)

B &B

And

i s a b i j e c t i o n then the

K

c a r r y o v e r when

f

f,

is

S5

B = {(5,,51,...,5y)ly

where is

11(S5,B).

(K-closed,

S

(K-closure o f <

guarantees

5

f(B6) = 6 for a l l

K,

The r e m a i n i n g r e s u l t s o f s e c t i o n 1 a l s o

zfg

p r o v i d e d uses o f

&g.

a r e r e p l a c e d by

To summarize: THEOREM 10.

Suppose

Then there i s and the dl

-

is

5K

A

2

@degrees

and isomorphism

j(dl)

.

6-degree(A)

K-RE r e l a t i v e t o

Moreover

d2

-

If

iff

d3

such that

-

is

j

K,

dl, d 2 _ -

-

are

i s (K-closed.

S5

between the

j(dl)

i s @-RE r e l a t i v e t o

-

d -1 < d and

has regular cardinality

5

K-degrees

i s tamely

&RE

iff

is

j(d2) -

dl

-

d2.

K-RE r e l a t i v e t o

Thus

-

2 K-degree(A)

K-degrees

K-degree@)

r e l a t i v e to

then

j(d2).

-

K-RE r e l a t i v e t o some

j(5')

weak B-jump(j(d))

=

j(4") = ( j ( d ) ) ' .

5

I n case

t o be a

< ( K ' ) ~ = l e a s t L-cardinal greater than

master code f o r . S

An

A (S ) n y

there i s a

bijection

Ifwe assume

5 such t h a t

S5

V = L, is

@ <-->

y

A

can be chosen

i s the l e a s t ordinal > 5

such t h a t

K.

Friedman I1981bJ p r o v i d e s a c h a r a c t e r i z a t i o n o f those

5.

I n the

i s (K-closed i f f t h e c r i t i c a l p r o j e c t a o f

5 all

(K-ClOSed i n terms o f t h e c r i t i c a l p r o j e c t a

t e r m i no1 ogy o f t h a t paper, have c o f i n a l i t y

where

Y

then

K

s5

of

We s h a l l have g r e a t use f o r t h e c r i t i c a l p r o j e c t a i n t h e n e x t

K.

section. SECTION THREE

SINGULAR CnkDINALITY

I n t h i s case we do n o t o b t a i n isomorphisms, o n l y monomorphisms. cases where an isomorphism does n o t e x i s t . ) K

and assume

B = (S .B)

5

15 amenable,

Thus l e t

l,proj(B)

5 have s i n g u l a r c a r d i n a l i t y

= K.

Our f i r s t o b s t a c l e i s t h e

f a c t t h a t t h e Maass c o l l a p s e does n o t a p p l y i n t h i s c o n t e x t u n l e s s T h i s l a s t c o n d i t i o n i s a l o t t o hope f o r when be s i n g u l a r i n s i d e

s5.

However we show t h a t i f

n e v e r t h e l e s s t h e r e does e x i s t a such t h a t f o r

C

C_K,

structure

A = (LK,A>

LEMMA 11.

If

i s s i n g u l a r as

K

C

is

A1 master code

I1 (LK,A)

iff

C

llcf(B)

-

COf(K),

=

for

B;

is

ll(S).

i n s t e a d o f t h e Maass c o l l a p s e o f

Ilcf(B) = I,(S)

l 1proj(B)

(We know o f

= K

K

z,(B)

i.e.,

Ilcf(B)

-

=

K.

may v e r y w e l l cof(K)

a set

A

then

5 K,

We can t h e n use t h e

B. then there i s a

The Turing Degrees and the Metadegees have Isomorphic Cones g:

11(8)bijection z

E

I->

z

e; i.e.,

which i s

K

g-l[zI

f : S ->

and a

K

B

ll(B)

many subsets o f o r d e r t y p e

h: y ->

ll(B)

is a

K

let

fy be t h e

$(x,y)>).

y < yo,

Z!

for each

S6

E

(P / y < yo)

Y

F i x a ll(B)

of

-

h(y + 1)

f

Y’

( sk(y)’

6

B

n

where

h ( 0 ) = 0.

Also

and f o r each

( t h u s i f graph(f)

graph ( f y ) = I(x,Y)I

I 1 ( B ) sequence

into

K

h(y) 5 P

k : yo ->

ll(B-)-function

=

sk(y)>k

has o r d e r y p e

f o r each

K

such t h a t f o r each

(E Iy < yo) Y

+1 i s a b i j e c t i o n between Range(fY )-Range(fY)

Y

[z]

rlcf(B).

yo=

We can assume t h a t Range(fY+’)-Range(fY) Thus we can choose a

y < yo.

partition

approximation t o

yth

I1 then

{(x~Y)IBF+(x,.v)~.+

-1

o r d e r p r e s e r v i n g , unbounded f u n c t i o n and

choose an o r d e r p r e s e r v i n g , unbounded y < yo

Let

We assume t h a t

K.

g

ll(6).

is

Much l i k e Lemma 4 o f Friedman [1981c].

injection

-

<->

6

and the function

LK

Proof. yo

S

153

and

P

We a r e

Y‘

f i n a l l y prepared t o d e f i n e : +1 g ( x ) = Ry o fy ( x ) , Then

is a

g

ll(B)

from t h e f a c t t h a t

->

g-1 [ z l

where

y = least

b i j e c t i o n between S6 z E LK ->

z

n

K

s.t.

y

and

c_ h ( y + l )

x E Oom(fY+’). The tameness o f

K.

f o r some

y < yo ->

follows

zn

K

fY+l. -1

can be computed from

Under the hypothesis of Lemma 11 there i s a

COROLLARY 1 2 .

g

A1

t_ P Y

master code for

B.

Proof.

Let

$(e,x)

n So)]

T = I(e,x,o)/(Su,B

A = g[T].

and l e t

the sequence which i s X

E C <->

(T

Then

n

ll(LK,A)

n yIy

(A

g- 1 [y] Iy <

<

is

K)

l,(B).

jo’((g(e),g(x),o’)

E A).

g(y) = y + y)

f o r m u l a and s e t

By Lemma 11 choose a tame

= $(e,x)l.

i s also

example, r e q u i r e

2,

be a u n i v e r s a l

is

K)

ll(B)

ll(B). But i f

g: S

<->

K

s i n c e b y t h e tameness o f

From t h i s i t f o l l o w s t h a t any C CK

is

As we can assume

t h i s shows t h a t

C

is

I1(B) g

r

K

ll(LK,A).

t h e n f o r some is

ll(LK,A)

g

C e (for

-1

Our second o b s t a c l e t o overcome i s t h a t Lemma 2 may n o t a p p l y i n t h e s i n g u l a r case.

(In fact i f

B-bi-immune.)

K

i s singular i n S

8

and

V = L

t h e n no s u b s e t - o f

K

is

I n s t e a d we need a d i f f e r e n t way o f choosing r e p r e s e n t a t i v e s of t h e

‘A-degrees so t h a t

>

and

2

agree on these s e t s .

T h i s method i s o b t a i n e d as

S.D. FRIEDMAN

154

an e x t e n s i o n o f Lemma 11 o f Friedman [1981a]. Definition. {(X,Y)

C

i s B-reduced i f

F K

5 c,y n c

E SB.lX

=

01 2 f B u x , y ) E LKIX 5

I t i s c l e a r t h a t if C1 ,C2

any

L)

=

(V

Lemma 13.

Let

Proof. bounded

f: y

c a r d i n a l f o r each

y < yo.

w i t h parameter

and f o r each

inside

8 = (Sf(y),B Y

not i n

H

Y'

Given of

LK.

some

C

5K

Thus

have t h e same

C*

both

ll(A)

C*

i s that if

x

n C* $ 0.

g(y)

5

6, u ( x n 6')

E

H

Y

Moreover, i f

C*.

x

Thus

sy + g c ( y )

n C*

=

= Hy

Y

where

=

A

K

define

is a

Al

8-degree(D)

where

THEOREM 14.

If

j

A

.

Y'

x

n

0, x E H .-> is

C*

l,proj(B) j

4

is

u

{p}

= l e a s t ordinal

K

I

For

The s e t s

(syly < yo)

C

are

f o r some

=

8

llcf(B)

Then

is defined as above then

A-degrees and the regular 8-degrees.

j

=

I1(B)

V = L.

-

then

j

A-degree

Suppose

cof(K).

Othenrise l e t

and f o r any

8-reduced.

g(y)

-1

B-reduced.

K,

2

6

c LK. As any x c Sg belongs t o

as i n s e c t i o n 2.

master code f o r D E

ll(B)

g(y)

C* = {6 <

( g ( y ) I y < yo),

[6,&+) f 0 x

Y

and

hull of

n (6,6+)+ @ ) I .

Hy

We now see how t o d e f i n e t h e monomorphism j. Assume

llcf(B)

are

H ,x

E

x E H

i s amenable and

is a

f,g

n g(y)+

Finally,

or

g(y)

i n the canonical wellordering

g(y))

.

such t h a t

B

un-

E SE t h e n whether o r n o t x i s a subY C* o n l y depends on C* n g ( y ) . For, i f x E H Y So, x 5 C* i f f I x n 6'1 < 6' f o r each 6 and

x

f o r some y < yo we see t h a t

B = (SB.B)

Then

code t h e same i n f o r m a t i o n .

C

o r i s d i s j o i n t from

n

y < yo, s

n

such t h a t

l1 Skolem

A-degree as t h e sequences

The p o i n t of

x

Y

(6 <

and

then for a l l

< C 1 4 2'

functions.

K

p E S Hy =

y < y

f o r each

gc

C*

let

F o r each

and

s e t of

H

y < y

g c ( y ) = rank (C

and

6

6 and g: y ->

->

Sf(y)).

g c ( y ) < g(y)+

5

C

Choose o r d e r - p r e s e r v i n g ,

K.

Choose a parameter

we l e t

y < yo, g ( y )

and

n

< C <-> 1 4 2

is singular with respect to

= 1 1 ( ~ )- c o f ( K ) <

functions

p

C

rJ1

B-reduced set.

yo = & c f ( B )

I1(B)

K

Suppose

A-degree contains a

a r e E-reduced t h e n

5K

c

c,y 17

4

Then if

A = (LK,A) define

j(4)=

i s a w e l l - d e f i n e d monomorphism.

is an isomorphism between the

The Turing Degrees and the Metadegrees have Isomorphic Cones

Proof.

Notice t h a t i f

and c o f i n a l as

(yo = r l c f ( B ) )

n g(y))

{f(y),D

B, g:

f : yo ->

t h e n any

yo

->

D c_

8-reduced s e t

and t h i s s e t i s r e g u l a r .

Iy < yo}

l1 (B),

are

K

15s

orderpreserving

has t h e same

K

Thus

Range(j)

8-degree

5 regular

8-degrees. Conversely, suppose

E

E* = { z E SEtIz

1) define

5 E)

*

from Lemma 11,

i s r e g u l a r and ( s i m i l a r l y t o t h e proof o f Theorem

5 SB

v Iz 6 S B , / z n

E

2

E.

But conversely

E LKilx 5 E**,y n E** = 0) &8 E* E.

-1

K =

Nu

regularity o f I n case

1

(and

A-degrees a r e we1 1-ordered.

= L)

V

present i f

= g[E*]

where

E

3

As b e f o r e i t i s

E**,

by t h e tameness o f

i s o n t o when

E*

g,

2

Es by t h e

t h e r e s u l t s o f Friedman [1981al show t h a t t h e

not w e l l - o r d e r e d

K

However

i n t h i s case and thus t h e

o f Theorem 14 i s n o t o n t o a l l t h e 8-degrees.

j

j

*

comes

g

Thus t h e r e g u l a r 8-degrees a r e we1 1 - o r d e r e d

i t i s k n o w n t h a t t h e B-degrees a r e

monomorphism

E**

E a 8-reduced s e t o f t h e same A-degree as E**.

easy t o show t h a t {(xJ)

01,

E

I t i s n o t known a t

= N u . ( I n t h i s case t h e A-degrees a r e n o t w e l l -

ordered. )

As i n e a r l i e r s e c t i o n s one can determine t h e Range o f and t h e i n v e r s e image under Tamely

8-RE degrees and

e 5 $.

A-RE

t o an

construction o f

j

o f the

8-RE

degrees.

j - ' [B-RE degrees] =

!I

on t h e

j

Thus

j [A-RE degrees] =

E A-degrees/!

The p r o o f uses t h e tameness o f

relative

A-RE

8-reduced s e t i n a

A-RE degree).

given

B o f cardinality

F i n a l l y we determine under what c o n d i t i o n s g i v e n an o r d i n a l K,

A-RE

is

and a s l i g h t l y m o d i f i e d

g

8-reduced s e t s (so as t o produce an

A-RE degrees

there e x i s t s

B c_ SB

above hypotheses; i . e . ,

8 = (S ,B)

such t h a t llproj(B)

=

K,

B llcf(B)

i s amenable and s a t i s f i e s o u r =

1 (8)-cof(K).

seen t h a t t h i s i m p l i e s t h e e x i s t e n c e o f an i n j e c t i o n 9 - l r y € SB jection

f o r each

S ->

can d e f i n e

B

K

(if

llcf(B)

o t h e r w i s e use Lemma 1 1 ) .

K;

B

y <

=

K

g: S ->

B

g

then l e t

But i f

g

B

f o l l o w s from t h e key p r o p e r t y o f

K

such t h a t

be any

I,(B)

in-

i s any such i n j e c t i o n we

c_ SB such t h a t o u r c o n d i t i o n s a r e s a t i s f i e d :

The r e g u l a r i t y of

We have a l r e a d y

g.

B = 1g-l Clearly

yIy < Z,proj(B)

K).

=

K

S.D. FRIEDMAN

156

as

g

I1(B).

is

function

8:

->

K

And,

&cf(B)

Let

= I1(B)-COf(r)

6

f(y) = least

I,(B)

as there i s a

1 by F g-

s.t.

cofina

s6.

Now the existence o f g can be completely analyzed using the techniques of (see the p r o o f o f Theorem 9 o f t h a t paper).

Frienman I1981cJ

This y i e l d s t h e

next r e s u l t . THEOREM 15.

(V

=

L)

Suppose

5 a l l have c o f i n a l i t y -1

such t h a t

g

there is a s e t

= cofinality(K).

r y F S5' A

5K

5-degrees

dl

relative to

is

K-RE

j ( d l ) is

-

d3

is

6-RE

K-RE

(j (9 1' .

for a l l

y <

K.

B-degree(A). dz

relative to

If

2

K-degree(A)

dl, d2

_ -

i s tamely

j(d2)

iff

dl i s K-RE -

dz.

-

Thus

-

g: S

B

->

K

a r e isomorphic t o

a r e K-degrees

j(dl)

-

and t h e c r i t i c a l p r o j e c t a of

Moreover t h i s l a s t condition implies t h a t

iff

-

relative to

K

Then t h e r e is an i n j e c t i o n

such t h a t t h e K-degrees

the regular

-

8 has c a r d i n a l i t y

&RE

2

K-degree&)

relative to

r e l a t i v e t o some

j ( 6 ' ) = weak 5-jump

(j(4))

and

then

j(d2)

and

d < 3 -

d s.t. 1

-

j(g") =

The Turing Degrees and the Metadegrees have Isomorphic Cones

157

References Feiner, L.,

The S t r o n g Homogeneity Conjecture, JSL 35, Pages 375-377 (1970).

Friedman, S.D.,

B-Recursion Theory, T r a n s a c t i o n s AMS 255, Pages 173-200 (1979).

, Negative S o l u t i o n s t o P o s t ' s Problem 113, Pages 25-43,

11,

Annals o f Mathematics

(1981a).

, Uncountable Admissibles I :

Forcing,

t o appear, T r a n s a c t i o n s AYS

(1981 b ) .

, Uncountable Admissi b l e s

11: Compactness, t o appear, I s r a e l J o u r n a l of Mathematics, ( 1 9 8 1 ~ ) .

Jockusch, C., Maass,

and Simpson, S., A Degree-Theoretic D e f i n i t i o n o f t h e Ramified A n a l y t i c H i e r a r c h y , Annals o f Math. L o g i c 10, Pages 1-32, (1976).

W., I n a d m i s s i b i l i t y , Tame RE Sets and t h e A d m i s s i b l e Collapse, Annals o f Math. L o g i c 13, Pages 149-170,

,

(1978).

On t h e a- and B-RE Degrees, H a b i l i t a t i o n s c h r i f t , U n i v e r s i t y 6f Munich, ( 1 979).

PATRAS LOGIC SYMPOSION GMetakidex (ed.) @North-Holland Publishing Cornpony. 1982

159

SYMMETRIC GROUPS AND THE OPEN SENTENCE PROBLEM Verena Huber-Dyson U n i v e r s i t y o f Calgary Calgary, A l b e r t a Canada

1.

INTRODUCTION The elementary t h e o r y o f a c l a s s

w

w i l l be denoted by TK.

and

K

o f structures o f f i x e d s i m i l a r i t y type

w i l l stand f o r the sets o f q u a n t i f i e r f r e e

3K

formulas whose u n i v e r s a l o r e x i s t e n t i a l c l o s u r e s b e l o n g t o

IQ

TK,

w h i l e we w r i t e

f o r t h e s e t o f t h o s e t h a t a r e s a t i s f i a b l e i n some K - s t r u c t u r e .

problem f o r

VK

i s what T a r s k i c a l l s t h e open sentence problem f o r

e q u i v a l e n t t o t h e d e c i s i o n problem f o r

The'decision

K.

t h e e q u a t i o n problem f o r

K3,

K, i n

Macintyre's terminology.

If

all finite

t h e n b o t h 'dK and Kf 3 a r e r e c u r s i v e l y enumerable, and

K-structures,

TI(

i s f i n i t e l y a x i o m a t i z a b l e and Kf

It i s

Xf 3 i s d i s j o i n t f r o m t h e s e t

consists o f

o f a l l n e g a t i o n s o f WK-formulas.

If K

is

closed under d i r e c t p r o d u c t s i t s e q u a t i o n problem reduces t o simultaneous s a t i s f i a b i l i t y of f i n i t e systems o f e q u a t i o n s t o g e t h e r w i t h one i n e q u a t i o n . Let

6, F

and B

s t a n d f o r t h e c l a s s e s o f a l l groups, o f a l l f i n i t e groups

and of a l l f i n i t e symmetric groups,

Sn, n

E

N,

denote t h e group of a l l permuta-

t i o n s o f f i n i t e s u p p o r t on a c o u n t a b l y i n f i n i t e s e t by c l a s s o f a l l i n f i n i t e models o f lem

TK.

Su,

and w r i t e

K,

for the

The u n d e c i d a b i l i t y o f t h e open sentence prob-

f o l l o w s f r o m t h e u n s o l v a b i l i t y o f t h e word problem f o r group t h e o r y .

In

1961, 111, Cobham proved t h e h e r e d i t a r y u n d e c i d a b i l i t y o f t h e t h e o r y o f f i n i t e symnetric groups and w i t h i t t h e u n d e c i d a b i l i t y o f f i r s t o r d e r f i n i t e group t h e o r y . I have been i n t r i g u e d by t h e open sentence problem f o r t h a t t h e o r y s i n c e 1963, [21. Since e v e r y f i n i t e group i s embeddable i n a s u f f i c i e n t l y l a r g e s y m n e t r i c one and

Su i s t h e u n i o n o f an ascending c h a i n o f c o p i e s o f t h e Sn's

we have

V. HUBERDYSON

160

More u s e f u l t h a n

So

i s i t s e x t e n s i o n by an automorphism t h a t c y c l i c a l l y p e r -

mutes a g e n e r a t i n g s e t o f t r a n s p o s i t i o n s . t r a n s p o s i t i o n and an i n f i n i t e c y c l e .

T h i s group

S

i s generated by a s i n g l e

T h a t i t s open t h e o r y c o i n c i d e s w i t h

\If: was

Here I s h a l l show how elementary a r i t h m e t i c can be d e f i n e d i n

proved i n [ 3 1 .

S

u s i n g e x i s t e n t i a l p r e d i c a t e s o f t h e language o f group t h e o r y e n r i c h e d by two constants for the generators.

From t h e u n s o l v a b i l i t y o f H i l b e r t ' s t e n t h problem, [51,

follows t h e u n d e c i d a b i l i t y o f t h e open sentence problem f o r S i n t h e extended language.

The f i r s t group t o which t h i s t y p e o f argument was a p p l i e d was t h e two [6].

g e n e r a t o r r e l a t i v e l y f r e e group o f n i l p o t e n c y c l a s s two,

Observe however t h a t

t h e open t h e o r y o f t h a t group i s a p r o p e r e x t e n s i o n o f t h e open t h e o r y o f n i l p o t e n t groups o f c l a s s two, which i s i n f a c t d e c i d a b l e , s i n c e i t i s a x i o m a t i z a b l e and n i l p o t e n t groups a r e r e s i d u a l l y f i n i t e .

The open sentence problem f o r t h e c l a s s o f

a l l n i l p o t e n t groups on t h e o t h e r hand i s s t i l l open t o o . Although I have n o t been a b l e t o e l i m i n a t e t h e two c o n s t a n t s w i t h o u t s p o i l i n g t h e s i m p l i c i t y o f t h e i n t e r p r e t a t i o n , t h e r e a r e a few i n t e r e s t i n g c o r o l l a r i e s , and a r e c o n s t r u c t i o n o f Cobham's r e s u l t , structure o f

S

[ l ] .I hope t h a t a deeper a n a l y s i s o f t h e

and of how i t encodes a r i t h m e t i c and t h e t h e o r y o f f i n i t e symmetric

groups w i l l improve t h e r e s u l t s below.

2.

THE GROUPS I t i s w i s e here t o d e a l w i t h c o n c r e t e groups r a t h e r t h a n w i t h isomorphism

types.

Our p e r m u t a t i o n s range o v e r t h e group

of t h e i n t e g e r s and



fi o f a l l b i j e c t i o n s on t h e s e t

stands f o r t h e subgroup o f

We s h a l l need t h e t r a n s p o s i t i o n

T =

(O,l),

for

i

E

2,

the basic cycles o f order

for

n

E

N,

t h e successor

p:i

H

n+l

generated by t h e s e t

i t s s p e c i a l conjugates 5, = (0,1,

i+land t h e groups

....n ) ,

c-,

T~

Z UC 0.

= (i,i+l),

= (0,-1.

...,-n),

161

Symmetric Groups and the Open Sentence Problem

The p e r t i n e n t p r o p e r t i e s o f these groups and permutations a r e c o l l e c t e d in the next t h r e e lemnas in an o r d e r t h a t leads r i g h t u p t o t h e lemnas of the next s e c t i o n . They a r e easy t o prove, [ 3 ] , by d i r e c t c a l c u l a t i o n s and inspection of diagrams. We use t h e abbreviation [x,y] f o r t h e commutator x-'y-'xy, the derived subgroup of c e n t r a l i z e r of

x

write U '

U generated by i t s commutators and w r i t e Z(G,x) W e c a l l an extension R '

in t h e group G.

for for the

o f a presentation

by a s e t of negations of equations complete i f t h e diagram o f th8 group

<XIR>

gp<XIR> c o n s i s t s of consequences of

R'.

For b r e v i t y we indulge i n t h e abomination

o f concatenating equations. LEMMA 1.

(i)

S

i s the extension of

phism induced by the cyclic permutation (ii) S'

= S ,;

S' =

-

Su 'li

by its outer automorT

{[x,y] ]x,ycS}, S u = S ' U T S ' , and

normal subgroup of Sw (iii) z(s,p) = < p > , Z(S,pcil)

=

o f ~the generators,

~

+

S'

and of S, xSn, and Z(S,pkr)

(iv)

z(sn+l,c.n) = <en>, Z(Sn+l,
(v)

every permutation in

=

is a mhimal = < PkT >

for k # 0,

< ckn ~ > ,for O
S has finite o r cofinite support and

finitely many orbits.

LEMMA 3. The following presentations hold for

T,

p,

a,b,c: S = gp, Sn+l

=

gp
5, interprerhg

,..., [a,cn-1I2,cn+l~t,)-n>.

V. HUBER-DYSON

162

The addition of the inequation [ a , b ] # l or [a,cl#l t o t h e above sets of relations makes the presentations complete.

3.

THE PREDICATES Extending t h e language

and

b

L

o f group t h e o r y by adding two c o n s t a n t symbols

t o t h e p r i m i t i v e s 1, * and

-'

we o b t a i n a language

L'

a

i n which we s h a l l

use t h e f o l l o w i n g terms and p r e d i c a t e s x'=b-'xba,

'x=bxa-lb-',

vO(x,y)'1, vk+l(x>Y)'(vk(x,y))' k=vk(a,b)=b- k (ba) k ,

I

vk,1(x3Y)"(vk(x>Y))

-

D(x,y) : [x,bl =[xy,bal= [ y , ~ - ~ a x ~[y,(xb)]= 1a x b l = l , C(x): (3z)D(z,x),

N(x) : (3z)D(x,z),

E(x ,y) : (3 z)(D(z,x)

& [y,x'b-ll = [ ~ , z - ~ a z ~ l = l ) ,

( 3 u ) ( 3 v ) ( D ( u S x ) & D(v,Y) & D(uv,z)),

Z(X,Y.Z):

: (3u) (3 v ) (3w) ( 3 r ) ( 3 s ) (3 t ) ( ( x = z = l ) v ( y = z = l )V(D(u ,x) & D(v,y)

n(x,y,z)

& D(w,z)

& D ( v 2 , r o ) & E ( r , s ) & t r , s l = [ v w t - 1s t , u b a l = l ) . The r e s t o f t h e p r e d i c a t e s belong t o

L

itself:

T ( x ) : x 2 =1 & ( W Z ) ( [ x , z ] = ~ ~ [ x , z ] ~ = ~ ~ [ x , z ] ~ = ~ ) , S(x,y):

T ( x ) & [x,y] 3 =1 & [ x , y ] # l & (Wz)( [ z , y l = l

G(x,Y):

S(X,Y) & ( W t ) - S ( t , y t ) ,

I(x,y):

(3t)(S(t,x)

D(x,Y):

I(Y,xY-'),

& S ( t i Y t ) & (Ws)(S(s,ts)

=> z=yVZ=y- 1V [ x , Z I 2 = l ) ,

& -S(s,ys)

& t s , y l # l => S(S,XS))),

Sm(x,y,z):

( 3 r ) ( 3 s ) ( 3 g ) ( I ( r - 1xr,s- 1y s ) & z - l = g - l z g = r - 1x r s - 1y s ) ,

Pr(x,y,z):

( 3 t ) ( 3 g ) ( 3 r ) ( 3 ~ ) ( 3 ~ ) ( 3 ~ ) ( 3 hk ))((S3( t , z ) & S(t,r-'xr)

8 S(t,s-'ys)

& O(z,r-'xr)

& z-'=g-lzg & o(z,s-'ys)

& [u,z] =[v,z]= [ u x ' , z t I = [ v y ' , z t ] = [ u x ' t , z - ' k - ' v y '

R2(x,y): Rn+l(x,Y):

2 3 x =1 & [x,yl =1 & I x , y l # l , Rn(x,y)

'where x '

, y'

k ] = [ v y ' t , z - l h - ' u x ' h ] =1) 1

are short f o r

r - l x r , s- 1y s .

& tX,Y n 1 2

A s t r a i g h t f o r w a r d a p p l i c a t i o n o f t h e lemmas o f 12 e x p l a i n s t h e meaning of t h e s p r e d i c a t e s f o r t h e groups

Sa, where

t e g e r s l a r g e r than 6, and

s,=~.

CY

ranges o v e r

w, w + l

and a l l p o s i t i v e i n -

163

Symmetric Groups and the Open Sentence Problem

I f the constants

LEMMA 4. t,,,

a

and

b

a r e i n t e r p r e t e d by

and

T

t h e n , i n t h e g r o u p S a + l , t h e e l e m e n t s 5 , 0 and 5

6 < a 5 w,

satisfy the predicate D(x,y)

iff

E = c 9 and

C(X)

iff

E=cq,

E(x,y)

iff

c=< 9

C(x,y,z)

iff

5=cp,

n=cq

5=cP,

9

f o r some qcN w i t h 2 q < a , 9' f o r s o m e qcN w i t h 2 q < a , rl=<

and r l ~ S ~ + f o~r ,some qcN w i t h Zqca, and c = <

and II(x,y,z)

iff

f o r some p , q ~ Nw i t h 2 p , 2 q < a ,

P+9'

and < = < P'9'

f o r some p , q ~ Nw i t h 2 p , 2 q < a .

The next two lemmas deal with the language

of a permutation

IT

o f groups.

The support

consists of a l l points t h a t are moved by i t .

a single o r b i t o f cardinality larger than one. i s a subset of

L

151 on which the action of

A cycle

The o r b i t o f a sub cyc1e;rl coincides with t h a t of

rl

for a t the single point in which the supports o f

rl

and

En-'

,€

5 has of

5

5 except

intersect.

of length one has t r i v i a l action, the length of a non t r i v i a l cycle

In[

A cycle

coincides

with the cardinality of i t s support and with i t s order, the predecessor of which we denote by LEMMA 5.

v(E).

For 6 < a 5 o + l :

(i)

S a I= T ( S )

(ii)

Sa b S ( c , r l )

€,ES is a transposition, i.e.,

iff

iff

=s

B+1'

€, i s c o n j u g a t e t o

f o r some B < a , by a n i n n e r a u t o m o r -

phism i n c a s e B<w, ( i i i ) S a + l I= G(5,rl) (iv)

Sa C I ( € , , r l )

iff

<S,rl>=Sa+l,

( i i i ' ) So k (WX)(WY)-G(X,Y),

iff

t h e s u p p o r t s o f t h e c y c l e s 5 and

iff

rl

i n one p o i n t , (v)

Sa C O(€,,rl)

LEMMA 6.

5= 1

iff

If 6 < a

5

i s a sub c y c l e of t h e c y c l e w + l and c,q,c

E

Sa+l,

€, i s a f i n i t e c y c l e w i t h v(€,)=O,

( I t ) ( 3 g ) (3h) ( S ( t , t ( )

& n=g-'t€,g=h-'n-'h)

then

5.

n intersect

V. HUBER-DYSON

164

4.

COBHAM'S THEOREM Lemma 6 shows how t o define predicates

for ncN, Sc, Sm*, Pr* and O* in Cn, the language of groups so t h a t the axioms of the fragment Ro of arithmetic become true in every i n f i n i t e model of the theory of f i n i t e symmetric groups.

Ro

encodes

the diagram of the arithmetic of the natural numbers under successor, addition, multiplication and the natural ordering, using predicates rather t h a n terms and has the additional axioms

(Vx)

- O*(x,l)

and

(Wx)(Yy)(Vz)(Cn(x)& Sc(x,z) & O*(y,z) => Cn(y)VO*(y,x)), f o r Every f i n i t e subset of

i s undecidable.

Ro

has f i n i t e models, b u t every theory compatible

embeds a copy o f

S

Using the f a c t t h a t every s u f f i c i e n t l y large alternating group

n v i a . t h e diagonal Sn

nating groups where the cycles

<,

with Ro

For a discussion and a proof yielding stronger r e s u l t s the reader

i s referred t o [8].

of

ncN.

+

SnxS,

Lemma 6 can be proved f o r a l t e r -

a r e replaced by products of two d i s j o i n t copies

and we arrive a t

THEOREM 1 (Cobham). Every c l a s s o f g r o u p s t h a t c o n t a i n s i n f i n i t e l y many f i n i t e s y m m e t r i c o r a l t e r n a t i n g g r o u p s h a s a n u n d e c i d a b l e element a r y theory. I t follows t h a t the theories of f i n i t e groups, simple groups, many more.

of monotlithic

of complete groups,

of f i n i t e

and o f periodic groups are a l l undecidable, and so are

Note however t h a t Mal'cev's r e s u l t [4] of the hereditary undecidability

o f the theory of f i n i t e groups of exponent p , f o r prime p > 2 , and of nilpotency

Symmetric Groups and the Open Sentence Problem

165

class 2 i s not covered by Cobham's theorem.

5.

THE UNDECIDABILITY OF Neither of t h e groups

AND

S

Sw.

In

i s a model f o r t h e theory of f i n i t e groups.

S , Sw

f a c t t h e i r t h e o r i e s a r e compatible with t h e f i n i t e l y axiomatizable e s s e n t i a l l y undecidable fragment Q

of a r i t h m e t i c .

Thus t h e u n d e c i d a b i l i t y of t h e i r elementary

theories follow by t h e c l a s s i c a l methods o f [ 7 ] . relativize t o the predicate N

For

S

i t i s most convenient t o

and use Lemma 4 t o obtain an i n t e p r e t a t i o n of

in the i n e s s e n t i a l extension of the theory TS by t h e constants Sw

relativize t o the predicate

Lemma 6.

a

Q

and b.

For

( 3 z ) S ( z , x ) and use t h e predicates introduced in

The c r u c i a l d i f f e r e n c e between i n f i n i t e models of t h e theory of f i n i t e

symnetric groups and

lies i n Lemma 5 ( i i i ) , ( i i i ' ) .

Sw

i n t e r p r e t a t i o n s of the axioms of f i n i t e group theory and

Ro

Q

The conjunction of the

i s of course t h e negation of a theorem of

has t o be used f o r Cobham's proof, but i n

So

every

cycle has a "successor", and a l l "sums" and "products" e x i s t .

THEOREM 2.

The e l e m e n t a r y t h e o r i e s o f b o t h t h e g r o u p o f a l l p e r -

m u t a t i o n s o f f i n i t e s u p p o r t o n an i n f i n i t e s e t a n d i t s e x t e n s i o n b y a f i x p o i n t f r e e cycle are h e r e d i t a r i l y u n d e c i d a b l e .

6.

THE EQUATION PROBLEM FOR SYMMETRIC GROUPS WITH A DISTINGUISHED PAIR OF GENERATORS Lemma 4 shows how t o a s s o c i a t e w i t h every polynomial equation

integral c o e f f i c i e n t s and v a r i a b l e s $ ( a , b ) of t h e language

L'

f r e e v a r i a b l e s x1 ,...,xk,yl

nl

,...,n k

and only i f

S = Su+l.

t h e equation

x1 ,. . . ,xk,y,

,. . . ,yh

an e x i s t e n t i a l formula

of groups enriched by two constants

,...,y h

P(nl

a, b

and w i t h

such t h a t , f o r every choice o f natural numbers

,..., nk,yl ,...y h )

$(nl,...,n+,yl ,...,y h )

P w i t h positive

has a n o n t r i v i a l s o l u t i o n i n

N

if

i s s a t i s f i a b l e i n t h e permutation group

The u n s o l v a b i l i t y of H i l b e r t ' s t e n t h problem, [51, thus e n t a i l s t h e un-

d e c i d a b i l i t y of t h e equation problem f o r

S

in terms of the generators

T

and

B u t a b i t more can be s a i d upon s c r u t i n y of the language introduced i n 53. All

p.

166

V. HUBER-DYSON

terms t h a t occur in I) will be of t o t a l exponent zero in words of the form v(a,b) s a t i s f i a b l e in

S

t h a t in f a c t belong t o Sw,

so t h a t our formula i s

iff

Sw+l k ~ ( n , ( a , b ),...,~ , ( a , b ) q, ( a , b ) f o r some ml, ...,m h

and the solutions are

b

E

N.

,..., rn+(a,b))

B u t then, there i s an integer

q

(d

such t h a t

Sq+l k I ) ( q ( a , c ),. . . , s ( a , c ) , q ( a , c ) ,. . . ,rn+(a,c)).

q

(9)

d i l l be a p a r t i a l recursive function of the n ' s with the property t h a t i f

(w)

holds then the sentence on the r i g h t will be a logical consequence of the f i r s t q

if

defining relations of

(9)

r > q,

[a,b] # 1 .

and the inequation

So+l

holds f o r a formula of the form indicated, then so will including

w.

On the other hand, (r)

for a l l

Noting the recursiveness of the presentations involved and

modifying the notation of I1 by writing

W', 3'

where the language

i s con-

L'

cerned and attaching a subscript 0 i f we mean r e s t r i c t i o n t o formulas in which only terms of t o t a l exponent 0 i n b

occur we find t h a t

5 30' = 5m3O' = 3$

m

= 3;

s

and t h a t a l l these s e t s are recursively enumerable b u t n o t recursive.

I t follows

t h a t the s e t s 53', 3'5, and 3's a r e recursively enumerable b u t n o t recursive and

t h a t none of the s e t s in the following sequence i s recursive S3' 3 Sm3'

2

3 ' S m 3 3'S3 3 '

V ' g c V'SmC W'S

To see t h a t the inclusions are proper note t h a t b

b9 = 1

in only f i n i t e l y many

Sn's,

i s a commutator in i n f i n i t e l y many b u t n o t in almost a l l of them, for fixed

n , m , c(n,m,y) b

.

s

i s s a t i s f i a b l e i n almost a l l symmetric groups b u t n o t in a l l , and

i s conjugate t o i t s inverse in a l l f i n i t e symmetric groups b u t n o t in

S.

THEOREM 3. The s e t s o f q u a n t i f i e r f r e e f o r m u l a s o f t h e l a n g u a g e o f

g r o u p t h e o r y e n r i c h e d by t w o c o n s t a n t s - - i n t e r p r e t e d by a t r a n s p o s i t i o n and a maximal c y c l e

--

t h a t a r e s a t i s f i a b l e i n some, i n a r b i t r a r -

i l y l a r g e o r i n almost a l l f i n i t e symmetric groups a r e u n d e c i d a b l e , and so i s t h e e q u a t i o n p r o b l e m f o r t h e g r o u p S,

S.

The o p e n t h e o r i e s o f

o f a l m o s t a l l f i n i t e s y m m e t r i c g r o u p s and o f a l l f i n i t e s y m m e t r i c

Symmetric Groups and the Open Sentence Problem

167

groups in the extended language are not recursive. The p r e d i c a t e

E

o f 53 a l l o w s a r e l a t i v e i n t e r p r e t a t i o n o f t h e t h e o r i e s o f

a l l f i n i t e symmetric groups, o f a l l f i n i t e a l t e r n a t i n g groups and o f almost a l l f i n i t e symmetric groups i n t h e t h e o r y o f t h e group

A p p l i c a t i o n s w i l l be i n v e s -

S.

t i g a t e d elsewhere,

7.

REDUCTION TO THE LANGUAGE OF GROUP THEORY I t should be p o s s i b l e t o sharpen theorems 1, 2 and 3, b u t a t t h i s p o i n t t h e

f o l l o w i n g remarks w i l l have t o s u f f i c e .

U s i n g Lemna 5 ( i i i ) and t h e a s s o c i a t i o n o f

86 between p o l y n o m i a l e q u a t i o n s and f o r m u l a s b u i l t up from t h e p r e d i c a t e s and terms

o f 83 one f i n d s t h a t t h e s a t i s f i a b i l i t y i n some f i n i t e symmetric group of t h e 3W3formula

(3a)(3b)(G(a,b)

,... ,n+.y ,,... ,yh))

i s equivalent t o t h & s o l v a b i l i t y

& $J(~I~

of t h e corresponding d i o p h a n t i n e problem.

Moreover, due t o t h e form of

$J.

the

formula i s s a t i s f i a b l e i n some f i n i t e symmetric group i f and o n l y i f i t i s so i n almost a l l o f them and a l s o i n t h e group

THEOREM 4.

S.

The W3W-theories of finite symmetric groups, of almost

all finite symmetric groups and o f the group

S

of permutations gen-

erated by a transposition and a fixpoint free cycle on an infinite set are not axiomatizable. o f 83 and t h e f a c t t h a t 9 t h e u n i v e r s a l t h e o r y o f a l l f i n i t e groups c o i n c i d e s w i t h t h a t o f 5, [3], one o b t a i n s

W i t h t h e p r e s e n t a t i o n s o f Lemma 3, t h e p r e d i c a t e s

R

THEOREM 5. For any quantifier f r e e formula

H

groups let

of the language of

abbreviate the W3-sentence 9 (Vx)(Vy)(Rq(x,y) =' ( 3 2 1 ) ...(3zk)H(x,y,zl,...,zk))'

(i)

H

For each positive integers

q , either

symmetric group of degree greater than of group theory for all

q,

H

9

fails in some finite

or e l s e

Hr

is a theorem

r 2 q.

(ii) The set of quantifier free formulas

H

for which

H is, for 4

V. HUBER-DYSON

168

q, a theorem o f finite group theory is recursively enumerable

some

but not recursive. Finally, the presentation f o r

S

on

exponent z e r o i s an elementary p r o p e r t y o f troducing the predicate (WZ)(X

2

and

T

and t h e f a c t t h a t b e i n g o f

p,

as an element o f

p

S

justifies in-

A(x,y):

2 2 1 = l & [ x ,y13=1& [x ,y1 f l & y f z & ( [ z ,y1 = l & z = l = > z = l ) & ( [z, y1=1=>z=yz=y- v [ z , x l 2 = 1 ) f .

Any two elements o f a group isomorphic t o

S

and

G

that satisfy

A

in

G

w i l l generate a subgroup

A ( T , ~ ) holds i n the f r e e product o f

S

w i t h any group.

From 56 and Theorem 2 o f [3] one o b t a i n s a theorem t h a t , amusingly, has t h e undec i d a b i 1 it y o f Wsgrouptheory as a c o r o l l a r y . THEOREM 6.

by the axiom arithmetic.

The extension (3x)(3y)A(x,y)

T

of the elementary theory of g r o u p s

is compatible with the fragment

Q

of

Its universal theory coincides with that of all groups

and is undecidable while the existential closures of exactly those quantifier f r e e formulas that are finitely satisfiable are theorems of

T.

The set of quantifier free formulas

(Wx) (Wy) (A(x,y) = > (3 zl).

,

H

for which

. (3zk)H(x,y,zl,. . . ,zk))

grouptheory is not recursive.

is a theorem o f

Symmetric Groups and the Open Sentence Problem

169

REFERENCES

A . Cobham, Undecidability in group theory, AMS Notices, vol 9 (1962), 406. V. Huber-Dyson, The word problem and r e s i d u a l l y f i n i t e groups, AMS Notices, vol 11 (1964), 743. V. Huber-Dyson, A reduction of the open sentence problem f o r f i n i t e groups, t o appear in t h e B u l l e t i n of t h e LMS, vol. 13.

A. I . Mal'cev, The u n d e c i d a b i l i t y of t h e elementary theory of f i n i t e groups,

Dokl. _ _ Akad. Nauk, SSSR 138 (1961), 771-774.

Y.

v.

Dokl. Akad. Matiyasevich, Enumerable s e t s a r e Diophantine, - Mauk. -

SSSR 191 (1970), 279-282.

C . F. M i l l e r 111, Some connections between H i l b e r t ' s 10th problem and t h e theory of groups, i n m p r o b l e m s , ed. W . W . Boone, F. 8. Cannonito and R . C . Lyndon, North Holland Co. 1973. A . T a r s k i , A . Mostowski and R. M . Robinson, Undecidable Theories, North Holland Co. 1953. R . L . Vauaht. On a theorem o f Cobham concernina undecidable t h e o r k s . i n t h e Philosophy of Science, ed. E . Nagel, P . Suppes and A. T a r s k i , Stanford Univ. Press 1962.

w ,Methodology and

PATRAS LOGIC SYMPOSION G. Meinkides (ed.) @North-HollandPublithing Company, I982

171

ITERATED INDUCTIVE FIXED-POINT THEORIES: APPLICATION TO HANCOCK'S CONJECTURE Solomon Feferman-11 Department o f Mathematics Stanford University S t a n f o r d , C a l i f o r n i a 94305

Denoting t h e p r o o f - t h e o r e t i c o r d i n a l o f a t h e o r y

ABSTRACT.

A

(IDn(

result gives

T

by

IT(,

t h e main

A

5

where ( i )

an

IDn

i s a t h e o r y o f n-times i t e r a t e d i n d u c t i v e

d e f i n i t i o n s i n which o n l y t h e f i x e d - p o i n t p r o p e r t y i s a s s e r t e d and (ii)a. =

eO,

( 0 ) . M a r t i n - L o f ' s c o n s t r u c t i v e t h e o r y o f types w i t h n u n i v e r ? e s (ML,) an t h u s s e t t l i n g Hancock's c o n j e c t u r e : lMLnl = an. can be i n t e r p r e t e d i n an ID,

an+l = @

A

To

Analogous r e s u l t s a r e proven here f o r p a r t s o f o u r c o n s t r u c t i v e t h e o r y f u n c t i o n s and c l a s s e s .

PART I.

(The r e s u l t s f o r

THE THEORIES

arithmetic

(PA)

A1, and

&(P l,...,Pi) and

...,An Ai

Ai(Pi,x)

,. ..,Pn).

n

t h e language o f

augmented by unary p r e d i c a t e symbols

order language, denoted C (P, by a sequence

n

F o r each

ID,. 1

Ai(Pf,x)

a r e p r e v i o u s l y due t o Aczel.)

UPPER BOUNDS. h

1.

n= 1

of

The axioms o f

IDn

P1,

...,Pn '

for

Pi

Ai.

It i s a f i r s t -

IDn(
o f f o r m u l a s w i t h one f r e e v a r i a b l e

contains

i s t h a t of Peano

a r e determined

x where each Ai

o n l y i n p o s i t i v e occurrences.

is in

We w r i t e b o t h

F o r any g i v e n sequence o f formulas,

'i5n'

i s dropped f r o m t h e d e s i g n a t i o n o f t h e system. PA = PAo + I n d where

Write

PAo

the p r i m i t i v e r e c u r s i v e f u n c t i o n s and language

2'

of

PA,

Ind(P1 ,.

1 formulas o f t h e language S (Pl, (i) PAo (ii) (iii)

Ind(P l,...,Pn) The axioms o f

.. ,Pn)

... ,Pn).

i s t h e u s u a l s e t o f axioms f o r Ind

O,',

i s t h e i n d u c t i o n scheme f o r

and

N i n the

denotes t h e i n d u c t i o n scheme extended t o a l l n

The axioms o f

IDn

c o n s i s t of:

172

S. FEFERMAN

-' P n ( x ) l .

( i v ) Vx[An(Pn,x)

The f i n a l axiom i s denoted

We assume c l a s s i c a l 1 s t order p r e d i c a t e c a l c u l u s

FP,.

as the basic l o g i c .

(The s u p e r s c r i p t (i)

t u i t i o n i s t i c logic:

HA = PA(i)

denotes Heyting a r i t h m e t i c . )

...,An

A1,

D i f f e r e n t choices o f

take t h e usual closure c o n d i t i o n s O1 ,...,On,

i s used t o i n d i c a t e r e s t r i c t i o n t o i n -

A

give r i s e t o d i f f e r e n t

IDn(0)

(and s i m i l a r l y f o r o t h e r

. IN f o r o r d i n a l s a t l e a s t up t o and

including

r0.

a,B.y,...

a r e used t o range over t h e domain o f t h i s ordering.

This i s provided by an o r d e r i n g

{BIB < a } .

r e s t r i c t e d t o the i n i t i a l segment

PA

(indeed i n

$,

E ~ . )

HA); c f . Feferman [1968].

ro

'To.

IN o f type

Greekletter

We w r i t e

The o r d i n a l f u n c t i o n s

Here

c r i t i c a l f u n c t i o n s (elsewhere denoted

( I ~ ( B )=

in


+, exp and $

common s o l u t i o n i s the l e a s t a

and

5 of with

$

Y

i s t h e Veblen hierarchy o f

$,

Ba):

$,(B)

= mB

and f o r

(5) = 5 f o r a l l y < a.

(Thus

= a.

$,(O)

Given (a presentation o f ) a c l a s s i c a l o r i n t u i t i o n i s t i c axiomatic theory we denote by

T

(Con,)

IT1

the least ordinal

can be proved i n

f i n i t e i n d u c t i o n up t o

REMARKS. (i)

I n practice, f o r

ConT provable i n

HA

a

+ TI(<,)

IT1 = a ,

PRA + Q F - T I ( d )

t i f i e r - f r e e t r a n s f i n i t e i n d u c t i o n up t o a r i t h m e t i c a l theorem o f i s i n d i c a t e d below by

B < a

T

where

TI(6)

PA

Eo,

a

5

r0.

( p r i m i t i v e r e c u r s i v e a r i t h m e t i c w i t h quan-

a ) , and ( i i ) f o r PA

+ {TI(
T containing

+ {TI(
Jt we thus have a conservative extension r e s u l t .

L e t a. =

i s the scheme o f t r a n s -

we have:

i s provable i n

T 5

T,

such t h a t the consistency statement f o r

This i s o n l y needed here f o r

a.

4

and t h e i r elementary p r o p e r t i e s a r e provable i n

0

a > 0, Q a ( B ) = Bth

When we

L.

We assume a standard n o t a t i o n system i n

are r e c u r s i v e l y defined on

,.

f o r t h e r e c u r s i v e o r d i n a l number classes

Ai

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

fami 1i a r i n d u c t i v e d e f i n i ti ons)

IDl,..,,IDn.

T

PA,

each

The r e l a t i o n ( i i )

proves

TI(
f o r each

an+, = $ (0). The main r e s u l t i s t h a t \ I D n \ 5 an and t h a t an IDn 5 PA + I T I ( B ) } B < a For n = 1 t h i s i s due t o Aczel [1976], where i t was n used t o show t h a t M a r t i n - L o f ' s c o n s t r u c t i v e type theory w i t h one universe has h

=,

.

173

Iterated Inductive, Fixed Point Theories (MLII = al = $c (0);

he a l s o used i t t o show

ITo-IGI =

(cf. below).

f o r my t h e o r y

(0)

$c

To

0

0

We s h a l l prove e x t e n s i o n s o f t h e s e r e s u l t s , t h u s s e t t l i n g what i s

c a l l e d Hancock's c o n j e c t u r e . A

Friedman [1969] had f o r m u l a t e d a t h e o r y theory

ID1

t i o n on

P1

o f one i n d u c t i v e d e f i n i t i o n .

ID?

In

l y i n g between

and t h e usual

ID1

t h e scheme f o r p r o o f by induc-

ID1

is

W x[Ai(B,x)

-+

B(x)l +V

X[Pl(X)

which i s t a k e n f o r a l l f o r m u l a s are permitted./

+

B(x)l

s

B o f t h e system.

Friedman showed l o c . c i t . t h a t

h i s r e s u l t i s s t r o n g e r t h a n A c z e l ' s above f o r

,.

In

only formulis

IDf

\ID;]

5

I n t h i i respect,

(0).

$

B(Pr,x)

€0

However, t h e p r o o f needed f o r

ID1.

t h i s s t r o n g e r r e s u l t i s somewhat h a r d e r and i s n o t needed f o r t h e a p p l i c a t i o n t o Hancock's c o n j e c t u r e o r t h e a p p l i c a t i o n t o subsystems of

paper I s h a l l show how t o g e n e r a l i z e Friedman's theorem t o : evident i t e r a t i o n o f t h e theories

ID:).

REMARK ON THE FORMULATION OF

ID,.

,

( f o r the

5 an

h

We c o u l d a l t e r n a t i v e l y t a k e i n f i n i t e l y

P

t i v e occurrences. n

define

2.

\ID;[

a t each s t a g e i where Ai contains only predicates P Ai Aj j < i ( f o r any A . ) as w e l l as t h e s p e c i f i c p r e d i c a t e P w i t h j u s t posi-

many p r e d i c a t e s for

I n a subsequent

To.

IIDnl

Then

J

O1 i s a c e r t a i n one o f t h e ,. = supC(a1: I D n I O l ( Z ) l .

P

A1

Ai p r e d i c a t e s and we c o u l d

n = 1. So f a r as I know, t h e p r e l i m i n a r y paper Aczel [1976] has

THE CASE

n

never appeared i n p u b l i s h e d form. preting

,.

It i s shown t h e r e t h a t

I D 1 i n t h e 2nd o r d e r t h e o r y

from Friedman's t h e s i s [1967].

z1 1

AC

jfDll

by i n t e r -

5 al

and u s i n g t h e r e s u l t

1

IZ1

-

A C I = al

We s h a l l go o v e r t h e argument here, as i t i s t o be

extended below. F i x any ID1

A1(PY,x).

We d r o p t h e s u b s c r i p t ' 1 ' i n t h i s f o r m u l a , t h u s t a k i n g

as h a v i n g t h e axioms: (FP)

VX[A(P,X)

Ind(P)

PAo. <-->

P(x)].

The usual 2nd o r d e r d e f i n i t i o n o f Vx[A(X,x)

-+

x

E

XI.

and

P

is

1 n1, namely as t h e l e a s t

To p r o v e t h a t t h i s s a t i s f i e s

induction t o t h e formula

B(x) = A(P,x);

(FP)

X

satisfying

we would need t o a p p l y

-

a t t e m p t s t o v e r i f y t h a t i n 1; AC

(or

S. FEFERMAN

174

even

1 Z 1 - DC)

solution

run i n t o trouble. 1

Indeed h i s s o l u t i o n i s 1

involving

Z1

n o t n e c e s s a r i l y the minimal one.

and i s obtained by a standard d i a g o n a l i z a t i o n argumen

El,

1

We c a n f i n d a E i -AC

z = O,l,Z,

AC,

Cl-

formulas.

LEMMA 1 (ACZEL).

PROOF.

1

o f ( i i ) can be defined i n

P

me

(Try it.) Aczel’s t r i c k i s t o show t h a t

Let

E(z,x,y)

...

1

all

A(F,x) I

be a

formula

c->

B(x).

F(x)

such t h a t

formula which provably enumerates f o r

C1

formulas

Cl

C1

Q(x,y).

Write

Qe(x,y)

for

E(e,x,y).

Given

A

any formula P(t)

B(x),

i n A(P,x).

(1 1

i n C,

1

P

-AC

f o r the r e s u l t o f substituting

t o one of t h e

rl Qe(F,x) i s

B(t)

for

E(z,z,u),x).

has o n l y p o s i t i v e occurences i n A(P,x),

(2)

But

A(uB(u),x)

Now consider t h e formula

A(;

Because

write

Qe(z,x),

with

t-

~(j~(z,z,u),x)

E(e,e,x),

so i f we take

-AC

e

the formula (1) i s equivalenl

found e f f e c t i v e l y from

A:

~,(z,x).

P ( x ) = Qe(e,x)

(3)

we have t h e desired conclusion.

REMARK.

I f one s t a r t s w i t h a formula

A(P,x, ...) w i t h i n d i v i d u a l and/or s e t

parameters the same argument can be r e l a t i v i z e d t o g i v e a

Ei

formula

p(x,

...)

w i t h t h e same parameters such t h a t

,...),x ,... )

A($(u

C i -AC

TDl

Now t o i n t e r p r e t

( C i -AC)

p(x

,... ).

unchanged and 1 Every instance o f Ind(P) i s provable i n C1 -AC since i n t e r p r e t Pl as pl. ,. we are assuming f u l l i n d u c t i o n on pll i n t h a t theory. Thus I D 1 i s contained i n 1 (z1 -AC) have

REMARK.

in

++

under t h i s t r a n s l a t i o n . ,. IDl 5

Obviously

=d

since t h a t i s provably

we leave t h e language

X’

Since a r i t h m e t i c a l formulas are unaffected we

1 Z1 - AC. c a n ’ t be e q u i v a l e n t t o t h e standard 1 TI1- complete.

1

I$d e f i n i t i o n of

However, we can d i a g o n a l i z e j u s t as w e l l

0.

175

Iterated Inductive, Fixed Point Theories w i t h II;

formulas since i f 1

alence i n 1

1

Z1- AC

AC).

C1-

A(F',x)

P'(x).

Z11 - AC

which symbols

i t i s denoted

by s u b s t i t u t i n g the C1 (Pl

,...,Pn)-AC

The

1 Z1

n t 1.

f o r t h e system

i s equivalent

2

C (P

,,..., Pn);

C1

t h i s c o n t a i n s t h e lankuage of

2 2 (P1

formulas o f

1

The 2nd-order language i n

When augmented by t h e unary predicate

f o r s e t parameters i n

Pi

5;

Ol.

2 i s formulated) i s denoted C-.

as a sublanguage.

such t h a t

I t i s n o t obviously excluded t h a t

I$d e f i n i t i o n o f

,...,Pn

P1

(up t o provable equiv-

A(B,x)

p'

formula

Ill

EXTENSION OF THE INTERPRETATION TO

3.

1

1

Thus we o b t a i n a

+P

so a l s o i s

II;,

1

t o the standard

IDn

is

B

1

,. . . ,Pn)

are j u s t thqse obtained

formulas o f C

C1

2

- AC

formulated i n C (P1

2

.

We w r i t e

,...,Pn);

thus i t s

axioms are: fi)

PAo

(ii)

Ind(E2(Pl

,. . . ,Pn))

( i i i ) W r 3 Y Q(x,Y ,...) + 3 X W x Q ( x , ( X ) , ,...) for each

1

Zl

LEMMA 2.

1 Z1(P1,

formula

Q o f c 2 (Pl,...,Pn).

We can find a formula

...,Pn) - AC h

interpretation o f 1

g (Pl,.

. . ,Pn)

pn+,(x)

in

1 C1(P1,.

, , ,Pn)

such that

[An(; P,+,(u),x) *+ Fn+l(x)l. Hence there is an 1 in Z1(P1, Pn)-AC + I D leaving

...,

unchanged and taking

Pn+l

into

Fn+l

PROOF. This i s immediate by t h e Remark t o Lemna 1, t r e a t i n g

P,

,...,P,

as t h e

parameters.

4.

PROOF-THEORETIC LEMMAS.

We s h a l l now extend t o t h e present s i t u a t i o n the

p r o o f - t h e o r e t i c methods f o r t h e e l i m i n a t i o n o f Sieg [1981] (Sec.2).

1

Z 1 - AC

given i n [F,S]

= Feferman,

F a m i l i a r i t y w i t h t h a t work ( o r i t s d e s c r i p t i o n i n t h e subse-

quent paper Feferman, Jager [198l])

i s assumed.

I t i s p l a u s i b l e from the [F,S]

n

approach and Lemma 2 t h a t i f an an+l.

then

1

Z1

- AC

I D n can be reduced t o a standard theory of strength

h

over

IDn

A

(and hence

can be reduced t o one of strength

However, t h e d e t a i l s o f t h e extension r e q u i r e some novel considerations which

are n o t e n t i r e l y r o u t i n e ,

S. FEFERMAN

176

n

Fix

throughout t h i s section.

2 CW1,w(Pl,...,Pn).

ments o f

dard techniques.

,...,Pn),

2

C (P1

We s h a l l c a r r y o u t t h e p r o o f t h e o r y i n f r a g -

T h i s may t h e n be e f f e c t i v i z e d and f o r m a l i z e d by s t a n -

lo2

The b a s i c language o f

( P l,...,Pn) 1 3w t h e atomic f o r m u l a s a r e tl = t2, t

i.e.

a r e p r i m i t i v e r e c u r s i v e terms and

tl, t2, t

b u i l t up b y

-,

M , W, WX m < w m<w

t i o n o f formulas o f length

and 3 X .

i s t h e same as f o r E

and

X

t

i s a s e t variable.

X

,...,Pn)

2 By ESa(P1

Pi

E

where

Formulas a r e

i s meant t h e c o l l e c -

< a ( s i m i l a r l y f o r < a).

The p r o o f system i s t h a t o f t h e Gentzen s e q u e n t i a l c a l c u l u s f o r 2 S~,,~(P~,...,P~)

including the cut-rule.

f i n i t e sequences o f formulas.

F

tl = t2 w i t h

false.

r

FA

i f 10

B

and l e n g t h o f

Q S a

and each

a < a, 1

and s i m i l a r l y f o r t3 :

r k
and s i m i l a r l y f o r

As u s u a l , f o r m u l a s F'

F

2 o f X < w (P l,...,Pn)

QFsa (P1

10 :

r FZEA. r ::A.

'B',

means 18:

r FEA

F'.

r

)-
for

B.ul

< a

then

and

'C' for

QF f o r m u l a s .

f o r some

r kE A )

r

(2x F ( x ) ) '

Each axiom o f

denotes t h e q u a n t i f i e r - f r e e f o r m u l a s o f E

l,mdh<w

(A

are translated

( y x F ( x ) ) ' = M F($' m <W F with

i f 10,10'

&(P1,...,Pn).

"1 i s w r i t t e n f o r 38(fi:

2 o f t h e language C ( P l,...,P,)

by t a k i n g

and10 E Q '

B belongs t o

r F<",A C:A

tl = t2 c l o s e d and

has a d e r i v a t i o n o f f i n i t e c u t - r a n k and l e n g t h

,...,Pn)

r,A a r e

A means t h a t t h e c u t - r a n k of

These a r e b u i l t u p e n t i r e l y f r o m atomic f o r m u l a s by letters 'A',

r

F i n a l l y , we w r i t e

Where t h e r e i s no a m b i g u i t y we i d e n t i f y

PAo + Ind(P l,...,Pn)

B: T

formula o f

d e r i v a t i o n o f cut-rank 0 i s cut-free).

with

t, = t2

i s a derivation o f

a r e d e r i v a t i o n s o f t h e same end-sequent.

19

where

The o n l y n o n - l o g i c a l axioms a r e t h e sequents

tl = t2 c l o s e d and t r u e , and

We w r i t e 10:

r k A,

Sequents a r e

and mw:.

2

< w + w.

(P1

,...,Pn).

We r e s e r v e t h e

1I1

Iterated Inductive, Fixed Point Theories We consider two schemes: (QF-CA)

Each u n i v e r s a l closure o f 3X

[E E

M

X

c-t

B(m)l

m<w with (QF-AC)

quantifier-free.

B

Each u n i v e r s a l c l o s u r e o f 3Y B(E,Y) + 3 X

A

m<w

B quantifier-free.

with (Here 2

2 (P,

t

E

i s written f o r

(X),

,...,Pm)

(m,t)

t r a n s l a t e s i n t o one of

o f instances o f

(QF- AC)

LEMMA 4. ([F.S])

QF QF.

in

I n our paper [F.S]

-

QF- AC

-

r I- A

- AC

in

we s h a l l a l s o

o n l y a f i n i t e number

Suppose

AC.

a

is a n E-
Then we can find a derivation of

(QF- AA)

t h e e l i m i n a t i o n i s given i n f a v o r o f

{ m ( BmI w i t h QF ,B

which uses a b s t r a c t s

(QF- AC),

Z:

...,P);,

2 i n 2 <w(P1,

are supposed t o appear.

Elimination o f

(i)

Each instance o f

X.)

E

I n sequents:

use extensions o f t h i s below.

REMARKS.

B(m,(X),)

M

m<w

as 2nd order terms and the axioms

M [k E {mlB,} Bk]. I t i s r o u t i n e t o then reduce (QF- AA) t o k<w ( i i ) Also one has the r e s u l t i n [F,S] more g e n e r a l l y f o r sequents 2 r where r i s i n 3 V form and A i s i n W3 form ( i n 2,,(P1

t<",A

(QF- CA)

.

(QF- AC),

,...,Pn)).

The above weaker statement s u f f i c e s f o r our purposes. LEMMA 5. A

E

Elimination of

QF

( Q F - CA)

and that

PROOF.

Q F - CA. FCaA;

Suppose

a

is a n €-number and

<,A.

then

We s h a l l f i r s t c a r r y o u t t h e argument assuming

A

then be r e l a t i v i z e d u n i f o r m l y t o any s e t parameters o f A.) many instances o f

Q F - CA

i n t h e antecedent o f B : ( Q F - CA)

i s closed.

( I t can

Suppose the f i n i t e l y

"1 a1

A(a, < a )

are of

the form (1)

WY 3X

M

m<w

[m

E

X

++

Bi(m,Y)],

(1

5

i

5

k),

where f o r s i m p l i c i t y we have collapsed the s e t parameters t o a s i n g l e one.

Consider

S. FEFERMAN

178

the formulas

C(x)

generated as f o l l o w s from t h e f o r m u l a

o f these t h e n each Co(x)

Bi(x,E

C(y))

,...,Cm(x) ,..., (m< w ) .

x = x:

i s a g a i n such a f o r m u l a .

F o r each

m,

i s one

We enumerate these as

lCml

the length

if C(x)

-

This follow

< a1 w.

from t h e i n e q u a l i t y (2)

IF(x,

1

C(Y))

ICI

2

which h o l d s f o r any f o r m u l a then a l s o

/ C m I < a1 * w, a2 <

+

IF(x,Y)l,

F(x,Y),

by i n d u c t i o n on

i C m ( y ) ) ( < a1

(Bi(x,

-

IF\.

Let

w.

I n p a r t i c u l a r if

a2 = al

-

so a l s o

w,

a. The n e x t s t e p i s t o t r a n s l a t e each 2nd o r d e r f o r m u l a

formula (3)

p r e s e r i n g a t o m i c formulas,

F*,

('IX F(X, ...)

*

=

...)

*

= w

(3X F(X,

A

m<w

m<w

I t i s seen by i n d u c t on on

F * ( i Cm(x) that

case of ( 3 ) ,

I W X F ( X ) I = ( F I + 1,

IF*(;

I lCm(

C,(X))\

(WX F(X)I*

5 a2

-

w

+ IF*\

The same h o l d s f o r each

+

I u2

+ IF1 + 1.

w

,

into a (first-order)

by:

( i c m ( X ) ,...) ,

F*

F

and

-',M

F

,... ) .

(F*l

-

(x), ml f o r free set variables

+ IF\. F o r example, i n t h e f i r

-

Thus f o r

F(X^ C

w

0

/ W X F ( X ) ( * = sup ( F * ( i C m ( x ) ) 1 + 1. m<w w + / F I = a2 w + I F \ so

and ct2

2 a2

...,X^

F

€6: ~ u l we have

(x))

C,

F* t S s a 2

.

+

But

1

(I

.

o b t a i n e d by s u b s t i t u t i n g

9

...,

X1, X o f F. These t r a n s l a t i o n s and 9 cml y.-.ycm 9 s u b s t i t u t i o n s a r e extended t o sequents i n t h e obvious way. The f o l l o w i n g i s now proved by i n d u c t i o n on s u b d e r i v a t i o n s 9 ' If

tion (4)

,,...,Xo)

8' : r(X

a'*

,..., X U )

+A(X]

o f t h e same l e n g t h as

a ' * : r*(xc,

1

t h e n f o r any

a*

:

( x ) ,..., xcm ( x ) ) q

(QF - CA)*

Now t h e ith instance o f

A * ( X C ~ (x) 1 8

*

,.... XC,,

we o b t a i n a d e r i v a

(XI). U

o f t h e same l e n g t h as

A.

(QF - CA)*

,...,m q

' with

I t f o l l o w s f r o m ( 4 ) t h a t we have a d e r i v a t i o n

(5)

ml

of 9.

translates i n t o

B

with

Iterated Inductive, Fixed Point Theories

(6)

W

M

j < w

M

m < w

e c w

[c,(m)

I79

Bi(lil,Gcj(V))].

i +

i s a C,(x), so ( 6 ) i s t r i v i a l l y p r o v a b l e . Ri(x,FC.(y)) J 8 Thus a p p l y i n g c u t t o each i n s t a n c e ( 6 ) o f (OF - CA)* we g e t a d e r i v a t i o n B But by d e f i n i t i o n each

of

A.

a*

Note t h a t each c u t - f o r m u l a o f

i s o f t h e form

( x ) ,...,^x Cm ( x ) ) where F(X1 ,...,X ) 9 ml * 9 cut-rank o f fi i s a l s o bounded by a2 w+al, F*(;

C

-

we have

I-

fie :

A

where

a3

-

- a2

To p r o v e t h e statement when A

t o obtain

CA)*

c o n t a i n s s e t parameters Ui

improve t h e bound i n t h e c o n c l u s i o n 1(QF - CA)*I

1

:a".

Also

Thus f i n a l l y

< aA.

U1,

....Ut

we

t o appear as parameters i n each (QF

Even i f we s t r e n g t h e n t h e h y p o t h e s i s t o

o f complexity

-

( O t l ) < a.

simply r e l a t i v i z e t h r o u g h o u t a l l o w i n g t h e REMARK.

(QF

i s only f i n i t e l y greater than the lenqth o f 8 . < a3

Thus t h e

u s i n g t h e bounds c a l c u l a t e d above.

This bound i s n o t i n c r e a s e d when we a p p l y c u t t o the l e n g t h o f &I@

a.

i s a c u t - f o r m u l a of

-

Cm.

lo< aA we cannot

CA)

The reason i s t h a t i n t h e end a c u t

i s introduced.

The f o l l o w i n g i s now t h e main p r o o f - t h e o r e t i c r e s u l t t o be used helow. COROLLARY. and

A

PROOF. 5.

of

(Complete e l i m i n a t i o n QF

E

and

(QF - AC) ]-&A

QF - AS)

then

If

a

i s an

€-number

b O <9 , ( 0 ) A .

By Lemmas 3, 4 and 5.

P r o o f - t h e o r e t i c bounds f o r t h e

DEFINITION.

6,.

@ ( n + ll a ) = 0

$(Ola).=a ,

.

1

$ (n a ) ( O )

Informally,

(0)

.

(0)

n

The f o l l o w i n g i s immediate.

LEMMA 6.

( M a i n Lemma)

and t h a t n o p r e d i c a t e

Suppose

Pi

CY

is an < - n u m b e r and t h a t

o c c u r s in

A.

If

FP,,

A

E

...,FP1 I-< " A

cL

QF

S. FEFERMAN

180

1.; n.

n.

By induction on

PROOF.

and the f a c t t h a t

A

n = 0,

For

the conclusion

i s an E-number.

a

k:

a A

follows from

Suppose the statement i s true for

To prove i t f o r n + 1 , consider any derivation of

(11

FP ,..... FP1

FPn+ls

k <<wa

A.

F"pn+l

By Lemma 2 of 13 we can find a proof of 1

-

C1(P l,...,Pn)

Y

where FPn+l

AC,

in the axiom FPn+l

of

1:

QF - AC

(2)

2 in $<w.2(P1

A

,...,P n ) ,

i s the r e s u l t of substituting

wWs 2

Pn+l;

< w

throughout f o r

FPi(l 5 i 5 n )

leaves each

Pn+l

and A

-

the r e su l t i s

for Pn+l

En+l

tion (of i t s t r a n sl a ti o n ) with bounds Now s u b st i t u t e

-Pn+l

This t r an s l at es into a derivation of

since each axiom of

-Pn+l

in the system

FPn+,,

FP,,

FPn

A...A

PAo + Ind(P1 ,..., P n )

has a deriva-

on cut-rank and < w - 2 on length. in the derivation of ( 1 ) .

This

unchanged, since none of these contain

...,FP1 1;

:

A.

Applying cut t o t h i s and (2)

now yields

Hence

(41

QF

-

AC

k;

FP1

-+

Since a

i s an E-number i t follows t h a t

(5)

QF

-

AC

k:

a FP

n

FP1

A...A

+

A.

A.

We can now apply the complete elimination of

preceding section: (6)

1 :

Hence, cutting with (7)

FPn

A...A

FP1

FP,,

.,.,FPn

-+

A.

we obtain

F p n ~ * * ' s F pk:l T a ( 0 ) A .

QF - AC obtained a t the end of the

181

Iterated Inductive, Fixed Point Theories Now i n d u c t i o n h y p o t h e s i s can be a p p l i e d t o g i v e

This g i v e s t h e c o n c l u s i o n s i n c e

fDn

If (i)

THEOREM 1.

5

(ii)I~D,I

$(nl$ (0))

t-A

with

$(n+lla).

=

A

0

arithmetical then

A.

I--< $(nlEol

$(nIco)

h

(iii) IDn 5

< $(nlEo).

PA+ ITI(
:n PROOF.

translation i n t o

f(p

,,..., FP1 1.'

A.

FP

< Eo

(ii)

If

PAo + Ind(P1

By h y p o t h e s i s

(i)

,,...,pn)

we have

k:

Then

,...,Pn)

,..., F P ~~2

F P ~

9(nlEo) A

A,

A

that

must be t r u e .

A

P of

(tl;=

t2).

A

of l e n g t h

Here we cannot p r o v i d e any s p e c i f i c bound be-

on t h e h y p o t h e t i c a l p r o o f , so t h i s uses

low $ ( n l e o )

SO by

so c e r t a i n l y

any atomic sentence

we p r o v e by i n d u c t i o n a l o n g a c u t - f r e e p r o o f

A

< $(nlEo)

2.:

I-A,

by t h e Main Lemma.

We f o r m a l i z e t h e argument f o r (i)and

hn1

,...,FP1}

+ IFP,

TI(.(

.

$ ( n IEO)

)

t o prove

The whole argument i s c o n s t r u c t i v e , so can be f o r m a l i z e d i n ConIDn A t TI(< $ ( n I c o ) 1. F o r each s p e c i f i c d e r i v a t i o n o f an a r i t h m e t i c a l sentence A

(iii)

associate an a < $ ( n l c o )

and a fi

o f which we can show t h a t 8 :

a p a r t i a l t r u t h d e f i n i t i o n from formulas o f complexity PA + T I ( < a )

that

THEOREM 2.

S u p p o s e g i v e n any

A

i s true.

<

-:I

I^nn

If

Using

IDnFO(a)

A.

where then

$(nlcol.

PROOF. \In[(e)(n)&

L e t so +

(e)(n)

be t h e r . e . r e l a t i o n generated by: q o

3-5e]

a and (iii)

Let A (x) = [x= 1 v x = 2 0

A

(xo)

v x =


3.5

b


c

(x), A

+

(i)

a - 2a,

a co c .

\Iu([(X)2~(U)+)

wu([(x)21(u)~o c(x)2}(u+l))l.

we

we show i n

fE,,(o),

starting with

fDn

A.

Then by T a r s k i ' s T r u t h Scheme we d e r i v e

i s t h e s e t of r e c u r s i v e o r d i n a l n o t a t i o n s .

1.

LA

from

(if)

0

S. FEFERMAN

182

(Thus

A.

is

(1 1

Ao(x)

We t a k e

0

TI;.)

i s i n f o r m a l l y d e f i n e d as t h e l e a s t

Vy[y

A

Al(Pt,x)

XI

+

x

~ y [ . yq u x

+

P1(x)],

q0 X

= Ao(x)

A

y

-+

E

A 2 (P 1 ,P+,x). 2

c o n s i d e r any f u r t h e r

E

X

satisfying:

0

for

X. and w r i t e

e t c . d e t e r m i n i n g an

CDn.

P1.

Now

&-U(z)

If
t h e n we have

. . ,FP2,

FP,,.

(2)

2:sk

FP1

O(H),

E~

i n place o f

-~ a f o r t i o r i t h e same w i t h

W.

2.

We can r e p e a t t h e argument of

t h e Main Lemma 6 u n t i l we come t o

s i n c e a t each s t a g e we a r e s u b s t i t u i n g d i f f e r e n t tack from before.

1

theory (4)

3(a) = W X [ C ( X )

+

1

Z1-DC

1

a

XI

E

i s straightforward.

where

proves

1. Ao(x)

C1 - D C

(5)

0

for

Pi

with 1 111

i t s natural

i 2 2.

Now we t a k e a

b

definition

i n the

following ( l ) , i.e.

L, - D C ,

We c l a i m t h a t

Give

Pi

A

FP1

C(X) = Wx[Ao(x)

A

Wy(yqO x

f o r this translation.

WY(Y*u x

i(Y))

+

+

+

y

E

X))

+

x

E

X

The proof of

6(x)

The p r o o f o f t h e converse i m p l i c a t i o n i s a l i t t l e l e s s so,

and hinges on t h e f a c t f r o m ( 5 ) t h a t Z,

(6) Since

1

1

b

- DC

is

k

C:

7

b(x)

+

we can use

~ Y ( Y <0 x

Li - D C

t h e r e i s a -CO-descending c h a i n f r o m then

Vy(y q0 x

+

bfy)),

x.

C(Y)).

A i

t o prove t h a t f o r any

x

with

1

T h i s a l l o w s us t o show t h a t if 0 ( x )

f o r if n o t we c o u l d s t a r t a descending c h a i n i n

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

i(x)

7

6.

( T h i s i s r e a l l y a s p e c i a l case o f a more

general s i t u a t i o n a p p l y i n g t o t h e t r e a t m e n t o f any a c c e s s i b i l i t y i n d u c t i v e d e f i n i tion i n

1

Z1 - D C .

We s h a l l go i n t o t h a t i n t h e subsequent paper on t h e

Once i t i s e s t a b l i s h e d t h a t

(7)

1

C1 - D C

1. Wx[Ao(X)

A

Wy(y <0 x

+

O(U)) * O(X)].

ID:.)

183

Iterated Inductive, Fixed Point Theories We can use ( 3 ) t o show [C(X)

(8) Now we can a p p l y

[F,S]

der t h e €-number

$(n-1,

(9)

1.: $ ( n l e o ) [C(X)

QF-CA

+

a

XI.

E

t o get the elimination o f E

QF-DC

i n favor o f

QF-CA

un-

): +

a

X!

E

then a p p l y t h e complete e l i m i n a t i o n r e s u l t - f r o m §4 t o g e t

Applying t h e argument of Sch’irtte [19771 p. 210 we may conclude claimed

l a ( < $(n(Eo)

as

.

PART 11.

EXACT BOUNDS

6. HANCOCK’S CONJECTURE FOR MARTIN-LOF‘S CONSTRUCTIVE TYPE THEORIES.

Martin-Lof

has developed s e v e r a l v e r s i o n s o f a k i n d o f l o g i c - f r e e c o n s t r u c t i v e t h e o r y of t r a n s f i n i t e types.

The f o r m a l d e s c r i p t i o n of t h e s e i s somewhat e l a b o r a t e and we The r e a d e r i s r e f e r r e d f o r such t o

s h a l l n o t t r y t o p r e s e n t t h e d e t a i l s here. M a r t i n - L o f [1975],

[19791.

While t h e l a t t e r paper c o n t a i n s a number of improve-

ments i t i s a l i t t l e e a s i e r t o i n d i c a t e t h e n a t u r e of t h e system i n t h e former one. T h i s i s done h e r e v e r y r o u g h l y and w i t h d e l i b e r a t e o v e r s i m p l i f i c a t i o n . The b a s i c system o f M a r t i n - L o f [1975] i s denoted g i v e n l o c . c i t . make use o f u n i v e r s e s reflection principles. M Ln.

The

Vo

,...,V n,. ..

system w i t h

n

M Lo.

Extensions of

M Lo

which s a t i s f y c e r t a i n

universes

Vo,...,V

n-1

i s denoted

There a r e terms f o r o b j e c t s and terms f o r types. A l l d e r i v e d statements

a r e o f t h e form:

a

E

A

(a-

i s o f type

A)

special form o f natural deduction calculus. and when A

i s a proposition,

To e s t a b l i s h t h a t a term

A

a

E

A

and

a = b;

d e r i v a t i o n s a r e by a

P r o p o s i t i o n s a r e i n t e r p r e t e d as types

g i s a p r o o f o f A.

i s i n t e r p r e t e d as:

denotes a t y p e one e s t a b l i s h e s

Each o b j e c t has a u n i q u e t y p e .

The c o n s t a n t types o f

i n t e r p r e t e d as t h e n a t u r a l numbers and

(1

,. ..,k)

?r Lo

resp.

A

E

Vn

are

(k=0,1,2..

f o r some

N and

..).

Nk, TO

n.

S. FEFERMAN

184

describe these we use o b j e c t terms f o r

and the successor o p e r a t i o n s .

0

x

i s e s t a b l i s h e d t o be a type under t h e hypothesis (ZxcA) B ( x )

E

then

A

a r e types, w i t h t h e i r usual i n t e r p r e t a t i o n .

B(x)

If

(IIXEA) S ( x ) and

Corresponding t o t h e

f i r s t of these we have formation of o b j e c t terms by a b s t r a c t i o n and a p p l i c a t i o n , while corresponding t o the second we have formation by p a i r i n g and p r o j e c t i o n s . (When B ( x )

(Vx

E

x

i s a proposition f o r

A) B(x)

and

t i o n f o r each type A proposition t h a t

E

A ) B(x)

and

a,b

(Ex

For

a = b.

E

E

A

then

i s written A

Y Ln

a type

(nx (3x

E

A ) B(x)

i s written

A) B(x).)

E

There i s i n addi-

I A ( a , b ) which i s i n t e r p r e t e d a s the

each V i ( i < n )

i s conceived of a s a type

a l l of whose members a r e t y p e s , s a t i s f y i n g t h e same axioms and r u l e s a s t h e informal "universe of a l l types". Systems

Furthermore, we take

Vi

for

Vi+l

E

i + 1 < n.

w i t h e s s e n t i a l l y t h e same f e a t u r e s a r e presented i n Martin-

Y L,,'

In a d d i t i o n , t h e r e a r e extensions given by r u l e s f o r inductively gen-

LSf [1979].

Such extensions a r e not

erated t y p e s , e.g. t h e type Qf c o n s t r u c t i v e o r d i n a l s . considered here.

According t o Martin-Lof [1975], Peter Hancock conjectured t h a t f o r each M Lo

ML1

with

n.

This was known f o r HA.

n

For

=

n

= 0

[!I L n l = an

by the p r o o f - t h e o r e t i c equivalence of

1 i t was e s t a b l i s h e d by Aczel [1976] a s follows.

was i n t e r p r e t e d ( o r simulated, i n Aczel ' s terminology) i n a c e r t a i n

Rext fD, was i n t e r p r e t e d i n E;

( w i t h classical logic).

- AC,

First,

f0,

a s described

above i n 12. Using the r e s u l t of Friedman [1967] ( c f . [1970]) by which 1 IZ1-ACI 5 Q E O ( O ) , one concludes IML1l '< a l . In the r e v e r s e d i r e c t i o n , Aczel f u r t h e r used Friedman's r e s u l t ( l o c . c i t . ) t h a t equivalent t o

RA,

1

Z1

(ramified a n a l y s i s in l e v e l s

-

AC <

i s proof-theoretically

E ~ ) .

Now

<

ML1,

Jervel 1 [19761

11

0

and the l a t t e r i s

translatable into RA(i) interpretable i n

is

RA<

0

€0

giving a1 5 IML,

1.

o u t l i n e d a proof t h a t

an 5 lMLnl

d i r e c t construction of o r d i n a l s and ordinal functions within

f o r each

n,

these systems.

by a

3/ -

Thus t o complete t h e proof of Hancock's c o n j e c t u r e , i t is s u f f i c i e n t by t h e main

-

Iterated Inductive, Fixed Point Theories

I IDnl 5 an A

result

fDn.

Aczel's i n t e r p r e t a t i o n o f

in

s i v e codes o f types.

then (Zx

e

If

model

Vi(i

further

< n)

T h i s can be done by a d i r e c t e x t e n s i o n of f0,.

(O,O),

F i x codes

Roughly speaking, one d e a l s w i t h r e c u r (1,k)

and { f } ( x )

A

i s a code f o r

A)B(x);

E

ML1

i s a code f o r

(3,e,f)

can be i n t e r p r e t e d

o f P a r t I o f t h i s paper t o show how MLn

(or "simulated") i n a s u i t a b l e

resp.

185

(nx

E

(5,e,(a,b))

A)B(x)

and

(2,i)

for

i s a code f o r and

(4,e,f)

i s a code f o r

as t h e l e a s t s e t c o n t a i n i n g

B(x)

(5,e,(a,b))

a,b

E

(where

A

whenever

e

codes

A).

e

E

and

Vi

{fl(x)

and

Vi

x

f o r each

IA(a,b).

N o r m a l l y we would

(O,O), (1,k) (k=0,1,2. ..:)

E

Vi

A

E

i s a code f o r

( 2 , j ) (j < i ) and c l o s e d under t h e code f o r m a t i o n o p e r a t i o n s and

FI, Nk

f o r each

(3,e,f). x

E

However, f o r t h e i n t e r p r e t a t i o n o f

and

(.4,e,f)

A

and

ML,

it is

s u f f i c i e n t t o use o n l y t h e f i x e d - p o i n t p r o p e r t y o f t h e corresponding i n d u c t i v e

Vi(i

d e f i n i t i o n f o r each

< n).

Hence t h e p r i n c i p l e s o f

fDn

s e r v e the'required

purpose. REMARKS. (i)I n h i s 1976 paper, Aczel s a i d h i s main r e a l i z a t i o n f o r

lMLl( = a,

was t h a t t h e f i x e d - p o i n t p r o p e r t y o f a s u i t a b l e i n d u c t i v e d e f i n i t i o n s u f f i c e d t o interpret

ML1.

A c t u a l l y , one needs a r e f i n e m e n t o f t h e g i v e n argument t o deal w i t h t h e

(ii)

types " a t l a r g e " i n ML,,

which a r e n o t r a n g i n g o v e r some Vi

i.e.

The whole " u n i v e r s e " o f types a t l a r g e a c t s l i k e

Vn

i n MLn+l,

with

i < n.

and t h e d i r e c t

A

i n t e r p r e t a t i o n o n l y p u t s us i n

fDn.

in

of

I n s t e a d , we i n t e r p r e t

each p r o o f o f

MLn

W h i l e such a p r o o f can r e f e r (by v a r i a b l e terms) t o a r b i t r a r y members

Vi(i

< n),

i t i n v o l v e s o n l y e s s e n t i a l l y a f i n i t e number o f types a t l a r g e .

For these we can use an i n t e r p r e t a t i o n which i s a r i t h m e t i c a l i n t h e i n t e r p r e t a t i o n s taken f o r (iii) IML<

Let

u1

Vo

,...,Vn-l.

ML<

=

u

n
ML,.

I t f o l l o w s from t h e r e s u l t

lMLnl = an

that

= sup an which, as we know, i s t h e l e a s t i m p r e d i c a t i v e o r d i n a l n

Aczel announced t h e r e s u l t i n Oxford i n A p r i l , 1980.

IML<

uI

=

ro

ro.

a t an i n f o r m a l p r o o f - t h e o r y meeting h e l d

However t h e methods he d e s c r i b e d d i d n o t y i e l d Hancock's

S.FEFERMAN

186

( l o c a l ) conjecture. (iv)

Reeson I19801 has g i v e n r e c u r s i v e code models f o r t h e v e r s i o n s

systems i n M a r t i n - L o f [1979].

o f the

,',

t'

Using these, t h e obvious i n t e r p r e t a t i o n of

is

h

A

in

M,L',

IDnt1,

b u t one can reduce

YL,',

to

IDn

by t h e k i n d o f r e f i n e m e n t o u t 1 ned

i n ( i i ) above.

T h i s was t h e outcome o f a d i s c u s s i o n which Beeson and I had

January, 1981.

Presuming t h a t , one a l s o concludes

(ML,,'I

= an

and

IML:

n

ro,

w( =

as would be expected. 7.

BOUNDS FOR SUBTHEORIES OF To WITH UNIVERSES.

versions o f t h e constructive type-free theory

There a r e two ( m i l d l y d i f f e r i n g

To

o f f u n c t i o n s and c l a s s e s :

Feferman [1975] and [1979].

We r e f e r t o t h e l a t t e r v e r s i o n here.

d u c t i v e g e n e r a t i o n axiom o f

To,

f a l l i n t o f o u r groups: hension)

111.

To-IG

denotes t h e r e m a i n i n g axioms.

To-IG

and

IV.

ECA (Elementary compre-

J (Join).

I - 111 i s a l s o denoted

EMN + J ; t h i s system i s denoted

i s t h e same as

These

11.

APP ( A p p l i c a t i v e axioms)

( N a t u r a l numbers)

N

l o c . c i t . so

EMN

I.

so

IG i s t h e i n -

TA

in

the following. To e x p l a i n ables

a,b,c.

TA

i n more ( b u t n o t f u l l ) d e t a i l :

...,x,y,z

and c l a s s v a r i a b l e s

A,B,C,

t h e r e a r e two s o r t s o f v a r i -

...,X,Y,Z.

Basic r e l a t i o n s

a r e those of e q u a l i t y (between o b j e c t s o f any s o r t ) , t h e a p p l i c a t i o n r e l a t i o n xy

2

z

and membership

x

A.

E

There i s a b a s i c O n t o l o g i c a l Axiom

and t h e l o g i c i s t h a t o f t h e i n t u i t i o n i s t i c p r e d i c a t e c a l c u l u s .

WX3x(X=x)

Axiom group

I

g i v e s a p a r t i a l combinatory s t r u c t u r e w i t h p a i r i n g and p r o j e c t i o n f u n c t i o n s and, on

N: successor, predecessor and d e f i n i t i o n by cases.

f o r each elementary

0

Axiom group I 1 a s s e r t s

(no bound c l a s s v a r i a b l e s ) e x i s t e n c e o f a c l a s s

{xlfl(x,y,L)],

g i v e n u n i f o r m l y as t h e v a l u e o f a f u n c t i o n

parameters o f

0.

Axiom group I11 g i v e s c l o s u r e o f

cgl(x,L)

applied t o the

N under D and

N ( f o r a l l f o r m u l a s o f t h e system).

f u l l scheme o f i n d u c t i o n on

and t h e

Finally, the

J o i n Axiom I V i s Wx

A3Y(fx = Y)

+

3X j ( A , f )

hWz{Z

E

x

X -3x,y[Z=(X,y)

A

X E

A

A

y

E fX]~],

Iterated Inductive, Fixed Point Theories where TA

i s a c o n s t a n t o f t h e system.

j

We w r i t e

Z fx XEA

187

for

j(A,f).

The

theory

i s ( p r o v a b l y ) n o n - e x t e n s i o n a l , b o t h f o r f u n c t i o n s and f o r c l a s s e s . From ECA

{f(Vx

E

A3y

we o b t a i n t h e o p e r a t i o n s B(fx = y)}).

E

From

and

A

B (=lflf:A+B1 =

+

n fx, XEA A sequence o f c l a s s e s (Rx)xEA is

and J

ECA

E

taken s i m p l y t o be an

satisfying this l a t t e r condition with

A

i s a c l a s s o f c l a s s e s i f Wx

i s also w r i t t e n

Z X.

We have

XEA

1

E

Y).

we o b t a i n t h e o p e r a t i o n

which i s a c l a s s when Wx f

A 3Y(fx

A x S

A3X(x=X).

(X,y)

ZX

E

XEA

++

Then X

1 x XEA

A

E

y

A

E

f x = Ex.

i s w e l l d a i n e d and X,

X X

i.e.

is

XcA

j u s t t h e membership r e l a t i o n r e s t r i c t e d t o c l a s s e s i n A . There is b y

(ECA)

a universe

V

C

o f a l l o b j e c t s b u t no c l a s s

of all U

By a u n i v e r s e o f c l a s s e s

classes, as i s seen by a f o r m o f R u s s e l l ' s paradox. we mean one s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s : (i)

U

i s a c l a s s of c l a s s e s

(ii) N E U (iii)U

i s c l o s e d under

Z1

then

U

(iv)

,...,Zn

E

U

+

AcUh(f:A+U)+ We w r i t e

Uni(U)

t h e system Uni(Vi)

TA

if

0(x,y,

{x\0(x,y,

Z1

,...,Z n ) l

J,

i s c l o s e d under

Z1

i.e.

ECA,

i s elementary

U

E

i.e. 1 f x ~ U . XEA

f o r t h e c o l l e c t i o n o f statements ( i ) p l u s axioms f o r

n

i + 1 < n, Vi

and f o r each

,...,Zn)

Vo

universes E

Vitl.

-

,...,Vn-l,

Let

TA

(iv).

By

i.e.

TA,n

we F a n

f o r each

1

< n,

, < u = UTA,,. n

We s h a l l p r o v e t h e f o l l o w i n g i n t h e n e x t s e c t i o n . THEOREM 3.

n = 0

For ITAI =

(0),

$E

=

(i)

(ii)

IT:,

<

u1

ro.

=

p a r t ( i ) o f t h i s theorem was a s s e r t e d i n Aczel [19761 as w i t h p r o o f o u t l i n e d as f o l l o w s :

0

much t h e same way as C11 - A C

and

@(n+llco)

i s , thus g i v i n g

MLl

i s reducible t o

T;

IT;/cal

by t h e f a c t t h a t

i s interpretable i n

Ti

=

RA,iL (

@E

(n).

in

On t h e o t h e r hand

is'interpretable 0

fD, i n TA;

S. FEFERMAN

188

thus

al

5 IT:\.

(See Feferman, Sieg [1981] f o r another and f u l l p r o o f of these

results. ) Remark.

Note t h a t w h i l e

TA

and

,n

MLn a r e analogous i n c o n t a i n i n g axioms f o r

I = an+l w h i l e IMLn( = an. The reason f o r the d i f ,n ference can be ascribed t o t h e f a c t t h a t t h e ML systems a r e l o g i c - f r e e w h i l e

n

universes, we have

IT:

f u l l i n t u i t i o n i s t i c p r e d i c a t e c a l c u l u s i s assumed i n the use the universe o f types a t l a r g e i n MLn+l

systems.

TA

We must

t o do l o g i c i n a universe

vo, ...,v n-1.

"over"

Vn

i n mind one might w e l l have With t h i s and A c z e l ' s r e s u l t f o r Ti conjectured t h e r e s u l t o f Theorem 3 i n analogy t o Hancock's conjecture. 41

8.

To prove

PKOOF OF THEORE!' 3 . A

in

ITA,nl zan+l

= $(n+llEo)

For t h i s we l o o k a t the r e c u r s i v e code model o f

[I9751 o r i n 119791 pp. 197-198, i g n o r i n g the clauses f o r take t h e domain o f i n d i v i d u a l s t o be APP

the

axioms.

(0,O) f o r tary

A

0

b

and f o r each

codes and

c

J

and

fx

y

for

~ ~ ( 1 . z when )

(l,L),

i s a code f o r x

E

A,

fx

n

IG. To begin w i t h we

If}(x)

L.

to satisfy

y

and

i s the G8del number o f an elemen-

(2,a,f)

codes some c l a s s

( A c t u a l l y we generate

E

for

Z fx

xeA

when

codes

Now we i n d u c t i v e l y generate

Bx.

o f classes and the membership r e l a t i o n axioms.

++

given i n Feferman

Then t o t e l l which numbers are codes o f classes we use a code

N, (l,n,b)

and

o

To

we i n t e r p r e t TA,r

x

E

c

t o s a t i s f y the

and i t s complement

N. ECA

simultaneously.)

However, m i n i m a l i t y i s n o t needed t o e s t a b l i s h a model, o n l y a f i x e d p o i n t o f t h i s inductive d e f i n i t i o n .

( R e l a t i v e t o t h i s we i n t e r p r e t the f i n a l c l a s s

o f classes as c o n s i s t i n g o f those

c

such t h a t

Wx[x

Formalizing t h i s argument gives t h e i n t e r p r e t a t i o n o f done by Aczel).

Vo

Next t o i n t e r p r e t Ti,l

o f codes s a t i s f y i n g

Ti

using

peat t h e procedure over t h a t . number of times, t o reduce The proof o f

an+l

in

ID2, A

E

c Ti

++

1 (x

in

o f codes

c)].)

'6,

(as had been

we f i r s t determine a c l a s s

and then p u t i n a code f o r

?D,,

C

Vo

and r e -

This process may e v i d e n t l y be repeated any f i n i t e

T'

o ,n

5

duced i n Feferman [1964] ( i n 6.19,

to

A

and thus e s t a b l i s h

Ian+,.

u t i l i z e s as i t s main component t h e method i n t r o 6.21) t o prove t h a t

ro I I I R (

where

IR

is

189

Iterated Inductive, Fixed Point Theories

an unramified system of predicative analysis; cf. also Feferman 119681 14.

HOW-

ever, some novel considerations are required. Write Prog(0) Prog(0)

+.

Wx

for

VX[

a0(x).

<

xP(y)

0(x)]

+

Then put I(cx,X)

i

I ( a ) = V.XI(a,X).

y

and TI(a,P) for

for WX

I(A)(a)

Finally, we write

where P(X)

= TI(a,fl)

E

is (x

A.I(a,X).

E

X)

and

(Sihce our

notation system i is contained in N we are using side by side lower case Latin letters as natural number variables and lower case Greek letters as variables ranging over the domain of

< .)

The method of well-ordering proof uses the ramified analytic hierarchy We first recall the matter informally. By definition,

(RAa)a.

(1)

(i)

RAo

(ii)

RAa+l

Bi RA

(iii)

consists of all Ix

consists of all Cx

=

y

Each RAa

( RAa 1

E

NI 0

(x,R l,...,Bm)l

and fl(x,Y, ,...,Ym) is in 2 ( Z ) , 2

RAa

E

with P arithmetical,

NI fl(x)]

E

LJ

n
RAa

for limit

wh'er5

and

y.

is thus a class of subclasses of N. A s usual W

( RAa

in (l)(ii)

indicates that all 2nd order quantifiers in P are relativized to RAa. The hierarchy may be directly relativized to any subclass X of N; that case it is denoted (RAa(X))cx. a < B

+.

E

RAo(X)

and

RAa(X) 5 R A B ( X ) .

For any fl(x), shown that Prog(b) (2)

Of course X

TI(E.P*)

Note that 0*

let pl*(x)

*

+

+

= vy[wz

4

yfl(z)

+

y + wxfl(z)l.

Vz

It is easily

Prog(P ) and so

i1(2,p).

results from 0 by use of propositional operators and numerical

quantifiers. Hence if

U is

any class o f subclasses of N closed under arithmetic

definability, we have I ( " ) ( p )

+

1(")(oB).

In particular,

for each 6 and U = R A 6 ( X ) we have I(')(B) I(')(wB). wn(B) Let wo(E) = B I w ~ + ~ ( B )= o ; thus sup(on(B) + 1) = f(E), n (3)

in

-+

and since

S. FEFERMAN

190 E ( E ~ )= E

(4)

+we~

~

f o r each

have U = RA6(X)

if

6,

It f o l l o w s t h a t t h e p r o p e r t y

then

I(')(E~+~).

-+

i s p r o g r e s s i v e . We now use t h i s t o

0(B) =

show t h a t

(5)

f o r each

For,

6 > 0

(RAV+l ( X I 1 I (8)

-+

(RAV(X)) 1

RAV(X))

I61 1

( E ~ ) )E

f o r each

(For f i n i t e a

=

and

U = RA

&a+' .6

(X)

explicit definition o f

$a+l

$ o . = X6.w'

from

Y

from

$a

We show

I(')(@)

+

I(u)($a(f3)).

the s h i f t t o

~ ~ ' ~ - 6

The p r o o f of ( 6 ) i s based on t h e

( j u s t as

= [email protected]

C$l

y = l i m y[n],

for

o f the d e r i v a t i o n o f (5) from (3).

we have

U = RA (X); 2.6

i s n e c e s s i t a t e d t o t r e a t t h e l i m i t cases.)

$

RAv(X).

Now t h e general s i t u a t i o n i s as f o l l o w s :

i t i s s u f f i c i e n t t o take

and o f

WO.6

+

T h i s simply comes f r o m t h e f a c t t h a t

(E~).

RAV+,(X).

a, 6 > 0

I(''(6)

we have

i s a l i m i t number, so RAw.,(X)

0.6

(6)

U = RAw.,(X)

and

n

i s d e f i n e d from

B

and f o l l o w s t h e i d e a

F o r f u l l d e t a i l s c f . Feferman [1968] 83.

The

f o l l o w i n g i s a c o r o l l a r y o f (6): (7)

f o r each

a

and

VX

U,

E

U [ R A a+l(X)

For, under t h e h y p o t h e s i s , g i v e n U1 = RA

a+l(X),

w

X

E

5 U]

I(')($a(0)).

U we c e r t a i n l y have

(U1 1 I ( c $ ~ ( O ) ) by (6).

SO

+

But

X

6

U1

I

(U),

(0)

where

so t h i s i m p l i e s

0

I ( $ (O),X); NOTE. a,6

we conclude

I(')($,(O))

since

X

i s arbitrary i n

U.

F o r t h e a p p l i c a t i o n ( 7 ) t o a g i v e n G we need t o e s t a b l i s h ( 6 ) o n l y f o r with

uses o n l y

a

5

I(')(;)

and for

~ " ~ - 46 wa+l.

w

= RA oa+'+l

T h i s i s done by i n d u c t i o n on

TA,,(

= TA + n u n i v e r s e s ) .

a.

and

(XI.

Flow we c o n s i d e r t h e s e statements i n t h e f o r m a l framework o f of t h e

a4

TA

and t h e n

F i r s t of a l l we a p p l y ( 2 ) t o any formula

13

191

Iterated Inductive, Fixed Point Theories of $ ( T i ) .

T h i s a l l o w s us t o show

(8)

kTI(i,0)

Ti

f o r each

TI(wn(0),O)

a<

f o r each s p e c i f i c

To proceed f r o m ( 7 ) and ( 8 ) we must see f o r which RAa(X)

l i s h existence o f

we can e s t a b l i s h WX

X

and

a

we can estab-

as a c l a s s o f c l a s s e s and t h e n f o r which

U[RA,(X)

E

i n our theories.

5 U]

examine t h e d e f i n i t i o n o f t h e

Thus

0.

and each

E~

n.

a

and

U

And f o r t h a t we must r e -

Ma. F i r s t o f a l l , i t i s t o be n o t e d t h a t c l a s s i -

cal l o g i c p l a y s no e s s e n t i a l r o l e i n t h e p r e s e n t a p p l i c a t i o n o f t h e r a m i f i e d The d e f i n i t i o n o f ( 1 ) o f

hierarchy.

,

@A$

," ~

can be r e a d j u s t as w e l l con-

s t r u c t i v e l y p r o v i d e d we have v a l i d a t e d t r a n s f i n i t e r e c u r s i o n up t o

v;

moreover

However, we must

the arguments f o r ( 2 ) - ( 7 ) a r e a l l w i t h i n i n t u i t i o n i s t i c l o g i c .

look c l o s e r a t t h e d e f i n i t i o n (1) t o see how o u r q u e s t i o n s a b o u t e x i s t e n c e of t h e as c l a s s e s of c l a s s e s may be answered.

RAa(X)

I n o u r n o t a t i o n system we a s s i g n t o each l i m i t number y mental sequence

y[nl

with

y = sup y [ n ] .

F!ow use t h e r e c u r s i o n theorem ( i n t h e n t o d e f i n e an o p e r a t i o n r ( a , n ) = y such t h a t f o r

applicative structure o f

Ti)

each a, r ( a , n ) ( n

... )

= 0,1,2,

w i l l be an enumeration o f

r(0,n)

of

N. F o r l i m i t

we can t a k e

y

and t h a t

Wn 3 Y r ( a , n )

Denoting

Ir(a,n)ln

then f o r m

such t h a t

( RAa 1

for s u i t a b l e

n

E

i t s range f o r

N1 b y RA,

we t h u s have =

N and x

(x,B 1,...,8m)

bi,

r(a+l,n).

i.e.

= j(N,Xn.r(a,n))

Ja

0

E

Y,

r ( y [ ( n ) o ] $ (n),).

r(y,n)

r e q u i r e s some work i s t h e d e f i n i t i o n o f

formula

RA,.

The d e f i n i t i o n

s i m p l y uses t h e s t a n d a r d enumeration o f t h e a r i t h m e t i c a l subclasses

of

(n,x)

a c a n o n i c a l funda-

1 r(a,n),

nE N

r(a,n).

Suppose

RA,

Ri

i n RA,,

now s u b s t i t u t i n g

t

E

Ra-~(Y).

elementary f o r m u l a O'(x,bl,

f o r each occurrence ( RA, ) Thus 9 (x,R l,...,Bn)

...,bm,J a) .

Y

we have t

RA,.$(Y)

E

We can

whose members a r e j u s t those p a i r $

E

formulas WY

classes.

Now i n any r e l a t i v i z e d 2nd o r d e r

w i t h parameters

,J

i s defined

a c l a s s of c l a s s e s .

f u r t h e r we r e p l a c e each q u a n t i f i e d subformula 3 Y E

r(cr,n)

... c o n s i s t s of

n = 0,1,2,

and so we can r e p l a c e each occurrence

(n,t)

The o n l y case t h a t

E

Ri

by

Bi = r(a,bi) (bi,t)

E

Ja;

by 3n$(r(a,n)),

and s i m i l a r l y f o r sub-

becomes e q u i v a l e n t t o a n

The t r a n s f o r m a t i o n o f

0 i n t o 8'

is

S. FEFERMAN

192 e f f e c t i v e and uniform i n Ja; t i o n An r ( a + l , n ) . chy i n To

using t h a t one now sees how t o d e f i n e the enumera-

(For more d e t a i l s about the treatment o f the r a m i f i e d h i e r a r -

cf.

Feferman, Sieg [1981] 1.2.)

A l l o f the foregoing may be d i r e c t l y r e l a t i v i z e d t o any define

= y w i t h the intention t h a t

r(a,n,X)

enumerate

RAa(X).

This r e c u r s i v e d e f i n i t i o n o f

i n g conclusions. (9) (i)For n = O,l,Z, (ii)

If

(iii)

U[RAa(X) 5 U]

E

If 4

If

E

then V X

U[RA,(X)

RAa(X) t U

(We here w r i t e (10)

y WX

E

Vn 3Y

E

0(')(;)

we must know t h a t

U[RAy(X) 5 U ] .

If a

(ii)

then

E~

For each a < a.

= $JE (0)

fl(')(a)

{alfl(")(a)I

8'(a,J)

Y].)

2

E

Thus:

c-

U[RAa(X)

where

Ul

is

0 is

2

defines a c l a s s .

E

O*

J =

x

This w i l l exists,

X€U

f o r s u i t a b l e elementary

We a r e now nearing the conclusion.

(iii)

E

i s a c l a s s o f classes, f o r then

U

f ~ ( ~ ) ( a )i s e q u i v a l e n t t o

(i)

and

Thus t o apply t r a n s f i n i t e i n d u c t i o n on i t i n order t o

= Y].

be so i f we know t h a t

(11)

(ECA)

i s a l i m i t number and

U[r(a,n,X)

The p r o p e r t y i n question may be w r i t t e n as

and

WX

and

X.

(in a).

progressive

YX Yn 3Y[r(a,n,X)

then y i e l d s t h e f o l l o w -

U].

i s a universe o f classes, the p r o p e r t y Y X

U

establish

(J)

U[RAa+,(X)

then

5 UI

for

will

enumerates the classes a r i t h m e t i c a l i n

i s a c l a s s o f classes and y

U

Va

..., r(O,n,X)

N so as t o

L= 9 s 1 s 2 s . . .

for

r(a,n,X)

i s a c l a s s o f classes closed under

U

YX

r(a,n,X)

X 5

8'.

F i r s t o f a l l we obtain:

= y.

TA F Y n Y X 3Y r(a,n,X) TA k I ( $ a ( 0 ) ) .

ITAI.

0

The proof o f (i) uses

of classes"

of

TA

TI(;)

in

TA

i s closed under

0 ( a ) i s progressive j u s t as i n ( 9 ) . existence o f t h e c l a s s o f classes RA

oa+l+l

(X).

given by (8), and t h a t t h e " f u l l universe

(J) and (ECA),

so t h a t the p r o p e r t y

Thus f o r each

and

RAa(X).

X

a

.(

E~

we have

But then the same holds f o r

As we have seen f o r (7), t h i s guarantees

I($Ja(0),X).

193

Iterated Inductive, Fixed Point Theories NOWl o o k a t (ECA)

Tl,;

= TA

we can a p p l y ( 1 0 ) . W

a class

+

Uni(Vo).

i s c l o s e d under

B u t as we have seen t h e p r o p e r t y Thus if a s

i n t h e universe o f a l l classes.

t o conclude t h a t

Vo

Since

1(1$~(0),W) holds.

E~

U0)

( J ) and

(a), defines

la10

we can a p p l y ll(ii)

I t f o l l o w s t h a t f o r each

B <

(0)

we

0

can prove (12)

0

(V0)

(B).

T h i s now g i v e s us p a r t ( i ) of t h e f o l l o w i n g .

If B <

(i)

(0)

@€

then

1 VX

TA,l

(ii)F o r

6

(0)

@€

we have

'Val.

Vo[RAB(X)

c

Po)

0

1

(@B(0)).

TA,ll

(V0) I (a)

TA,lI.

0

(iii) F o r

(iv)

al

a < a =

2

@

1

we have

(0),

I.

IT;,1

P a r t (ii)f o l l o w s from ( i ) and (7); t h e n ( i i i ) i s an immediate consequence. RAo

Since

cVo,

every

TI(a,J,)

with arithmetical

i s p r o v a b l e from

J,

(Vo) I (a) so we t h e n o b t a i n ( i v ) .

-

The g e n e r a l s t a t e m e n t i s as f o l l o w s . (13)

(if

If B

(ii)F o r

an t h e n

an+l

Vx

,n

B < an we have TA

(iii)F o r a < an+l, (iv)

TA

2

we have

for

TA

1

fV0) I (@B(0)).

.n

1

(Vo)

I

(a).

lT;,nl*

This i s e s t a b l i s h e d by i n d u c t i o n on ntl

,n

vo[mB(x)cvoI.

E

universes o f classes

Vo

E

n t o t h e n u n i v e r s e s Vl

V1

n.

Given t h a t i t h o l d s f o r

E . . . ~ Vn

E...E

Vn

n,

and g i v e n

we a p p l y t h e i n d u c t i o n h y p o t h e s i s

i n p l a c e of

Vo

E...E

Vn-l.

Then f o r

(V1) (V0) a < an+l, we have I (a). Now l o o k a t t h e p r o p e r t y 0 (B) = (Yo) @)I which belongs t o V X E VoIRAB(X) 5 Vo]. T h i s d e f i n e s a c l a s s {BIB (V0) s i n c e V E V1 and s i n c e ( x ) E V,. Hence we o b t a i n 0 ( B ) from xcvo

I

(V0)

(B)

for

P < an+l.

d e r i v e (ii) - (iv)

This gives ( i ) f o r

f r o m ( i ) j u s t as i n ( 1 2 ) .

n

+

V1

1 and we can t h e n proceed t o

T h i s completes t h e p r o o f of Theorem 3.

194

S. FEFERMAN

FOOTNOTES 1/

This paper takes the place o f my p r e s e n t a t i o n a t the Patras meeting which was e n t i t l e d 'Some t h e o r i e s o f s t r e n g t h

To'.

I t r e s u l t e d from t h i n k i n g about

some of the open problems i n t h a t subject.

The research was supported i n p a r t

by a g r a n t from the National Science Foundation.

2/

The t h e o r i e s

IDn

were f i r s t e x p l i c i t l y formulated by K r e i s e l i n 1963 ( i n un-

pub1 ished r e p o r t s o f a seminar on the foundations o f a n a l y s i s a t Stanford). There was a question as t o t h e i r c o n s t r u c t i v e significance, and w i t h i n t u i t i o n i s t i c l o g i c . d e f i n i t i o n t e l l s you what i s

n = 1

even f o r

As p u t i n K r e i s e l [1965] (p. 139) "an i n d u c t i v e

&

PA

n o t what i s

consideration o f r e s t r i c t e d forms o f

in

PA".

i n particular

ID1,

*

This motivated However o n l y

ID1.

p a r t o f t h e idea i s captured by t h a t r e s t r i c t i o n , since negation can s t i l l be applied t o

3/

i n arguments w i t h i n

(classical o r i n t u i t i o n i s t i c ) .

ID;

Peter Aczel has informed.me t h a t Hancock had shown how t o c o n s t r u c t the o r d i n a l < an

i n ML,

made an

4/

PA

5

i n h i s M. Sc. t h e s i s some time before J e r v e l l ' s work.

lMLnl

This

plausible.

Since w r i t i n g t h i s I have learned from Aczel t h a t he knew t h e reductions

5 TA,n-l < YOn

MLn

i n 11r76 and had conjectured t h a t these t h e o r i e s a l l have

t h e same p r o o f - t h e o r e t i c o r d i n a l

F u r t h e r r e l a t e d conjectures he had made

an.

a t t h a t time w i l l be discussed elsewhere.

Added i n proof.

i n t e r p r e t a b l e i n the system

(II: -

i s already i n t e r p r e t a b l e i n

k(0) gotten.

.

Apropos Theorem 2, S . Simpson pointed o u t t o me t h a t CA)

and hence t h a t

ID(0)

.

lfD(0)l

is

I n fact

€0

PA, a f a c t I knew a long time ago b u t had f o r -

This goes back t o Kleene's mistake i n t h e e v a l u a t i o n o f t h e predicate

x € 0 i n h i s 1944 paper 'On the forms o f predicates i n t h e theory o f c o n s t r u c t i v e o r d i n a l s ' (Amer. J. Mathematics); he claimed t o show t h e r e t h a t 0 i s i n TI;.

The

e r r o r was corrected i n h i s 1955 paper f o r t h e same j o u r n a l . A l l t h a t had been showr i n t h e 1944 paper was t h a t t h e r e i s some where

A

I$

i s t h e usual c l o s u r e c o n d i t i o n f o r

solution P o f

0

Wx[A(x,P)

( b u t o f course such

P(x)]

~--3

P

i s not

195

Iterated Inductive, Fixed Point Theories the minimal s o l u t i o n ) .

The p r o o f i s by a s i m p l e d i a g o n a l i z a t i o n .

Write

A(x,P)

as (XIo

x = o v [x = 2

(4,

A

P ( X ) ~ ]v ( x = 3-5

y

m

A

k - m]

A

on

0),

T T ~ in

P.

Ifa

combination o f

JI;

a i'Ii P

f o r m u l a s and f o r m u l a s

Note f u r t h e r t h a t i f we d e f i n e r e l a t i v i z e s t o show t h a t each

A.

h

IDn

an.)

for predicates

{(x)2}(n')

P(k)l).

so

A

i s a positive

I$ formula:

B

is

A l l o f t h i s may be c a r r i e d o u t i n ' PA.

U1 = 0,

On+l

=

h

00'

t h e n t h e p r e c e d i n g argument

i s interpretable i n

IDn(~l,-..On)

r e s u l t s may make i t a b i t s u r p r i s i n g t o t h e r e a d e r t h a t s u i t a b l e have s t r e n g t h

A

so d i a g o n a l i z a t i o n w i l l ' p r o d u c e

t h e r e s u l t i s p r o v a b l y I$,

s a t i s f y i n g FP1 f o r t h i s

+.

c

t o be r.e. b h i c h agrees w i t h

P

Vn 3 k 3 m [ { ( x ) 2 } ( n ) = k

Wn Wk[C(x)2}(n) = k

We can t a k e <

substituted f o r

A

PA.

fDn

(These theories

,

I n any case, Theorem 2 i s o n l y o f i n t e r e s t when d e a l i n g w i t h P2

,...,P n

other than

O2

,...,On.

S. FEFERMAN

196

REFERENCES P. Aczel [1976], "The s t r e n q t h o f M a r t i n - L o f ' s i n t u i t i o n i s t i c t y p e t h e o r y w i t h one u n i v e r s e " , ( u n p u b l i s h e d MS). M. Beeson [1980], "Recursive models f o r c o n s t r u c t i v e s e t t h e o r i e s " , U t r e c h t , Dept. o f Maths., P r e p r i n t No. 179.

University o f

S . Feferman [19641, "Systems o f p r e d i c a t i v e a n a l y s i s " , J. Symbolic L o g i c 30, 1 - 30. S . Feferman [1968], "Systems o f p r e d i c a t i v e a n a l y s i s , 11: n a l s " , J. Symbolic L o g i c 33, 193-220.

representations o f ordi-

S. Feferman [1975], "A language and axioms f o r e x p l i c i t mathematics", and L o g i c , L e c t u r e Notes i n Maths. 450, 87-139.

i n Algebha

S . Feferman [1979], C o n s t r u c t i v e t h e o r i e s o f f u n c t i o n s and c l a s s e s , i n L o g i c C o U o q L L i ~ n1978 (N.H. Publ. CO.) 159-224.

S. Feferman and G. Jager [1981] , "Choice p r i n c i p l e s , t h e b a r r u l e and autonomousl: i t e r a t e d comprehension schemes i n a n a l y s i s " , ( t o appear). S . Feferman and W . S i e g [1981], " P r o o f t h e o r e t i c equivalences between c l a s s i c a l and c o n s t r u c t i v e t h e o r i e s f o r a n a l y s i s " , i n L e c t u r e Notes i n Maths. 897, 78-142.

H. Friedman [1967],

"Subsystems o f s e t t h e o r y and a n a l y s i s " , D i s s e r t a t i o n , M.I.T.

H. Friedman [1969],

"Theories o f i n d u c t i v e d e f i n i t i o n " , (unpublished notes).

H. Friedman [19701, " I t e r a t e d i n d u c t i v e d e f i n i t i o n s and and Phood Theohy (N.H. Pub. Co.) 435-442. H.

Z:

-

AC", i n I n M o n i s m

R. J e r v e l l 113781, " C o n s t r u c t i v e u n i v e r s e s I " , i n H i g h m S e t Theohy, L e c t u r e Notes i n Maths. 669, 73-98.

G. K r e i s e l [1965], "Mathematical l o g i c " , ( W i l e y ) 95-195.

i n L e m e b on modem muXhem&a,

P. M a r t i n - L o f 119751, "An i n t u i t i o n i s t i c t h e o r y o f types: p r e d i c a t i v e p a r t " , Logic C O U O 1973, ~ ~ (N.H. Pub. CO.) 73-118.

111

in

P . M a r t i n - L S f [1979] , " C o n s t r u c t i v e mathematics and computer programming", Proc. 6 t h I n t . Cong. L o g i c , Method., P h i l . S c i . (Hannover) ( t o appear).

K. S c h i i t t e 119771, Phood Theohy ( S p r i n g e r - V e r l a g ) x i i + 302 pp.

PATRAS LOGIC SYMPOSION

C. Memkides (ed.) @ North-Holland Publishing Company, 1982

197

A FINITE COMBINATORIAL PRINCIPLE WHICH IS EQUIVALENT TO THE 1-CONSISTENCY

OF PREDICATIVE ANALYSIS Harvey M. Friedman* Mathematics Department Ohio S t a t e U n i v e r s i t y Columbus, Ohio 43210

Kenneth McAloon** Department o f Computer & I n f o r m a t i o n Science City U n i v e r s i t y o f New York Brooklyn College Brooklyn, New York 11210

Stephen G. Simpson*** Mathematics Department Pennsylvania S t a t e U n i v e r s i t y U n i v e r s i t y Park, Penksylvania 16802

51. INTRODUCTION G o d e l ' s incompleteness theorem, p u b l i s h e d i n 1931, says t h a t i n any g i v e n formal system f o r mathematics t h e r e must e x i s t u n d e c i d a b l e number-theoretic s t a t e ments, i . e . sentences i n t h e language o f a r i t h m e t i c which a r e n e i t h e r p r s v a b l e n o r r e f u t a b l e i n t h e g i v e n system.

T h i s i s a remarkable r e s u l t , b u t i t i s w e l l known

t h a t G o d e l ' s examples o f undecidable statements a r e somewhat a r t i f i c i a l i n t h a t they i n v o l v e numerical coding o f n o t i o n s from l o g i c .

Therefore, e v e r since- 1931,

l o g i c i a n s have l o o k e d f o r s t r i c t l y mathematical examples,

i . e . number-theoretic

statements which a r e undecidable b u t a l s o m a t h e m a t i c a l l y s i m p l e and i n t e r e s t i n g .

A b r e a k t h r o u g h came i n 1977 when P a r i s I141 d i s c o v e r e d a simple, mathematic a l l y i n t e r e s t i n g , undecidable statement o f f i r s t o r d e r Peano a r i t h m e t i c .

I n fact,

i n t h e p a r t i c u l a r formal system

S1

El

i s a transparent generalization o f

the f i n i t e Ramsey theorem and so i s e a s i l y f o r m a l i z a b l e i n t h e language o f P a r i s showed t h a t

i s t r u e b u t n o t provable i n

S1

P a r i s ' work see McAloon [13] . )

PA

PA.

PA.

( F o r an e x p o s i t i o n o f

Thus P a r i s ' achievement was t o b r i n g t h e incomplete-

ness phenomenon i n t o t h e r e a l m o f f i n i t e c o m b i n a t o r i c s . The purpose o f t h i s paper i s t o b u i l d o n P a r i s ' work bv f o r m u l a t i n g a t r a n s parent c o m b i n a t o r i a l statement stronger.

ATRO

We show t h a t

o f *metical

S2

S2

which c l o s e l y resembles

S1

b u t i s v e r y much

i s t r u e b u t n o t p r o v a b l e i n Friedman's formal system

t r a n s f i n i t e r e c u r s i o n w i t h r e s t r i c t e d i n d u c t i o n [41.

t h a t t h i s r e s u l t i s i n t e r e s t i n g because i n v e s t i g a t i o n s o f Friedman 131, [41, and S t e e l [191 have r e v e a l e d t h a t

ATRO i s a much s t r o n g e r system t h a n

the s t a n d p o i n t o f mathematical p r a c t i c e .

We f e e l I51

PA f r o m

Namely, a g r e a t deal o f c l a s s i c a l

H.FRIEDMAN, J. McALOON and S.G. SIMPSON

198

mathematics involving n o t only continuous functions b u t also countable ordinals, Bore1 and analytic s e t s , e t c . , can be formalized in (conservative extensions o f ) ATRO b u t cannot be formalized in (conservative extensions of)

PA.

We should mention t h a t , using e n t i r e l y different considerations, Friedman I61 has discovered other f i n i t e combinatorial statements able in systems much stronger t h a n ATRO, that

e.g.

ZF

are n o t quite as transparent as

S3, S4,...

S 3 , S4,...

which are undecid-

s e t theory.

However, we feel

S2.

We now proceed t o a detailed description of our r e s u l t s . Let

X

be a subset of

o, the s e t of natural numbers. We write

XI.

[ X I c w = { f i n i t e subsets of

W e say t h a t a subset C

an element of (Co,Cl)

of

[w]<~

i s an element of

C

such t h a t

Co U C1 =

set

(Co,C1)

homogeneous

for

i s downward closed i f every i n i t i a l segment of

A p a r t i t i o n of

C.

Co

and

is a set

Y

and

X

C1

i s an ordered pair

A

are downward closed.

c_ X such t h a t

[YI
C_ Co

or

[Yl<w~C1

The following statement i s an immediate consequence of the GalvinlPrikry theorem 171:

f o r every p a r t i t i o n of an i n f i n i t e s e t

f i n i t e homogeneous s e t .

We denote t h i s statement by

X

c_ o there e x i s t s an in-

RT(<w,2)

since i t i s a gen-

eralization, of (the i n f i n i t e version o f ) Ramsey's theorem. We imitate Paris by considering the following " f i n i t e miniaturization" of RT(
i s 0-dense i f

1x1

2 and

1x1

X

c w

min(X).

i s denoted

We say t h a t

X

1x1. is

for every p a r t i t i o n of X there e x i s t s an n-dense homogeneous s e t . undecidable statement dense f i n i t e s e t .

S2 reads as follows: f o r every n

This can be formalized as a

We say t h a t (n+l)--

Our strong

there e x i s t s an

consistent with a l l true

ATRO,

n-

0 112 number-theoretic sentence.

O u r main r e s u l t i s t h a t , provably i n primitive recursive arithmetic, equivalent t o the 1-consistencyof

if

S2 i s

i . e . t o the statement t h a t ATRO

is

0

TI, sentences. From t h i s i t also follows t h a t , provably

S2 i s equivalent t o the 1-consistency of IR of predicative analysis 121. Thus the proof-theoretical

in primitive recursive arithmetic, Feferman's system

ordinal associated with

S2 i s none other t h a n the Feferman/Schutte ordinal

ro.

A Finite Combinatorial Principle

199

We a l s o obtain estimates f o r t h e r a t e s of growth of t h e functions least y

fn(x) =

such t h a t the i n t e r v a l [x,yl = I x , x + l , ...,y-1 ,y}

i s n-dense.

ro.

up t o

Namely, these functions turn o u t t o be c o f i n a l i n the "Wainer hierarchy" (This r e s u l t may be compared w i t h t h e growth r a t e estimates i n Ketonen/

Solovay [9] . ) . The plan of our paper i s as follows.

In 92 we d e f i n e and d i s c u s s a c e r t a i n

system of second order a r i t h m e t i c w i t h r e s t r i c t e d induction. Precise sense i n which

S 2 i s a " f i n i t e m i n i a t u r i z a t i o n " of

We then explain t h e

RT(<w,2).

In § 3 we

define and d i s c u s s t h e system ATRO and show t h a t i t s principal axiom i s equivalent to

RT(<w,2).

This gives our main r e s u l t .

In §4 we discuss the connection

with p r e d i c a t i v e a n a l y s i s and o t h e r t r a d i t i o n a l proof-theoretic c o n s i d e h t i o n s . In 55 we consider reformulations of

S2 which avoid t h e awkward concept of n-

n.

density defined by recursion on

For helpful comments r e l e v a n t t o t h i s paper, we would l i k e t o thank Peter Clote, Carl G . Jockusch, J r . , Angus Macintyre, Richard B . Mansfield, James H. Schmerl, Robert M . Solovay, and each o t h e r .

A suggestion of Jockusch helped t o

simplify t h e proof of Theorem 3.2.

12.

A MINIATURIZED PARTITION THEOREM The formal systems which we s h a l l consider use c l a s s i c a l l o g i c and a r e formu-

lated i n t h e language of second order a r i t h m e t i c . arithmetic contains number v a r i a b l e s w,

plus set v a r i a b l e s

X, Y,

...

i , j, m , n , x , y , ... intended t o range over

intended t o range over subsets of w .

ical terms of t h e language a r e a s follows:

t.

t2 where

tl

tl = t2, t l < t2, and a s e t variable.

0, 1 , number v a r i a b l e s ,

t2 a r e numerical terms.

and

tl

E

X

The language of second order

where

t,

and

The numer-

tl + t2 and

The atomic formulae a r e

t2 a r e numerical terms and X

The formulae of t h e language a r e b u i l t up from atomic formulae

by means of number q u a n t i f i e r s propositional connectives

A,

vx and j x , v , -, +, <->.

s e t quantifiers

yX

and 3X,

and

A formula i s c a l l e d a r i t h m e t i c a l i f

is

H. FRIEDMAN, J. McALOON and S.G. SIMPSON

200

i t contains no s e t q u a n t i f i e r s ( b u t may contain f r e e s e t v a r i a b l e s ) .

If:

( r e s p e c t i v e l y I$) i f i t i s of t h e form vX3Y@ ( r e s p e c t i v e l y VX@) where

i s arithmetical.

A formula i s

i s a numerical term.

@

i f i t i s b u i l t u p from atomic formulae by means

II;

v x < t and 3 x < t where

of propositional connectives and bounded number q u a n t i f i e r s

t

A formula i s

1

A formula i s

0

(respectively

@

is

nz

form vx3y@ ( r e s p e c t i v e l y vx@) where

0

nl)

i f i t i s of the

I#.A sentence i s a formula with

no f r e e v a r i a b l e s . The axioms of our b a s i c system, +, -, 0, 1 , <;

semiring axioms f o r 0

E

X

A Vn(ncX

n+l

-f

a r e a s follows:

ACAO,

the usual ordered

the induction axiom E

X)

-+

Vn(nrX);

and comprehension axioms 3X vn(neX <-> @ ( n ) ) where @ ( n ) is any a r i t h m e t i c a l formula letters

ACA

The s u b s c r i p t

in

a r e a n acronym f c

ACAO

n which X

does not occur f r e e l y .

The

" a r i t h m e t i c a l comprehension axiom."

0 denotes the f a c t t h a t our systems have only r e s t r i c t e d induction,

i . e . we do not assume t h e induction scheme @(O)

f o r a r b i t r a r y @.

vn(@(n)

A

+

@(n+l)) vn@(n) +

Note however t h a t a l l a r i t h m e t i c a l instances of t h e induction

scheme a r e provable i n Clearly any model

ACAO. M

of

PA

can be expanded t o a model of

the set v a r i a b l e s range over parametrically d e f i n a b l e s u b s e t s of follows t h a t ACAO

is a conservative extension of

f i n i t e l y axiomatizable.

PA.

ACAO

by l e t t i n g

M.

From t h i s i t

Note too t h a t ACAO

is

Further information on ACAO may be found i n 141.

We assume t h a t some coding of the f i n i t e subsets of remainder theorem) has been formalized in the language of

w

(e.g. via the Chinese PA.

T h u s t h e statement

0 "yn3n-dense f i n i t e s e t " ( c a l l e d S2 i n 51) i s formalized a s a n2 sentence. Let 1 a l s o RT(<w,2) be the llz sentence formalizing t h e statement t h a t f o r a l l p a r t i -

tions of

w

( a s defined i n 51) t h e r e e x i s t s an i n f i n i t e homogeneous s e t .

We now define a p r e c i s e sense i n which

"vn3n-dense" i s a f i n i t e miniaturiza-

A Finite Combinatorial Principle

tion o f

F + T1

A f i n i t e s e t o f sentences

RT(<w,2).

i s c o n s i s t e n t where

T1

20 1

i s s a i d t o be 1 - c o n s i s t e n t i f

F

Ily

i s t h e s e t of a l l t r u e

the statement of 1 - c o n s i s t e n c y o f

F

sentences.

Clearly

0

can be f o r m a l i z e d as a

TI2 sentence.

The

main r e s u l t o f t h i s s e c t i o n i s :

THEOREM:

2.1

0 112

The f o l l o w i n g

sentences a r e provably equivalent

in p r i m i t i v e r e c u r s i v e a r i t h m e t i c : (i) Wn 3 n - d e n s e f i n i t e s e t ; ACAO + RT(<w,Z).

(ii) 1-consistency o f

We s h a l l g i v e an i n f o r m a l , m o d e l - t h e o r e t i c argument f o r t h e equivalence o f (i) From t h i s i t f o l l o w s t h a t ( i )

and ( i i ) which can e a s i l y be f o r m a l i z e d i n ACAO. and ( i i ) a r e p r o v a b l y e q u i v a l e n t i n the equivalence p r o o f w i t h i n

PA.

The m o d i f i c a t i o n s needed t o f o r m a l i z e

p r i m i t i v e r e c u r s i v e a r i t h m e t i c a r e r o u t i n e and w i l l

be l e f t t o t h e r e a d e r . and l e t F i r s t , assume (ii)

n

be g i v e n .

We w i s h t o prove t h a t t h e r e e x i s t s

By 1 - c o n s i s t e n c y i t s u f f i c e s t o p r o v e t h a t t h e e x i s t e n c e o f an

an n-dense s e t .

n-dense s e t i s p r o v a b l e i n ACAO + RT(<w,Z). more.

Let

prove t h a t n,

...,.jl

ACAO + RT(<w,2) ACAO + RT(<w,2)

does n o t p r o v e

Vn 6 ( n )

i s n-dense. proves proves

X

dense.

Then f o r a l l

j

that

6(n).

ACAO proves

6(n+l)).

-+

assume t h a t j,

d(n)

. . ,j}

From t h i s i t f o l l o w s t h a t ,

ACA

X

holds b u t

n {O,l, X

0

+ RT(<w,2)

n

...,j 1

{O,l,

6(n+l) i s not

...,j l

B u t , by

RT(<w,2)

(n+l)-

w i t h no nX

t h e r e e x i s t s an i n f i n i t e s e t

i s n-dense.

6(n)

there e i i s t s

y n In-dense.

j

T h i s c o n t r a d i c t i o n completes t h e p r o o f t h a t

( i i ) implies ( i ) . Conversely, assume

fails.

Hence by K i j n i g ' s lemma [ill t h e r e e x i s t s a p a r t i t i o n of

which i s homogeneous f o r t h i s p a r t i t i o n , and by Y rl IO,l,.

j

6 ( 0 ) . We s h a l l

(Note however t h a t

there exists a p a r t i t i o n o f

w i t h no n-dense homogeneous s e t .

5X

Wn(6(n)

be an i n f i n i t e s e t such t h a t f o r a l l

dense homogeneous s e t .

Y

Clearly

there exists

s i n c e we do n o t have t h e f u l l i n d u c t i o n scheme.)

Reasoning i n ACAO + RT(
X c_w

be t h e s t a t e m e n t t h a t f o r e v e r y i n f i n i t e

X rl {O,l,

such t h a t

for a l l

6(n)

We s h a l l a c t u a l l y prove a l i t t l e

We w i s h t o p r o v e 1-consistency of

such

202

H. FRIEDMAN, J. McALOON and S.G. SIMPSON

ACAO + RT(<w,2).

lrle s h a l l be sketchy s i n c e o u r argument c l o s e l y resembles t h a t o f

K i r b y l P a r i s [I01 and P a r i s [141; see a l s o McAloon [13]. Let

M

Wn 3n-dense p l u s a l l t r u e

plus in M

PA

be a c o u n t a b l e nonstandard model o f a l a r g e f i n i t e fragment o f

and l e t

X

sentences.

TI;

Let

n

be a nonstandard i n t e g e r

be an M - f i n i t e s e t which i s n-dense.

an enumeration of a l l M - f i n i t e p a r t i t i o n s of

Let

i < w)

((C6.C;):

{ O , l , . ..,max(X)I.

be

C o n s t r u c t a de-

scending sequence o f M - f i n i t e s e t s

x

=

x0 3 x 3 - 1 -

I = {a

I

3 1 -

i, Xi+l

so t h a t , f o r a l l

Clearly

2 x.

i s (n-i-1)-dense

and homogeneous f o r

i (Ci,Cl).

Put

M: 3 i ( a < m i n ( X . ) ) } .

E

i s t h e l o w e r p a r t o f a c u t i n M.

We c l a i m t h a t f o r each by c o n s i d e r i n g a p a r t i t i o n

i, X .

1

multiplication.

McAl oon [13]

such t h a t f o r a l l

N e x t we c l a i m t h a t

I

a

T h i s i s e a s i l y seen

5 max(X),

Ia,b}

E

i f and o n l y i f

C6

{a}

E

C i

i s c l o s e d under a d d i t i o n and

a' < b.

(Cb,Cl)

i

such t h a t

F o r d e t a i l s see

.

A subset such t h a t

I.

i s unbounded i n

T h i s i s e a s i l y seen by c o n s i d e r i n g a p a r t i t i o n

5 max(X),

a < b

n I

(Ci,C:)

i f and o n l y if a > min(Xi).

for all

..a

H

of

I

n I.

H = H^

i s s a i d t o be

M-coded

i f t h e r e e x i s t s an M - f i n i t e s e t

We c l a i m t h a t , i f s e t v a r i a b l e s a r e i n t e r p r e t e d as r a n g i n g

I,

o v e r M-coded subsets o f

then

I

becomes a model o f

c l a i m w i l l complete t h e p r o o f of (ii) since c l e a r l y

ACAO + RT(<w,Z).

This

i s a t i s f i e s a l l t r u e :X

sentences. The n o n t r i v i a l p a r t o f t h e v e r i f i c a t i o n t h a t showing t h a t i f

H

I

satisfies

ACAO

reduces t o

c_ I i s M-coded t h e n so i s

where we employ t h e p a i r i n g f u n c t i o n (x,y) To see t h a t

K

=

1 7

(x

*

H

+

y ) ( x + y + 1) + x.

i s M-coded, c o n s i d e r a p a r t i t i o n

(Cg,Ci)

such t h a t , f o r a l l

203

A Finite Combinatorial Principle

a

< b < c .(max(X),

{a,b,clEC;

wx
Xi+l

Of course

i f and only i f

H* <-> 3z
E

i s homogeneous f o r t h i s p a r t i t i o n .

x such t h a t x

clearly

K

*

< a < b

i s M-finite, and

and 3y
K = K

*

so K

I

n

The verification t h a t M-coded p a r t i t i o n

C1 = C i fl (C,,C,).

I

satisfies

(Co,Cl) of

Xi+l

Then

I,

Let

i s M-coded.

given i s closely related t o t h a t of Kirby and Paris;

Xi+l.

E

Then

see McAloon [131.

be such t h a t

i

be the s e t

K*

The argument j u s t

Namely, given an

RT(<w,Z) i s t r i v i a l .

let

> min(Xi+l) + 1

f o r some a , b

H*

E

I

IXi+l

Since

[ X i + l l < w c_ C;.

i t follows by a simple counting argument t h a t of a l l

.

H*l

E

Co = C; n

and

n I i s M-coded, unbounded in I , and homogeneous f o r

This completes the proof of Theorem 2 . 1 .

W e conclude t h i s section by defining another precise sense in which $ i n In-dense" i s a f i n i t e miniaturization of cursive function f : w

n

+

w

by

n

For each

RT(<w,2).

E

w

define a re-

such t h a t the interval

f n ( x ) = the l e a s t y

[x,yl = {x,x+l, ...,y-1,yl

From the proof of Theorem 2.1 i t i s clear t h a t f n i s provably recursive in ACA, + RT(<w,2), i . e . f o r each n , ACA, + RT(<w,2) proves wx(fn(x) i s n-dense.

i s defined).

Conversely we have the following r e s u l t which shows t h a t any function

which i s provably recursive in ACAO + RT(<w,2) i s majorized by some f n .

other than

@ ( x , y ) be a

Let

2 . 2 THEOREM.

x

and

vx 3y @ ( x , y ) .

y.

0

f o r m u l a w i t h no f r e e v a r i a b l e s

Z1

Suppose t h a t

Then t h e r e e x i s t s

n

ACA, E

w

+

proves

RT(<w,Z)

such t h a t

ACAO +

RT(<w,2)

proves

wx 3 y [ @ ( x , Y ) A Y < fn(x)I.

PROOF.

If the conclusion f a i l s , l e t M be the arithmetical part of a countable

model of

ACAO + R T ( < w , 2 ) + {Gly [$(a,y)

least y

in V

such t h a t

Hence by overspill

[a,b]

A

@ ( a , y ) holds.

y < fn(a)l : n

Then

[a,b]

E

I , b ,d I ,

and

I

d.

Let

b

be the

i s n-dense f o r a l l

i s n-dense f o r some nonstandard

Theorem 2.1 shows t h a t there e x i s t s an i n i t i a l segment I

a

E

n. of

i s the arithmetical p a r t of a model of

n

The proof of

M

such t h a t

ACAO + RT(<w,2).

E

w.

H. FRIEDMAN, J. McALOON and S.G. SIMPSON

204

Clearly

I

satisfies

~ l $y( x , y ) .

Hence

ACAO + RT(<w,Z)

does n o t prove

T h i s completes t h e p r o o f o f Theorem 2.2.

y x 3 y $(x,y).

F o r more i n f o r m a t i o n on models o f a r i t h m e t i c t h e r e a d e r may c o n s u l t t h e a r t i c l e s i n [O].

93.

ARITHMETICAL TRANSFINITE RECURSION We s h a l l d i s c u s s t h e formal system ATRO which c o n s i s t s o f

w i t h axioms a s s e r t i n g t h a t subsets o f

i n which ordering

X

w.

F o r each a r i t h m e t i c a l f o r m u l a

does n o t o c c u r f r e e l y , we have t h e f o l l o w i n g axiom:

-< o f x

there exists

w,

= {(y,n):

d(y,{(x,m):

X

c_ w

m
together

can be d e f i n e d by a r i t h m e t i c a l t r a n s -

w

f i n i t e recursion along w e l l orderings o f

ACA,

+(y,Y)

given a w e l l

such t h a t (x,m)

A

xI)I.

Here we employ t h e p a i r i n g f u n c t i o n (x,y)

+ y + 1) ( x + y) + x .

= $x

T y p i c a l examples o f s e t s d e f i n e d by a r i t h m e t i c a l t r a n s f i n i t e r e c u r s i o n a r e I n fact,

t h e Davis/Flostowski H-sets o f h y p e r a r i t h m e t i c t h e o r y ( S p e c t o r [ 1 8 ] ) . can e a s i l y show t h a t tence:

for all

r e l a t i v i z e d H-set

ATRO i s e q u i v a l e n t t o

B c_w

B

Hb

ACAO

plus the following

and f o r a l l r e l a t i v i z e d o r d i n a l n o t a t i o n s exists.

It i s t h i s f o r m u l a t i o n o f

b

E

one

1 112 senB 0 , the

ATRO which w i l l be

used i n t h e d e t a i l e d arguments below. The system

ATRO t u r n s o u t t o be a v e r y n a t u r a l v e h i c l e f o r f o r m a l i z i n g t h e

p r o o f s o f many theorems o f o r d i n a r y mathematics.

( I n t h i s respect i t contrasts

f a v o r a b l y w i t h t h e w e l l known systems o f " h y p e r a r i t h m e t i c a l a n a l y s i s , " w h i c h cont a i n e s s e n t i a l l y no s i g n i f i c a n t theorems o f o r d i n a r y mathematics beyond those a l r e a d y p r o v a b l e i n ACAO

(Kreisel [12]).)

For instance,

n a t u r a l f o r m a l i z a t i o n o f t h e K l e e n e / S o u s l i n theorem b o t h s e t s o f i n t e g e r s and s e t s . o f r e a l s . orderings o f theorem f o r

w

1

can be proved i n ATRO.

ATRO proves t h e

1 (Al = e f f e c t i v e B o r e l )

for

T h i s i s because c o m p a r a b i l i t y of w e l l Similarly,

ATRO proves t h e r e d u c t i o n

I$ r e l a t i o n s as w e l l as t h e f a c t t h a t e v e r y

1

Z1

set o f reals i s

A Finite Combinatorial Principle

20s

Lebesgue measurable, has the property of Baire, and e i t h e r i s included in a countable s e t or includes a perfect s e t . Some o f the r e s u l t s j u s t mentioned have converses saying t h a t a given formalized theorem of ordinary mathematics i s in f a c t equivalent t o ATRO. 3.1

Provably i n

THEOREM.

Thus we have:

ACAO, t h e f o l l o w i n g a s s e r t i o n s a r e p a i r -

wise e q u i v a l e n t : (1)

ATRO;

(2)

comparability of well orderings of

(3)

every uncountable c l o s e d s e t of r e a l s has a p e r f e c t subset;

(4)

every uncountable a n a l y t i c ( i . e . boldface

w;

1 : )

s e t of r e a l s

has a p e r f e c t subset; (5)

e v e r y c l o p e n game i n

(6)

e v e r y o p e n game i n

i s determined;

ww ww

i s determined.

Here the equivalence of ( 1 ) t h r o u g h ( 4 ) i s due t o Friedman [4] while the equivalence of ( l ) , ( 5 ) and ( 6 ) i s due t o Steel [19].

Further information may be found

in the writings of Friedman [31, [41, [51. The purpose of t h i s section i s t o prove y e t another equivalence r e s u l t of the above type:

provably in ACAO.

ATRO

i s equivalent t o

RT(<w,2).

Here of course

RT(<0,2) i s the p a r t i t i o n theorem which was stated i n 91 above and miniaturized in 82.

From t h i s new equivalence r e s u l t , the main theorem of the paper will follow.

W e shall actually prove a l i t t l e more. = { f i n i t e subsets of

For X

c w

we write

XI

and

[XIw

=

{ i n f i n i t e subsets of

W e work in the space

[ w ] ~endowed with the usual (Baire space) topology.

well known t h a t a subset there e x i s t s

R 5 [wfW

XI.

P

of

[wIw i s open ( i . e . boldface 1;)

such t h a t

P = I X c [ w I w : 3n(Xn10.1,.

..,n-1 I E R ) ~

I t is

i f and only i f

H. FRIEDMAN, J. McALOON and S.G. SIMF'SON

206 Silch

an R

A 0l l

i f a n d only i f b o t h

P 5- [ w I w

i s called a & f o r

A

i s clopen ( i . e . boldface

P

and i t s complement [wlw\P a r e open.

P

i s Ramsey i f there e x i s t s

[A]' n P = 8. Such an

We say t h a t

P.

A

[AIw 5 P or

such t h a t e i t h e r

[0Iw

t

i s said t o be homoaeneous for

P.

We say t h e t

The Galvin/PriCry

By wans of codes. the weaker

theorem [7] says t h a t every Pore1 s e t i s Pamse.y.

statement "every clopen s e t i s Ramsey" can he formalized as a

n21

sentence.

The

mlin theorem of t h i s section, which was announced in [16]. reads as follows: 3.2

Provably i n

THEOREM

ACAO. t h e f o l l o w i n g a s s e r t i o n s a r e p a i r w i s e

equivalent: (1;

ATF$,;

(71

every clopen subset of

(8)

e v e r y open s u b s e t o f [ w I 0

i s pamsey

[wlw

i s Ramsey.

Before proving Theorem 3.2 l e t us d e d c e from i t the rain theorem of t h i s pape r. 3.3

THEOREM

The f o l l o w i n g

n 02

sentences a r e provably equivalent i n

primitive recursive arithmetic: ( i ) vnan-dense f i n i t e s e t ; (ii) 1 - c o n s i s t e n c y of

ATRO.

By Theorem 2.1 i t suffices t o show t h a t the 1-consistency of ACAO + RT(
valent t o the 1-consistency of

valent, via codes, t o the statement t h a t i f are closed. then there e x i s t s A

I n t h i s version,

RT(<w,2)

E

But,

[@Iw

=

ATRO

i s eaui-

RT(
C o u C1

[ o ] ~such t h a t e i t h e r

where Co

[AIW 5 Cg

and

or

C1

[ A I w 5 C,.

t r i v i a l l y implies t h a t clopen s e t s are Ramsey and i s

implied by the statement t h a t open s e t s a r e Ramsey. follows t h a t ACAO + RT(<w,2)

i s equivalent t o

ATR".

Hence, by Theorem 3.2, i t This completes the proof

of Theorem 3.3. The r e s t of t h i s section i s devoted t o the proof of Theorem 3.2. We begin by establishing our notation f o r hyperarithmetical theory (Harrison 1231, Spector [18]).

If

B cw and e

E

w

we denote by

( e l B the function

207

A Finite Combinatorial Principle partial recursive i n B' = l x :

{xlB(x)


t i o n such t h a t a fined

B

{elB(n)

B

b

2a 3.5e.

c0

t o mean t h a t a

(ii)i f a <: (iii)i f

0,

Clearly

b

is

B L e t <0 be t h e s m a l l e s t t r a n s i t i v e r e l a -

X

or

b

then

2

a

a = b.

Let

B { e l (n)

5w

n,

0:

{elB(n)

if

be t h e s e t o f a l l B

b

w i t h l e a s t element

We d e f i n e

i s w e l l o r d e r e d by

i s d e f i n e d and <: OB B <0

lelB(n+l)

t o be t h e s e t o f a l l

i s de-

We w r i t e

i such

that

1;

b

E

for all

n.

0 9 such

, i . e . e v e r y nonempty subset of We i n t r o d u c e a p a i r i n g firnction

has a l e a s t element under cO.

+ n + 1) + m

(m + n ) (m

=

and

%B b;

then B.

e

i s r e c u r s i v e l y enumerable i n 8.

B

(m,n) and f o r

B

Clearly


and, f o r a l l

i s l i n e a r l y o r d e r e d by

I$ i n

B l a : a <e b)

a

for all

3B b

3-!je

B

B {a: a<0 b}

jump o f B i s t h e s e t

i s defined)

B (i)l a : a L~ b l

that

The

T i s an a p p r o p r i a t e T - p r e d i c a t e .

where

&,

e.

~~T(B,x,x,Y)~

= IX:

a

w i t h index

B

write

(X)n = l m :(m,n),Xl. Using t h e r e c u r s i o n theorem we can w r i t e down a that

B

0,

and

(i)

if

b = 1 then

(ii)

if

b = 2a

b

E

(iii) i f

b = 3

It i s p r o v a b l e i n ACAO that

H(B,b,X).

then

B

H(B,b,X)

which says

X = B;

then ~

0

112 f o r m u l a

t5h e n~

X

i s t h e jump o f a s e t H(B,{e}R(n),(X)n)

t h a t f o r each

T h i s s e t i s denoted

b B Hb

E

Y

for all

such t h a t

H(B,a,Y);

n.

UB t h e r e i s a t most one s e t

i f i t exists.

If

B

X

such

i s t h e empty s e t

i s c u s t o m a r i l y o m i t t e d f r o m a l l o f t h e above n o t a t i o n .

Our f i r s t main g o a l i s t o p r o v e t h a t (7) i m p l i e s (1), i . e .

t o prove ATRO

f r o m t h e assumption t h a t c l o p e n s e t s a r e Ramsey. The p r i n c i p a l B B ATRO i s t h e a s s e r t i o n t h a t Hb e x i s t s f o r a l l B 5 w and b E 0

( w i t h i n ACA )

0

axiom o f

.

208

H. FRIEDMAN, J. McALOON and S.G. SIMPSON B

By r e l a t i v i z a t i o n we may safely assume t h a t

i s t h e empty s e t .

Thus the

a s s e r t i o n which we s h a l l prove reads: Wb[baQ

3X

-+

H(b,X)I.

Our s t r a t e g y w i l l be t o define, f o r each

E

Pb

i s homogeneous f o r

[w]'

0,

E

Pb 5 [ w I w

i . e . r e c u r s i v e l y coded clopen) s e t

A

b

f o r which we can prove t h a t i f

i s r e c u r s i v e i n A.

Hb

then

was c a r r i e d o u t s t r a i g h t f o r w a r d l y i n [16];

0 l i g h t f a c e Al,

a recursive (i.e.

This s t r a t e g y

however, because we are now working i n

ACAO, an a d d i t i o n a l d i f f i c u l t y a r i s e s , as w i l l be seen. We denote by functions from

t h e s e t o f a l l i n f i n i t e sequences o f n a t u r a l numbers, i . e .

w"

into

w

We denote by

W.

o f n a t u r a l nmbers, i . e .

f u n c t i o n s from a f i n i t e i n i t i a l segment o f

u

The l e n g t h o f a f i n i t e sequence

-

nX f o r t h e p r i n c i p a l f u n c t i o n

X

elements o f

A

s

majorizes

E

A

X,

tree i s

a set

f(i)

and

lh(o).

Given

E

A

f

E

for all

i < 1st.

T ~ w " such t h a t every i n i t i a l segment o f an element o f

f

i s an element o f

I X : H(X)I

T

A

path

t h a t every f i n i t e i n i t i a l segment o f

recursive tree

predicate

i s a path through

and t h e corresponding path Ta

H(X)

f

TI.

1-1

ww

such

correspondence between

a r e r e c u r s i v e i n each o t h e r .

ha

E

I t i s w e l l known

Furthermore, i f

be the t r e e associated i n t h i s way t o t h e

has a t most one path, denoted

T.

f

T

we may e f f e c t i v e l y associate a p r i m i t i v e

such t h a t t h e r e i s a canonical

and { f : f

major-

s

we say t h a t

E

i s an i n f i n i t e sequence

0 [8] t h a t t o each n2

we w r i t e

we say t h a t

w',

T

T.

w.

i. The c a r d i n a l i t y o f a f i n i t e s e t

[ o ] < ~and u

ns(i) L o ( i )

and

[a]'

E

through

i s an element o f

cw

X

For

into

w

i . e . t h e f u n c t i o n which enumerates t h e

for all

Is\. Given s

Is1 = l h ( o )

if

2

vA(i)

if

[ w ] < ~ i s denoted a

of

i s denoted

E 0'"

i n i n c r e a s i n g order. f

t h e s e t o f a l l f i n i t e sequences

0

TI2

i f i t exists.

H(X)

For each

predicate

H(a4X).

Furthermore

ha

X:

holds then a

E

0,

let

Ta

Thus

e x i s t s i f and

only i f

Ha

e x i s t s , i n which case they are r e c u r s i v e i n each o t h e r .

that i f

ha

e x i s t s then i t i s r e c u r s i v e i n every s e t which majorizes i t [8].

Note a l s o

All

o f t h i s i s provable i n ACAO. The f o l l o w i n g n o t a t i o n w i l l be convenient.

Given A

E

[wIw

and

s

E

[01<~

A Finite Combinatorial Principle

209

we w r i t e A/s = IkeA: k > i f o r a l l Pa 5 [o]'

Our r e c u r s i v e s e t s decide whether o r n o t

icsl.

A

E

Pa,

a r e d e f i n e d as f o l l o w s .

let

s1

to

[w]',

E

be t h e s m a l l e s t i n i t i a l segment o f

majorized by

s1

tence o f

f o l l o w s from K o n i g ' s lemma and t h e f a c t t h a t

s1

one path.

have a c m o n i n i t i a l segment

s2

Let

> min(A/sl)

T~

and any two f i n i t e sequences i n

and

o f l e n g t h min(A).

T~

b e t h e s m a l l e s t i n i t i a l segment o f

have a comnon i n i t i a l segment only i f

A

A

Isl\ > min(A) and such t h a t any two f i n i t e sequences i n Ta which a r e

such t h a t

Is21

Given

b o t h e x i s t and

T~

A/sl

Ta

'cl

5

T

~

.

has a t M s t one such t h a t s2

which a r e m a j o r i z e d by

o f l e n g t h min(A/sl).

T~

Ta

The e x i s -

A

Put

into

i f and

Pa

T h i s completes t h e d e f i n i t i o n of

'a' I t s h o u l d be c l e a r t h a t i f

[A]'

and c o n v e r s e l y i f

A.

Thus

Ha

i n which case

5 Pa

A

majorizes a path through

t h e n t h e r e e x i s t s a p a t h through

e x i s t s i f and o n l y i f t h e r e e x i s t s Ha

Ta

A

A.

i s r e c u r s i v e i n e v e r y such

E

[wfil

then

Ta

Pa,

[A]'c

recursive i n

[ A I M 5 Pa

siich t h a t

These statements a r e p r o v a b l e

i n ACAO. I n order t o prove t h a t

Ha

e x i s t s , we need t h e f o l l o w i n g c o m b i n a t o r i a l

1enma.

3.4

LEMMA.

subsets o f

The f o l l o w i n g i s p r o v a b l e i n [wIw

a r e Ramsey.

c l o p e n s u b s e t s of a l l

k

PROOF. R 5 [w]"

E

A Let

and Ri

i<.k, A/lkl

b e a sequence of

A

E

is homopxeous f o r

be codes f o r

Si

Assume t h a t c l o p e n

(Qi: i < w)

Let

Tben t h e r e e x i s t s

[wI0.

and

ACAO.

Qi

and

of g e n e r a l i t y t h a t i s k-decisive i f

Ri

and

R Si

Wi
i t s e l f belongs t o a r e upward c l o s e d .

such that for

Qi.

[ulw\Qi r e s p e c t i v e l y .

i s s a i d t o be upward c l o s e d i f e v e r y element o f

i n i t i a l segment b e l o n g i n g t o

[0Iw

R.

[w]"

which has some

We may assume w i t h o u t l o s s

X.

s

L e t u9 say t h a t

I t s h o u l d be c l e a r t h a t f o r each

there e x i s t s a k-decisive i n i t i a l s e g m n t o f

A set

k

E

[wlco

and

X

t

[wIw

H. FRIEDMAN, J. McALOON and S.G. SIMPSON

210

Q 5 [wJw. Given A

We now d e f i n e a c l o p e n s e t not

A

put

Q,

E

k = min(A)

i n i t i a l segment o f B/s,.

Put

A

Wick ( s l

Let

B.

into E

and s2

B = A/{kl.

Let

E

s1

t o d e c i d e whether o r

[w]',

be t h e s m a l l e s t k - d e c i s i v e

be t h e s m a l l e s t k - d e c i s i v e i n i t i a l segment of

i f and only i f

Q

s2

R . <-> 1

Ri).

E

Q.

T h i s completes t h e d e f i n i t i o n o f

A

Using t h e assumption t h a t c l o p e n s e t s a r e Ramsey, l e t

Q.

eous f o r B = A/lk}.

Let

so

i n i t i a l segment o f

To see t h i s , p u t

[ A I W 5 Q.

We c l a i m t h a t

be t h e empty s e t and l e t

B/s

n

.

E

[wIw

k = min(A)

be homogenand

s ~ +be ~t h e s m a l l e s t k - d e c i s i v e

By t h e p i g e o n h o l e p r i n c i p l e t h e r e must e x i s t

n > m > 0

such t h a t w i c k (sm

E

sn

Ri <->

Ri).

E

Put

C

{k}

=

Then c l e a r l y

C

u (B/sn).

L S ~ ( I S

A

Since

[AIWnQ.

E

i s homogeneous f o r

i t follows that

Q

t A I W 5 Q. F i n a l l y we c l a i m t h a t f o r a l l i < k.

Put

B = A/fk}.

Wi
E

f o r a l l k-decisive

Ri <->

t

Then C1,

J

C2 t. 1

t

= Ik3 u t . E

E

A, A/{k}

i s homogeneous f o r

Qi

for all

E

2

R.) 1

t 2 ~ [ B ] < W . To see t h i s , l e t

C

= B/(tlu$,)

and f o r

s

j = 1,2

be t h e s m a l l e s t kput

uC.

[AIW 5 Q R . <->

s

<->

t

1

E

I t s u f f i c e s t o show t h a t

and 1 d e c i s i v e i n i t i a l segment o f

C. J

k

so f o r a l l R

E

2

E

i
we have

i Ri.

T h i s completes t h e p r o o f o f t h e lemma. We a r e now ready t o p r o v e ( i n ACA

0

and assuming clopen s e t s a r e Ramsey) t h a t

21 1

A Finite Combinatorial Principle

Ha

exists f o r a l l

a

E

0.

Suppose not.

c

Fix

Consider t h e countable c o l l e c t i o n of clopen s e t s a i, c

be such t h a t f o r each

A

Pa.

eous f o r

Note t h a t f o r each

3 k [A/{klIu 5 Pa

i n which case

&

Ha

all

a

c

such t h a t A

%3i H(a,Ci) ) .

z # 1.

does not exist.

Ha,

H

a i, c

z

Ha

exists.

2

Z

exists b u t

B u t then, f o r each H

exists for all

a

E

n,

H2a,

A.

i s recursive

,

This c o n t r a d i c t i o n completes t h e

0.

I t remains t o show

t h i s s e c t i o n we work within

ATRO.

Our proof t h a t

P

For the rest of

i s Ramsey w i l l employ a kind o f

Barwise compactness argument involving Mathias forcing conditions.

s

E

U, V, W

[ o ] ' ~ and

X 5s u U

c P

and 1

i s IT1

range over U = U/s.

s

t h e jump of

e x i s t s ; i n f a c t , by Enderton/Putnam [ I ] ,

ATRO implies open sets a r e Ramsey.

ables

Also, Z

The only remain-

t h a t ( 1 ) implies (8), i . e . ,

P 5 [wlW be open.

A,

under :O.

Hrel(,,)

We have now shown t h a t ( 7 ) implies ( 1 ) i n Theorem 3.2.

Let

be t h e ' s e t of

Z

i s not r e c u r s i v e I n

Ha

such t h a t

3. 5e i s r e c u r s i v e i n t h e double j u m p of

Ha

Let

We can d e f i n e Z a r i t h m e t i c a l l y ( i n

then we have t h a t

= 3-5e.

i s homogen-

e x i s t s i f and only i f

be t h e l e a s t element of

z

From this i t follows t h a t

3. 5e proof t h a t

By t h e lemma, l e t

This v i o l a t e s a r i t h m e t i c a l comprehension.

ing possibility i s t h a t in A.

Ha

does not e x i s t .

A/{kl

such t h a t

k

i s r e c u r s i v e i n A.

Ha

Let

c.

z = Za

If

-r, c ,

Hc

(Pa: a k c ) .

Hence, by a r i t h m e t i c a l comprehension,

i s nonempty by choice of

Clearly

a

does not e x i s t .

parameter A ) a s t h e s e t of a l l i.e.

there exists

0 such t h a t

E

Let t h e v a r i -

[wlW. A condition i s an ordered p a i r (s,U) W e denote by

[s,U] t h e s e t of a l l

i s an i n i t i a l segment of

X

E

[w]'

Note t h a t t h e a s s e r t i o n

X.

so i t s truth depends on a c e r t a i n ordinal number.

where such t h a t [s,Ul

We s h a l l need t o

The idea of using an assignment of

discuss t h e s e o r d i n a l numbers e x p l i c i t l y .

ordinals t o prove t h a t open s e t s a r e Ramsey i s taken from Solovay [17].

See a l s o

h n s f i e l d [24]. By r e l a t i v i z a t i o n we may s a f e l y assume t h a t (i.e. lightface

0

zl)

with r e c u r s i v e code

cursion t o d e f i n e a p r e d i c a t e

R(b,s,U)

R.

P

i s r e c u r s i v e l y enumerable

Use a r i t h m e t i c a l t r a n s f i n i t e re-

whose i n t u i t i v e meaning i s t h a t

b

6

0

H. FRIEDMAN, J. McALOON and S.G. SIMPSON

212 and

[s,U]

5

P via ordinal B = Ibl

(i)

R(l,s,U)

(ii)

R(Za,s,U)

s

<->

111.

{ x : x cO

Namely

R;

E

yk

<->

= o r d e r t y p e of

U R(a,s u { k l , U / { k l ) ;

E

(iii)R ( 3 ~ 5 ~ , s , U ) <--> I n R(Ie}(n),s,U). We can t h e n p r o v e

(iv)

R(a,s,U)

A

(v)

R(a,s.U)

A V 5 U +. R(a.s,V).

Note t h a t if R(b,s,U)

a <0 b

R(b,s,U);

+

h o l d s f o r some

f u t u r e r e f e r e n c e , t h a t even i n ACAO

b

0

E

then

[s,U]

Note a l s o , f o r U b c 0 and Hb e x i s t s

5 P.

we can p r o v e t h a t i f

t h e n so does R:

=

I s : R(b,s,U)}

w i t h p r o p e r t i e s c o r r e s p o n d i n g t o ( i ) - ( v ) above. in

b

Y c w

0 and

E

Y Ha

i s recursive i n

(

we denote by

f o r some

a c0 b.

the set o f a l l

#e say t h a t

posable i f i t i s c l o s e d under o r d i n a l a d d i t i o n , i . e . , {x: x c0 b l

an o r d e r isomorphism o f

onto

{x: a <

X, Y , Z,

if X

E

Y Mb

and

Y

E

2 Mb

We say t h a t a c o n d i t i o n (s,U) that

R(b,s,V)

holds.

i s b-good f o r each

X.

If

E

E

X

E

U

b

5w

such t h a t

i s a d d i t i v e l y indecom-

x c0 b l .

there exists

a c0 b

f o r each

Note t h a t if b

i s transitive, i.e.

i s b-good i f t h e r e i s no

is

for a l l

V

E

Mn:

[UIu

such

i s s t r o n g l y b-good i f ( s u { k } , W I k l )

(s.U)

U. ACAO.

Assume t h a t

i s b-good,

then there exists

e x i s t s for a l l

H:

is a d d i t i v e l y i n d e c o m p o s a b l e and t h a t

(s,Lr)

X

2 Mb.

The f o l l o w i n g i s p r o v a b l e i n

LEMMA,

a c0 b

k

We say t h a t

then

P

Mi

X

a d d i t i v e l y indecomposable t h e n t h e r e l a t i o n

3.5

i s uniformly recursive

U Hb. For

X

R:

The s e t

V 5 IT

such t h a t

(s,Q

i s s t r o n g l y a-good.

PROOF. \'E

I n f a c t we c l a i m t h a t we can f i n d

PI: n [ U I w

there e x i s t

k

E

V

and

I.1

E

V \I

E

U

Ma.

Suppose n o t .

!la n [V/{kl]w

such t h a t

Then f o r a l l R(a,suIk},W).

213

A Finite Combinatorial Principle Put

Wo = U.

wi+l

c

Wi

Having defined

U Ma n[kJi/Cki}lw

that

SO

H.:

Out r e c u r s i v e l y i n belongs t o

WbU n [UI'.

R(Za,s,V),

hence

b

V = {ki:

U

E

[ x : xC0 a1

into

b # 1,2

b

U

E

a


b

and

b

[wIo

E

(0.w)

i s b-good f o r a l l

all

a


c-number


b

y(b).

b b

X

and

U

E

and M

b

Since

i < o)

(ai:

field i n M in

E"

field i n M

E

Let

0 b

U1

5 P

exists.

b

0 'there

E

f o r some Hence t h e c o n d i t i o n

,

Hi

i s an c-number,

i s b-good.

U1

such t h a t

1

is

exists for

I t i s c l e a r t h a t f o r any

ACAO + y ( b ) ;

be t h e s e t of a l l

does n o t belong t o M

and

a.

= b

i n fact, f o r

a

E

0,

such t h a t

t o be s t r o n g l y

M, c o n s t r u c t sequences

0

U1. but

i s complete

1

ITl,

hence n o t

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

b

E

0,

F' i s a c o u n t a b l e o-model o f

b.

0,

and

t o be an E-number,we

ai-good,

U

Since

El.

i s a p r o p e r subset o f

does n o t belong t o

t o be

a

0, such t h a t t h e r e e x i s t s a c o u n t a b l e o-model o f ACAO +

t h e r e e x i s t s an i n f i n i t e d e s c e n d i n g sequence ai+l

c U ai

for all

i. Since

may s a f e l y assume t h a t each

V 5 U

a

i

b

i s satis-

i s satisfied

i f (s,U)

i s satis-

such t h a t ( s , V )

i s satis-

Hence, b y Lemma 3.5,

then there e x i s t s ai+l-qOOd.

Now, r e c u r s i v e l y i n t h e sequence for

0,

E

(0,d

t o be a d d i t i v e l y i n d e c m p o s a b l e .

field i n M

b

and f i n a l l y

0 5 111 and

F i x such b

I t i s provable i n

0.

i s such an u-clmdel.

such t h a t

ACAO + y ( b ) .

E

U

t h e r e e x i s t s a c o u n t a b l e o-model o f

fbr some Clearly

b

5 o,

i t follows t h a t

Z;,

i s Ramsey. If [U]'

be t h e a s s e r t i o n t h a t

Y 5 o, M i

any a

y(b)

i s a fixed point o f

i.e. f o r a l l

t h e n we a r e done, so assume t h a t no such

Let

b

i s an E-number.

P

We a r e now ready t o p r o v e t h a t

U

and

{xcU b: x i s a d d i t i v e l y indecomposable}.

such t h a t

i, hence

0 b t h e r e e x i s t s an o r d e r i s q o r p h i s m

ATRO t h a t t h e r e e x i s t a r b i t r a r i l y l a r g e c-numbers, exists

and hence

T h i s proves t h e lemma.

i s n o t b-good.

f o r a l l a<

HaU

i s recursive i n h o l d s f o r each

i s an E-number i f

a d d i t i v e indecomposables, i . e . , of

i < o}

(s,V)

Thus

and

bli

E

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

CkilWi+l).

L

R(a,su'kil,V/{ki})

R(b,s,V).

We say t h a t

R(a,s

Hence Also

M: n R]Iw choose ki

E

(Ui:

i
(ai:

i<w)

and (ni:

and t h e s a t i s f a c t i o n r e l a t i o n i
as f o l l o w s .

Recall f i r s t

214

H. FRIEDMAN, J. McALOON and S.G. SIMF'SON

t h a t t h e c o n d i t i o n (0,w)

i s satisfied i n

assume i n d u c t i v e l y t h a t we have chosen s 5 {n,,,nl,---,ni-lj,

j = 2i.

Apply the lemna Zi

k = j + 2' =

A = {ni:

zi+l

.

Put

A

E

i s satisfied i n M

ni = min(Vi)

and

satisfies

s

R(l,s,Ui)

a,-good

Vi

0

and

= w

t o be s t r o n g l y ak-good where

= Vi/{nil.

Ui+l

U

so t h a t i-1 < min(Ui) M t o be a.-good where J 5 Ui such t h a t f o r a l l

E

R.

Let

[ A I w n P = 0.

i be such t h a t

Finally, put

j.

I f not, t h e r e e x i s t s

s 5 {no,nl,***,ni-l>.

c o n t r a d i c t i n g t h e f a c t t h a t ( s,Ui)

f o r some

J

and we c l a i m t h a t

[wIw

[A]<W such t h a t

be

t o get

M

Put

i
Clearly s

i s satisfied i n

times w i t h i n

<s,Vi)

s 5 {no,nl,-*-,ni-ll,

... <

no < n 1 <

c5,Ui)

for a l l

P t o be a al-good.

Then

i s satisfied i n

This c o n t r a d i c t i o n completes t h e p r o o f t h a t

P

M

P to is

Ramsey. For f u r t h e r refinements see the

The p r o o f o f Theorem 3.2 i s now complete. appendix below. For the f o l l o w i n g c o r o l l a r y , r e c a l l t h a t

= least

fn(x

y

such t h a t

[x.yl

i s n-dense.

3.6

COROLLARY.

For each

Conversely, let

@(x,y)

n

w,

E

be

Zy

ATRp

ATRO

proves

yx 3y @(x,y).

n

ATF.(,

proves

vx 3y [@(x,y)

w

PROOF.

sucb that

is defined).

with only the free variahles shown,

and suppose that E

y x (fn(x)

proves

TI-en there exists A

y < fn(x)l.

Imnediate from Theorem 2.2 and t h e discussion j u s t preceding i t .

APPENDIY TO 13.

P A R T I T I O N CALCULI'S I N

ATRO

The f o l l o w i n g p a r t i t i o n theorem has already played a basic r o l e i n t h i s paper. 3.7

THEOREM (ATRO).

s u c h t h a t either

PROOF.

Given a closed s e t [ A I W n C = 9,

or

C 5 [o]' there exists

[AIw

c

A

E

C.

This r e s u l t i s j u s t a restatement o f t h e i m p l i c a t i o n (1) i m p l i e s ( 8 ) i n

Theorem 3.2 which was proved above. The purpose o f t h i s appendix i s t o e s t a b l i s h c e r t a i n g e n e r a l i z a t i o n s of Theorem 3.7 i n which t h e closed s e t c o l l e c t i o n o f closed sets.

C

i s replaced by a ( f i n i t e o r i n f i n i t e )

The t r u t h o f these g e n e r a l i z a t i o n s w i l l be f a i r l y

[wl'

A Finite Combinatorial Principle

215

obvious, b u t t o p r o v e them i n ATRO we s h a l l need some s p e c i a l devices. 3.8

THEOREM (ATRO).

there exists

A

G i v e n f i n i t e l y many c l o s e d s e t s [ 0 1 ] ~

E

sucb that e i t h e r

i < k,

5 [w]',

Ci

3 i < k [Afn

Ci

fl

=

or

V i c k [ A I W 5 Ci. PROOF.

The p r o o f w i l l be o b t a i n e d as a m o d i f i c a t i o n o f t h e p r o o f o f Theorem 3.7.

By r e l a t i v i z a t i o n we may s a f e l y assume t h a t t h e open s e t s [ O ] ~ \ C ~ a r e u n i f o r m l y lightface

0

i.e.

C1,

u n i f o r m l y r e c u r s i v e l y enurnenable.

be a r e c u r s i v e code f o r proof of Theorem 3.7.

and d e f i n e a p r e d i c a t e

[wIw\Ci

there e x i s t s

li c k [U]"

such t h a t

[w]'

assume t h a t no such

U

A

Theorem 3.7 y i e l d s

as i n t h e

(s.U)

exists.

E

n Ci =

If

t h e n we a r e done, so

0

Then t h e same c o n s t r u c t i o n as i n t h e pro,of

W i < k [ A I W 5 Ci.

6uch t h a t

[o]'

E

is

t$ n [ w ] ~ such t h a t l i < k Ri(b,s,V).

V

E

Ri(b,s,U)

Ri

let

M o d i f y t h e d e f i n i t i o n o f b-goodness t o say t h a t

b-good if and o n l y i f t h e r e i s no U

i< k

F o r each

of

T h i s completes t h e

proof. 3.9.

5 [wIw,

Ci

MIw n Ci

i =

w,

there exists

0

for

some

C

be t h e s e t of a l l

c

PROOF.

Let

i < k = min(X).

Clearly

such

E

[wIw

i, o r e l s e

[A/Ik3]w

X

E

i s closed.

such that e i t h e r

[ A / { k l l w 5 Ci

for a l l

or

n Ci

[wI0

such t h a t

[AIW 5 C. =

Put

k

E

A

0.

Hence

i k 1 LI B

E

E

Ci

C.

k = min(A).

for all

A

E

[wIw

If there exists

fl t h e n we a r e done, so assume t h a t no

Then by Theorem 3 . 8 t h e r e e x i s t s =

X/Ik1

By Theorem 3.7 t h e r e e x i s t s

such t h a t 3 i < k [ A I W n Ci

B exists.

W i < k[B]'

C

[ A I w n C = fl

such t h a t e i t h e r E

A

i < k.

and

B

Given a n i n f i n i t e s e q u e n c e o f c l o s e d s e r s

COROLLARY (ATR,,).

Hence

B

E

[A/{k1lw

[ A I W 5 C.

such t h a t

T h i s proves t h e c o r -

ollary. We do n o t know whether t h e c o n c l u s i o n o f Theorem 3.8 can be strenathened ( i n ATRO o f course) t o say t h a t f o r a l l

[A]'

5 Ci.

i< k

either

[A]O

n Ci

=

fl or

We c o n j e c t u r e * t h a t t h e c o n c l u s i o n cannot be so strengthened.

* T h i s c o n j e c t u r e has been v e r i f i e d b y Simpson [30].

However,

216

H. FRIEDMAN, J. McALOON and S.G. SIMPSON

we do have t h e f o l l o w i n g v a r i a n t o f C o r o l l a r y 3.9 which w i l l be proved be a somewhat d i f f e r e n t method. G i v e n an i n f i n i t e s e q u e n c e o f c l o s e d s e t s

3.10 THEOREM (ATR,,).

j < w,

there exists

j,

some

PROOF.

o r else Let

E

X

and

zw.

for a l l

P c

E

0,

A

[ O ] ~

Vk

such t h a t

assume t h a t t h e open s e t s r e c u r s i v e l y enumerable.

F o r each Rj(b,s,U)

o u r c o u n t a b l e w-model

M

and

A

Y

of

ACAO

j < k.

plus

W j VU

0

i.e.

Z1,

uniformly

be a r e c u r s i v e code f o r

[Olw

E

B(c).

We s h a l l c o n s t r u c t

as i n t h e p r o o f o f Theorem 3.7.

satisfies

= 9 for

C.

By r e l a t i v i z a t i o n we may s a f e l y

0.

R. J

let

j

c

- 1

i s a l i m t o f E-ndnbers, and

are uniformly lightface

[w]@\C. J

and d e f i n e p r e d i c a t e s

E

g j [ A ] w 5 Cj.

B W j < k [B] n C j =

E

[A]@

As i n t h e p r o o f o f Theorem 3.7 we can f i n d

[ o ] ~such t h a t

c

k

c

0+\0 such t h a t t h e r e e x i s t s a c o u n t a b l e w-model Assume t h e r e i s no

B

such that e i t h e r

[A/Ckllw n Cj =

a <0 c

a

E

[wIw

E

!?(c) be t h e assertion t h a t

HX e x i s t s f o r a l l c

A

C . c [w]', 3 -

3b <0 c 3V

[wIw\C.

J

We c l a i m t h a t

U

E

Mb n [Ulw R.(b,$,V). J

I f t h i s were n o t so t h e n we c o u l d r e p e a t t h e argument f o r Theorem 3.7 t o g e t

A

[UIw

[AIW 5 C.

c o n t r a r y t o o u r i n i t i a l assumption. Now u s i n g the 3 c l a i m r e p e a t e d l y d e f i n e a sequence o f s e t s ( U ' j < w) as f o l l o w s . Begin w i t h E

such t h a t

j'

U =

Having d e f i n e d

O.

such t h a t

t o be any such

Put B

E

M n [wIw

E

b. J

be t h e c 0 - l e a s t €-number

b <0 c

R.(b,q,V) f o r some v E MUbj n [Ujlw, and choose Uj+l J This e n t i r e construction i s recursive i n the satisfaction re-

V.

M.

B = {ki:

[ulw and

i

B/Ckl

E

where

u}

cUj+l

s u f f i c e s t o show t h a t

bj

which o f c o u r s e i m p l i e s

E

ko = 0

0,

[Uj+l]w

o u r i n i t i a l assumption. F o r each

L

E

w

k

for all

and

= min(Uk /{ki?).

Clearly i To complete t h e p r o o f , i t

j < k.

B,

E

ki+l

f o r t h e n b y c o n s t r u c t i o n we have n C. =

J

t h e argument f o r Theorem 3.7 t o g e t

[ w I w = Uj,lCj

let

satisfies

M

lation for

U J.

A

E

0.

But, i f

[Uj+,]w

b . i 0,

3

such t h a t

R.(b O,Uj+l) J j'

t h e n we c o u l d r e p e a t

[AIW 5 Cj

T h i s completes t h e p r o o f o f Theorem 3.10. let

w i t h each

RT(<w,L)

C. J

be t h e s t a t e m e n t t h a t whenever

closed, there e x i s t s

A

E

[wIw

such t h a t

contrary t o

217

A Finite Combinatorial Principle

3j <

II[A]O c

C..

- J

We already know that ATRO proves RT(<w,2).

We shall now

point out that ATRO proves V l l RT(<w,9.).

This result will be needed in 45.

3.11 COROLLARY (ATR").

j < 1 , sucb j' [wIw such that 3 j < II [Alw c C