Patras Logic Symposium: Proceedings

PAWS LOGIC SYMPOSION Proceedings of the Logic Symposion held at Patras, Greece, August 18-22,1980 Edited by GEORGE MET...

Author: George Metakides

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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.

ISBN: 0 444 86476 8

Published by:

North-Holland Publishing Company- Amsterdam New York Oxford

Sole distributors for the U.S.A.and Canada:

Elsevier Science Publishing Company, Inc. S2 Vanderbilt Avenue New York, N.Y. 10017

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 15
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


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) & Wnn1c < 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


* 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

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 -


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)
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


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


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

&


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

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
:

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)
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
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

B

f o r each y

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)
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

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

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

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,

for k # 0,

< ckn ~ > ,for O=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

2 , and of nilpotency


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


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


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(

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 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
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
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 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.

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 0

imply

For background material on V of [ 2 6 ] .

(a)=

K ~ ,y

For each ordinal

< B.

r0

Then

define a

B

B > 0.

and, for

ma

K

B

(a)= the a t h

i s the l e a s t ord-

(a) < y. B we assume familiarly with [2] or with Chapter

ro

K

I n particular we assume t h a t a canonical promitive recursive system of

notations ( N ,

f o r the ordinals
1,

are c o u n t a b l e intersections o f differences o f o p e n i n d e x s e t s . ) I t i s open (and d o u b t f u l ) whether even

by open i n d e x s e t s f o r t h e case e.g.,

I$

index sets.

X = ww.

Gg

i n d e x sets 5 o - a l g e b r a generated

Analogously, no normal f o r m i s known f o r ,

231

The Addison Game Played Backwards: Index Sets in Topology

REFERENCES J . W . Addison, The Theory o f h i e r a r c h i e s , L o g i c , Methodology and Philosophy o f Science (Proc. I n t e r n a t . Congr., 1962, pp. 26-37.

1960), S t a n f o r d U n i v . Press, Stanford,

J . W . Addison, Some problems i n h i e r a r c h y t h e o r y , Proc. Sympos. Pure Math., v o l . 5, Amer. Math. SOC.,

Providence, R . I . ,

1962, pp. 123-130.

J . W. Addison, C u r r e n t problems i n d e s c r i p t i v e s e t t h e o r y , Proc. Sympos. Pure Math., v o l . 13, P a r t 11, Amer. Math. SOC., Providence, R . I . , 1974, pp. 1-10. Y . L. Ershov, On a h i e r a r c h y o f s e t s 11, t r a n s l a t e d i n Algebra and.Logic 7 (1968), pp. 212-232. L. Hay, Isomorphism types o f i n d e x s e t s o f 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 , Proc. Amer. Math. SOC. 17 (1966), 106-110.

,

Index s e t s o f f i n i t e c l a s s e s 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 34 (1969), 39-44.

K. Hrbacek and S . Simpson, On Kleene degrees o f a n a l y t i c s e t s , t o appear i n Proc. o f Kleene Conference (Madison 1978), N o r t h H o l l a n d , Amsterdam.

A. K e c h r i s , On a n o t i o n o f smallness f o r subsets o f t h e B a i r e space, Amer. Math. 229 (1977), pp. 191-208.

x.

Trans.

D. M i l l e r , Index s e t s and Boolean o p e r a t i o n s , t o appear.

J . Mohrherr, Ph.0. Thesis, U n i v . o f Ill. a t Chicago C i r c l e , 1981. Y. Moschovakis, D e s c r i p t i v e S e t Theory, N o r t h H o l l a n d , Amsterdam, 1980. H. G. Rice, Classes o f r e c u r s i v e l y enumerable s e t s and t h e i r d e c i s i o n problems, J . Symbolic L o g i c 74 (1953), pp. 358-366. H. Rogers, Theory o f R e c u r s i v e F u n c t i o n s , McGraw-Hill, New York, 1967. R. L. Vaught, pp. 269-294.

I n v a r i a n t s e t s i n t o p o l o g y and l o g i c , Fund. Math. 82 (19741,

PATRAS LOGIC SYMPOSION

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

239

A N A L Y T I C EQUIVALENCE RELATIONS AND COANALYTIC GAMES

Jacques Stern UniversitC de Caen Caen, France

§

0

INTRODUCTION The present paper i s an attempt t o cast a bridge between two subjects of

descriptive s e t theory which have been investigated in the past years:

the deter-

minacy of coanalytic games and the study of analytic equivalence relations.

The

f i r s t topic culminated with the following r e s u l t due t o Martin ( 5 ) and Hhrington

(3).

THEOREM 0.1:

The d e t e r m i n a c y of c o a n a l y t i c games i s e q u i v a l e n t t o t h e

hypothesis

Va

(a'

exists).

As f o r the other topic, i t was s t a r t e d by the following deep r e s u l t o f Silver (7):

THEOREM 0.2:

Any c o a n a l y t i c e q u i v a l e n c e r e l a t i o n on

has countably

many c l a s s e s o r a d m i t s a p e r f e c t s e t o f p a i r w i s e i n e q u i v a l e n t e l e m e n t s S i l v e r ' s original proof used an elaborate forcing argument; a simpler proof has been found by Harrington ( 4 ) which yielded the following e f f e c t i v e version of the above theorem: THEOREM 0.3:

Let

E

be a

1 111

e q u i v a l e n c e r e l a t i o n on

w

w

which d o e s

not admit a p e r f e c t s e t o f p a i r w i s e i n e q u i v a l e n t e le m e n ts; t h e n

i s t h e union of t h e

A;

Ow

sets included i n a s i n g l e equivalence c l a s s .

Following Silver, Burgess ( 1 ) found a similar r e s u l t on analytic relations:

THEOREM 0.4: N1

Any a n a l y t i c e q u i v a l e n c e r e l a t i o n on

ww

h a s a t most

many c l a s s e s o r a d m i t s a p e r f e c t s e t o f p a i r w i s e i n e q u i v a l e n t

elements.

J. STERN

240

zi

The proof of t h i s theorem r e l i e s on a structural analysis of relation.

Let

be such a r e l a t i o n and l e t

E

f

equivalence

be a continuous function such

that

then, i f

5

E,(a,B)

iff

5

i s a countable ordinal, one can define a binary relation f(a,B) i s n o t a well-ordering of type < 5.

Clearly E

=

n

E5,

L'H,

furthermore

PROPOSITION 0.5 (Burgess ( 1 ) ) : that

by

EL

The s e t o f c o u n t a b l e o r d i n a l s

5

such

i s a n e q u i v a l e n c e r e l a t i o n i s a c l o s e d unbounded s u b s e t o f

EL

N1'

W e now come t o the main definition of the paper; equivalence r e l a t i o n , we define an analytic s e t A ( E )

A(E) where y'

E 8;

1 I B : V Y E A ~ ( B ) 3 ~ 2' T B

=

zTB

means

as usual

-

of Turing degrees by

E(Y,Y')~

t h a t y' i s recursive in

8. We now attach t o

a coanalytic Turing game G ( E ) : two players I and I1 together produce a real I1 wins i f

B

THEOREM 0.6: if

-

be a given analytic

E

let

E

belongs t o

Player

I

A(E).

The basic r e s u l t on

has a winning s t r a t e g y i n

G(E)

i s the following i f and o n l y

G(E)

admits a p e r f e c t s e t of pairwise inequivalent elements.

The proof of t h i s basic theorem will be given i n J 1 . between

E

and

G(E)

Then the correspondence

will be used in b o t h directions.

I n 5 2 , we will use analytic equivalence relations studied by R. Sami and the 1 games; especially we will prove the a u t h o r in order t o obtain new examples of I$

following THEOREM 0.7

-

i):

F o r any r e c u r s i v e o r d i n a l

5

there is a

game whose d e t e r m i n a c y i s e q u i v a l e n t t o t h e s t a t e m e n t

I$

Turing

fkk i s c o u n t -

able" i i ) : There i s a

1 111

T u r i n g game whose d e t e r m i n a c y i s

24 1

Analytic Equivalence Relations and Coanalytic Games equivalent t o t h e statement.

L

We a l s o d e f i n e a existence o f

; ' 0

TIl

1

is a w e a k l y compact cardinal i n

is countable".

L

w h o s e s u c c e s s o r in

"There

T u r i n g game whose determinacy i s e q u i v a l e n t t o t h e

t h i s game i s d i f f e r e n t f r o m H a r r i n g t o n ' s o r i g i n a l one.

Thus,

theorem 0.7 ii)shows t h a t t h e r e a r e c o a n a l y t i c games whose determinacy i s i n t h e gap between t h e p e r f e c t s e t theorem f o r c o a n a l y t i c s e t s and t h e e x i s t e n c e o f sharps. I n 53, we use o u r correspondence between i.e.

assuming

1 nl-determinacy,

E

and

G(E)

i n the othec d i r e c t i o n

we i n v e s t i g a t e t h e consequences o f t h e e x i s t e n c e o f

a w i n n i n g s t r a t e g y f o r p l a y e r I 1 i n t h e game.

We t h u s o b t a i n a new p r o o f o f a

theorem o f Burgess on t h e enumeration of c l a s s e s of a

1

C1

equivalence r e l a t i o n

w i t h no p e r f e c t s e t o f i n e q u i v a l e n t elements. B e f o r e I c l o s e t h i s i n t r o d u c t i o n , l e t me p o i n t o u t t h a t t h e i d e a s f o r t h i s paper grew o u t a t a t i m e when A . S . K e c h r i s and R. Sami were i n P a r i s and obviousl y t h e i r presence was e x t r e m e l y f r u i t f u l f o r me.

§l.

PROOF OF THE B A S I C THEOREM We assume t h a t

E

is a

Cl

one t o c o v e r t h e general case. elements, we p i c k a code w i n t h e game real

6,

G(E)

TI

1

e q u i v a l e n c e r e l a t i o n ; m i n o r m o d i f i c a t i o n s enable If

P

i s a perfect set o f pairwise inequivalent

f o r this perfect set

by s i m p l y p l a y i n g

TI.

P

and we c l a i m t h a t

Indeed, i f

I

can

I 1 answers b y p l a y i n g a

then

1

and c o u n t a b l e , t h e r e f o r e i t has measure z e r o w i t h r e s p e c t t o t h e 1 s e t o f p o s i t i v e measure c a n o n i c a l measure on P; i t s complement i s a nl(B.a)

is

Z1 (B,TI)

and t h e r e f o r e , by t h e Sacks-Tanaka b a s i s theorem i t has a member shows p r e c i s e l y t h a t t h e r e a l

(B,TI)

i s n o t a member o f

1 A,(R,n).

This

A(E).

We now c o n s i d e r t h e converse i m p l i c a t i o n (whose p r o o f was s u p p l i e d b y Kechris). real

u

Note t h a t a w i n n i n g s t r a t e g y f o r p l a y e r such t h a t

I i n t h e game

G(E)

is a

242

J. STERN

We pick such a real u and we choose a continuous f such that E(a,B) holds if and only if

i(a,B)

is not a well-ordering, Now, if 6 is a real such

that u

is recursive in B , then, B does not belong to A ( E ) 1 a real y which is Al(B) and satisfies the following

@(y,B)

If we let

:

so that there is

be

5 is a linear ordering and, for some real y' recursive in 6, 5

is an initial segment of f(y.y') then, the relation 5 E

@(y,B)

is C11 and given B with u

8, the set

is an initial segment of the set of well-orderings at least for some y 1 in A1(5). From this it follows that

@(y,B)

is a set of well-orderings. This is true also of the set U

1 set of well-orderings to which we may apply the boundedness which is a C1

theorem. We may actually pick a bound 5 for the well-orderings in U,

such

that the corresponding approximation EL of the relation E is an equivalence relation (by proposition 0.5). In order to achieve the proof, it is enough to show that E

5

admits a perfect set of pairwise inequivalent elements or equiva-

lently, by theorem 0.2, that E

5

has uncountably many classes; if this is not

true, one can find a real a such that i) ii)

any real is E -equivalent to some (a),, 5

o

is recursive in a

1 Because o f condition i i ) there is a y which is Al(a), such that

243

Analytic Equivalence Relations and Coanalytic Games i s an i n i t i a l segment o f

$(y,a)

we may assume t h a t t h i s i n i t i a l segment

WO;

i s m i n i m a l ; then, i t c o n s i s t s o f w e l l - o r d e r i n g s o f t y p e

< 5;

from t h i s i t follows

that bn so t h a t

< 5

f(u,(a),)

y

is

E -inequivalent t o a l l reals

5

(a)n;

t h i s g i v e s t h e r e q u i r e d con-

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

52.

SOME NEW

IIi

GAMES

B e f o r e we s t a r t d i s c u s s i n g theorem 0.7 l e t us make t h e f o l l o w i n g remark. REMARK:

I f an a n a l y t i c e q u i v a l e n c e r e l a t i o n

E

h a s c o u n t a b l y many

c l a s s e s t h e n p l a y e r I1 h a s a w i n n i n g s t r a t e g y i n t h e game

G(E).

A c t u a l l y p l a y e r I 1 can ?!in by p r o d u c i n g any r e a l enumerating a s e t o f r e p r e sentatives for the equivalence r e l a t i o n . Obviously, then i s no converse t o t h e above remark; n e v e r t h e l e s s p a r t i ) of theorem 0.7 w i l l f o l l o w f r o m a p a r t i a l converse which a p p l i e s t o s p e c i a l a n a l y t i c r e l a t i o n s i.e.,

t o a n a l y t i c r e l a t i o n s whose c l a s s e s a r e Bore1 o f bounded rank;

these e q u i v a l e n c e r e l a t i o n s have been c o n s i d e r e d and s t u d i e d by t h e a u t h o r ( [ 9 1 ) . One o f t h e main t o o l s which we w i l l use i n o r d e r t o g e t t h e r e s u l t s i s a n o t i o n of f o r c i n g i n t r o d u c e d by S t e e l and we f i r s t r e c a l l some b a s i c f a c t s on t h i s n o t i o n of forcing.

2.1

REVIEW OF STEEL FORCING Given any o r d i n a l

consists o f a l l p a i r s i) T

one can d e f i n e a s e t o f f o r c i n g c o n d i t i o n s

5,. (T,f)

f

i s a function:

long t o

T

and

A condition

t

T

+ W -

5

J

such t h a t

{m}

i s a proper extension o f

(T',f')

which

where

i s a f i n i t e t r e e on o

ii)

Pc

i s smaller than

s,

(T,f)

f(0) =

f(t) < f ( s )

if

T

or

and i f

s,t

f(s) =

i s a subset o f

be-

-.

T' and

J. STERN

244

f'

extends

5 which i s

has countably many classes.

L ( o ) , we prove a lemma.

J. STERN

246

Let

LEMMA 2 . 2 . 4 :

then

a

be s u c h t h a t

there is a condition

in

p

such t h a t , f o r any S t e e l g e n e r i c t r e e a r e a l inequivalent t o a l l r e a l s

i s inequivalent t o a l l r e a l s

ordinal,

i s a Steel generic t r e e over

(a)n. If

(0,~)

say

Now, working in

5‘

belongs t o

a t level

L(a,a)

strategy f o r player

cursive in

extending

p,

11,

e,

Iel(T’al

is

(a)n.

5’ > 5, we may assume t h a t y

i s a winning

T

From the hypothesis, i t follows t h a t some real

PROOF OF THE LEMMA: L(a)

and a r e c u r s i v e i n d e x

IPy

y

c’,

y

in

i s a large enough a-admissible

L , ( a ) , and therefore, i f

5

y

1 A1(a,T).

is

T

Because u

i s actually equivalent t o a real re-

{e)(T’u). L(a,a),

we may find a Borel code

for

8

~x : { e j ( X ’ a ) i s a real inequivalent t o a l l (a)’,, s~ which i s a IIo -5+1 9

Borel s e t and we can find a condition q

such that

A

It

Ah5

3

B);

by proposition 2 . . 2 , we get

s5 Taking

p =

/tA(TC

4‘.

I

i t i s easy to cheek t h a t given a Steel generic t r e e T extending i s inequivalent t o a l l r e a l s

p.

( a ) n , t h i s proves the l e n a .

We now go back t o the proof of the theorem, s t i l l assuming t h a t ably many classes in

L(a).

We consider the product of H~

consists of functions from a f i n i t e subset of standard way.

H~

into

E

copies of

has uncount-

P5 which

P5 , ordered in the

This s e t of forcing conditions s a t i s f i e s the countable chain condi-

tion and adds a sequence

o f Steel generic t r e e s .

We claim t h a t one can

241

Analytic Equivalence Relations and Coanalytic Games mutually inequivalent reals recursive i n u

f i n d N1 A

and one o f t h e t r e e s

TX,

< N 1'

PROOF OF THE CLAIM:

I f t h e c l a i m does n o t h o l d , t h e r e i s a c o u n t a b l e o r d i n a l

such t h a t any r e a l r e c u r s i v e i n a

A.

u and one o f t h e t r e e s

equivalent t o a real recursive i n be f o r c e d by a c o n d i t i o n

q

i s a subset o f

q.

q.

and one o f t h e t r e e s

We p i c k an o r d i n a l

e a s i l y seen t h a t t h e r e i s a r e a l

a

(a),.

TX X < Xo.

is

T h i s may

such t h a t t h e domain of

(P5 )"

which extends

s a t i s f y i n g c o n d i t i o n s i )and ii)of femna

By t h e lemna, t h e r e i s a c o n d i t i o n

f o r any S t e e l g e n e r i c t r e e e x t e n d i n g

X A .

5

I n o r d e r t o prove t h a t o

we show t h a t lemma 2.2.4 holds w i t h

then, by proposition 2.4.2 i ) any

E -inequivalent t o each

5

tree player

such t h a t

Such a

T over some l a r g e enough ordinal I1

B admits an

X in place of 5.

MB

If

CI

i s given and

i s isomorphic t o L HCtl

can be found 5 ; because

i s strong w.r.t.

A;

is

in some Steel generic

u i s a winning s t r a t e g y for

Em-equivalent element r e c u r s i v e in

T,o

say

J. STERN

254

Ie}(Tyu) i s s t i l l we can find a Bore1 code y

E5

classes of


It

s e t (remember t h a t the

f o r the following L!y+5+2

E -inequivalent t o a l l

{ x : {e}(xyu) i s a real

q

L(a,o)

no 1 -1+5+1

are

Some condition q

( a ) n . Working in

5

(aln’s}

5

i s such t h a t

A(TI;’Y)

by proposition 2.1.2, we get -A

Taking p

A

11 =

A(T~.Y)

$,

i t i s easy t o see t h a t given any Steel generic t r e e T

p. { e ~ ( T ~ ias ) E -inequivalent t o a l l reals

5

PROOF OF PROPOSITION 2 . 5 . 1 i i ) : L > A;

by proposition 2.4.5,

if

a

(a)n.

We write NL

L

is %+2

L,

cardinal in

we have

5

f o r the l e a s t cardinal in

5+ 1

i s a real such t h a t

then, i f there i s no weakly compact cardinal in i s isomorphic t o

L < N L ~ + ~any , B such t h a t Mg


( a ) n . Because A

5

< A

PROOF OF PROPOSITION 2 . 5 . 1 i i i ) :

Let

a

I f the conclusion does not hold then

i s the f i r s t weakly compact cardinal in

L.

be a real such t h a t

We need the following l e m a . LEMW.: T h e r e i s a c o n s t r u c t i b l e t r a n s i t i v e s e t i)

X

ii)

w

i s a n e l e m e n t a r y e x t e n s i o n of

u Iwl

c

X

i s not a

and we can argue exactly as above.

w < A < (W+)L

where w

extending

L

W

X

such t h a t

255

Analytic Equivalence Relations and Coanalytic Games iii)no o r d i n a l b e t w e e n

We l e t

PROOF OF LEMMA: a map f r o m

v

onto

( c x ) XELV *

(crl

)rlzp.

v

and

A+1

< ( v + ) ~ such t h a t i n

be an o r d i n a l

p

is a c a r d i n a l i n

X.

L

P

there i s

Then, we c o n s i d e r a language w i t h c o n s t a n t symbols

A.

By weak compactness we can f i n d a model o f t h e f o l l o w i n g s e t

o f ( i n f i n i t e ) formulas

-

The t h e o r y o f

- wv

-

c

-

Lv

VECx < - > w v YEX rl'

< c

rl

= c

& On(cn)

6 i s w e l l -founded

Such a model i s i s o m o r p h i c t o a t r a n s i t i v e (because

If

p

X

morphic t o

E X)

w i t h the required conditions

. B

i s g i v e n b y t h e lemma, and i f then, because

X,

X

X

i s any r e a l such t h a t

MB i s i s o -

satisfies

'There i s no weakly compact c a r d i n a l ' , ~ ( € 3 )=N1.

Also,

v

i s a cardinal i n

X one i s e x a c t l y Nv+l,

therefore

B is

X

and t h e n e x t one i s €,+,-inequivalent

>'A;

t o each

t h i s next (a),,.

The end

o f t h e p r o o f i s s i m i l a r t o t h e above.

13

THE EFFECT OF SHARPS Throughout t h i s 5,

E

is a

p a i r w i s e i n e q u i v a l e n t elements. a winning strategy

1

C1

e q u i v a l e n c e r e l a t i o n w i t h no p e r f e c t s e t of

We assume t h a t '0

e x i s t s so t h a t p l a y e r

I1 'has

u i n t h e game G ( E ) .

generic extensions o f

L(u).

As i n §§1,2, we i n t e r p r e t f r e e l y E i n 1 O f course what i s done below f o r C1 e q u i v a l e n c e

r e l a t i o n s can be extended t o a n a l y t i c e q u i v a l e n c e r e l a t i o n s p r o v i d e d t h e axiom

~ a ( a #e x i s t s ) i s t r u e .

3.1:

The f o l l o w i n g r e s u l t i s t h e key s t e p towards o u r a n a l y s i s o f t h e equivalence

classes o f

E.

J. STERN

256

PROPOSITION 3.1.1: is a condition

Let of

p

be a r e a l and

a

P A and a r e c u r s i v e i n d e x

any S t e e l g e n e r i c t r e e r e a l equivalent to

and i f

EL, 5 < N 1 ,

p , {e3(T'a)

extending

is a

a.

i s a recursive function such t h a t

f

f ( a , B ) i s n o t a well-ordering of type < 5

pick such a countable ordinal Steel generic t r e e of Level

a Al(6)

such t h a t f o r

i s an equivalence r e l a t i o n f o r uncountably many 5 ;

E5

of type 1

L(a,u)

e,

there

uia");

i s defined by

E (a,B) = i f f 5

then,

over

T

He recall t h a t , i f

PROOF:

the ordinal

X

such t h a t

5,

5 >

c',

S;(6) < 5'

uial').

We l e t

t h e n , in

L5,(T),

5'

therefore we may ++ (5 ) L ; i f T i s a

be

there i s a well-ordering

( j u s t take an L-generic well-ordering).

6 is

EL

equivalence relation which has countably many c l a s s e s , (otherwise E

has perfectly many c l a s s e s ) ; from t h i s i t follows, by Harrington's theorem (0.3), 1 t h a t any equivalence class w . r . t E i s the union of i t s A1(6) subsets; t h i s i s 5 true in particular of the equivalence class of the given a ; hence, by the usual basis theorem a i s EL equivalent t o some A12 ( 6 ) r e a l . Because N:(&) < 5' we

1 A2(T); b u t u i s a winning strategy f o r player I 1 in the game G ( E ) , therefore any A1l ( T ) real i s E-equivalent ( a n d subsequently E

get that any

1

A2(6) real i s

equivalent) t o a real recursive in u

5

and 1.

From the above argument i t follows t h a t f o r some index

{el(Tva) i s a real

e

E -equivalent t o a , 5

or equivalent t h a t f ( { e l ( T ' a ) , a ) i s not a well ordering of type < 5

and therefore (*)

f ( { e l ( T ' u ) , a ) i s not a well-ordering of type
T h e

-+

sequences a r e g i v e n l e x i c o g r a p h i c a l o r d e r i n g and extended t o n - t u p l e s of such sequences.

We l e t

cedures and l e t

RE

{Elx,

< a, be an a - r e c u r s i v e enumeration o f r e d u c t i o n pro-

E

denote t h e

Eth - a-r.e.

set.

We w i l l h e n c e f o r t h r e f e r t o t h e f o l l o w i n g c o n s t r u c t i o n as t h e s t a n d a r d construction.

I t i s a m i n o r s i m p l i f i c a t i o n o f t h a t i n [6], t a k i n g i n t o account

t h e o b s e r v a t i o n o f Posner and E p s t e i n [5, Theorem 11 t h a t i n any c o n s t r u c t i o n of a set

6 o f minimal degree u s i n g s p l i t t i n g t r e e s and f u l l t r e e s , t h e r e i s no

6 # Re.

need t o make s p e c i a l e f f o r t s t o ensure t h a t

The non-r.e.

ness o f

B is

a u t o m a t i c a l l y achieved. We c o n s t r u c t o u r s e t Stage 0:

Set

6 i n stages: Bo = 0.

TE = I d e n t i t y t r e e and

Stage u = p + 1.

Let

nu

be t h e l e a s t

Case 1.

nu

= p

f o r which t h e r e i s

I f such an

6,

on

T.!

Then

B,

has p r o p e r e x t e n s i o n s on

no p r o p e r e x t e n s i o n o f exist, l e t 1 ',

< p + 1

E

B;

let

f

'1U

t o be

does n o t

= p + 1.

+

1.

and t h e r e f o r e has two i n c o m p a t i b l e e x t e n s i o n s Take

E

3f,

= Sp(T:,p,Bu).

= Bp+l.

Let

TZ = TE

for

0

Bp

E

and

< 1 ',

TE,

Bk* and

328

C.T.CHONG

no 5

Case 2. T+:l

As i n [ 6 ] , 'I, = v + 1

p.

i s of t h e f o r m

Sp(Tt,v,B)

1

B;,

Bp

for

v.

extending

B on T t ) r e s p e c t i v e l y w h i c h P

for

P

6,

Tt

I f however t h e r e a r e two e x t e n s i o n s o f

6; on TE which s p l i t 6 not, take

on

Then

By assump-

P'

T ~ T~ ,

(Immediate e x t e n s i o n s o f

B,

split

B 56

f o r some

t i o n t h e n t h e r e a r e no e x t e n s i o n s

v.

f o r some

0

Bp.

t o be

v,

B,

take

6,.1

t o be

If

The argument i n 1.8 o f 161 shows

t h a t such a s e l e c t i o n can be made t o achieve o u r o b j e c t i v e . TZ = T:

Let Stage X

if

l i m i t ordinal.

$

Let

and s e t

< rl,,

E

T'

= Fu(T~,B,).

be t h e l e a s t o r d i n a l

q

5

6

changes unboundedly o f t e n a t stages below

BX = UB,

+ 1,

rl = v

let

0

T:

x,

E.

By t h e f a c t t h a t

or i f

K2

J(

sup J(w) < J ( 5 ) . F o r any o r d i n a l x within;this xv J'(x,v)

i s a splitting tree f o r a l l

Hence we o b t a i n :

E.

U(G,E),

for a l l

x

such

u < 5. By Lemma 1

It follows t h a t

TZ,

b e i n g a sub-

u such t h a t sup J(u) 2 u < J ( 5 ) . By

V x > J'(x? s Z c f ( a ) =

i s s a i d t o be

K,

I t i s c l e a r t h a t t h e p r o o f o f Theorem 1 depends

J.

We now show t h a t i f c o n t i n u i t y i s preserved,

t h e n an a l t o g e t h e r d i f f e r e n t p i c t u r e emerges, a t l e a s t when

i s uncountable.

K

Me w i l l f i r s t i n t r o d u c e t h e f o l l o w i n g obvious e f f e c t i v i z e d v e r s i o n s o f well-known notions.

An a - f i n i t e subset

whenever

K c C and K i s K - f i n i t e .

be s t a t i o n a r y i f

S

C

of

K

i s s a i d t o be c l o s e d i f An a - f i n i t e subset

S

of

sup K E C K

i n t e r s e c t s e v e r y c l o s e d and unbounded subset o f

i s said t o K.

334

C.T.CHONG Lemma 3

S

Let

C K

be stationary.

If

g

v

in

S,

K

S

0.

such

such that

is stationary.

I f n o t , then f o r each

Proof.

a

Let

there i s a closed and unbounded s e t C

whenever w

g(w) # 5

next a-cardinal a f t e r

is in

S

n

C . Since

5 5* C

the map

5 i s a - f i n i t e , hence

g

i s K + - f i n i t e . Then 5 i s a K t - f i n i t e closed and unbounded set. Then any element of K),

C' = {

u5 u(u,x)

=

0,

and s i m i l a r l y f o r t h e function

v.

Now l e t T ( < , E )

denote

the r e l a t i o n

u and v

Using t h e Z2-definition f o r as well.

Thus t h e r e e x i s t s a Z2-selection function

that for all S

in

S,

Clearly

T(S,t(C)).

l o s s of g e n e r a l i t y we may assume t h a t by the c o n t i n u i t y of t ( 5 ) < J(5').

such t h a t

is

f o r each

Now t h e r e l a t i o n

But T'

c l e a r l y C2.

J,

T( s 2 c f ( < ) =

K

is

we know

S choose t ' ( 5 ) t o be the l e a s t 5 '

is a-finite.

By Lema 3 t h e r e i s a 5*

Global and Local Admissibility such t h a t

c*

t'(

=

W.

$,

and no

K

UlJ(c)

U,

take

La

-

U

W

U

S c K be s t a t i o n a r y

for all

5

in

S.

If

U

and

U.

W

such t h a t

V

W >a U.

For

such t h a t

Furthermore t h e r e l a t i o n

z2 - r e l a t i o n .

As

s2p(a) >

K,

p o s s i b l e t o choose an a - f i n i t e sequence n < w.

satisfies

Let

> w.

i s minimal o v e r

V

W

UlJ(a)).

s2p(a) > s 2 c f ( a ) =

i s n o t minimal o v e r

there i s a

where we have

U

R( Us,

For each

5

5

let

To show t h a t

{Ail n < w , a < w l l

and

Then NA!a)la

$I

E '0.

by:

i(n)

then f o r every n ,

= maxI$(ai)I i

E

Let

under

nl =

a < wl}

By making adjustments i f necessary, we can assume t h a t the creasing and t h a t

If

f a ( i ) < f ( i ) f o r uncountably many a's. B u t

i s countable by hypothesis.

( a ) , suppose t h a t

- A?.

An;'

E

w l is an

E

then there would be a

t h i s i s a contradiction, since { a / f a ( i ) < f ( i ) f o r every

n{Af(:)l

{All n , i

sl

5

-+

w

be

Set i s an w l =

< ail i

E

w >

5 n+l}. Notice t h a t

we have

g g ( n ) < max{h (a ) , ...,h5(an,,)} < $ ( n ) . Thus, i f T were uncountable, then given 5 0 any 11 < w1 there would be a 5 E T with 11 < 5 and so g, < g5 < $, contradict ing the hypothesis t h a t the

and

{At[ n < w, a < w l l

g 's

have no upper bound.

5 i s an w,-matrix.

Hence,

T

i s countable,

In view of t h i s r e s u l t , Baumgartner asked the following question:

If there

i s an w-matrix, i s there a 2W-matrix? An independence r e s u l t may be hard t o achieve, since the standard ways of a d d i n g r e a l s a l s o seem t o add a 2W-matrix. With t h i s , the survey of the variations of sections, consistency r e s u l t s are discussed.

A(K,A)

i s a t an end; in the next

A. KANAMORI

3 46 52.

A+(K,X)

CONSISTENCY OF

This s e c t i o n discusses t h e question of consistency f o r

with f i r s t

A+(K,X),

a d i r e c t c o n s t r u c t i o n , then a forcing argument, and f i n a l l y t h e s i t u a t i o n i n

L.

The following enumeration argument i s a natural one i n t h e present c o n t e x t , b u t i t has an antecedent already i n Braun and S i e r p i n s k i [ B S ] , Proposition (Q). If

THEOREM 2.1:

i s a successor cardinal and

K

2K- =

2K- =

s

< s , p such t h a t

K, E

satisfying A+(K,K), every a

forcing extension i n which A + ( K , X )

K

in

i s a successor cardinal

there is a

V,

where (a)

F

i s a function:

axy

(b)

S

i s a s u b s e t of

{<s,pl

f o r some y
E Sn+i+l}-U{sl<s,@> define hi+l by

E

Sn+i},

352

A. KANAMORI

Clearly,

2 hi

hi+l

was changed.

i s a g a i n a c o n s i s t e n t map f o r

fi

fi i s a c o n s i s t e n t map f o r T = j&Sj. we have ~ ( c L ~ =+ ~ ) so t h a t fi i s a c t u a l l y a

Finally, set

Moreover, f o r each

i

c o n s i s t e n t map f o r

TU

w,

E

= Uhi,

so t h a t

1

I<s,$>l. T h i s e s t a b l i s h e s t h e Claim.

A l l t h a t remains i s t o e s t a b l i s h t h e

II K < ~ = K

a y
=

1 p

t

,

E

F &

i

say t h e former.

E

{O,ll

1.

E

F &

IJJ~

F

i

i s regular,

be t h e con-

as above and

E

Either

f: '1-1

{0,11 1.

-+

f-l({01)

By h y p o t h e s i s , t h e r e i s an

Then

s

2

were a con-

or

f '({I}))

E

F with

B u t t h i s i s a c o n t r a d i c t i o n , as t h e r e s h o u l d be i n f i n i t e l y many f ( a ) = 1.

As mentioned i n J1, Chudnovsky had s t a t e d t h a t i f

QK K+

p

have a common lower bound.

There i s one s i t u a t i o n where i t i s c r u c i a l t h a t

If

s, l e t

Given a f a m i l y

{)>~ s

D =

any

F o r any s e t

{i}.

and

such t h a t f o r e v e r y

i t s e l f cannot have a l o w e r bound, f o r suppose

s i s t e n t map f o r

E

and range

consider

K,

i s directed; i n fact,

D F :Q

a

1,

and

is a cardinal.

F(a,O,l). V

F(a,O,O)

PROOF:

We use i n d u c t i o n on

(1) f o r

a+l

v

v

v

= F(a.O.0).

a. Suppose a t f i r s t ( 1 ) - ( 3 ) h o l d f o r

f o l l o w s f r o m Lemma 11.

F o r (2), n o t e t h a t i f

i n case 1 above, t h e n v v v

Ra

be t h e i n v e r s e l i m i t o f

The o r d e r i n g o f

R denote t h e d i r e c t l i m i t o f t h e sequence

LEMMA 13:

a r e de-

= >.

Rv as u-sequences

and o r d e r e d c o o r d i n a t e w i s e .

such t h a t f"(p)

R(h)

be t h e W - l e a s t s e t s u c h t h a t

a < u = uv,

i s defined f o r a l l

u,

and a l l elements of

Ra 8 P ( A,a).

=

We may c o n s t r u e elements o f a

f

i s d e f i n e d , c o n s i d e r t h e f o l l o w i n g two cases:

Ra

p":

>

V

Otherwise: Ra

A)

and d e f i n e

Ra+l

If

E

I n this situation

{@I, = >.

R(a)

CASE 2:

t(a,B)

and o r d i n a l s < X.

Ib g

Ra

5

Ra

A

Ro =
1.

f o r y < 6 and a l l 6 < v:

CASE 1 :

p

then i n

V'

Z(LK)-sentence.

T h i s w i l l end t h e proof o f Theorem 7 .

z c y i n V'.

R(a)

E

as above can be expressed by a

i s an a r b i t r a r y s e t i n

M

x = P(y).

and show t h a t

M

E

But x,

and

< a I F ( ~ , B , W ) ~i s n o t a l i m i t c a r d i n a l ] .

z

i s definable from

as d e s i r e d .

M

A

and some o r d i n a l < a ,

V

by

R,

N

and

whence

Q.E.D.

denotes an e x t e n s i o n o f

then

M

satisfies

the following condition: (*)m

If

N cM

i s a t r a n s i t i v e model o f

cardinals, then Note t h a t PROBLEM 16:

L

Does

and

M

ZFC

and

M

have t h e same

N = M. L[O # I =

'L

have t h i s p r o p e r t y .

satisfy

(*),?

The method o f McAloon [2] y i e l d s a model

M

of

2w = u2 w i t h t h e f o l l o w i n g

J. VANANEN

310

s t r o n g e r property: (*)

If

5

N c M i s a t r a n s i t i v e model of ZFC containing a l l o r d i n a l s and

uN = uM fur. u u

where

MF

5

4

5 5, then N

i s the l e a s t

= M,

5 such t h a t 5

2w = u l .

PROBLEM 17:

Is

(*),

= w

I t i s obvious t h a t

5'

consistent with

MI= Z u

=

(*)1

implies

u,?

Also the following problem remains open:

PROBLEM 18:

Is

(*)-

c o n s i s t e n t w i t h a supercompact c a r d i n a l i n

I t i s not even known t o the author whether w i t h a supercompact c a r d i n a l .

2 A(L1) = A ( L )

However, the construction of [3] f o r a model of

V = HOD + ' a supercompact c a r d i n a l ' a l s o gives a model in which

A(L2),

where G i s the q u a n t i f i e r

GxA(x)

i s consistent

2 I A ( ' ) I = I A ( .)

I+,

and t h e r e i s a supercompact c a r d i n a l .

A(L(1,G)) =

M?

Generalized Quantifiers in Models of Set Theory

371

REFERENCES

[11

P. Lindstrom, F i r s t o r d e r l o g i c w i t h g e n e r a l i z e d q u a n t i f i e r s , T h e o r i a 32

[21

K. McAloon, Consistency r e s u l t s about o r d i n a l d e f i n a b i l i t y , Ann. Mafh. L o g i c 2 (1971) pp. 449-467.

131

T. K. Menas, Consistency r e s u l t s concerning supercompactness, Trans. Amer. Math. SOC. 223 (1976) pp. 61-91.

(1966) pp. 186-195.

[4] J. Vaananen, Eoolean v a l u e d models and g e n e r a l i z e d q u a n t i f i e r s , Ann Math. L o g i c 18 (1980) pp. 193-225.

PA TRAS LOGIC SYMPOSION G. Metakides fed.) 0North-Holland Publishing Company, 1982

313

€-THEOREMS AND ELIMINATION THEOREMS OF UNIQUENESS CONDITIONS Nobuyoshi Motohashi Department o f Mathematics U n i v e r s i t y o f Tsukuba Sakura-Mura, Ibaraki

Nihari-gun

300-31

. JAPAN

I n t h i s paper, we s h a l l show t h a t some forms o f "E-Theorems" ( c f . D e f i n i t i o n 6 below) and some forms o f " E l i m i n a t i o n Theorems o f Uniqueness C o n d i t i o n s " ( c f . Defin i t i o n 12 below) a r e e q u i v a l e n t f o r many l o g i c s i n c l u d i n g t h e c l a s s i c a l p r e d i c a t e l o g i c w i t h e q u a l i t y LK ( c f . Theorem 15 below).

Before g i v i n g a precise explanation

o f o u r main r e s u l t i n t h i s paper, we w i l l g i v e a h i s t o r i c a l and i n t r o d u t t o r y exp l a n a t i o n o f E l i m i n a t i o n Theorems o f Uniqueness C o n d i t i o n s i n t h e l o g i c LK. Let

P be an ( n + l ) - a r y p r e d i c a t e symbol.

Then, P - f r e e ( P - p o s i t i v e ,

f o r m u l a s a r e f o r m u l a s which have no ( n e g a t i v e , p o s i t i v e ) occurrences o f existence condition o f

P, denoted by

uniqueness c o n d i t i o n o f WiWxWy(P(i,x) A P(2,y).

P, 3

THEOREM A .

provable in

denoted by

x=y).

o f Uniqueness C o n d i t i o n s i n

ExP,

LK

i s t h e sentence

UnP,

P.

WijyP(i,y)

The

and t h e

i s t h e sentence

Then, t h e most s i m p l e form o f E l i m i n a t i o n Theorems i s t h e f o l l o w i n g statement.

F o r any P-positive formula

if and only if

LK

P-negative)

Exp

3

A

A,

UnP

A

ExP.

3

is rrovable in

A

is

LK.

T h i s theorem i s a b y - p r o d u c t o f a t r i a l t o reduce F u j i w a r a ' s i n t e r p o l a t i o n theorem (Theorem C below) t o O b e r s c h e l p ' s i n t e r p o l a t i o n theorem (Theorem B below) s y n t a c t i cally.

Suppose t h a t

o f a l l t h e formulas i n polants o f such t h a t

C

si(C)

LK,

...,sm

and whose ranges a r e f a m i l i e s o f s e t s .

i n the logic

s

si(A)

a r e p r o v a b l e i n LK.

a r e f u n c t i o n s whose domains a r e t h e same s e t

LK w i t h r e s p e c t t o

r l si(B)

s , , ~ ~..., , sm a r e formulas

i = 1,2, ...,in,

F r ( a ) , Pr+(A), Pr-(A),

and

o f f r e e v a r i a b l e s i n A, t h e s e t o f p r e d i c a t e symbols o c c u r r i n g i n A

positively,

t h e s e t of p r e d i c a t e symbols o c c u r r i n g i n A

and

and b o t h A 3 C

be t h e s e t

2

Let

f o r each

Then, i n t e r -

Fun(A)

C

B

A 3 B

s1,s2,

n e g a t i v e l y , and t h e s e t of f u n c t i o n

N. MOTOHASHI

314

symbols i n A,

respectively.

THEOREM B (OBERSCHELP

[lo],

In

[lo]).

I f a formula

then t h e r e i s an i n t e r p o l a n t Fr, Pr+, Pr-,

(#)

I

If

of

C

A

3

A

13

in

B

is provable i n

B

LK,

with respect t o

LK

which s a t i s f i e s t h e f o l l o w i n g a d d i t i o n a l c o n d i t i o n ( # ) ;

h a s a t l e a s t one p o s i t i v e ( n e g a t i v e ) o c c u r r e n c e o f t h e

C

1 equality

1 tive)

symbol =, t h e n

A(B)

has a t l e a s t one p o s i t i v e (nega-

occurrence of i t .

[lo],

I n a f o o t - n o t e o f p.271 i n Fun

Oberschelp proved t h e f o l l o w i n g theorem.

i n t h i s theorem.

Oberschelp s a i d t h a t he f a i l e d t o add t h e f u n c t i o n

A f t e r w a r d s , P r o f . F u j i w a r a r e a d h i s paper and proved t h e

f o l l o w i n g f a c t s e m a n t i c a l l y i n [l]. THEOREM C (FUJIWARA [lJ). If a formula

the n t h e r e i s an i n t e r p o l a n t

of

C

A

A 1 B

is provable i n

B

2

in

LK,

with respect t o

LK

F r , Pr',

P r - , Fun,

A t first,

t h e a u t h o r c o n s i d e r e d Theorem C as an immediate consequence o f Theorem B

which s a t i s f i e s t h e c o n d i t i o n

i n Theorem B .

(#)

and t h e usual technique o f r e p l a c i n g " f u n c t i o n symbols" by " p r e d i c a t e symbols", and Then, he found an o b s t a c l e t o i t .

t r i e d t o show t h a t .

I n order t o explain t h i s

o b s t a c l e , we have t o remind o u r s e l v e s o f t h a t t e c h n i q u e i n d e t a i l s ( c f . Kleene [ 3 ] , p.417).

Sub(A,f,P)

any n - a r y f u n c t i o n symbol

P which does n o t o c c u r i n

i c a t e symbol

(i)

A,

F o r any f o r m u l a

A,

f, and any ( n + l ) - a r y pred-

we can a s s o c i a t e a non-empty s e t

o f formulas, which s a t i s f i e s t h e f o l l o w i n g f i v e c o n d i t i o n s ( i ) - ( v ) :

Sub(A # B,f,P)

= IC #

D

I

C E Sub(A,f,P),

D E Sub(B,f,P)I,

where

#

is

3 , A , V.

( i i ) Sub(A,f,P)

=

{A}

if

( i i i ) F o r any formulas LK,

and

A

5

B, B '

B

in

in

Sub(A,f,P),

F o r any f o r m u l a

B

in

iff

LK

Sub(A,f,P),

P r + ( A ) U { P I , Pr-(B)

occurrence o f = o n l y i f (v)

does n o t o c c u r i n A .

i s provable i n

( i v ) F o r any f o r m u l a Pr+(B)

f

A

5

UnP A ExP.

UnP A ExP.

3

13

B

B

=

B'

i s provable i n

i s provable i n

LK.

F r ( B ) = F r ( A ) , Fun(B) = Fun(A)-{fI,

Pr-(A) U {P};

and

B

has a p o s i t i v e (negative)

has a p o s i t i v e ( n e g a t i v e ) one o f i t .

Sub(A,f,P),

P A :B [ f ]

i s provable i n

LK,

where

P BLf1

€-Theorems and Elimination Theorems of Uniqueness Conditions i s the formula obtained from P(?,s)

o f t h e form

by

B

by r e p l a c i n g e v e r y occurrence o f

in

i s provable i n

LK.

Suppose t h a t a

For t h e sake o f convenience, we assume t h a t

Fun(A)-Fun(B) = { f } , Fun(B)-Fun(A) = { g } ,

f

i s n - a r y , and

g

i s m-ary.

5

Also,

we should remark h e r e t h a t we can always add t h e c o n d i t i o n

Fun(C)

i n t h e c o n c l u s i o n o f Theorem B.

B1 E Sub(B,g,Q),

and

Q

nor i n

B.

P

Let

A1 E Sub(A,f,P)

and

Fun(A) U Fun(B) where

a r e ( n + l ) - a r y and ( m + l ) - a r y p r e d i c a t e symbols which o c c u r n e i t h e r i n A (UnQ A ExO.

2

B1)

2

is

By a p p l y i n g Theorem B t o t h i s formula, we have an i n t e r p o l a n t C

LK.

o f t h i s formula w i t h respect t o

C

UnP A ExP A A1

Then, by ( i ) , ( i i ) , ( i i i ) ,

provable i n

that

B

Now, we

Sub(A,f,P).

Sub(A,f,P).

a r e g o i n g t o prove Theorem C from Theorem B by u s i n g A 2 B

P

f(f)=s.

Then, we can e a s i l y see t h a t t h e r e e x i s t s such non-empty s e t

formula

375

A 1 B

Pr-.

By ( i v ) , ( v ) , we can e a s i l y see

A 2 B w i t h respect t o

i s an i n t e r p o l a n t o f

i s an i n t e r p o l a n t o f

F r , Pr',

w i t h respect t o

Fun,

F r , Pr',

Pr-.

Moreover,

C

because

Fun(UnP A Exp A A1) = Fun(UnQ A ExQ. 2 B1) = Fun(A) fl Fun(B).

C

But t h i s

does n o t always s a t i s f y t h e c o n d i t i o n ( # ) , because

UnQ A ExQ.

has a t l e a s t one p o s i t i v e occurrence o f =, and

A

n e g a t i v e occurrence o f =, even i f no n e g a t i v e occurrence o f =.

T h i s i s an o b s t a c l e .

examining t h e usual c o n s t r u c t i o n o f t h e s e t Sub(A,f,P)

P-negative formula.

F o r example, i f

2

A

S(,?,y))

Sub(A,f,P)

from

Sub(B,g,q).

the formula

ExP A

Since the formula

A1.

3

has

By

A,

we

(ExQ 2 B1)

where

S

from

A1

Sub(A,f,P) B1

3

tence i s guaranteed by Theorem B.

Then, t h i s

C

F r , Pr',

Sub(A,f,P).

and a Q - p o s i t i v e formula

i s P - p o s i t i v e and Q - p o s i t i v e ,

i s provable i n

i n LK w i t h r e s p e c t t o

i s an n-.ary

i s a P-positive formula i n

LK w i t h r e s p e c t t o

A 3 B

Sub(B,g,q).

from the formula

i s vXS(x,f(x)),

an i n t e r p o l a n t o f t h i s f o r m u l a i n

formula

and

i s a P-negative f o r m u l a i n

A1

Now, we choose a P-negative f o r m u l a B1

B

To a v o i d t h i s o b s t a c l e , we have

Sub(A,f,P)

p r e d i c a t e symbol, t h e n W a y ( P ( i , y ) A S(x,y)) dxVy(P(x,y)

has a t l e a s t one

has a t l e a s t one P - p o s i t i v e f o r m u l a and one

can e a s i l y see t h a t

and

B1

has no p o s i t i v e occurrence o f = and

t o make more c a r e f u l s e l e c t i o n s o f formulas i n

Sub(A,f,P),

2

UnP A ExP A A1

LK

by Theorem A. Fr,

+ Pr , P r - ,

Let

C be

whose e x i s -

i s a l s o an i n t e r p o l a n t o f t h e

Pr-, Fun,

which s a t i s f i e s ( # ) i n

376

N. MOTOHASHI

Theorem 8. This gives a syntactical proof of Theorem C from Theorem B by using Theorem A (cf. [8] for details). On the other hand, Theorem A is an obvious consequence of Hilbert-Bernays' second €-Theorem (cf. [ZJ), and the following obvious, but important fact: POSITIVE LEMMA.

P

If

and

Q

is a P-positive formula, then the formula is provable in

LK,

B

where

placing some occurrences of

Vi(P(2)

2

Q(X))

is a formula obtained from

P

A

are n-ary predicate symbols and

in

A

by

A

A

A.

3

B

by r e -

Q.

Then, the author tried to give a direct syntactical proof of Theorem A without using Hilbert-Bernays' second €-Theorem, and found a syntactically simple proof of HilbertBernays' second €-Theorem from Theorem A. This shows us that Theorem A, the most simple form of Elimination Theorems of Uniqueness Conditions, is an equivalent expression of Hilbert-Bernays' second E-Theorem with using neither Skolem functions nor the €-symbol (cf. [ 4 ] ) .

This is an origin of our main theorem of this paper,

which is a generalization of the fact mentioned above.

In 5 1 of this paper, we shall define "logics" and "elimination theorems" in a general setting. As examples of elimination theorems, we shall explain two types of them, one of which is "E-Theorems" introduced 52 below, and the other is "Elimination Theorems of Uniqueness Conditions" introduced in 5 3 , 54 below. In 55, we shall state our main theorem, which shows us some equivalency between E-Theorems and Elimination Theorems of Uniqueness Conditions. An outline of the main theorem will be given in 56 below.

51.

LOGIC AND ELIMINATION THEOREMS.

In this paper, we shall consider first order languages with equality, or first order languages with equality and €-symbol. In order to express our results as general as possible, we adopt the following two definitions. DEFINITION 1. A logic in

L

over a language

L

is a set of formulas

L which is closed under modus ponens, generalizations, and substi-

E-Theoremsand Elimination Theorems of Uniqueness Conditions

311

tutions o f predicate symbols by formulas, and contains all the formulas LJ,

provable in the intuitionistic predicate logic (i) (ii) (iii)

A

3

B E L

A(a) E L A

E L

A E L

and

implies

and

B

implies

B E L.

VvA(v) E L.

is a formula obtained from

some occurrences of predicate symbols in L,

(iv)

A A

by replacing by formulas in

B E L.

then

LJZ L.

DEFINITION 2. S

i.e.

is a set of formulas in

L

be eliminable in for any formula

L

Suppose that

B

L.

Then, a formula

with respect to in

L,

is a logic over a language

S

if

A

3

A

in

B E L

L

and

is said to

implies

B E L

S.

The last paragraph of Definition 1 is included here for the sake of convenience only. A formula A in L

is said to be provable in a logic L if A E L.

ination theorems are statements o f the form: "Any

.....formula

Elim-

is eliminable in,

,,,,,,logic with respect to the set of ------formulas." In this paper, we shall introduce two types of Elimination Theorems, one of them is "c-Theorems" and the other is "Elimination Theorems of Uniqueness Condition$'.

5 2.

E

-THEOREMS.

For each E-free language L , adding the €-symbol

E

let LE be the language obtained from L by

as a new logical constant and modifying the formation .rules

and formulas as usual. DEFINITION 3 . LE

For each logic LE,

L

over a €-free language

L,

let

L

as a sublogic.

Since every formula in LE is obtained from a formula in L '

by applying modus

be the least logic over

which includes

ponens and generalizations, we have the following remark, where L '

i s the set of

formulas obtained from formulas in L by replacing predicate symbols by formulas in lE.

N. MOTOHASHI

378 REMARK 4.

formula A

L€

is a conservative extension of L, A E

in

L,

i.e. for any

implies A E L.

L€

Next, we d e f i n e f o u r types o f €-axioms

E-axioms of type ( 0 ) - (3) are sentences of the fol-

DEFINITION 5.

lowing forms (0) - (3), respectively: (0)

Vx(SyA(G,y)

(1)

vGvy(X=y 3

(2)

VxVy(Vz(A(2,z)

=

(3)

VGby(Vz(A(x,z)

: B(y,z)

-

where

x

=

-

y

means

3

A(%, EYA(~,Y))

EVA(~,V) = EUA(~,U))

x1 = y l A x2

Then, k - t h €-Theorems (k=1.2,3)

-,

A(y,z))

3

EvA(~,v)

2

i n a logic

L

=

EuA(~,u)) EuB(~,u))

.......

= y2 A

a r e d e f i n e d by:

The "k-th €-Theorem in L"

DEFINITION 6.

=

EvA(~,v)

is the statement "Any

finite conjunction of E-axioms o f type ( 0 ) and (k), is eliminable in LE

with respect to the set of all the €-free formulas", where

k=l,2,3. By t h i s d e f i n i t i o n , we can e a s i l y o b t a i n t h e f o l l o w i n g f a c t s .

REMARK 7.

in

L,

(i)

The 3-rd €-Theorem in

and 2-nd €-Theorem in

(ii)

L

The 1-st €-Theorem in

Theorem and the 3-rd €-Theorem in

L

implies the 2-nd €-Theorem

implies the 1-st €-Theorem in

L.

LK

E-

LK

is Hilbert-Bernays' second

is Maehara's €-Theorem (cf.

Maehara [S]). The f i r s t i m p l i c a t i o n o f ( i ) o f Remark 7 i s obvious because e v e r y €-axiom o f t y p e (2) i s a l s o o f t y p e ( 3 ) , b u t t h e second i m p l i c a t i o n i s n o t so obvious, because sentences o f t h e form;

V i W y ( i = y ~Wz(A(i,z) z

A(y,z)))

are n o t generally provable i n r e p l a c i n g A(x,z)

by

L'.

(If these formulas are provable i n

z = E v B ( ~ , v ) , we have

LE,

by

€-Theorems and Elimination Theorems of Uniqueness Conditions

~ V j ( i = j EvB(X,V) z

=

EvB(Y,v)) E

319

LE.)

UNIQUENESS CONDITIONS.

53.

I n t h i s s e c t i o n , we s h a l l d e f i n e "uniqueness c o n d i t i o n s " and s t a t e some o b v i -

ous p r o p e r t i e s about them. DEFINITION 8.

pair

(A,a)

of a formula

free variables of length occurs in

L

An n-ary formula in a language A

L

in

n,

is an ordered

5

and a sequence

o f distinct

such that every free variable in

A

a.

An n - a r y f o r m u l a

w i l l be denoted by

(A,:)

i s l i k e l y t o occur.

A(:)

or

i t s e l f , i f no c o n f u s i o n

A

.

Also, we sometimes i d e n t i f y t h e n - a r y p r e d i c a t e symbol

w i t h the n-ary formula

(R(al

,. . . ,an),),

where

al

,. .. ,an

R

are d i s t i n c t

f r e e variables. DEFINITION 9.

A(a,a) ,B(6,b) ,E[a,a') ,G(a,6) , H ( a , c ) ,

Suppose that

K(a,a) are (n+l)-ary formula, (m+l) -ary formula, 2n-ary formula, (n+m)-ary formula, (n+p)-ary formula, (n+q)-ary formula, respectively. Then, ExA

is

UnA

is VxVxVy(A(x,x)

V&yA(x,y), A

Un(A;E)

is

Co(A;E)

is VxVy(E(x,y)

A(x,y). A

V%VyVxVy(E(x,y)

is

VxVyVxVy(G(x,y)

Co(A,B;G)

is

VxViVz(G(x,y)

is

ax(~(x,c)

x=y),

A(x,x)

A A

A

A(y,y).

A(2,x)

B(y,z).

A 3

B(y,y).

Ex(A,ab),

A,

Un(A;E)

2

x=y),

K(x,a)).

A

or

tence c o n d i t i o n o f t h e n - a r y f o r m u l a of

x=y),

A(x,z)),

Note t h a t o u r expressions i n D e f i n i t i o n 9 a r e v e r y rough. be w r i t t e n i n t h e f o r m

3

:A(y,z)),

Vz(A(%,z)

3

Un(A,B;G)

H"K(S,;~)

2

Ex(A(ab)),

A,

UnA

etc.

I n fact, ExA

ExA

i s called the exis-

i s c a l l e d t h e uniqueness c o n d i t i o n

i s c a l l e d t h e uniqueness c o n d i t i o n o f t h e n - a r y formula

r e s p e c t t o t h e 2n-ary f o r m u l a

E,

Co(A;E)

should

A

with

i s c a l l e d t h e congruence c o n d i t i o n o f

N. MOTOHASHI

380

A with respect to E, Un(A,B;G) is called the uniqueness condition of the n-ary formula A and the m-ary formula B with respect to the (n+m)-ary formula G , and Co(A,B;G) i s called the congruence condition of A and B with respect to G. By Definition 9, we have;

REMARK 10. The following sentences are all provable in

14.

(1)

ExA

A

Un(A;E).

(2)

ExA

A

Un(A,B;G).

(3)

Co(A; a = 6 )

(4)

UnA

(5)

k'xVy(E(x,y)

(6)

Un(A,A;E)

(7)

ExA

LJ.

Co(A;E)

3

Co(A,B;G)

3

Un(A; a = 6 )

A

E E(y,x)) E

3

Co(A,A;E) : Co(A;E)

Un(A;E)

Un(A,B;H)

A

Un(A,C;K).

3

Un(B,C;HnK)

ELIMINATION THEOREMS OF UNIQUENESS CONDITIONS. Suppose that R i s an n-ary predicate symbol in a language 1 and L is a

logic over 1. In the following of this section, we assume that; E is an R-free En-ary formula, Q is an R-free m-ary formula, G is an R-free (n+m)-ary formula,

Ro is R,

ko is k, R.1 are R-free k.-ary formulas (i=l 1

R-free (ko+ki)-ary formulas ( i = l ,...,N ) V%y(E(x,y) V%y(Eo(Z,Y)

3

V?(G(X,Z)

3

5

,...,N),

Ei are

such that all the sentences

G(y,?)))

Vfi(Ei(Z,fi) F Ei(j.z.1

are provable in L. DEFINITION 11. Uniqueness conditions their associative sets following table for

S(A)

A

in the logic

k=1,2,3,4,5.

of

L

R

of type

k and

are defined in the

e-Theorems and Elimination Theorems of Uniqueness Conditions

I 1

I

1

UnR

2

Un(R;E)

3

Un (R,Q;GI

14 1

B

I

B

ExRACo(R;E).>B

1

Un(R;E) AUn(R,Q;G)

where

=I

ExR

38 1

ExRA Un(Q;GnG)

A

Co(R,Q;G(. ~

E x R A Un(Q;GnG)

A

3

B

~~

Co(R;E)

A

Co(R,Q;G)B-I

is an R-free formula.

For example, uniqueness conditions of R of type 2 (in the logic L) a+-eformulas for some R-free 2n-ary formula E, and S(Un(R;E))

of the form Un(R;E)

set of formulas of the form ExR

A

Co(R;E).

2

is the

for some R-free formula 6 ,

B

and

uniqueness conditions of R o f type 4 in the logic L are formulas of the form Un(R;E)

A

Un(R,Q;G)

for some R-free m-ary formula Q, some R-free (n+m)-ary

formula G, and some R-free 2n-ary formula E such that the sentence ViVi(E(i,y)

3

VZ(G(j2,Z)

i s provable in L, and S(Un(R;E)

G(y,Z)))

E

A

Un(R,Q;G))

is the set o f formulas of the form ExR

A

Un(Q;GnG)

A

Co(R;E)

A

Co(R,Q;G).

2

B

for some R-free formula B. DEFINITION 12. The "k-th Elimination Theorem o f Uniqueness Conditions in the logic

L"

(abbreviated by "k-ETUC in L")

ment "Any uniqueness condition A

L

is eliminable in

of

R

L with respect to

o f type

S(A)"

k

is the statein the logic

for each

k=1,2,3,4,5.

Then, clearly;

REMARK 13. (ii)

(i)

5-ETUC = > 4-ETUC = > 2-ETUC = > 1-ETUC. 93 - ETUC

In the 5-ETUC in

L,

we can assume VkE(x,x)

E L

without

N.MOTOHASHI

382

l o s s of generality.

J5.

E ( a , 6 ) by

(If not, replace

E(a,6)

V

a=b.)

MAIN THEOREM. DEFINITION 14. A logic over a language

L

is said to be closed

under function substitutions if A E

L

ExR

A

UnR.

for any n-ary function symbol

A,

and any R-free formula A

f E L, A[R]

3

f,

any (n+l)-ary predicate symbol R, f is the formula obtained from A[R]

where

by replacing every occurrence of

f

in

A

by

R

in the usual

manner (cf. I 3 1 ) . A [ Rf] i s obvious, b u t t e d i o u s work.

To g i v e an e x a c t d e f i n i t i o n o f if

g i v e an example here, i . e .

A

is

Wx3y(f(y)=x),

then

Wx3y3z(R(y,z) A z = x ) , ( c f . I n t r o d u c t i o n o f t h i s paper). l o g i c s c l o s e d under f u n c t i o n s u b s t i t u t i o n s , and

f A[R]

So, we o n l y

is

LJ,LK

a r e examples of i s an example

LJ + W x3y(f(y)=x)

o f l o g i c s which a r e n o t c l o s e d under f u n c t i o n s u b s t i t u t i o n s . THEOREM 15.

1,

L

Suppose that

is a logic over an

which is closed under function substitutions.

of the form; Vxay(ZzA(x,z)

L

then 1-st €-Theorem in €-Theorem in

L

2

A(x,y))

-----(*)

€-free language If every sentence

is provable in

is equivalent to 1-ETUC in

L.

is equivalent to 2-ETUC in

L,

L, and 2-nd

If every sentence

of the form; 6[~y(A(x,y)X B(x,y)) is provable in in

L,

A

2yB(%,y).x

~Y((~ZA(~,Z)XA(~,~)) A B(%,y))l---(@

then 3-rd E-Theorem in

L

is equivalent to 5-ETUC

L.

Note t h a t e v e r y sentence o f t h e f o r m (*) o r ( t ) i s p r o v a b l e i n LJ.

A l s o , i f e v e r y sentence o f t h e form

tence o f t h e f o r m ( * ) i s p r o v a b l e i n REMARK 16.

(2)

i s provable i n

L,

LK,

but not i n

t h e n e v e r y sen-

L.

An approximation theorem of uniqueness conditions by

existence conditions in

LK

(cf. [ 6 1 ) gives a proof-theoretic proof

383

€-Theoremsand Elimination Theorems of Uniqueness Conditions Hence, we obtain a new proof-theoretic proof o f

LK.

of 5-ETUC in

Maehara's €-Theorem by Theorem 15 (cf. [ 7 ] , [ 9 ] ) . REMARK 17.

As pointed out by Prof. T. Uesu, 2-ETUC in

L

5-ETUC in

are equivalent for many logics

L

L

and

which satisfy some

natural, but complicated conditions, which will be obtained from a close examination of the proof of 5-ETUC in [ 7 ] .

A PROOF.

16.

I n t h i s s e c t i o n , we o n l y g i v e a p r o o f o f t h e e q u i v a l e n c y between t h e 3-rd €-Theorem i n

L

and t h e 5-ETUC i n

L

i n Theorem 15.

S o , we o m i t them.

15 a r e s i m i l a r l y proved. assume t h a t

L,

i s a l o g i c over a

I n t h e f o l l o w i n g o f t h i s s e c t i o n , we

€ - f r e e language

L, which i s c l o s e d y n d e r

f u n c t i o n s u b s t i t u t i o n s , and e v e r y sentence o f t h e f o r m in

L.

I\ N Un(Ro,Ri;Ei) i=O

i n Theorem 15 i s p r o v a b l e

[ExRoA

2

L h o l d s and t h e sentence;

AN Un(Ri,R.;EJ i, ~ = 1

i

Co(Ro,Ri i=1 i s p r o v a b l e i n L, k 1. - a r y f o r m u l a

.. , N ) ,

WiWY(Eo(i,Y) Let

(2)

[A p r o o f o f t h e 5-ETUC from 3 - r d €-Theorem]

Assume t h a t t h e 3 - r d E-Theorem i n

(i=O,l,.

The o t h e r p a r t s o f Theorem

3

where

(i=1,2, and

Ro

;Ei

1.3 C

l

i s a ko-ary p r e d i c a t e symbol,

...,N ) ,

C

koEj) A Co(RO;EO) A

Ei

i s an

Ro-free (ko+ki)-ary

Ri

i s an Ro-free

formula

i s an R o - p o s i t i v e f o r m u l a such t h a t t h e sentences

Gi(Ei(i,?i)

=

Ei(Y,Zi)),

1=1,2,

...,N,

are a l l provable i n

L.

LE be t h e l o g i c ; LE

+

c - a x i o m o f t y p e (0) and ( 3 ) .

S i n c e t h e 3 - r d €-Theorem i n Remark 3.

L

holds,

LE i s a c o n s e r v a t i v e e x t e n s i o n

A l s o , we use t h e f o l l o w i n g a b b r e v i a t i o n s :

Ex

for

ExRO,

Un

for

N iaOUn(Ro,Ri

Um

for

i,!=lUn(Ri,Rj;EikOEj),

co

for

c ~ ( R ~ , EA ~ i !)j l ~ O ( ~ o , ~ i

B(a,a)

for

ip13ii ( Ei ( a,ii)

;Ei

1,

A Ri

;E~),

(i ,b)), i

L

by

N. MOTOHASHI

384 D(a,a) Since

for

CoAEx.

sentence

3

L.

i s provable i n

3

Wzi(Ei(i,'fi) L,

Co 2 Co(D;E ) 0

t h e sentence

able i n

LE.

able i n

L,

Since

3

Hence,

LE

i s provable i n F o r each

and

L, l e t F*

(2)

LE,

and

3

C*

LE.

L,

then

Un 3 (Ex A Co A Um. L.

Also,

Un(RO,RO;EO)*

Un*

i s prov-

C)

a r e prov-

(Un

So,

i s provable i n

3

F*

A

Ex.

3

C)*

LE.

is

i s c l e a r l y provable from

Co

L.

...........................

(3)

LE.

i,j=1,2

i s provable i n

,..., N, L,

where

t h e formula;

J

ao,al

O

,;.)

R . ( i . , ~ v D ( a ~ , v ) ) . 3 E v D ( ~ ~ , v= )c J J ,..., ;N,61,62 6,c a r e m u t u a l l y d i s j o i n t se-

J

quences o f d i s t i n c t f r e e v a r i a b l e s . U m A Ei(a 0 ,6.) 1 A Ri(fii,c)

i s provable i n

A

,...,

Hence, t h e formula;

A B ( z O , ~ ~ D ( z O , ~ )3 ). EvD(~~,v)=c

LE. T h e r e f o r e , t h e formula;

Urn A Ei(ao,Li)

A Ri(bi,c)

A D ( ~ , , E V D ( ~ ~ , V ) ) .3 E V D ( ~ ~ , V ) = C

LE. By t h e f a c t t h a t t h e sentence (1) i s p r o v a b l e i n L, t h e f o r -

i s provable i n Ex A Co A

um.

Hence, t h e sentence

...,N.

L.

be t h e f o r m u l a i n

i s provable i n

;Ei )*. 3 C*

Co A i!$Un(Ro,Ri

UrnAEi(iO,fii)ARi(fii,c)AE.(i

i=1,2,

i s provable i n

by t h e f a c t t h a t t h e sentence ( 2 ) i s p r o v a b l e i n


mula

...,N

i s provable i n

C

3

i s provable i n

Ex*

F

Urn

U n * A Ex*.

Un(RO,RO;EO)* A i!lUn(RD,Ri;Ei)* in

in

F

. Co A

Un A Ex.

t h e sentence

But, c l e a r l y

Co(B;EO)

Note t h a t i f

Un A Ex.

LE.

i=l,

F by r e p l a c i n g e v e r y occurrence o f Ro o f t h e form

EvD(t,v)=t.

i s provable i n

(1 1

.........................................

F o r each f o r m u l a

L.

which i s o b t a i n e d f r o m by

by t h e f a c t t h a t e v e r y sentence o f t h e

L

E Ei(y,?i))),


Ro(f,t)

the

L,

So, t h e sentence


i s provable i n

i s provable i n

Since t h e sentences;

LE.

WiWi(Eo(i,y)

RO(x,y)) A 3yRO(x,y)],

.................................................

V i D ( i , ~ v D ( i,V ) )

2

3

i s provable i n

ExD

i s provable i n

Ex A Co.

B(a,a)) A Ro(a,a).

3

Wi[Wy(B(i,y)

3

Co A Ex.

(2)

form

(3vB(a,v)

3

(Ei(a0,Li)

A Ri(fii.c).

Ex A Co A Um.

3

13

Un(Ro,Ri;Ei)*

~ v D ( 0a ,v)=c)

i s provable i n

i s provable i n

By t h e f a c t t h a t t h e sentence ( 3 ) i s p r o v a b l e i n

LE,

LE

LE.

f o r each

t h e sentence

385

€-Theorems and Elimination Theorems of Uniqueness Conditions Ex A Co A U m . 2 C*

i s provable i n

t i o n o f t h i s paper h o l d s i n i s provable i n in

But

LE.

tence

in

Ex A Co.

Ex A Co A C*.

L.

t h e f o r m u l a V ~ ~ X ( E V D ( ~ ~ ,2V R ) =O X( i 0 , ~ ) ) A C * . 2 C

LE,

Hence, t h e sentence w i R O ( ~ , ~ v D ( i , v ) A ) C*.

LE.

i s provable i n

Since t h e p o s i t i v e lemma i n t h e i n t r o d u c -

LE.

2

i s provable i n

C

2

LE.

WiRo(i,~~D(,i,v))

Since

LE.

Ex A Co A Urn.

T h i s means t h a t t h e 5-ETUC i n

L

i s provable i n

2

LE.

Therefore, t h e sen-


C

i s provable

C

2

Ex A Co A U m . I C

i s E-free, t h i s sentence i s p r o v a b l e

[A p r o o f o f t h e 3 - r d €-Theorem

holds.

f r o m t h e 5-ETUC] Assume t h a t t h e 5-ETUC i n LE.

Then, t h e r e a r e f o r m u l a s

(0)

every f r e e v a r i a b l e i n

(i)

e v e r y subexpression

L

holds, and an € - f r e e f o r m u l a

Bo(io,a),B1(il Bi

occurs i n

EVB(V)

of

,a),. ai,a

.. ,B(iN,a) f o r each

has t h e form

B

i s provable i n

C

such t h a t : i=O,

...,N,

.

€vBj(f,v)

f,

and some

C

(ii)t h e f o r m u l a

i s provable i n

WiiPuBi(ii,u)

13

LE

from t h e sentences ) ] , i=O,l,

Bi(iii,~viBi(iii'vi

...,N

and t h e sentences

=

B.(y x ) ) 2 €viBi(iii,vi) = ~v.B.(y.,v.)], WiiWy.pjx(Bi(~i,x) J J j' J J J J L e t fo,fl ,. . . ,fN be d i s t i n c t f u n c t i o n symbols such t h a t each fi rences i n

BO,B1,

length o f

ai,

..., Bn,C,

f o r each L

the formula i n subexpression o f

and t h e number o f argument p l a c e s o f

...N.

i=O,l,

o b t a i n e d from F

o f the form

F

j > i

f o r some

For each f o r m u l a

F

in

,...,N.

has no occurfi

LE,

by t h e f o l l o w i n g procedure:

E V ~ B ~ ( ~ ~by, Vf o~( t )o )

i,j=O,l

is

ki,

let

F'

the be

We r e p l a c e every

throughout

F

first,

t h r o u g h o u t t h i s f o r m u l a second, and so on up t o

(N+l)-

t h e n r e p l a c e e v e r y subexpression o f t h e above r e s u l t f o r m u l a o f t h e form E V ~ B ~ ( ~ ~by, V f l~( f )l ) steps.

Then, each

t i o n symbols the formula dii[3uBi(ii,u)'

Bi(ai,a)'

has no occurrences o f t h e o f t h e E-symbol and func-

f . ( j 5 i ) by ( i ) . A l s o , C ' i s C because C J C i s p r o v a b l e i n LE from t h e € - f r e e sentences 2

Bi(ii,fi(ii))'],

i=O,l,

...,N

By (ii),

and t h e € - f r e e sentences

:B.(Y x ) ' ) I f i ( i i ) = f . ( 4 . ) ] , WxiWy.~x(Bi(xi,x)' J J j' J J i s p r o v a b l e i n L f r o m these sentences. L e t D /N\ V x-i p uBi(ii , u ) ' 2 Bi ( i i , f i ( i i ) ) ' 1

i=l

i s €-free.

i,j=O

,..., N.

be t h e formula;

By Remark 4 ,

C

N. MOTOHASHI

386 N diiWy.[Wx(Bi(ii,X)' i,j=1 J

!B . ( ~ . , x ) ' ) 3

J

c.

3

Then,

D

i s provable i n

L

from

fi(ii)

2

f o ( i o ) = fi(yi)],

be t h e ( k i + l ) - a r y

Ei(ai,6i)

be t h e (kotki)-ary

...,N.

i=O,l,

Then,

i=O,l,

( i ,x).

A R

by

we have t h a t

Ro,

0 0

Un(Ro,Ri;Ei),

2

i

)

i s provable i n

. ,N,

i=O,l,..

L,

in

and

L i s c l o s e d under f u n c t i o n s u b s t i t u t i o n s . L,

i s provable i n

WxWY(EO(Z,'Y)

2

L

from

Z I

2

Bo(i,x)'))

is

Ro-positive,

E Ei(j,?i)))

..., N.

and

W~DW~iWxIWu(Bo(~o,u)' i=1,2

,...,N.

(3uBo(i,u)'

i=O,l,

By r e p l a c i n g ExRO,

fo

UnRO.

Bo(i,x)')),

3

because

Un(RO,RO;EO)

...,N.

2

UnRO

A l s o , t h e sentences


A Co(RO;EO) A

i, j = 1

2

L

(3uBo(x,u)'

by t h e d e f i n i t i o n 3

Bo(i,x)'))

AN Co(Ro,Ri;Ei).

=I

.

3

D

i=l V%x(R,(i,x)

i s provable i n

Let

t h e formula

AN Un(Ri,Rj;EikoEj)

ExROA

D.

D

3

Since t h e f o r m u l a Wkx(R,(i,x)

EO,E l,...,EN.

be a

f o r each

Since t h e sentence

ExRO, Un(Ro,Ri;Ei),

Wzi(Ei(i,,Zi)

of

i=1,2,

Ro

the formula

WiWx(Ro(i,x) 2 ( 3 u B o ( i , u ) ' i s provable i n

nor i n

from t h e sentences;

WxWx(Ro(x,x)

Let

5 Bi(6i,x)1),

f o r each L

..., N.

Bo

a, f o r each

i s equivalent t o

x=fi(yi)] D

=

f o r m u l a Wx(Bo(ao,x)'

Un(R ,R.;E O i

Bi(yi,u)

fi(ai)

formula

J

B o ( ~ o , f o ( ~ o ) ) ' l and

new ( k O + l ) - a r y p r e d i c a t e symbol which occurs n e i t h e r in Ri(ai,a)

f.(y.)] J

dx0[3uBo(io,u)'

W~oW~i[Vx(Bo(~o,x)' E B i ( y i , x ) ' ) 2

=

J

L

3

by 5-ETUC i n

( k o + l ) - a r y f o r m u l a 3uBo(i,u)'

3

L.

(3uBo(x,u)'

2

BO(i,x)'))

3

D

By r e p l a c i n g ( k O + l ) - a r y p r e d i c a t e

Bo(a,a)',

which w i l l be denoted by

Ro(a,a) Ao(a,a),

by we

have t h a t t h e f o r m u l a

k Co(A0,Ri;Ei). 2 D Un(Ri,Rj;Ei OEj) A Co(AO;EO) A i, ~ = 1 i=1 i s p r o v a b l e i n L. But, c l e a r l y ExAO and Co(AO;EO) a r e p r o v a b l e i n ExAO A

AN

Wii(3uBi(xi,u)' B.(y x ) ' ) J j' i,j=1,2,

=I

2

fi(ii)

...,N.

that

C

Bi(ii,fi(ii))')

Hence

i s provable i n

Co(Ao,Ri;Ei)

3 Un(Ri,Rj;EikoEj) J i s prov.able i n L .

= f.(y.))

J D

=I

and

L.

=

WiiWyj(Wx(Bi(Ei,x)'


Also,

L

f o r each

By c o n t i n u i n g t h i s process, we see

L. T h i s shows t h a t t h e 3 - r d €-Theorem i n L h o l d s .

€-Theorems and Elimination Theorems of Uniqueness Conditions

381

REFERENCES

T. Fujiwara, A generalization of the Lyndon-Keisler theorem on homomorphism and its applications to interpolation theorem, J. of Math. SOC. Japan, V O .~ 30 (1978), 287-302. Hilbert & Bernays, Grundlagen der Mathematik, vol.1, v01.2, 1934, 1939. S.C. Kleene, Introduction to Metamathematics, Van Nostrand, Princeton, 1952. A.C. Leisenring, Mathematical Logic and Hilbert E-symbol, Gorden & Beich, New York, 1969.

S. Maehara, Equality axioms on Hilbert esymbol, J. of the Faculty of Science, Univ. o f Tokyo, Sect. 1 , vol. 7 (1957), 419-435. N. Motohashi, Approximation Theory of Uniqueness Conditions by Existence Conditions, to appear.

N. Motohashi, Elimination Theorems of Uniqueness Conditions, to appear. N. Motohashi, Some ‘proof-theoretic results on equivalence conditions,’congruence conditions, and uniqueness conditions, to appear. N . Motohashi, Elimination, Axiomatization, and Approximation.

[I01 A. Oberschelp, On the Craig-Lyndon interpolation theorem, J.S.L., vol. 33 (1968), 271-274.

389

LIST OF PARTICIPANTS

ACZEL, P e t e r

GAIFMAN, H a i m

ALVES, C a r l o s S e r r a

GALVIN, F r e d

ANAPOLITANOS,

GANDY, R o b i n

Dionisis

ARGYROS, S p i ros ASH

, Christopher

GROSZEK, M a r c i a GUILLAUME, M a r c e l

BALDWIN, John

HADJILAZAROU

BARWISE, K e n n e t h

HADLEY, M a r t i n

BAUMGARTNER, James

HAJNAL, A n d r a s

BENDS, A n a s t a s i o s

HARRINGTON, L e o

, J.

BUONCHRISTIANI

HARTLEY, John HAY, Louise

BURGESS, J o h n

HERRERA, Jorge CARLSON, T i m

HIRSHFELD, J o r a n

CHONG, Chi T a t

HOOPER, M a r t i n

CICHON, Adam

HRBACEK,

CLOTE, S t e p h e n

HYLAND, M a r t i n

Karel

COOPER, B a r r y IVANOV, L i u b o m i r - L a l o v

COX, Jonathan CROSSLEY, J o h n

JAMBU-GIRAUDET,

Michelle

D E I L , Thomas

JECH, Thomas

DEVLIN, K e i t h

JOHANNESEN, K y r r e

DIETZFELBINGER, M a r t i n DIMITRAcOPOULOS, C o n s t a n t i n o s

KALAMIDAS, N i c h o l a o s

DONNADIEU, M a r i e - R e n e e

KALANTARI, I r a j

DRAKE, F r a n k

KANAMORI, A k i

DYSON-HUBER,

Verena

EBBINGHAUS, H e i n z - D i e t e r

KASTANAS, I l i a s

KECHRIS , Alexander KEISLER, J e r o m e KESSEL, C a t h e r i n e

FEFERMAN, S o l o m o n

KLEENE, S t e p h e n

FENSTAD, J e n s

KLEIJNEN, L e t t y

FIRARIDIS, A n e s t i s

KOLAITIS, P h o k i o n

FRIEDMAN, H a r v e y

KOUMOULIS, G e o r g e

FRIEDMAN, S y

KDYMANS, K a r s t

List of Participants

390

KRANAKIS, E v a n g e l o s

PAPADOPETRAKIS, E f t i c h i o s

KRASNER, M a r c

PAPADOPOULOS PAPAGEORG IOU

LAVAULT, C h r i s t i a n

PARIGOT, M i c h e l

LAVER, R i c h a r d

PELZ, E l i z a b e t h

LENDOUDIS, P a u l

PHIDAS, A t h a n a s i o s

LERMAN, M a n u e l

PHILLIPS, Lain

LEVY, A z r i e l

P L A CARRERA, Josef

LILLIE, Gordon

POGORZELSKI, H e n r y PORTE, J e a n PRIKRY, K a r e l

MAASS, Wolfgang MAGIDOR, M e n a c h e m MAKKAI, M i c h a e l

RAISONNIER, Jean

MAKOWSKY, J o h a n

RAMBAUD, C h r i s t i a n e

MATHIAS, A d r i a n - R i c h a r d D a v i d

REMMEL, Jeff

MEISSNER, W i l f r i e d

RIMSCHA, M i c h a e l

MERKOURAKIS, S o f o k l i s

RODENHAUSEN, H e r m a n n

METAKIOES, G e o r g e

ROTHACKER, H e n r y

MICHAILIDES, T e f k r o s

ROUSSAS, G e o r g e

MIGNONE, R o b e r t

R U I Z , Jose

MIJAJLOVIC, Z a r k o MIKULSKA, M a l g o r z a t a

SACKS, G e r a l d

MILLER, D o u g l a s

SAMI, Ramez

MITCHELL, W i l l i a m

SAPOUNAKIS, A r i s t i d i s

MOLDESTAD, G o r d o n

SCOTT, D a n a

MONRO, G o r d o n

SGOUROVASILAKIS

MORAN,

SHELAH, S a h a r o n

Gadi

MOSCHOVAKIS , Y ianni s

SHEPHERDSON, John

MOTOHASHI, N o b u y o s h i

SHORE, R i c h a r d

MOUTAFAKIS, N i c h o l a s

SIEG, W i l f r i e d

MUELLER, G e r t

SIMCO, N a n c y

MY T I L I N A I O S , M i c h a e l

SIMPSON, S t e p h e n SKORDEV, D i m i t e r SLAMAN, T e d

NAGY, Z s i g m o n d

SMITH, Jan

NEGREPONTIS, S t y l i a n o s

SOARE, R o b e r t

NERODE, A n i l

SPREEN, D i e t e r STANLEY, Lee

NIANIAS, G e o r g e NICOLACOPOULOS, NINO, J a i m e NORMA",

Dag

Pantelis

STEEL, John STERN, J a c q u e s STOLTENBERG-HANSEN,

Viggo

List of Participants

39 I

THIELE, E r n s t - J o c h e n

WEISSPHENING, Wolker

THOMASON, S t e v e n

WILLIAMSON, John

THOMPSON, Simon

WOODS, A l a n ZACHARIADIS, Theodosis

.... ..

ZACHARIOU, A n d r e a s

VAANANEN , Jouko

ZACHOS, S t a t h i s

VISSER, A l b e r t

ZIEGLER, M a r t i n

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Logic Colloquium 1984: Proceedings

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Logic Colloquium 1976: Proceedings

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