INTENSIONAL MATHEMATICS
STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 113
Editors J. BARWISE, Stanford D. KAPLAN, LosAngeles H. J. KEISLER, Madison P. SUPPES, Stanford A. S.TROELSTRA, Amsterdam
NORTH-HOLLAND AMSTERDAM 0 NEW YORK 0 OXFORD
INTENSIONAL
m"71mmrc
Edited by
Stewart SHAPIRO The Ohio State University at Newark
Ohio
U.S. A.
1985
NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD
OELSEVIER SCIENCE PUBLISHERS B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 87632 4
Published by: Elsevier Science Publishers B.V. P.O. Box 1991 1OOOBZ Amsterdam The Netherlands Sole distributors for the U.S.A.and Canada: Elsevier Science Publishing Company, Inc. 52VanderbiltAvenue NewYork, N.Y. 10017 U.S.A.
Library of Congmar Cat.1ogi.g In Publlcatlon Data
Main entry under title: Intensional mathematics. (Studies i n logic and the found&Sons of mathematics ;
v. 113)
Bibliography: p; 1. Modality (Logic)--Addresses, essays, lectures.
2. Constructive mathematics--Addresses, essays, lectures. 3. Intuitionistic mathematics--Addresses, essays, lectures. I. Shapiro, Stevart, 195111. Series. QA~.46.157 1985 511 04-10056 ISBN 0-444-87632-4 (U.8. I
.
PRINTED IN THE NETHERLANDS
V
TABLE OF CONTENTS 1. Introduction: Intensional Mathematics and Constructive Mathematics
Stewart Shapiro
1
2. Epistemic and Intuitionistic Arithmetic Stewart Shapiro
11
3. Intensional Set Theory John Myhill
47
4. A Genuinely Intensional Set Theory
Nicholas D. Goodman 5. Extending Gadel’s Modal Interpretation to Type Theory and Set Theory Andrej SZedrov
63 81
6. Church’s Thesis is Consistent with Epistemic Arithmetic Robert C. Flagg
121
7. Calculable Natural Numbers vladimir Lifschitz
173
8. Modality and &&Reference
Raymond M. Smullyan
191
9. Some Principles Related to Lab’s Theorem
Raymond M. Smullyan
213
This Page Intentionally Left Blank
Intensional Mathematics S. Shapfro (Editor] @ Elsevier Science Publishers B. V. (North-Holland), 1985
1
INTRODUCTION--INTENSIONAL MATHEMATICS AND CONSTRUCTIVE MATHEMATICS
S t e w a r t Shapiro
The Ohio S t a t e U n i v e r s i t y a t Newark Newark, Ohio 1T.S .A.
Platonism and i n t u i t i o n i s m are r i v a l p h i l o s o p h i e s of mathematics,
the
former h o l d i n q t h a t the s u b j e c t matter of mathematics c o n s i s t s of a h s t r a c t o b j e c t s whose e x i s t e n c e i s independent of the mathematician, t h e l a t t e r t h a t the s u b j e c t matter c o n s i s t s of mental c o n s t r u c t i o n .
Intuitionistic
mathematics i s o f t e n c a l l e d " c o n s t r u c t i v i s t " while p l a t o n i s t i c mathematics is c a l l e d " n o n - c o n s t r u c t i v i s t "
.
The i n t u i t i o n i s t , €or
example, rejects c e r t a i n n o n - c o n s t r u c t i v e i n f e r e n c e s and p r o p o s i t i o n s as i n c o m p a t i b l e w i t h i n t u i t i o n i s t i c philosophy--as
r e l y i n q on the
independent e x i s t e n c e of mathematical o b j e c t s .
The m o s t n o t a h l e of these
i s the l a w of excluded middle,
AV
7A_
, which
the i n t u i t i o n i s t t a k e s as
a s s e r t i n q t h a t e i t h e r the c o n s t r u c t i o n correspondinq to
A
can he
e f f e c t e d or t h e c o n s t r u c t i o n c o r r e s p o n d i n q to the r e f u t a t i o n of effected.
Another example is
iVg(x)
1 ZI~Z(X) which,
c a n he
i n the c o n t e x t
o f a r i t h m e t i c , the i n t u i t i o n i s t t a k e s as a s s e r t i n q t h a t i f n o t a l l numbers have a p r o p e r t y
,
t h e n one can c o n s t r u c t a numher which l a c k s
P l a t o n i s m and i n t u i t i o n i s m are a l l i e d in the r e s p e c t t h a t tmth views
are i m p l i c i t l y opposed to materialistic a c c o u n t s of mathematics which t a k e t h e s u b j e c t matter of mathematics to c o n s i s t ( i n a d i r e c t way) of
material o b j e c t s .
Perhaps it is f o r this r e a s o n t h a t p l a t o n i s m i s
sometimes c a l l e d " o b j e c t i v e idealism'' and i n t u i t i o n i s m is sometimes c a l l e d "subjective idealism".
Both views hold t h a t mathematical o b j e c t s
are " i d e a l " a t l e a s t i n the s e n s e t h a t t h e y are n o t material.
The
2
S. SHAPIRO
P l a t o n i s t holds t h a t the mathematical " i d e a l s " do not depend on a mind f o r their e x i s t e n c e , the i n t u i t i o n i s t t h a t they do. The two views are p h i l o s o p h i c a l l y incompatible.
Indeed, t h e
e x i s t e n c e of any mentally constructed o b j e c t depends on the mind t h a t c o n s t r u c t s it, and cannot he s a i d to e x i s t independent of t h a t mind. Nevertheless, matters of i n t u i t i o n i s t i c a c c e p t a b i l i t y a r e o f t e n r a i s e d i n non-constructive mathematical contexts.
I t may be asked, i n p a r t i c u l a r ,
whether a c e r t a i n proof is c o n s t r u c t i v e (or can he made c o n s t r u c t i v e ) or whether a c e r t a i n part of a non-constructive proof is c o n s t r u c t i v e can he made c o n s t r u c t i v e ) .
(Or
One does not have to he an i n t u i t i o n i s t , f o r
example, to p o i n t o u t t h a t Peano's theorem on the s o l u t i o n of d i f f e r e n t i a l equations d i f f e r s from P i c a r d ' s i n that t h e former is not c o n s t r u c t i v e , or t h a t the Friedherq-Munchnik s o l u t i o n t o Post' s problem c o n s i s t s of the c o n s t r u c t i o n of an alqorithm, followed by a non-construct i v e proof t h a t t h i s alqorithm r e p r e s e n t s a s o l u t i o n to t h e prohlem. One of the purposes of the f i r s t f i v e papers i n this volume is to formalize the c o n s t r u c t i v e a s p e c t s of c l a s s i c a l mathematical d i s c o u r s e . Each of these papers contains both a non-constructive
lanquage which can
express statements of p a r t i a l or complete c o n s t r u c t i v i t y and a deductive system which can express c o n s t r u c t i v e and non-constructive proofs.
My
own paper and t h e L i f s c h i t z paper concern a r i t h m e t i c , while the Goodman paper, the Myhill paper, and t h e Scedrov paper concern set theory, the l a t t e r a l s o s t u d i e s type theory. I n this Introduction,
I propose a conceptual l i n k between the
i n t u i t i o n i s t i c c o n s t r u c t i o n processes and the c l a s s i c a l epistemic processes.
This l i n k , i n t u r n , provides the p h i l o s o p h i c a l hacksround
f o r c o n s t r u c t i v i s t i c concerns i n non-constructive c o n t e x t s and, t h e r e f o r e , the motivation f o r my c o n t r i b u t i o n to this volume.
Althouqh
t h e o t h e r authors do not ( n e c e s s a r i l y ) share the presented view, t h e i r work is h r i e f l y discussed i n Liqht of it.
Intensional Mathematics and Constructive Mathematics
3
I t w i l l be u s e f u l here t o b r i e f l y r e c o n s t r u c t t h e development of
extreme s u b j e c t i v e idealism i n the c o n t e x t of qeneral epistomoloqy.
Of
c o u r s e , I do not subscribe to the conclusion of the next paraqraph. Probably the most basic epistemoloqical questions are "What i s t h e source of knowledqe?" and "What i s the qround of t r u t h of p r o p o s i t i o n s known?"
Descartes a s s e r t e d t h a t the source of a person's knowledqe i s
s o l e l y h i s own expreience (excludinq, f o r example, the pronouncement of a u t h o r i t y as a source of knowledqe).
This discovery led to a study of
experience and i t s r e l a t i o n to knowledqe.
The qround of t r u t h of a
p r o p o s i t i o n known must l i e i n t h e s u b j e c t matter of the p r o p o s i t i o n .
It
follows t h a t the qround of knowledqe lies i n what our experience is
of. Althouqh we experience of an o u t s i d e
experience
seem compelled to b e l i e v e t h a t our experience
is
world, we have no d i r e c t l i n k with t h i s w o r l d
e x c e p t throuqh our senses. n o t the o u t s i d e world.
The content of sense experience, however,
is
If one s t a n d s c l o s e to and f a r from the same
o b j e c t , he w i l l have d i f f e r e n t sense imaqes.
(For example, i n one of
them, the o b j e c t w i l l occupy more of the f i e l d of vision.)
Thus, t h e r e
seems to be a permanent epistemic qap between knowledqe-experience and t h e o u t s i d e world.
The problem i s t h a t d e s p i t e our s t r o n q conviction t h a t
t h e qround of t r u t h of our b e l i e f s is e x t e r n a l t o us, we are not a h l e t o transcend both our experience and i t s qround t o v e r i f y t h i s . cannot know t h a t our experience is experience
of an
That is, w e
o u t s i d e world.
Since
what we know i s based e n t i r e l y on experience and s i n c e t h e o u t s i d e world
i s a c t u a l l y not a c o n s t i t u e n t of experience, an a p p l i c a t i o n of Ockham's r a z o r seems i n order.
Not t h a t the r e a l i t y of t h e o u t s i d e world is
o u t r i q h t l y denied, but rather it is noted that, so f a r a s w e know, the o u t s i d e world does not f i q u r e i n anythinq we know--we know anythinq ahout it.
do not know t h a t we
Hence, we do not t a l k ahout i t l i t e r a l l y .
On
t h i s view, the whole of the o u t s i d e world is reduced to a supposition t h a t orders our experience.
S. SHAPIRO
4
There is a r a t h e r s e r i o u s d i v e r q e n c e between ( 1 ) p e r c e p t i o n / t h o u q h t
as conceived by such an extreme s u b j e c t i v e i d e a l i s t and ( 2 ) p e r c e p t i o n / t h o u q h t as conceived by t h o s e who hold on to the e x i s t e n c e of the o u t s i d e 1
world--its
e x i s t e n c e independent of p e r c e p t i o n .
The l a t t e r have t h e
( a t l e a s t i m p l i c i t ) p r e s u p p o s i t i o n t h a t part of the e x t e r n a l world i s r e p r e s e n t e d more o r less a c c u r a t e l y i n p e r c e p t i o n .
For example, it is
presumed t h a t correspondinq t o o n e ' s p e r c e p t i o n of a pen i s t h e a c t u a l o b j e c t , t h e pen. perception.
There i s no such presumption i n s u b j e c t i v i s t
On the basis of these p r e s u p p o s i t i o n s , the n o n - s u b j e c t i v i s t
makes c e r t a i n i n f e r e n c e s which may n o t be s a n c t i o n e d by extreme s u h j e c t i v e idealism.
For example, i f a n o n - s u b j e c t i v i s t
sees a b a s e b a l l
s a i l over a f e n c e and o u t of s i q h t i n t o some bushes, he h a s the b e l i e f t h a t t h e b a s e b a l l s t i l l e x i s t s and i s i n the bushes.
Furthermore, he can
make p l a n s t o r e t r i e v e t h e b a s e b a l l and f i n i s h the qame.
Such an
i n f e r e n c e does n o t seem t o be j u s t i f i e d i n s u b j e c t i v i s t thouqht.
It is
n o t hard to imaqine a s u b j e c t i v e i d e a l i s t who a r q u e s t h a t p l a n s a b o u t unperceived baseballs are w i t h o u t f o u n d a t i o n . I n t h e mathematical s i t u a t i o n , a similar d e s i r e t o e x c l u d e presumptions of an o u t s i d e world from d i s c u s s i o n m o t i v a t e s i n t u i t i o n i s m . The o h j e c t i v e r e a l i t y of t h e mathematical u n i v e r s e i s d e n i e d by the i n t u i t i o n i s t i n the same s e n s e t h a t the o u t s i d e world is denied by the subjectivist.
I n p a r t i c u l a r , the i n t u i t i o n i s t does n o t c l a i m o u t r i q h t
t h a t t h e e x i s t e n c e of the mathematical u n i v e r s e depends on t h e mathematician's mind.
R a t h e r he p o i n t s o u t t h a t a l l mathematical
knowledqe i s based on mental a c t i v i t y .
T h i s mental a c t i v i t y is
apprehended d i r e c t l y , t h e ( s o - c a l l e d ) mathematical u n i v e r s e is n o t .
The
m e n t a l a c t i v i t y of mathematicians, t h e n , i s t a k e n to be the s u b j e c t matter of mathematics-questions
a r e n o t t o he c o n s i d e r e d . called "constructions".
of an o b j e c t i v e mathematical u n i v e r s e
Correspondinq t o s e n s e imaqes are what are The i n t u i t i o n i s t Heytinq once wrote:
Intensional Mathematics and Constructive Mathematics
...
5
...
Brouwer's proqram c o n s i s t e d i n the i n v e s t i q a t i o n of mental mathematical c o n s t r u c t i o n as s u c h , w i t h o u t r e f e r e n c e t o q u e s t i o n s r e q a r d i n q the n a t u r e of the c o n s t r u c t e d o b j e c t s , such as whether these o b j e c t s e x i s t i n d e p e n d e n t l y of o u r knowledqe of them a mathematical theorem e x p r e s s e s a p u r e l y e m p i r i c a l In fact, f a c t , namelv the s u c c e s s of a c e r t a i n c o n s t r u c t i o n mathematics, from the i n t u i t i o n i s t p o i n t of view, is a s t u d y of c e r t a i n f u n c t i o n s of t h e human mind
...
...
.*
As i n d i c a t e d ahove, the same term " c o n s t r u c t i o n " a l s o o c c u r s i n
classical, non-constructive contexts. " s u b j e c t i v i s t perception", d i f f e r e n t contexts. is adopted:
As w i t h " p e r c e p t i o n " and
t h e word has d i f f e r e n t meaninqs i n t h e
To avoid a c o n f u s i o n of terminoloqy, t h e f o l l o w i n q
The a d j e c t i v e " c o n s t r u c t i v e " and the noun " c o n s t r u c t i o n " are
l e f t to the i n t u i t i o n i s t s .
Whenever these words are used i n the s e q u e l
( i n this i n t r o d u c t i o n ) , t h e y are taken t o mean what t h e i n t u i t i o n i s t s mean by them.
The pair " e f f e c t i v e " and " c o n s t r u c t " are used t o r e f e r to
t h e correspondinq c l a s s i c a l e p i s t e m i c p r o c e s s e s . From the p r e s e n t p o i n t of view, the main d i f f e r e n c e between the
c l a s s i c a l e f f e c t i v e mode of t h o u q h t and t h e i n t u i t i o n i s t i c c o n s t r u c t i v e mode is t h a t the former presupposes t h a t there is an e x t e r n a l mathematical world t h a t qrounds o u r c o n s t r u c t s .
c l a s s i c a l v i e w , the c o n s t r u c t d e s c r i b e d by
t o i t s e l f t h r e e times" c o r r e s p o n d s t o the u n i v e r s e expressed by
'I
32 = 3
+
3
+
3
"
32
fact
'I.
For example, on the is o b t a i n e d hy addinq
3
i n the mathematical
As w i t h non-subjectivism,
supposition allows c e r t a i n inferences--precisely
this
the n o n - c o n s t r u c t i v e
p a r t s of mathematical p r a c t i c e r e j e c t e d by the i n t u i t i o n i s t s .
For
example, i f a classical mathematician proves t h a t n o t a l l n a t u r a l numbers have a c e r t a i n p r o p e r t y , he c a n t h e n i n f e r the e x i s t e n c e of a n a t u r a l numher l a c k i n q this p r o p e r t y .
a number i s n o t known--even
This i n f e r e n c e can be made even i f such
i f t h e matematician does n o t know
n u m e r a l s it d e n o t e s such a number.
an e x a c t
An i n t u i t i o n i s t d e n i e s t h i s
i n f e r e n c e because he h e l i e v e s t h a t it relies on the independent, o b j e c t i v e e x i s t e n c e of t h e n a t u r a l numbers.
For an i n t u i t i o n i s t , each
6
S. SHAPIRO
a s s e r t i o n must r e p o r t a c o n s t r u c t i o n .
I n the p r e s e n t example, he would
c l a i m that the e x i s t e n c e of a n a t u r a l numher with the s a i d p r o p e r t y c a n n o t he a s s e r t e d because such a numher w a s n o t c o n s t r u c t e d .
A classical
mathematician may wonder whether such a number can be c o n s t r u c t e d - whether he can know of a s p e c i f i c numeral t h a t d e n o t e s such a number-h u t t h e l a c k of a c o n s t r u c t does n o t p r e v e n t the i n f e r e n c e . Accordinq t o the p r e s e n t a c c o u n t , then, both the c o n s t r u c t i v e mode o f t h o u q h t and the e f f e c t i v e mode of t h o u q h t are r e l a t e d t o e p i s t e m i c
matters.
That is, to a s k f o r a numher with a c e r t a i n p r o p e r t y to he
c o n s t r u c t e d is to ask i f there is a numher which can he known t o have this property.
I f this account i s p l a u s i b l e , t h e n the " c o n s t r u c t i v e " a s p e c t s
o € classical mathematics can be e x p r e s s e d i n a formal lanquaqe which
c o n t a i n s e p i s t e m i c terminoloqy.
T h i s i s t h e approach of the f i r s t f i v e
p a p e r s i n t h e p r e s e n t volume. K
I n my c o n t r i h u t i o n , a n e p i s t e m i c o p e r a t o r lanquaqe of arithmetic. mean
"
A
If
A
is a formula, t h e n
i s i d e a l l y o r p o t e n t i a l l y knowable".
a x i o m a t i z a t i o n e q u i v a l e n t t o t h e modal l o q i c i n this c o n t e x t .
As
suqqested,
i s added t o the K(A)
is taken to
I arque that a n S4
is appropriate f o r
K
35K(A(5)) i s taken as amountinq t o
" t h e r e e f f e c t i v e l y e x i s t s a numher s a t i s f y i n q
A
". The lanquaqe
of
i n t u i t i o n i s t i c a r i t h m e t i c is t h e n " t r a n s l a t e d " i n t o this e p i s t e m i c lanquaqe.
Followinq the i n t u i t i o n i s t i c r e j e c t i o n of non-epistemic
m a t t e r s , the ranqe of this t r a n s l a t i o n c o n t a i n s formulas which have, i n some s e n s e , o n l y e p i s t e m i c components.
S e v e r a l common p r o p e r t i e s of
i n t u i t i o n i s t i c d e d u c t i v e systems are o b t a i n e d f o r the e p i s t e m i c parts o f
mv d e d u c t i v e system (which i n c l u d e s t h e ranqe of the ahove t r a n s l a t i o n ) . The Flaqq paper develops a r e a l i z a h i l i t y i n t e r p r e t a t i o n for t h e lanquaqe of my system and, thereby, s h e d s l i q h t on i t s proof theory.
The Mvhill paper and the Goodman paper c o n t a i n e x t e n s i o n s of my lanquaqe and d e d u c t i v e system to set theory.
~ o t hlanquaqes c o n t a i n a
7
Intensional Mathematics and Constructive Mathematics
s e n t e n t i a l o p e r a t o r analoqous t o my
"K"
.
The lanquaqe i n M y h i l l ' s
p a p e r c o n t a i n s t w o sorts of v a r i a b l e s , one r a n q i n q over sets i n q e n e r a l ( c o n s i d e r e d e x t e n s i o n a l l y ) and one ranqinq o v e r " e x p l i c i t l y q i v e n h e r e d i t a r i l y f i n i t e sets".
The l a t t e r i n c l u d e s , f o r example, e x p l i c i t l y
q i v e n n a t u r a l numbers and e x p l i c i t l y qiven r a t i o n a l numhers.
In the
lanquaqe of Goodman's paper, a l l v a r i a h l e s range over i n t e n s i o n a l "set Althouqh s e t p r o p e r t i e s are n o t e x t e n s i o n a l , c l a s s i c a l
properties".
( e x t e n s i o n a l ) s e t theory can be i n t e r p r e t e d i n Goodman's system i n a s t r a i q h t f o r w a r d manner.
The %edrov paper p r o v i d e s a " t r a n s l a t i o n " of
i n t u i t i o n i s t i c t y p e t h e o r y i n t o a modal t y p e t h e o r y ( a l s o hased on
54)
and a " t r a n s l a t i o n " of i n t u i t i o n i s t i c set t h e o r y i n t o a modal s e t t h e o r y which employs the lanquaqe of Goodman's paper ( b u t h a s a s t r o n q e r Both t r a n s l a t i o n s are q u i t e similar t o t h e
deductive system).
t r a n s l a t i o n of i n t u i t i o n i s t i c a r i t h m e t i c i n my paper. The system developed i n t h e L i f s c h i t z c o n t r i b u t i o n i n v o l v e s a d i f f e r e n t u n d e r s t a n d i n q of t h e e p i s t e m i c i n t e r p r e t a t i o n of constructivity.
i s employed.
I n s t e a d of a n e p i s t e m i c o p e r a t o r , an e p i s t e m i c p r e d i c a t e
T is a v a r i a b l e , then
If
constructed".
K(x)
is t a k e n as
"
x
-
t Ktn)
f o r a l l numerals
t h e set of a l l n a t u r a l numhers. s e m a n t i c s of the paper.
-
fi ,
However,
t h e e x t e n s i o n of
VxK(x)
A(5))
K
would he
1 s u q q e s t t h a t the p r e c i s e meaninq of
K
is
For example,
i s t a k e n as amountinq t o " t h e r e e f f e c t i v e l y e x i s t s a
number s a t i s f y i n q " f o r any given
I f it d i d ,
i s f a l s e i n the
d e t e r m i n e d , i n part, by the c o n t e x t i n which i t o c c u r s .
35(K(x) &
K
I t is i m p o r t a n t t o n o t e t h a t t h e e p i s t e m i c p r e d i c a t e
d o e s n o t have a d e t e r m i n a t e e x t e n s i o n i n the n a t u r a l numbers. then since
can he
A "
2 ,
and
Vz(K(5)+ A ( 5 ) ) i s t a k e n as amountinq t o
A(x) ".
The formulas of i n t u i t i o n i s t i c a r i t h m e t i c
are i n t e r p r e t e d i n this lanquaqe as those formulas whose q u a n t i f i e r s are a l l restricted to
K
.
Althouqh f a i t h f u l n e s s of t h i s t r a n s l a t i o n is
open, s e v e r a l s u q q e s t i v e r e s u l t s are o h t a i n e d .
S. SHAPIRO
8
The systems i n t h e f i r s t f o u r p a p e r s of this volume b e a r a t l e a s t a s u p e r f i c i a l resemblance t o t h o s e developed i n some r e c e n t work by G. Roolos, R. Solovay and
other^.^
There are, however,
important
The l a t t e r systems c o n t a i n a modal o p e r a t o r 0
differences. i s taken a s
"
p
i s provable i n Peano arithmetic".
,
where
up
I n t h a t work,
i t e r a t e d modal o p e r a t o r s are understood a s i n v o l v i n q a r i t h m e t i z a t i o n . For example,
ocp
i s taken as
Bew( IEewrgll )
, where
is the
Bew
p r o v a b i l i t y p r e d i c a t e i n Peano a r i t h m e t i c and, f o r any formula i s t h e & d e l number of
A
.
The modal o p e r a t o r s i n t h e f i r s t f o u r p a p e r s
o f this volume c a n n o t be s i m i l a r l y i n t e r p r e t e d . example, t h e o p e r a t o r
K
I n my system, f o r
is i n t e r p r e t e d a s " p r o v a b i l i t y i n p r i n c i p l e " and
is thereby not r e s t r i c t e d to
Peano a r i t h m e t i c ) .
11 ,
any
p a r t i c u l a r d e d u c t i v e system ( s u c h as
For example, the " e x t e n s i o n " of
Contains n o t o n l y
K
formulas provable i n c l a s s i c a l Peano a r i t h m e t i c , h u t also formulas p r o v a h l e i n the system of my paper.
The o p e r a t o r
R
in M y h i l l ' s p a p e r
is i n t e r p r e t e d as p r o v a b i l i t y i n t h e s y s t e m of t h a t paper and, t h e r e f o r e , i s n o t r e s t r i c t e d t o p r o v a b i l i t y i n c l a s s i c a l s e t theory.
These
i n t e r p r e t a t i o n s of t h e modal o p e r a t o r s e l i m i n a t e t h e need f o r a r i t h m e t i z a t i o n t o understand formulas with i t e r a t e d o p e r a t o r s . M y h i l l ' s system, f o r example, provable".
-
BR(&)
i s simply taken as
"
B(A)
*
In
is
The p r e s e n t a u t h o r s s u q q e s t t h a t the b r o a d e r u n d e r s t a n d i n q of
t h e o p e r a t o r s f a c i l i t a t e s t h e i n t e r p r e t a t i o n of c o n s t r u c t i v e mathematics i n c l a s s i c a l modal systems. R.
Smullyan's f i r s t paper below can be seen as a s t u d y of t h e above
extended n o t i o n of p r o v a b i l i t y i n a more q e n e r a l s e t t i n q .
p
developed i n t h a t paper h a s a p r e d i c a t e e x p r e s s i o n s of t h e same lanquaqe. lanquaqe and as
"
'A1
If
a name of formula
A i s provable i n -
(p
i n which e v e r y theorem of
'I.
@
The lanquaqe
r a n q i n q over names of
(9
is a
d e d u c t i v e system on t h i s
A ,
then
prA1
c a n he i n t e r p r e t e d
Concern i s with those d e d u c t i v e systems
is t r u e under t h e i n t e r p r e t a t i o n of
p
as
Intensional Mathematicsand Constructive Mathematics
provability i n
8
.
9
Such d e d u c t i v e systems are c a l l e d " s e l f -
r e f e r e n t i a l l y correct". Smullyan's second paper, a s e q u e l t o the f i r s t , c o n c e r n s p r o v a h i l i t y i n a s t i l l more g e n e r a l s e t t i n q .
The r e s u l t s a p p l y t o a n y lanquaqe and
d e d u c t i v e system w i t h a ( m e t a - l i n q u i s t i c ) p r o v a h i l i t y f u n c t i o n s a t i s f y i n q t h e Hilbert-Bernays d e r i v a h i l i t y c o n d i t i o n s .
This i n c l u d e s , f o r example,
t h e systems of t h e f i r s t f o u r p a p e r s of this volume, t h e systems i n Smullyan's f i r s t paper and t h e systems i n , s a y , Boolos' work.
Concern i s
w i t h c o n d i t i o n s under which & d e l l s second incompleteness theorem and a " l o c a l i z e d " v e r s i o n of Lgh's theorem apply. I t s h o u l d he p o i n t e d o u t t h a t t h e a u t h o r s of t h e papers i n t h i s
volume do n o t completely s h a r e their p h i l o s o p h i c a l views and m o t i v a t i o n s . In p a r t i c u l a r , the p h i l o s o p h i c a l remarks i n t h i s I n t r o d u c t i o n e x p r e s s o n l y my views.
The disaqreements amonq t h e a u t h o r s are r e f l e c t e d i n p a r t
h v t h e mutual criticism c o n t a i n e d i n t h e f o l l o w i n s p a p e r s . I would l i k e t o thank John Mvhill and Ray Gumh f o r t h e i d e a of
c o l l e c t i n q papers on this s u h j e d t and t o thank John f o r encouraqinq t h e a u t h o r s to work on the project.
S p e c i a l t h a n k s to t h e e d i t o r i a l s t a f f a t
North Holland, e s p e c i a l l y D r . S e v e n s t e r , f o r t h e prompt and p r o f e s s i o n a l manner i n which the volume w a s handled. t h i s a l l t h e more.
Experience makes m e a p p r e c i a t e
S. SHAPIRO
10
Notes 1.
The word " p r e c e p t i o n "
( s i m p l i c i t e r ) i s used h e r e o n l y t o r e f e r
t o p e r c e p t i o n viewed w i t h t h e p r e s u p p o s i t i o n t h a t t h e r e i s a p e r c e i v e d e x t e r n a l world.
" S u h j e c t i v i s t p e r c e p t i o n " is t o r e f e r t o p e r c e p t i o n as
c o n c e i v e d by an e x t r e m e s u b j e c t i v e i d e a l i s t .
S i m i l a r for " t h o u q h t " and
" s u b j e c t i v i s t thouqht". 2.
A.
Heytinq,
Intuitionism,
Holland P u h l i s h i n q Company, 1956, pp.
3.
See, f o r example, G.
Boolos,
I n t r o d u c t i o n , Amsterdam, North 1 , 8 , 10.
llnprovahility
Camhridqe, Camhridqe D n i v e r s i t y P r e s s , 1979.
of C o n s i s t e n c y ,
In tensional Mathematics S. Shapiro (Editor) 0 Elsevier Science Publishers B. V. (North-Holland), 198.5
11
EPISTEMIC AND INTUITIONISTIC ARITHMETIC Stewart Shapiro The Ohio State University at Newark Newark, Ohio U.S.A.
Introduction. In this paper a language and deductive system of epistemic logic for arithmetic are developed.
In addition to the usual
connectives and quantifiers, the language contains an epistemic operator K. If
5
is a formula, then K ( A ) is taken to mean ,'
knowable"
.
is ideally, or potentially,
In addition to whatever intrinsic interest the presented system may have, I suggest that it can help illuminate the understanding and formalization of mathematical practice.
For example, even though the
underlying logic of the system is classical, it is shown that the language of intuitionistic arithmetic can be translated into the language, and thus, that the present system is capable of expressing formulas of both classical and intuitionistic arithmetic, as well as formulas of mixed constructivity. This indicates that the present system can contribute to an understanding of the difference between classical and constructive arithmetic, and, moreover, that it can account for and help understand the constructive and epistemic aspects of normal, non-intuitionistic mathematical practice. As a preliminary to the technical development, in section 1 below possible understandings and interpretations of the operator K are discussed.
Section 2 contains the basic details of the first-order
version of the language and deductive system. Section 3 concerns the possibility of interpreting K as "deducibility" in a particular deductive system. Although such an interpretation is developed, there are serious philosophical (and technical) limitations to it.
In the following section
4, certain properties of the deductive system are established (through the reinterpretation of section 3 ) .
Section 5 is a brief note on the
possibility of applying Hintikka's [71 semantics for ideal knowledge to the present language.
It is shown that although the present deductive
system is sound in Hintikka's semantics, his notion of "epistemic alternative" is not appropriate to arithmetic and, moreover, that this
S. SHAPIRO
12
shortcoming is shared by many semantics for (ideal) knowledge.
Section 6
contains the translation of the language of intuitionistic arithmetic and related matters.
In section I the present language and deductive system
are extended to include higher-order quantification. There are several interesting features of the expanded systems that are not shared by the first-order version.
Among these are technical counterparts of some well-
known problems in epistemology and the philosophy of language. In the final section 8 , some further applications of our language (and deductive system) to the formalization of mathematical practice are developed. 1.
The Epistemic
Operator. Knowledge, of course, involves a knower.
Thus, any epistemic operator must refer to the knowledge of a particular subjective being, such as a person or, perhaps, a community.
An
epistemic
language (such as that of [ 7 ] ) which involves more than one knower should have a different operator for each. The applications of the present deductive system, however, suggest that the added technical complications of the extra operators are not necessary.
Hence our single operator K.
The intended interpretation of K is not actual knowledge, but rather what may be called "ideal knowledge" or "knowability". It is assumed, in particular, that the "extension" of K is deductively closed:
If KQ)
and
.
The "knower" involved with K is taken to be an ideal then K ( B ) mathematical community. Informally, the preferred reading of K ( 5 ) is ''2 is
A FBI
knowable".
In short, everything known by the community is knowable and
anything that follows from knowable premises is knowable. other possible readings of K ( A ) are "it is possible to come to know community knows", (see [121).l
"5is
verifiable", and
"5 is
A
given what the
(informally) provable"
It is admitted, of course, that these readings are vague and,
perhaps, obscure.
It is hoped that the foregoing analysis will provide the
necessary precision to the present operator K and, derivatively, to the concepts involved in the above readings. It might be noted that the idealization of the present "knowability" is similar to that of other mathematical properties whose terms also have the suffix "-able". Examples include computability, decidability, solvability, definability, and even deducibility. The treatment of computability, for instance, is not made relative to the computation abilities of this or that computist, but rather involves computation ability as such.
Moreover, computability does not concern feasibility--no
(finite) bounds are placed on the memory, materials, life-span, etc. required for a computation. Here. there is only one "knower" and no bounds
Epistemic and lntuitionistic Arithmetic
13
are placed on the ability of the community to obtain the consequences of its knowledge. I return to the "-able" properties in section 8 below. Actual knowledge, of course, is time-dependent in the sense that the knowledge of a given person varies with time. At best, consideration of ideal knowledge only partially removes the time-dependence.
Indeed, even
if, say, the theorems of a deductive system are knowable simultaneously with the axioms, the possible discovery of knowledge through new axioms remains a time-dependent activity. Thus, here I do not envision the possibility of an absolute, time-independent concept of knowability. The operator K may be taken as referring to knowability at a fixed, but unspecified time. Finally, in ordinary language, it is both grammatical and semantically meaningful for an epistemic statement to occur within the scope of another epistemic operator. For a given sentence that it is known that then & is known.
A
&, for example, one can state is true
is unknown or that it is known that if
Moreover, some sentences like this are true and some are
false. Thus, in ordinary language, "knowledge" represents what may be called a (non-trivial) iterative concept. Because the present K is a sentential operator (and not a predicate) it also is iterative. There are well-formed formulas, for example, of the form K(-K(&)) and K ( A + K ( & ) ) . Formally, the operator K can be interpreted, or reinterpreted, as any iterative concept that does not apply to false propositions and is deductively closed.
(To be a sound interpretation vis-a-vis the present
treatment, the property should be closed in the present deductive system.) On the surface, at least, this rules out "deducibility in a particular deductive system" as an interpretation of K, because deducibility is not iterative. Such statements as c e &are usually ill-formed. As is well-known, however, this "surface" may be shallow. At least two attempts around this difficulty suggest themselves. The first is to invoke an arithmetization. If T is a deductive system for arithmetic, PrT the proof predicate for T, and
& a formula of arithmetic, then
be interpreted as PrT ( r&') and K (K@) ) as PrT ( 'PrT
( '&')')
.
K ( A ) might
This
possibility is explored extensively in the literature (see, for example, [2] and R. Smullyan's contributions to the present volume), but it is clear
almost at the outset that this interpretation will not do here. if
fi is
knowable, then & is true.
K(K(A)+;);
Thus, K(&)
-+Ais a
Informally,
correct scheme, as is
both are theorems of the present deductive system. However,
S. SHAPIRO
14
by L & ' s
theorem, PrT(f&7)
+fA
is not a theorem of T unless
5
is a theorem
of T. A second possibility, perhaps, would be to let T' be an extension of
the present deductive system and to relate K1A) to F
T'-A, K(Kf&)) to FT,K(A), etc. With this program, K is interpreted, not as deducibility-inarithmetic, but as deducibility-in-epistemic-arithmetic.
An interpretation
of the present system along these lines is developed in section 3 below. Philosophically, however, it is a =interpretation.
It is shown there that
besides the obvious circularity, there are serious (and insightful) philosophical and technical limitations to such an interpretation. 2.
The Basic Deductive System.
We begin with a standard, first-order
system for arithmetic. The language L has connectives
7,
v,
&,+,-++:
quantifiers V, 3 , and names for all of the usual primitive recursive 2
functions and relations, including the Kleene T-predicates and U-function. We employ a natural deduction system D which operates through the introduction and discharge of assumptions.
The details are routine.
To
note one example, the so-called "deduction theorem" is a rule of inference--the rule of arrow introduction: If F,A
+I:
kg then
F k A+B
-.
Dwill be abbreviated I-. D A new (sentential) operator K is added to L with the formation rule:
In what follows, If
A
is a wff then K ( A ) is a wff.
Parentheses are omitted when unnecessary for clarity. We call a formula ontic if it contains no occurences of K. -
The reason for this designation
is that such formulas do not concern knowledge--they involve only the natural numbers and the interrelations thereof. Moreover, the truth values of ontic formulas are independent of any actual or ideal knowledge. We call a formula epistemic if it is in the form K(A) for some formula
A.
Two rules of inference are added to D for the new operator, an "elimination rule" and an "introduction rule": KE:
K(A) /-
KI:
If F
t
A. A and
every formula in ?? is epistemic, then I?
+
K(A).
On the intended (informal) interpretation, KE is a correct rule simply because only true statements are knowable; KI is a statement of the deductive closure of knowability. Notice that the result of erasing all occurences of K from a theorem of D is itself a theorem of arithmetic. Hence, D is consistent.
15
Epistemic and lntuitionistic Arithmetic
The following theorem schemes have trivial proofs: TO:
K(5)
+A.
TI: K ( 5 ) -+KK(A). T2:
KK(&).
K(&)++
A
T3:
If
T4:
K(&-+B)
C
then -+
k K(A).
'
(K(A) - + K ( g ) ) .
The main theorem of 131 indicates that the logical (i.e., non-arithmetic) part of D is equivalent to the modal logic S4. Theorem T1 may be called a reflection principle. An informal justification for this scheme may be given in terms of a similar principle for actual knowledge. Assume, then, that if or at least knowable, that
A
A
is known then it is known,
is known. This seems plausible if "knowable"
is understood in terms of some sort of (informal) "provability". Suppose that
A
is knowable. Then, of course, A could become known.
By the
assumption, when Abecomes known, it becomes knowable that fi is known. Hence, the knowability of
is knowable.
Notice that a similar line of reasoning fails to justify the scheme If &is simply not known, then (perhaps) one can come to
-K(fi)-+K(lK(&)).
know that TK(_A)
A
means
is not known by self-reflection. This does not matter.
"11
Here,
is not knowable" and the unknowable formulas cannot be
determined by self-reflection even in principle. 3.
Comparison offand Deducibility. Proofs in axiomatic deductive
systems are sometimes thought to reflect, at least to some extent, the process of coming to know mathematical facts.
Thus, there seems to be at
least a similarity between formal deducibility and the intended interpretation of K.
In fact, later in this section an interesting and
fruitful metatheorem is obtained by partially interpreting K along these lines. Presently, however, it is shown that there is a serious logical and philosophical limitation to thinking of K as meanine "deducibility in a particular deductive system." 3.1.
In 141 a d e l presents and briefly discusses an axiomatization
equivalent to propositioned S4 (see section 6 below for the details). suggests that K(A) might be taken to mean
"A is provable", but
if so, "provable" must mean "provable in principle" and not
K(5)
cannot mean
''A is deducible in this
deductive system."
I-
He
adds that
--that is Concerning
the present system, the extent of the possible correlation between K and formal deducibility (or any arithmetic property) is born out in an
S. SHAF'IRO
16
interesting metatheorem suggested by some comments in Myhill 1121 on Gadel' s work. A fixed arithmetization of the formulas of L is assumed.
For each
natural number & let B be the formula with W d el number fl. Notice that -n_ if K is interpreted as deducibility in a particular deductive system, then (the set of Giidel numbers of) the extension of K would be recursively enumerable and, therefore, arithmetic. Suppose, then, that there is a formula
--
E(x) of L with one free variable
which is thought to represent the extension of K. sentence
s, suppose that g(E)++K ( B
simplicity, it may be assumed that the technical results.)
)
is true.
That is, for each (For conceptual
E is ontic, but this does not affect
It follows (under the assumption that nothing
knowable is false) that there are true, but unknowable sentences. This, of course, is no surprise. can be --
Our first metatheorem is that one such sentence
--
constructed from E.
{ g @ + + K(B
)
-n_ TA:
I
For this, let
(*E) be the set of sentences
is a sentence).
B
There is a sentence E of L such that
(*E) Proof:
f-
7
K(F).
Moreover, if
E
(*El
l-
and
is ontic, then so is
F.
Let d be the name in L of the diagonal function, the
-"_
primitive recursive function which assigns to each number E, the Wdel number of B (;I.
--
Let 2 be the Wdel number of
(*E) IK ( E ) and, hence, (*El I- F.
TE(d5) and let F be -$(dr~). It follows that From KE we have (*El I- 1
E++7K(E)
Notice that this result does not entail the inconsistency of (*El.
(*E) are not epistemic, the rule K I cannot be (*E) I- K ( F ) from (*E) I- F.) Indeed, in the next
(Since the formulas in invoked to produce
subsection it is shown that
(*g) is
in fact consistent for many formulas
-E.
Two corrolaries of TA, however, indicate philosophical and technical restrictions on the attempt to consider E as representing the extension
Of K. Notice, first, that if, for some formula
E ( x ) ,the
sentences in
(*E)
are all true, then one might think it consistent to add them as axioms of D.
This, however, is not the case.
Let D E be the deductive system
obtained from D by adding each instance of (*El as an axiom. CA1:
The deductive system DE is inconsistent.
Proof: CA2:
From TA, F D E 1 K ( F ) and kD&.
The set of formulas { K ( B )
By KI, J-
1-g~(*E) 1
K(E).
DE
is inconsistent with D.
Epistemic and Intuitionistic Arithmetic
17
That is, there is a sentence G E (*g) such that CD-,K(G) (and FDK(-,K(G)1 1 . One moral of these considerations is that it is not sufficient for new axioms of the deductive system to be true--new axioms should also not be unknowable. If there is such a concept as absolute, time-independent arithmetic knowability and if the deductive system 5 is sound for such a knowability, then theorem TA has a rather succinct interpretation. Suppose that there is a formula E(5) that represents the extension of absolute knowability. Then there is a sentence B unknowable that
E(5)
-n-
such that it is (knowable that it is) absolutely
and E ( B
are materially equivalent.
)
-”_
Thus, either the
extension of absolute arithmetic knowability is not arithmetic or the 4 extension is, in some sense, absolutely unknowable. As noted, however, the intended interpretation of K is not absolute knowability, but knowability-at-a-fixed-time, sat rule out the possibility that (at is represented by some formula
E.
g.
Theorem TA does not
z) the extension of K is arithmetic and The corollary CA2, however, indicates
G that 5 represents the in (*El is true, but
that for each formula E it cannot be known (or even knowable) at the extension of K is represented by extension of knowability at
g,
g.
In short, if
then every formula
some of these formulas are unknowable at
2.
Under these circumstances, it remains possible that it could become known at some later time, say represented by
E.
c,that knowability at g
is (or was)
This knowledge might be obtained, for example, by
reflecting on the epistemic processes available at
G.
CA2
shows that
this knowledge is genuinely new knowledge which was not available at In such a case, some of the formulas in the form
K ( E ( i ) *K(B
true only if the outermost K is interpreted as knowable-at
))
g.
would be
-n_ and the
inner K is interpreted as knowable-at 2. If K is completely reinterpreted as knowable-at 2, then some of the formulas of
-
false. An example of such a formula is g(dE) *K(%m). the extension of knowability at
--
hence, K ( B 3.2.
)
If E represents
2, then E ( d i ) (whicE is equivalent to
of TA) asserts that the sentence with a d e l number knowable-at g . This is false.
(*g) become
However, B
is true as reinterpreted.
-dE
7c
(i.e., F) is
is knowable-at &2- and,
It remains to be seen whether it is formally consistent for the
extension of K to correspond to that of deducibility in a particular deductive system.
For this, an interesting metatheorem is obtained by
S. SHAPIRO
18
adapting, simplifying and extending the method of Kleene 191 to our language and deductive system. A
relation D'] & between extensions D' of D and sentences A of L is
introduced. Roughly, D'I
5
may be thought of as "if K means kD,, then
is true".
A
I
If & is atomic, then D' & iff kD,&.
D'
I A&B iff D' I and D' I B. I fivg iff D'I 5 or D * I B. I A+g iff either D' ,j' A or D'I
D'
I &++B
D' D'
D'
I -, I~ ~
D'
I~xA(x)
D'
I K(A)
D'
g. B or
D' 1 5 and D' 1 g . 1 A. 1 iff 1 ) D *I ~ ( ifor ) every numeral i.
iff either D'I & and D'I
iff D'
(
iff
D'I ~ ( ifor ) some numeral i.
iff D'I & and kD,&.
In what follows, D I & is abbreviated
I A.
Under the assumption that D' is consistent, the following lemmas
A:
follow from the definition of D'I L1:
If
s
and t are closed terms and if s=t is true, then FD,
s=t and, hence, D' free, then D' I L2:
If
s
&(s)
s=t. Moreover, if
A(x) has only 5
++&(t).
is a closed term, then there is a unique numberal
such that f-.,,s=i L3:
I
and, hence, D'
I s=i. 1 A iff 5
If A is an ontic sentence, then D'
i
is true.
The proof of the following metatheorem is straightforward, but tedious. TB:
Let C be a set of sentences of L such that DUG is consistent and (DUC) I
g
for each
FEZ.
If
5 is any
I A.
sentence of L such that I-DUC-A, then (DUS) If the set C of additional sentences is empty, then TB is TB':
If
is a sentence of L and k &, then
I A.
The first corollary of TB is that it is consistent for the extension of K to correspond to that of any arithmetically definable, consistent extension D' of D which satisfies the premises of TB.
Let E ( x ) be any
ontic formula (with only one free variable) which expresses deducibility in D'.
That is, let CB1: Proof:
(*E)
g(i) be
B D'-n-' is consistent with D'. true iff f-
That is,
(*E)
--
D,O=l.
From the lemmas and TB, we have the following equivalences:
Epistemic and Intuitionistic Arithmetic
D' D'
19
I g(G) iff E ( i ) is true iff t= D-"_ ,B iff ( t ,B and -D'"_ I B ) iff D'I K(B ) . Therefore, I)' I (E(n)*K(S)).
?! That is, D'
n -
15 for
each G E
(*El. If (*El k,,;=i,then
it would follow (from TB and the rule of arrow introduction) that D'
I ;=i,a contradiction.
Combining this proof with corollary CA2, notice that for each appropriate deductive system D', there is a sentence 5 (in
(*E)) such that D' 15,
so
b(D17g,but kD,lK(g). Notice also that it is consistent for the extension of K to correspond to that of any arithmetically definable extension of D obtained by adding true ontic sentences. 4. Properties of the Deductive System.
This section focuses on
several corollaries of TB which, I suggest, correspond to important Let D' be any consistent,
properties of knowledge and knowability.
recursively enumerable extension of D which satisfies the premises of TB. 4.1. Disjunction @ Existential Quantification. On the present interpretation of K, there is a difference between K(5vE) and K@)VK(B). Consider, for example, an instance of these formulas in which g is the negation of 5. Notice first that K(Av75) amounts to the truism "it is
A
is either true or false". On the other hand, K(A)vK(~&) A is knowable. If 5 is any ontic says that either 5 is knowable or . sentence whose truth value is unknowable, then K(fiv~&) is true (and
knowable that
provable) but K(&)vK(~A) is not.
The next corollary to TB bears out the
difference between K (fivg) and K (5)VK (E) CB2:
If ;and either
.
are sentences of L and if kD,K@)vK(F)
then
F D I Cor kD,g.
If I-DIK(&)~K(g) then, by TB, D' I K(A)vK@). Therefore, A or kD,B. Hence, either +either D ' I K(5) or D' K@). D'1 . If 5 There is a similar difference between K(3g(x)) and 35K(5(5) Proof:
I
is ontic, for example, the former amounts to only "it is knowable that the extension of the property represented by 5 is not empty". The latter, however, is the stronger statement that there is a (particular) number such that it is knowable that x satisifes the property represented by
A.
This difference is born out by an analogous corollary: free and if kD,35K(5(5)), CB3: If A(x) is a formula with only
x
--
then there is a numeral
1 such that tD,5(;).
The proof of CB3 is similar to that of CB2. undecidable sentence and let g(&) be
For illustration, let
(s=?kg) v(z=i&Tg).
is a logical truth and, thus I-KdjxB(x) )
.
g be any
Of course, 35B(xf
However, if I-35(K(B(z) 1 , then,
20
S. SHAPIRO
by CB3, there would be a numeral however, entails that either
tg
--1 such that c(n=O&G) v (i=i&,g).This, -12, which
or
contradicts the assumption
of undecidability. 4.2.
Universal Quantification--The Barcan Formula.
the similar pair of formulas V g ( & ( x ) ) and K ( V & ( x ) ) . least, the two formulas are not equivalent.
&(x) is knowable
instance of
We next consider
On the surface at
The former asserts that each
(perhaps separately) while the latter is the
statement that it is knowable that & holds universally. is provable in D.
expected, K ( V g ( x ) ) + V g ( & ( x ) )
Barcan scheme, V g ( & ( x ) ) + K ( V S ( x ) ) .
As
might be
The converse is the
One might argue that this scheme
expresses a correct principle by appealing to a strong "reflection" Suppose that V g ( & ( x ) ) is true.
principle:
Then each instance of & could
become known.
If each instance of &did become known, then one could come
to know V&(x)
by a survey of knowledge. Hence, K ( V S ( x ) ) is true.
I
would suggest, however, that since the process of reflection involved here is infinite, the argument fails.
In the scenario, one does not realize
that & holds universally until after a survey of all of the numbers is completed. Such "procedures" are not legitimate even in the present context of ideal knowledge (in which no finite constraints are placed on The following corollary to TB shows that the Barcan
knowledge ability).
formula scheme is not derivable in D ' .
CB4:
There is a formula B ( x ) , with one free variable, such that j'D,Vg(B(x)
Proof:
Let
.
1 -+K(Vg(z)1 the primitive recursive predicate which
g ( x ) be
expresses "5 is not the G6del number of a proof in D ' of For each numeral D'
I
we have CD,B(i),D' I
Therefore, D' 1
K ( g ( Z ) 1.
Vg(g(x)1 .
second incompleteness theorem, Hence, by TB,
;=I."
B(E), and, hence, However, by W e l ' s
kD,Ve(x)and, so D '
,#K(Vg(x)
1.
VxK(B(5)) - + K ( V S ( X ) ) .
D' For contrast, notice the following:
-
FK(V+(A(x)))
+K(VS(x))
If t V s K ( A ( 5 ) ) then t K ( V g ( x ) ) That is, if it is knowable (or provable) that each instance of
is
knowable, then it is knowable (or provable) that &holds universally. 4.3.
Negation.
In L four types of negation can be formulated.
Classical negation
+, of course, simply anrounts to the
The stronger K(-&
says that the falsehood of
,K(&)
falsehold of &. & is knowable; the weaker
says that & itself is not knowable. Finally, the "intuitionistic"
21
Epistemic and Intuitionistic Arithmetic
K(,K(A))
says that the unknowability of
A
is knowable.
The following
implications are easily derived in D.
CB5:
In the deductive system D', the above diagram represents the only derivable implications among the four negations. That is, neither -&+K(lK(A))
nor K(TK(5)) +$nor
any of
the four converses is generally derivable in D'. Proof:
It suffices to show that neither of the two implications
which involve both classical negation (7A) and intuitionistic negation (K(-,K(:)))
are derivable in D'.
ontic sentence that is not refuted by D'.
(i) Let g b e any false Then D'
]E and
t/
b/D,+C(g).It follows from TB that D , (7g+K(lK(g) 1 ) . (ii) A s noted after the proof of corollary CB1, there is a sentence 5 such that D' by TB, 4.4.
A(g/g)
[ 5 and bD,7K(g). Thus, kD,K(lK(g)) and,
bc,, (K(-/K(G))+-ts).
Substitution of Equivalents.
If
A, g,Cz are
fOnIiukiS, let
be the result of substituting C2 for some (or all) of the
occurrences of
2 as subformulas of A. In ordinary first-order logic, all
instances of the substitutivity of equivalents scheme,
(2-z)
(&(~lJg)ttA),
are logically true and, thus, are theorems.
the case in D.
Pre-formally, notice that the
equivalence of
c.1 and 2 is not
truth of the
+
This is not (material)
sufficient for these formulas to be
intersubstitutable & epistemic contexts.
To illustrate, it is shown that
unrestricted substitutivity of equivalents would entail I-A+K(A):
A A +--f
premise
(o=o) (;=El)
-t
(K(~=O)+-+K(A)
substitutivity of equivalents
K(Z=Z)) : C K * K (A) The following theorems are the correct counterparts of the substitutivity of equivalents theorem. T5: b T6:
K(C1-E)
+
-5) l-A(GClC2) -11
(A(%/=)
If C g + - + g then
Generally, two formulas are intersubstitutable only if their equivalence is knowable.
S. SHAPIRO
22
4.5.
Substitution
x=y+((A(z) ++&(XI),
of Identicals.
The substitutivity of identicals,
is another scheme that ordinarily represents a logical
truth, but can fail in epistemic contexts. Let b and c be two constants. Even if b=c is true, it does not follow that b and c are intersubstitutable. The problem is that the identity may not be known or knowable. To take an example from ordinary language, "the-number-of-planets = 3x3" and "Hegel knew that 3x3 = 9" are both true, yet one would not conclude "Hegel knew that the-number-of-planets = 9 " .
Modifying the above discussion, perhaps
the substitutivity of identicals should be replaced in D by
K(x=y) (A(& ++&(y) -+
.
In the present case, however, this modification is not necessary. The reason is that in elementary arithmetic there are no true, but unknowable identities.
) provable First, the sentence YEV~( ~ = y - + K ( ~ =) ~ is
in D (by induction on 5 and l). Second, notice that all terms of the present language L are constructed from numerals, names of primitive recursive functions, and variables.
Thus, in principle, one can determine
the numerical value of any term (given the values of the variables), and, therefore, the truth value of any identity is knowable. The expanded deductive systems of section 7 below, as well as virtually any epistemic analysis, set theory, etc. do have true, unknowable identities.
In such cases, restrictions on the substitution of identicals
are in order. 5.
A
Note on Semantics.
There is no attempt here to provide a
philosophically correct semantics for the language L.
That is to say, I
do not attempt to codify a justified collection of truth conditions for the sentences of L.
There are several well-known semantics for the notion of
"ideal knowledge", but none (that I am aware of) are appropriate for mathematical knowledge all for more or less the same reason.
I discuss
here the work of Hintikka 171. Instead of the present single operator K, Hintikka has a class of operators
Y, in which
denotes a person. Because present concern is with
the ideal knowledge of a single person or community, the subscript is omitted. It is easily verified that if Hintikka's semantics is formulated on the present language L, then D is sound for it.
I suggest, however, that
the semantics has serious philosophical shortcomings in the context of arithmetic. The major technical artifacts of Hintikka's work are the "epistemic alternatives", possible worlds relative to a given subject's
Epistemic and Intuitionistic Arithmetic
knowledge. Let
5 be
23
any sentence such that both 2 and 75 are unknowable.
In Hintikka's semantics, this would amount to the subject having two epistemic alternatives, one in which
5 is true and one in which 5 is false.
Of course, at least one of these would have to be a non-standard model of arithmetic.
That is to say, in one of these alternatives, "the natural
numbers" would not denote a structure isomrphic with the natural numbers. An epistemic alternative, however, is supposed to be a world
consistent with present knowledge--a world which "might be the case for all the subject knows".
I submit that a non-standard model is not the sort of
thing that can constitute an epistemic alternative.
In arithmetic, all
that is unknown (or unknowable) are certain facts =the
natural
numbers, the very structure of the natural numbers &known.
Mareover, the
structure of the non-standard models is also known, along with the properties of these models that make them non-standard.
That is, it is
known why the non-standard models are not (isomorphic to) the natural numbers.
In short, my thesis is that non-standard models of arithmetic
are not possible given present knowledge and, thus, that they are not adequate candidates for epistemic alternatives. Of course, one might attempt to overcome this problem by using a second-order language or simply requiring that epistemic alternatives contain only standard models.
This would make the present semantics
useless, however, because under either condition, every ontic truth of arithmetic would be true-in-all-epistemic-alternatives and, thus, would be knowable. I suggest, in conclusion, that a semantics for the present language L (and its extensions) should allow the truth values of the ontic sentences to be fixed in advance by the mathematical structure under study, and should not rely on the possible truth values of such sentences in assigning truth conditions to sentences containing the epistemic operator. 6.
Intuitionism. Intuitionism is a philosophy of mathematics which,
in effect, denies the "ontic" aspects of mathematical practice in favor of what is called "the constructive". Concerning ontology, the intuitionist conceives of, say, the natural numbers as the result of a mental act of construction and thereby denies that these numbers have an existence independent of the mathematician.
Accordingly, theorems of arithmetic do
not represent objective facts about independently existing mathematical entities, but, rather, the results of construction. Concerning semantics, the intuitionist Heyting [61 once said that to explain a formula, one does
S. SHAPIRO
24
not give its
truth conditions, but
rather its proof conditions.
In
practice, the intuitionist rejects those classical laws and inferences which, when interpreted, are taken to be incompatible with intuitionistic philosophy. Notable among these are the law of excluded middle, &v-&, which is interpreted by the intuitionist as "either the construction corresponding to &has been effected or the construction corresponding to the refutation of & has been effected". Another example is the quantifier exchange + e - + 3 5 q A , property
which is interpreted as "if not all numbers have a
5, then one can construct a number that lacks i".
In recent years, several non-intuitionists have suggested that the logic of intuitionistic mathematics can be interpreted as a logic of justification or an epistemic 10gic.~ To a classical mathematician, then, intuitionistic mathematics might be understood as a "pure" epistemic mathematics--a mathematics with no non-trivial ontic component. In this section, the proposal is born out by a translation of the language of intuitionistic arithmetic into the present language L.
Intuitionistic
arithmetic is interpreted in L as the arithmetic of "purely" epistemic formulas. 6.1.
Translations. The following are taken to be intuitionistic
connectives and quantifiers:
2,
5, y, 2, T,t,?.
To paraphrase
Heyting [61, the meanings of these are:
-A&B :
AvB : A*:
A * : -., ":
Vz(x):
B. I can prove & or I can prove B, and I know which. I can prove &and I can prove
I have a method which, if given a proof of A, produces a proof of
B.
5
amounts to
(B+&).
amounts to ~ ;6=i. t I have a method which, if given a number 1, produces a proof of
A@.
3xA(x) : I know of a number 1 such that I can prove &(El. --
It is clear that the notion of provability in use here is pre-formal and does not refer to a fixed deductive
Moreover, intuitionistic
proof seems to be ideal in the sense that no finite bound is placed on the length of a proof.
Also,
"provability" here is at least prima facie
iterative. The intuitionistic "I can prove
If",
then, appears to be
clearly allied to the intended interpretation of the present KC;).
The
following translation of the intuitionistic connectives and quantifiers into L is proposed:
7
Epistemic and Intuitionistic Arithmetic
25
the clause "and I know which" has no translation. However, see CB2. this amounts to "it is knowable that knowledge of
(materially) implies
knowledge of
El'. An exact translation
is not possible. this amounts to "it is knowable that knowledge of
is impossible".
again, an exact translation is not possible. see CB3. Notice that if 4 is any formula in the range of the translation (i.e.,
--
ALB, AvB, etc.) then b+K($).
The following scheme for translating
intuitionistic formulas into L is proposed.
- be
For each ontic formula g , let
its intuitionistic translation in L. If
A
-
is atomic, then
ALB: a
-
AvB -:
-
A*:-
A+B
7
-
SrXA: -
>V
15:
3g: We say a formula
4 is K[&).
-E9I -
A :
formula
--AvBALB
$!
3 2 is intuitionistic if B is provably equivalent to a
& for some ontic A.
6.2.
Interpretation
Meta-theorems.
6.2.1.
Deduction Theorem.
There is an interesting difference between the interpretation of classical deductions and the interpretation of intuitionistic deductions.
In the
former, each line is only stated, but in the latter each line is asserted. Suppose, for example, that in the course of a classical deduction, a formula A is introduced as a premise or assumption. read "assume 5'' or, perhaps, "assume A is true".
That line might be
In an intuitionistic
deduction, a similar line would better read "assume A is known" or "assume
A is provable".
The difference is brought out in a single case of arrow
introduction (or deduction theorem) :
from
g,
infer b&+g. Formally,
the rule (or theorem) holds in both contexts, but the interpretation is
26
S. SHAF'IRO
different. Roughly, in classical deductive systems, the rule is taken to mean that if one can derive g after assuming A, then, in effect, one can derive "if & is true, then B is true".
In intuitionistic deductive systems,
on the other hand, the rule is taken to mean that if one can derive assuming that implies B ~-
--
g after
is provable, then one can infer that & intuitionistically
that "one can prove B if given a proof of
6".
To obtain this
conclusion, the assumption of & in the original deduction must be interpreted as involving the provability of that formula.
I conclude that
not only does each formula itself have a different meaning, but also that the
use of a
formula in a deduction has a different meaning in classical
and intuitionistic deductive systems. Thus, since the present language L and deductive system D is, after all, classical, one should not expect a general intuitionistic deduction theorem : from to hold.
r,i
I-
g
r t ++g
infer
A "counterexample" is readily obtained.
generally correct, but
A t E=E;t&
amounts to &
of course, does not generally hold.
A, E=E
KC&) or
1- -A is, of course, t +(&I, which,
The following intuitionistic
deduction theorem is easily verified:
r , K(&) f- B r t 5%. if
TC:
and every formula inr is epistemic, then
Faithfulness of the translation. The soundness of the present
6.2.2.
deductive system for intuitionistic arithmetic is tedious, but straightforward: TD:
For any ontic
A, if 5
arithmetic, then The converse of TD
--
is a theorem of intuitionistic
@.
the completeness of D for intuitionistic arithmetic--
was recently proved by Nicolas Goodman 151. weaker results are presented here.
Some easily obtained, but
Notice, first, that it is easy to see
that many instances of intuitionistic excluded middle, Indeed, it follows from CB2 that
provable in D. or I-,K(&).
I-
&vs
zg,are not only if either I-
The next theorem and corollary concern the completeness of
subsystems of D. Let LD be the logic subsystem of D.
That is, let I1D contain every
axiom and rule of D except the axioms of arithmetic. TE:
For any ontic
A,
-
if kL&,
then &is a theorem of
intuitionistic predicate calculus. Proof outline: Familiarity with the Kripke semantics for both
Epistemic and Intuitionistic Arithmetic
21
the modal logic S4 and the intuitionistic predicate calculus is assumed (see [lo] and 1111).
Let a be a Kripke structure for
intuitionistic predicate calculus. L, then a
If
fi is
an
ontic formula of
Il-A is taken as "& is satisfied by a viewed as a
Kripke structure for intuitionism". If then a f=B is taken as structure for S4".
"g is
B
is any formula of L,
satisfied by a viewed as a Kripke
TE follows from the following lemmas.
L6 is
due to Kripke.
g
is a theorem of ED, then a
kg.
L4:
If
L5:
If A is any ontic formula of L, then a
L6:
If a l h & for every Kripke structure a for intuitionism, then
8 f=ij iff a IF fi.
A is a theorem of intuitionistic predicate calculus. Let D- be the deductive system consisting of all the axioms and rules of D except the non-intuitionistic instances of the induction scheme. CE:
For any ontic
A,
if
kD-5,
then & is a theorem of
intuitionistic arithmetic. Proof:
It is easily verified that all the axioms and rules of D-
are satisfied by every Kripke structure for intuitionistic arithmetic.
The corollary follows from L5 and the appropriate
counterpart to L6. 6.3.
There are some interesting theorems of D which relate the
intuitionistic connectives and quantifiers to their classical counterparts. The proofs are straightforward. T?: T8:
T9:
(A&B) -+ (A&B) K(&&g) cf f;+ (A€iE)
28
S. SHAPIRO
T18:
(T&)
2 (&I
Theorems T9 and T12 indicate that classical conjunctions and classical universal quantifications are "intuitionisticly equivalent" to their intuitionistic counterparts.
That is, if an instance of one of them is
knowable, then the corresponding instance of the other is knowable.
It is
an easily verified corollary of TB that the converses of T7 and T10 do not generally hold. Theorems T13-Tl6 indicate that intuitionistic disjunctions and existential quantifications are at least as strong as their classical counterparts. Again, none of the converses hold (see, for example, CB2 and CB3).
Theorem T17 asserts that if the classical implication &*-is then (it is knowable that) if
A
is knowable, then
g
knowable,
is knowable.
This
suggests that a classical implication is "intuitionisticly no weaker than" the corresponding intuitionistic implication. Notice that it is a corollary of TB that the similar The latter formula amounts to "if knowable." 6.4.
(&+g)+ &+g is
does not hold generally in D. true and
A
is knowable, then
g
is
The converses are taken up in the next subsection. It is proposed that the present language and deductive system
is useful in understanding the differences between intuitionistic formulas and their classical counterparts. Several examples follow. 6.4.1.
In informal discourse it is often remarked that any formula
with an intuitionistic main connective or quantifier is at least as strong as the similar formula with a classical main connective or quantifier. At least in D, however, this is not the case. example, that
It is a corollary of TB, for
(e) 2 @+g), the converse of T17, is not derivable in D.
illustrate this, let
A(x) be ontic and
formula amounts to "if
notice that
/-&(z) ~ZX&(Z).
A(S) is knowable, then there is a number
A(x) - is knowable". It follows from this and The latter formula amounts to A(5) +@(E). true, then there is a number 5 such that
To
This such that
the converse of T17 that the implausible "if
&(x) is knowable".
&(s)is
similar
remarks apply to negation and the converse of T17. It is easily verified that if & is ontic and kg then
CA.
It may be
thought that each intuitionistic formula is no weaker than its classical counterpart. Another corollary of TB, however, indicates that this also is not the case.
Epistemic and Intuitionistic Arithmetic
CB6:
There is an ontic
Proof:
Let
29
such that/&g.
C(5) be a primitive recursive predicate such that Let g be - j V g ( ~ ) . The
V g ( r ) is true but not deducible.
sentence
is equivalent to K(,K(VS(tl)
)
.
Let D' be the
deductive system formed by adding this sentence to D as an axiom. Notice first that the result of erasing all occurrences of K from a theorem of D' is a theorem of the system consisting of Peano arithmetic plus
B.
It follows from the consistency of the latter
system that D' is consistent. Notice also that D' 1 therefore, that D' satisfies the premises of TB.
fD,E+g.
Therefore, by TB,
A
and,
B.
By L3 D'
fortiori,/&g.
It is well-known that the intuitionistic connectives and
6.4.2.
quantifiers are not interdefinable the same way their classical counterparts are. For example, although classically we have l-(-$vg) I-(&+€€)
c+l(A&,B),
++
(&-+El and
neither inference holds in intuitionistic logic.
Although the two schemes
(e) 2 (&+I and
intuitionistic logic, neither converse is.
(%+I
;t
~(32 are)provable
in
The differences are illustrated
The following are simplifications of the translations:
in L and D.
7AVB:
K(-iK(A)) v K(B)
A+B -7(A&7B): -_T_
K(K(&) -+K(E))
:
K(,(K(&)
&
K(7K(B))))
From T6 again, the following equivalences are derivable t+K(K(iK(&))VK(B))
(A) VK
c+ K (TK
)
-, 7(A&7B) ++ K(7K(&)V,K(-jK(B) 4 -
1)
Temporarily ignoring the o u t e m s t K in each formula, then, the implication
e g
amunts to "either A is unknowable or B is knowable".
-
The disjunction
YAvB amounts to the stronger "either the unknowability of 5
& knowable
or
B is knowable". The negated conjunction z(&&A&g) amounts to the weaker "either A is unknowable or it is not knowable that 6.5.
decidable". be known.
g
is not knowable".
Decidability. We list three possible interpretations of (1) &vz&--either
"A is
& is knowable or it is knowable that 5 can't
Because the truth value of &is not directly referred to here,
5 is 5 is true, then 5 is knowable; if & is false, then it is knowable that 5 is
this is called intuitionistic decidability. knowable or it is knowable that false.
A
is false.
(2) K(fi)vK(7A_)--either
(3)
(~K(A))Q(,(,A))--if
30
S. SHAPIRO
Fsrmula ( 2 ) is equivalent to Av~& and, therefore, implies formula (1). The converse does not hold generally.
It follows (from classical excluded
middle) that formulas ( 2 ) and ( 3 ) are equivalent. Markov's principle, which has caused much debate a m n g intuitionists, is the scheme 2%
(~V-I(~)Z 3 2 ( 5 ) ) . It asserts that if A is intuitionisticly decidable and if it is knowable that & does not universally k
fail to hold, then there is a number 5 such that t ( 5 ) is knowable. Informally, the number 5 is found by checking & ( G I ,
&(I),.
..
possible because & is decidable) until one is found that holds.
(which is It is
well-known that Markov's principal is not derivable in, but is consistent with intuitionistic arithmetic. There is a theorem of D which is a somewhat altered (and weakened) version of Markov's principle. k 3 & . & is strongly decidable and if the extension of & is (knowable to be) not empty, then there is a number 5 such that A(&) is
T19:
K(A)vK(@
T19 says that if
knowable. Notice that both the premise and the antecedent of Markov's principle are somewhat strengthened here (thus weakening the principle). First, the premise of decidability is strengthened and, second, under this premise, the antecedent of T19 entails that of Markov's principle. to say, K(A)vK(lE) I6.6.
That is
3=9V&.
Church's Thesis.
There is a formula scheme of intuitionistic
arithmetic that is sometimes thought to be an analogue of Church's thesis: CT:
v&(x,x).+
3 e v d Y "Tl(g,5,@)(x,U(y)) 1
For each intuitionistic formula
A,
on Heyting's interpretation of the
connectives, CT amounts to "if one has a method which given a number fi produces a number fi such that
&(i,i)is proved, then one knows of a is proved." &(&,r(&)
recursive function g such that, for each g,
It is well-known that the deductive system consisting of intuitionistic arithmetic together with all instances of CT is consistent. It follows from the faithfulness of the translation [ 5 1 that all instances of CT can be consistently added to D. Church's thesis itself says that if there is an algorithm which, given
m, produces f such -
such that
&(i,i)is true, then there is a recursive function
that, for each 2, A ( & G )
is true.
Stated this way, the existential
quantifier in "there is a recursive function" is classical
--
Church's
thesis does not entail that a name of the recursive function is known. Moreover, the restriction to intuitionistic formulas is artificial.
The
Epistemic and Intuitionistic Arithmetic
31
following formula scheme is a weaker version of CT which is closer to Church's thesis in these respects:
zlfi(&,x)-t3~Vx311(Tl(e,x.y)&A_(2,U(~)L)))
CT1: For each formula
of L.
1 It is tedious, but not difficult to verify that ICT It follows that 1 no contradiction can be derived in D from CT As above, however, this
.
.
does not entail that it is consistent to add every instance of CT1 to D. The consistency of CT1 with D is proved in R. Flagg's contribution to this volume. 7.
Second and Higher-Order Extensions.
In this section, the language
and deductive system are extended to include second and higher-order variables and quantifiers. We call the extended language L2 and the Several changes in the deductive system are
extended deductive system D2.
indicated by some problems that do not arise in the context of first-order arithmetic.
The new problems are technical variants of those in
epistemology and philosophy of language. 7.1.
Functions
Function Presentations. There are some well-known
difficulties that result when function names occur in epistemic contexts. To pursue an earlier discussion (section 4 . 5 ) , substitutivity of identicals fails.
it is often remarked that
For example, if r a n d
C J
are the same
function and Harry knows that f.is computable, it does not follow that Harry knows that
C J
is computable. The relevant fact here, of course, is
that Harry may not know that
g
and
C J
are the same function. If there are
two function names which denote the same function but which cannot be known
to denote the
same function, then this problem will occur even in the
present context of ideal knowledge.
This proves to be the case with the
extended language and deductive system. To overcome this difficulty, a distinction between number-theoretic functions and what may be called presentations of number-theoretic functions is in order.
For present purposes, a number-theoretic function
is a set of ordered pairs of natural numbers (containing no two pairs with the same first element).
A
presectation of a function is an interpreted
linguistic expression which denotes a function. For example,'Xx(x+l)2 ' and 1
'X5i;d2i+l)
' are two different presentations of the same function.'
In
general, each function has infinitely many presentations. Because functions are infinite abstract objects, human beings, as knowers, have no (epistemic) access to individual functions independent of their presentations. Modifying a proposal made by Frege, I suggest that
32
S. SHAPIRO
function names which occur in epistemic contexts be considered as denoting function presentations, and not functions themselves. To clarify this, consider the following sentence:
(1) Harry is given a function f and asked if he knows whether f is computable. Since Harry is a person, this can only mean: (1') Harry is given a function presentation 4 and asked if he
knows whether the function described by $ is computable. The connection between ( 1 ) and (1') holds simply because of the human dependence on language--no (non-physical) entity can be given to a person except with a verbal or written presentation.
More can be said about
functions, however, because functions are infinite abstract objects.
To
bear out the distinction, we compare the situation of (1) and (l'), which involves functions (and function presentations), to a similar situation involving numbers (and what may be called number presentations).
Consider
the following: (2)
Harry is given a nuntber
and asked if he knows whether
n is prime. Again, since humans have no access to individual numbers independent of language, this sentence must mean: (2') Harry is given a number
the number denoted by
5 and asked if he knows whether is prime.
There is, however, a clear epistemic difference between numerals and function presentations.
Numerals, of course, are standard canonical names
for individual numbers.
This alone, however, does not go to the heart of
the matter.
The point is that numerals are "transparent", in the sense that
when one is given a numeral in a standard notation (that is understood), one knows what number is denoted--at least in the sense that if one were given another numeral (in the same or another understood notation) one can tell (at least in principle) whether the numbers presented are the same or different and, if different, which number is larger. Function presentations, on the other hand, are not like this.
In the first place,
there is no canonical notation for functions and, even if there were, there is no uniform way of comparing the whole of the function presented by one presentation with the whole of the function presented by another (for example, to check if the functions are identical or if one eventually dominates the other)., Even if one begins with two effective presentations, the "comparison" is an infinite process.
Epistemic and Intuitionistic Arithmetic
33
To carry this further, consider the following sentences (la) Harry is given 'kz(~+l)~'and asked if he knows whether the function so described is computable. (lb) Harry is given 'ki&A+l)
and asked if he knows whether
I
the function so described is computable. (2a) Harry is given '5' and asked if he knows whether the number so denoted is prime. (2b) Harry is given 'the number of different Platonic solids' and asked if he knows whether the number so described is prime. The situations described by (la) and (lb) are instances of (1').
We
believe standard usage has both situations as instances of (1) as well. The situation described by (2a) is literally an instance of (2') and, we suggest, an instance of (2).
The situation described by (2b). however, is
not an instance of (2'), because 'the number of different Platonic solids' The situation is also not an instance of (2) (on the
is not a numeral.
usual meaning of that sentence).
In this case, one might say that Harry
is not given a number, but only a description of a number.
Of course, one
can back up and claim that in the situations described by (la) and (lb) Harry is not given a function, but only a description of a function. agree with this.
I
In the case of functions, however, function presentations
are the best one can do. The question for Harry depicted in (2) is, in some sense, independent of the numerals because numerals are canonical and transparent.
Two
different instances of (2') which contain different numbers for the same number (in different understood notations) would constitute the question for Harry.
same
The question depicted in (l), however, is not
independent of the function presentation.
Indeed, the two instances of (1)
depicted in (la) and (lb) represent different mathematical questions even though the function involved in each is the same. them would not do as an answer to the other.
An answer to one of
This is especially true if
Harry does not know the mathematical fact that the two presentations describe the same function. As above, the problem with function names in epistemic contexts Will
be carried over to the context of ideal knowledge if there are different presentations which describe the same function, but which can't be known to describe the same function.
For the present extended language, this
condition is equivalent to the existence of true but unknowable
34
S. SHAPIRO
propositions. Therefore, the first extensions of L and D contain terminology for function presentations. A later section concerns further extensions of L2 and D2 which contain terminology for functions themselves as well as terminology for other higher-order entities. For the expanded language LZ, then, a new second-order sort,
7.2.
called "function presentations", is added to L .
Capital letters from the
middle of the alphabet are used as variables ranging over function presentations.
In this subsection we discuss the changes from the previous
system which accomodate the new terminology. The first subsection 7.2.1 deals with additions to L and new axioms for D. while 7.2.2
deals with
actual changes required by the higher-order terminology. 7.2.1.
Formally, function presentations are, in some ways, like
functions. The first new axiom of D2 is a standard function axiom: F1: w z 3 ! y ( e = y ) . To avoid unnecessary complication, concern here is only with total function presentations. An important aspect of function presentations (unlike functions themselves) is that they are not extensional. That is to say, different presentations can have the same values at the same arguments, or, in other words, different presentations can describe the same function. To avoid a troublesome (and fruitless) technical problem, we do not introduce terminology for the identity of function presentations.
Indeed, there seem
to be no obvious criteria for determining whether two presentations are the "same" or, in other words, whether two presentations describe the same function the same way.
For present purposes, it suffices to introduce an
abbreviation to express extensional equivalence: abbreviation of Vx(Fx=Gx)
.
Ezc is taken as an
For the second addition to D2, an axiom scheme, let
&(z,x)be
any
ontic formula which has no second-order terminology. F2: V53!*(3y)
-+
33e(3pX)
This scheme entails that there is a function presentation corresponding to each "description" of a function. The introduction of an abstraction variable-binding-term-operator is thereby justified. If formula which has no second-order terminology and
&(z,x)is an ontic
xIz free, we
let
[ e ( f , 4 1 ) ] be a function presentation. In unambigious contexts, &(i,x)is abbreviated as & and [*(5,2)] added :
as 151.
The following axiom scheme is
35
Epistemic and Intuitionistic Arithmetic
(&(vpw) &‘Jz-($(~,E)- + E ~ E)v(~=~&V~Z(-&(~,~) ) ))
[ e ( z , xIF)-++
F3:
c
In more graphic language, this is equivalent to the least
rgv =
such that
$(v,w) is true,
if there is such a
w
0 otherwise
The restrictions on
5 in
F2 and F3 are, at least in part, for technical
and conceptual simplicity--it should be clear that there is a determinate
[A].
function described by
Moreover, the restrictions also facilitate the
extension of the “provability“ interpretation of K to the extended language. In section 7.4 the possibility of relaxing these restrictions is considered. In the first-order D, the axiom scheme of the substitutivity of
7.2.2.
identicals (for numbers) is
sr: (z=g+ (&(x)t t A ( y )
.
As noted above, substitutivity of identicals often does not hold in
epistemic contexts.
The problem is that true, but unknown (or unknowable)
identities cannot be inter-substituted in epistemic contexts.
In the
first-order case, however, there is no problem since in that case, there are no true, unknowable identities (at least not in L ) .
In the present
second-order L2, on the other hand, there are true, unknowable (numerical] identities.
Moreover, unrestricted substitutivity of identicals would
A-+K(A) for any first-order ontic formula 5. Indeed, let A be such a is free in A), let B be (z=z) & formula and (assuming that neither x nor (l=i)& A. Consider the following deduction: imply
A
premise
[BI5 =i K(i=i)
from axiom F3
c1. K(
[g]a=i)unrestricted Si
K ( [gG=i) K (5)
from axiom F3
It would seem natural at this point to maintain only those instances of SI in which
A (x)
x does not occur free within the scope of a K-operator in
and to add an “epistemic version“ of SI:
K(x=P)
-+
(A(x)++A(x))
This approach, however, fails because, as noted, VzVx(x=y+K(z=l)) provable in D.
is
This sentence, together with the epistemic S1, implies the
original unrestricted Sl. The present dilemma can be resolved by noting that, for similar reasons, D2 should also contain restrictions on the terms that can be interchanged with variables in epistemic contexts. Unlike the second-order
S.SHAF'IRO
36
variables (for function presentations), the first-order variables in L2 range over numbers and not what may be called "number presentations". A free number variable that occurs within the scope of a K-operator denotes an unspecified number rather than an unspecified number description.
Therefore, I propose that in an application of (first-order) universal instantiation or existential generalization, such a variable can be exchanged with a term t only if the denotation of t is knowable--only if RecaLl that there is a number ; such that it is knowable that t denotes the terms of the first-order L have this property: if s is a closed term
.;
of L, then there is a numeral
such that t-K(s=;).
Although it might seem
(at first glance) a bit drastic, the following quantifier rules are proposed : YE:
vs(s)ki(s) V%(x)fi&(t)
where s is a term of L where t is a term of L2 and 5 does not occur free within the scope of a K-operator
31: &(s)l-3g(x)where s is a term of L
A(t)f-3%(5)
where t is a term of L2 and ~f. does not occur free within the scope of a K-operator
To
L.
A(x) -
illustrate the change, let t be a term of L2 that is not a term of
That is, let t be a term which contains abstraction operators. Let be an ontic formula. The restriction of 31 may prevent
K(A(t) )CggK(&(s)).
Indeed, if A(t) is knowable, it does not follow
(automatically)that there is a number x such that &(s)is knowable, because it may not be knowable which number t denotes. However, we do have 1 . This inference only amounts to "if &(t) is knowable K(A(t))cK(3=(x) then it is knowable that the extension of A(@ is not empty". Similarly,
we do not generally have Vs(&(s))l-K(&(t)). that for each number n,
The formula Vg(&(x))
is knowable. That is, Vg(&(x))
says
entails that
each instance of & is knowable (perhaps separately), not that it is knowable that &holds universally. Therefore, if it is not knowable which number is denoted by t, then it cannot be inferred that A(t) is knowable. We do, however, have Vs(&(x))C&(t).
That is, it does follow from
VgC(A(5)) that A(t) is true.
We also have K(V=(x))kK(&(t)): if it is is knowable. knowable that &holds universally, then A(:) To return to the substitution of identicals, let s and t be two terms
of L2. with the restrictions on the quantifier rules, it is easily seen is directly that an instance of the formula scheme (s=t)+(A(s)*A(t))
Epistemic and Intuitionistic Arithmetic
derivable from SI only if either free in
s
and t are terms of L or 5 does not occur
&(x) within the scope of a K-operator.
restrictive.
31
As noted, this may be too
Intuitively, if s and t are known to denote the same number,
then (even if it is not knowable which number that is) they should be substitutable.
All is well, however.
It is straightforward (but tedious)
to see that SI and the quantifier rules imply an epistemic substitutivity scheme: KSI:
K(s=t)
3
(A(s) MA(t))
is a theorem for any (numerical) terms s,t and any formula
A
(provided that
no variables free in s or t occur in &I. It can also be seen that the above restrictions on the quantifier rules are not too harsh.
If t is a term of L2 (not containing 5 free),
then the formula 3lf(t=x) amounts to "there is a number x whose identify with t is knowable" (especially in light of the extension of CB3).
This is
the above informal condition for substitution (and corresponds to a similar requirement in [71). As might be expected, with KSI, VE, and31, the following are derivable in D2: Sg(t=x)
I
&(t) t 3 * ( y
t
axI
K(E=S)I7.3.
++
Theorems.
)
-
In this section, the provability interpretation is
extended to the present language L2 and deductive system D2. we define a property
]A of
sentences
5 of
To this end,
L2 which is an extension of the
property in section 3 . 2 used in the proof of TJ3.
It is straightforward to
extend this property to a relation between extensions of D2 and sentences of L . First,
IA is defined as in
3 . 2 for the cases in which
is an ontic
sentence of L. From this denotations to the closed terms of L2 are assigned.
Proceed by recursion on the complexity of such terms. The only
(even slightly) non-trivial case is that of assigning a denotation to [ ~ ( x , ~ ) ]when t the denotation of t has previously been assigned. For this, let t denote
m.
The restrictions on the abstraction operator entail
that Amust be an ontic formula of L with only x,y free. number
such that
IA(m,i),
then the denotation of IAlt is the smallest
such number 1. If there is no such number is 0.
If there is a
n,
then the denotation of @It
38
S. SHAPIRO
Returning to the extension of ( A , if recursive relation and if tl,
q,.
E is an ;-place
. . , -”_ m respectively, then IE(tl ,...,tn) iff
Finally,
I&
primitive
. . . , t- are closed terms of L2 denoting lg(Gl,--., 6 ). -”_
The only new
is extended to all sentences of L? as in 3.2.
IE(A(F))iff (A([xyB(x,x) I ) for all abstraction operators [El i and ]JF(&(g) ) iff (&([ e ( z , xI ) ) for some [El.
clauses in the definition are:
It is straightforward, but tedious, to extend the proof of TB to D2. Moreover, corollaries CB1-CB5 (of sections 3 and 4 ) also hold for D2. Corollary CB3 has an interesting extension: CB3a: 7.4.
If l-gFK(&(E)) then there is an abstraction operator
[El such that(!-I [ E l ) . The Restrictions 0” Abstraction. Recall that axioms F2 and F3
for function presentations (as well as the formation rule for abstraction) are restricted to ontic formulas with no second-order terminology. The possibility of relaxing these restrictions is now taken up. First, the restriction to ontic formulas is not necessary for TB and its corollaries.
That is to say, if both F2 and the formation rule for
abstraction are modified to allow occurrences of K in the formula involved (but no second-order terminology), then the definition of
15 and the proofs
of TB and corollaries remain intact. Second, if the restriction on epistemic operators is maintained, then it is possible to eliminate the restriction on second-order terminology. In this case, the methods of 1131 can be used to prove TB and corollaries CB1-CB5 in the expanded system. Corollary CB3a, however, is not a corollary of this version of T B .
It is an open problem to determine
whether CB3a holds under these circumstances. Finally, it is also an open problem to determine whether the corollaries of T B hold in deductive systems in which both restrictions are eliminated. 7.5.
A
Note on Further Extensions. The present system can be further
extended to include (additional) second-order variables and quantifiers which range over functions themselves. Following the discussion of section 7.1, however, function variables should not occur free in epistemic contexts. Therefore, in the re-expanded system it is stipulated that K(&) is not well-formed if & contains free variables for functions themselves. The stipulation, of course, applies throughout the recursive definition of well-formedness. For example, if K(&(X))
A(&)
has function variable 5 free, then
is ill-formed and, hence, so is 35K(&(E)). Indeed, the latter
Epistemic and Intuitionistic Arithmetic
formula entails "there is a function
5 such
39
that something about
x is
knowable". As above, nothing can be known about a function independent of a particular presentation of it.
It should be pointed out, however, that
K ( a E ( 5 ) ) is well-informed (as long as 5 does not occur in an epistemic context in A). In this case, the only function variable in 5 is bound. The latter formula entails only "it is knowable that the extension of a certain function property is not empty". Our language and deductive system can be extended even further to include terminology for sets of numbers and other higher-order entities. Again, however, terminology for, say, sets must be distinguished from terminology for set presentations. Only the latter is permitted to occur free in epistemic contexts. Myhill 1131 presents a different kind of extension of the present work to axiomatic set theory.
The ontic portion of his language and deductive
system is Zermelo-Fraenkel set theory, rather than the type theory suggested here.
Although Myhill does distinguish terminology for
"explicitely given hereditarily finite sets" from terminology for sets in general, he does not distinguish sets from set presentations. Either kind of variable can occur free in an epistemic context. On one hand, because his deductive system contains the axiom of extensionality, it seems that set variables and set names represent sets themselves when they don't occur
in epistemic contexts.
On the other hand, because, for example,
substitutivity of identicals fails in epistemic contexts, it seems that set variables and set names that occur in such contexts represent set intensions or
set presentations.
This, of course, is consistent with the
celebrated Fregean principle that the same name (or description) may have different denotations when it occurs in different contexts.
In particular,
when a name or description occurs in an epistemic context, then, according to Frege, it denotes its ordinary "sense" or its intension. Formal languages and deductive systems, however, are designed in part to eliminate the amhiguities of ordinary language. Accordingly, each denoting expression of an interpreted formal language should have a unique denotation.
We suggest, therefore, that the Fregean ambiguity is not
appropriate to the precision of formal languages and deductive systems. Myhill's work clearly demonstrates that in the case of set theory there are no technical problems with identifying terminology for set presentations which occur in non-epistemic contexts with corresponding terminology for
S. SHAPIRO
40 sets themselves.
This, however, might better be pointed out after analysis 10 and not set out in advance. 8. Applications.
8.1.
If the analysis of intuitionism in section 6
and the subsequent translation are plausible, then statements of arithmetic constructively can be expressed in the present languages L and L2. Because these languages also contain ontic (and mixed) components, however, the formalization of intuitionism takes place in a larger non-constructive context.
I submit, therefore, that the present language is more expressive
than either the usual language of intuitionism or the usual language of classical Peano arithmetic.
Indeed, the language of intuitionism conforms
to intuitionistic philosophy and, therefore, can express only purely constructive (or purely epistemic) statements; the language of classical Peano arithmetic, on the other hand, can express only ontic statements. A cursory survey of mathematical practice shows that, contrary to both of these, the pre-formalized language of mathematics contains both ontic and epistemic, and, moreover, even mixed terminology. this volume.)
(See the Introduction to
If this is correct, then the present deductive system is a
better formalization of mathematical practice than either classical arithmetic or intuitionistic arithmetic. To illustrate the combined expressive power of the present system, let
_A(x) _
be an ontic formula, with one free number variable, which represents a
property
of numbers.
a particular number the property
Suppose that a mathematician shows how to calculate
and goes on to prove non-constructively that
has
p. Because the latter proof is non-constructive, the result
cannot be expressed (directly) in the language of intuitionism. In the language of Peano arithmetic one can, of course, assertSe(5) but this amounts to only "the extension of p is not empty".
In the case at hand,
the mathematician has established m r e than this.
In particular, it has
been shown how to calculate a number with property
p.
was shown that a number with property p is knowable.
In other words, it In the present
language, 3g(A(x)1. An example of this is provided by the Friedberg-Muchnik solution to Post's problem (see [141, pp. 163-166). expresses
Let
B(x) be the predicate which
''zis the G6del number of a Turing machine which
enumerates a set
(of natural numbers) to which the halting problem cannot be reduced". Post's problem was to determine whether g g ( 5 ) is true. Muchnik solution does more than this.
The Friedberg-
Although the proof is ultimately
non-constructive, a particular algorithm whose a d e l number satisfies
g
41
Epidemic and Intuitionistic Arithmetic
is exhibited. That is to say,
3 g ( B ( x ) ) is established
non-constructively. 8.2.
A consequence of this gap between the language of classical
deductive systems and the pre-formal language of mathematics is a systematic ambiguity in certain mathematical terms such as "computable", "decidable", "separable", "countable", "measurable", "definable", and "axiomatizable". On one hand, in standard formulations--in the established formal languages--these items designate objective properties of mathematical objects.
That is to say, the definitions of these objective
properties do not involve reference to a knowing subject.
In this
connotation, such terms designate properties which are internal to mathematics.
On the other hand, in practice these same terms are sometimes
used to indicate pragmatic properties--properties that mathematical objects have or lack in virtue of human abilities, achievements or knowledge, often idealized. For example, a collection C of objects may be said to be either countable objectively, because there exists a function whose domain is the natural numbers and whose image is
c; or coclntable pragmatically, because
"one can count the elements of C using natural nwnbers". By way of illustration, since every subset of the natural numbers is objectively countable, the set of adel numbers of, say, the truths of arithmetic, is countable. This does not mean that the set is countable-in-the-pragmaticsense.
As is well-known, there is no formula of arithmetic which can be
used to define a "counting function" of this set. For a second example, a collection of strings may be said to be decidable either because there exists an algorithm which represents the characteristic function of the collection, or because "one can decide of an arbitrary string whether it is, or is not, in the collection". The standard languages used in formalizing mathematics can only express the objective readings of the above terms. To take just one aspect, the pragmatic properties are clearly intensional, while formal mathematical languages are thoroughly extensional. One may thus yo so far as to say that such languages cannot express anything concerning a relationship between mathematical objects and even ideal subjects.
It
should be noted, however, that most of the terms in question have common, non-mathematical meanings which are pragmatic and subjective. Moreover, it seems reasonable that even in mathematics, the pre-formal use of the terms is pragmatic.
It seems that when such concepts are given
42
S. SHAPIRO
mathematical treatment--when they become part of the subject matter of mathematics--they get redefined as objective properties. It might be added that in practice, pragmatic senses of the terms in question often occur as part of heuristics--the terms are used pragmatically to facilitate discovery or communication of the corresponding objective properties.
For example, one can come to know that a set is
countable-in-the-objective-sense by showing that it is countable-in-thepragmatic-sense--by showing how to count it.
One can show that a function
is computable-in-the-objective-sense by showing that it is computable-inthe-pragmatic-sense--by showing how to compute it. Of course, it is also possible to come to know mathematical facts concerning the objective properties without involving the corresponding pragmatic notions.
In such cases, one often cannot claim to have obtained
pragmatic knowledge. That is, in such a case, one cannot directly claim to have learned anything about the limits of human capabilities. Suppose, for example, that a mathematician proves that a certain set is countable or decidable (in the objective sense) but does not show how to enumerate the members of the set or how to decide memberships in the set. By establishing the objective mathematical fact in question, the mathematician cannot claim to have established very much concerning human "counting" capabilities or human decision capabilities. I suggest that the present languages can express pragmatic senses of at least some of the "-able" terms.
I focus on computability. The term
computable is restricted here to its objective sense and the term calculable is employed for the pragmatic. Pre-formally, the objective computability is defined as follows:
f is computable iff there is an algorithm that represents
r.
The existential quantifier in the phrase, "there is an algorithm" expresses only the existence of an algorithm, not knowledge thereof. variable
Thus, the
ranges over functions themselves, and computability is a property
of functions. Accepting Church's thesis, it is formulated as follows: CCE)
tfdf3E'dE3Y(Tl(EtE,X)
&
-
fx=U(x))
It is easy to see that computability is extensional: CCg) and Yx(fx=gx) together imply
.
C (9)
The pragmatic counterpart to computability--here called calculability--directly involves human computation ability: calculable iff
F is can be calculated, or, in other words, iff an algorithm
for F can be known. With a little more detail:
Epistemic and Intuitionistic Arithmetic
-F is
43
calculable iff there is an algorithm p such that it can
be established that P represents F. In this definition, the variable
F
occurs within the scope of an epistemic
operator "it can be established that" which, I suggest, is equivalent to the present "it is knowable that". section, the variable
Following the discussion of the previous
should be regarded as ranging over function
presentations and not functions themselves. Thus, calculability is a property of presentations. The following formulation is proposed: CB (E) * d$EK(V&(T1
(e,x.x)6i E = U (1) 1) .
Notice that calculability is not extensional in that CB(E) and rsgdo not imply CB(5). The epistemic calculability is discussed more discursively in my 1151 (in which calculability is called "quasi-effectiveness").
It is argued
that calculability is epistemically more basic than computability and, therefore, that calculability is closer to at least some of the prerecursive-function-theory connotations of computability. Moreover, it is shown that there are many occurrences of the epistemic calculability in the writings of logicians and that consideration of the above distinctions can contribute to the clarification and, perhaps, solution of some philosophical problems, confusions, and disputes.
ACKNOWLEDGEMENTS: I would like to thank John Corcoran, Alan Hazen, Nicolas Goodman, John Kearns, John Myhill, and George Schumm for many helpful comments on previous versions of this work.
44
S. SHAPIRO
1.
Hintikka [71 presents an elaborate semantics for a similar ideal
knowledge in ordinary, non-mathematical contexts and discusses the relevant epistemic notion at length. ideally known by knowledge of
p
Two interpretations are presented:
if and only if
.A
is
is inconsistent with the present
b, and 5 is ideally known by 2 if and only if it is possible
for b to come to know
A without obtaining new
facts about the world.
Kearns [El provides a semantics for what he calls "justification". Although his 2.
''A is
justified" is metalinguistic, it is similar to our K(A).
The (E+Z)-place Kleene T-predicate Tn(e,zl,
. . . , x ,y) holds 3 -
L is the a d e l number of a complete computation from the collection of .,x . recursion equations with GUdel number 5, on n-place input zl, iff
..
If y is the a d e l number of a complete computation, then U ( x ) is the
-E
resulting output: if y is not the G6del number of a complete computation, then U ( x ) is zero. 3.
Myhill appears to have changed his position on this issue in the
time since 1121.
His paper in the present volume 1131, which extends the
present system to axiomatic set theory, takes issue with my insistence that K cannot be interpreted as "deducibility". In support, he presents an interpretation similar to that of the next subsection. I would suggest that this apparent disagreement depends on an ambiguity in "can be interpreted". knowability-at
I show below that at a given time &, the extension of
can indeed correspond to that of deducibility in a
particular system, but that this cannot be known (or knowable) at 4.
g.
Notice the similarity between this conclusion and that of some
other works which relate &dells
incompleteness theorems to such areas as
knowledge vis-a-vis deducibility (e.9.. [41 and [121)and computability vis-a-vis mechanism (e.g., [l])
.
5.
See, for example, 141, [El, and the Introduction to the present
volume.
It should be noted that the word "interpretation" should not be
taken in a strong sense.
"Reinterpretation" might be a better term
because, philosophically speaking, intuitionism and platonism are not easily reconciled.
The former asserts that the subject matter of, say,
arithmetic consists of mental construction while the latter holds that the subject matter of arithmetic is a realm of numbers which exists independently of the mathematician. 6. Brouwer, for example, insisted that intuitionistic mathematics
45
Epistemic and Intuitionistic Arithmetic
cannot be captured in a deductive system. Heyting himself regarded his axiomatization of intuitionistic arithmetic as an approximation that stands ready to be modified as future methods of intuitionistic proof are developed. Gdel [ 4 1 proposed a similar translation of the intuitionistic
7.
propositional connectives. We disagree only with his interpretation of
A-tB - as
K(&)+K@).
A materially
unknowable or 8.
The latter formula asserts only that the knowability of
implies the knowability of
g.
This amounts to "either
A
is
g is knowable".
Lemma L5 does not hold if the above translation of the
connectives is replaced by that of [41. 9.
Here presentations are taken to be abstract objects.
Presentations, perhaps, correspond to the m r e popular and, perhans, m r e vague "function intensions". 10. These criticisms are largely overcome in N. Goodman's
contribution to this volume.
S. SHAPIRO
46
REFERENCES [l] [2]
Benacerraf, P., "God, The Devil and Gdel", The Monist 51 9-32. Boolos, George, The Unprovability
Cambridge University Press, 1979.
(1963)
of Consistency, Cambridge,
L3]
Corcoran, J. and G . Weaver, "Logical Consequence in Modal Logic 11: Some Semantic Systems for S4", Notre Dame Journal of Formal Logic 15 (1974), 370-378.
[41
Gdel, K., "Eine Interpretation des Intuitionistischen Aussagenkalkus", Ergebnisse cines mathematischen Kolloquiums 4 (19321, 39-40.
151
Goodman, N. D., "Epistemic Arithmetic is a Conservative Extension of Intuitionistic Arithmetic", Journal of Symbolic Logic 49 (19841, 192203.
[6]
Heyting, A., "Die Formalen Regeln der Intuitionischen Logik", Sitzungsberichte Preuss. Akad. mliss. Phys. Math. g . , 1930, 42-56.
[71
Hintikka, J., Knowledge and Belief, Ithaca, Cornell University Press, 1962.
[81
Kearns, J., "Intuitionistic Logic, a Logic of Justification", 37 (19781, 243-260.
Studia Logica --[91
Kleen, S., "Disjunction and Existence Under Implication in Elementary Intuitionistic Formalism", Journal of Symbolic Logic 27 (1962), 11-18.
[lo] Kripke, S., "Semantical Analysis of M o d a l Logic I. Normal Modal mathematische Logik und Propositional Calculi", Zeitschrift Grundlagen der Mathematik 9 (19631, 67-96.
fiir
Ill1
"Semantical Analysis of Intuitionistic Logic I" in J. Crossley and M. Dunnnett (ed.) Formal Systems and Recursive Functions, Amsterdam, 1965, pp. 92-130.
1121
Myhill, J., "Some Remarks on the Notion of Proof", Journal of Philosophy 57 (1960), 461-471.
[131
"Intensional Set Theory", this volume.
[141 Rogers, H., Theory of Recursive Functions and Effective
Computability, New York, McGraw-Hill , 1 9 6 7 7
[IS1
Shapiro, S., "On the Notion of Effectiveness", History and Philosophy 209-230.
Of Loqic 1 (1980),
In tensional Mathematics S.Shapiro (Editor) @ Elsevier Science Publishers B. V. (North-Holland), 1985
47
INTENSIONAL SET THEORY John Myhill Department of Mathematics S t a t e University of New York a t Buffalo Buffalo, New York U.S.A.
INTRODUCTION
My motivation o r i g i n a l l y was t o p r e s e n t a system, c o n s e r v a t i v e l y extending c l a s s i c a l mathematics, i n which c e r t a i n d i s t i n c t i o n s u s u a l l y a s s o c i a t e d with i n t u i t i o n i s m could be made i n a manner understandable by t h e c l a s s i c a l mathematician.
For example, I wanted t o d e f i n e a construc-
t i v e (numerical) e x i s t e n t i a l q u a n t i f i e r ( 3 x ) i n such a way t h a t ( ~ , x ) $ ( x ) I
(closed) would be provable i f f $ ( E l was provable f o r some numeral E; a l s o t o express (but n o t n e c e s s a r i l y t o prove) Church's t h e s i s i n t h e form
(Vx) (ZIy)$(x,y)
-f
(gel (Vx) (By) ( $ ( x ,( U ( y ) ) A T ( e , x , y ) ) ; l i k e w i s e t o d e f i n e
a constructive d i s j u n c t i o n VI i n such a way t h a t $V J, ( c l o s e d ) i s provable I
i f f e i t h e r 41 i s provable o r JI i s ; with such a d i s j u n c t i o n w e could express (but n o t n e c e s s a r i l y r e f u t e ) Bishop's l i m i t e d p r i n c i p l e of omniscience
( P I , p.
9 ) by (vx) (31!y)4 ( x , y l
-+
( ( v x ) $(x,o)VI (&IT+
( x , o ) ) ; and so on. MY
b a s i c t e c h n i c a l i d e a came from Shapiro 171 who introduced a system of t h i s kind c o n s e r v a t i v e l y extending Peano a r i t h m e t i c ; however, our philosophy i s d i f f e r e n t from S h a p i r o ' s , and it i s n o n t r i v i a l t o extend t h e methods and i d e a s of I71 t o ZF. Shapiro showed i n [ 7 1 t h a t i f w e restrict o u r s e l v e s t o a r i t h m e t i c alone w e can add an o p e r a t o r
with t h e axioms f o r p r o v a b i l i t y given i n
Gddel [ 3 ] , i n such a way t h a t
( ~ ) B $ ( x )(closed)
provable f o r some numeral E, and t h a t either
+
o r $I i s .
B$
V $ !
i s provable i f f
$(n)
is
(closed) i s provable i f f
I t then immediately follows t h a t t h e d e f i n i t i o n s
s a t i s f y t h e c o n d i t i o n s above.
However, f o r t h e i n t e r p r e t a t i o n of h i s
B
J. MYHILL
48
Shapiro introduced a notion of " p r o v a b i l i t y i n p r i n c i p l e " and i n s i s t e d ( f o l lowing Gfldel [31 and Myhill [ 5 1 ) t h a t it could n o t be i n t e r p r e t e d as formal p r o v a b i l i t y i n t h e system i t s e l f .
W e s h a l l show t h a t t h i s i s n o t so, and
i n f a c t we s h a l l so i n t e r p r e t it f o r a system of s e t - t h e o r y extending ZF. So much f o r my o r i g i n a l motivation.
I n f a c t , however, I had n o t gone
very f a r along t h e s e l i n e s b e f o r e I modified my plan.
T h i s was because of
t h e observation t h a t t h e r e a r e r e a l l y two d i s t i n c t n o t i o n s of " e f f e c t i v e e x i s t e n c e proof" used i n informal classical mathematics ( c f . e.g.,Sierpins k i 181, p. 41).
If
c e rt ai n property
w e say we have e x p l i c i t l y c o n s t r u c t e d a set w i t h a a continuous nowhere-differentiable real func-
(e.g.,
t i o n ) w e mean merely t h a t w e have proved
f o r some formula $.
But i f we say t h a t w e have given an i n t e g e r e x p l i c i t l y
a bound on t h e s o l u t i o n s of a diophantine e q u a t i o n ) w e mean w e have
(e.g.,
given it i n some canonical (e.g.,
decimal) n o t a t i o n .
W e do n o t mean t h a t
w e have proved a formula of t h e form (1);by using t h e least-number p r i n c i p l e we can make every numerical e x i s t e n c e proof " c o n s t r u c t i v e " i n t h i s t r i v i a l sense.
( P a r t o f ) t h e e s s e n t i a l c o n t e n t o f , e.g.,
Sturm's theorem
f o r polynomials i s t h a t i f w e a r e e x p l i c i t l y given a polynomial p ( x ) and two ( f o r s i m p l i c i t y r a t i o n a l ) numbers rl and r2 which are n o t z e r o s o f p ,
w e can e x p l i c i t l y c a l c u l a t e t h e number of z e r o s of p between rl and r2.
We
want a formalism t h a t w i l l enable one t o say a s much without dragging i n r e c u r s i v e f u n c t i o n s o r going i n t o t h e metalanguage.
W e t h e r e f o r e introduce
two kinds of v a r i a b l e s , small ones ranging over e x p l i c i t l y l i s t e d h e r e d i t a r y f i n i t e sets ( i n c l u d i n g e x p l i c i t l y given non-negative i n t e g e r s ) and l a r g e ones ranging over a l l sets (however given).
W e proceed t o t h e formal
d e t a i l s of our system c a l l e d IST.
1.
x,y,z,...
The symbols i n c l u d e t h e two sets of v a r i a b l e s
Formation R u l e s . and X , Y , Z ,
... j u s t
mentioned, t h e p r i m i t i v e r e l a t i o n s = and
t h e c o n s t a n t s 75 and 3, t h e f u n c t i o n symbols
of function c o n s t a n t s f , g , h , recursive functions f : F
-+
{I
E,
and U and an i n f i n i t e supply
... c a l l e d r e c u r s i v e f u n c t i o n
symbols (denoting
F , where F i s t h e set of h e r e d i t a r i l y f i n i t e
sets ( c f . B a r w i s e 111, p. 4 6 ) 1, t h e a b s t r a c t i o n symbol
13,
u n i v e r s a l quan-
tifiers v x and VX, t h e c l a s s i c a l connectives 1 and A , and t h e modal opera-
tor
?.
In what follows we s h a l l use informally t h e symbols 3, 8 ! , +, etc.,
Intensional Set Theory
49
with t h e i r c l a s s i c a l d e f i n i t i o n s .
-
Terms of the first kind are small v a r i a b l e s , 0, and a l l e x p r e s s i o n s of t h e form { t l ) ,
(t, U t 2 )and ft,,
where tl and t
2
a r e terms of t h e f i r s t
kind and f i s a r e c u r s i v e f u n c t i o n symbol.
T e r m s of t h e second kind a r e l a r g e v a r i a b l e s , 3, and a l l expressions of t h e form { X l $ ( X ) } where $ ( X ) i s a formula n o t c o n t a i n i n g
?.
A t o m i c f o r m u l a s a r e expressions (t, = t ) and ( t E t,) where t l and
1
2
t2 a r e terms of e i t h e r kind. The o t h e r formulas a r e b u i l t up i n t h e s t a n d a r d way, with t h e addit i o n a l clause t h a t
2.
?$
i s a formula whenever $ is.
Axioms. 2.1.
The axioms f o r t h e c l a s s i c a l p r o p o s i t i o n a l c a l c u l u s .
2.2.
The axioms f o r
2.3.
2.2.1.
B$
2.2.2.
Ij)$
2.2.3.
54
-+ -+
5 $
F33$
-
A B($-+$)
-+
?$
The q u a n t i f i e r axioms 2.3.1.
( v x ) $ ( x ) -+ $ ( t ) , where t i s a term of t h e f i r s t kind
2.3.2. 2.4.
(VX)$(x)
$ ( t ) , where t i s a t e r m of e i t h e r kind
-+
The i d e n t i t y axioms 2.4.1.
a=a, a=8
2.4.2.
$ ( a )h a = @
8=a, a=B A B=y
-+
-+
+
a=y
$ ( 8 ) where 4 c o n t a i n s no
B
(such
formulas are henceforth c a l l e d c l a s s i c a l ( S h a p i r o ' s "ontic") )
2.4.1
-
2.4.4,
$(y) f o r a r b i t r a r y
0
2.4.3.
$ ( x ) Ax=y
2.4.4.
$(a) A B ( ~ = B ) + $ ( B ) a l s o f o r a r b i t r a r y 4 .
-+
In
as henceforth, a, 6, e t c . , r e p r e s e n t v a r i a b l e s of e i t h e r
kind. 2.5.
Axioms f o r h e r e d i t a r i l y f i n i t e sets
$5
2.5.1.
X
2.5.2.
X 6 { x } ++
2.5.3.
X E (xuy)
2.5.4.
$
X=x
$ ( x U y ) ) -+ $ ( z ) 2.6. proved
6 x V X
iX+
6)A ( V x ) ( $ ( X I
-+
$
E
y
( { X I 1) A
( v x y )($ ( x )A 4 ( Y )
again f o r a r b i t r a r y $.
Defining axioms f o r r e c u r s i v e f u n c t i o n s .
I f w e have
+
J. MYHILL
50
(vxy)( @ ( x , y )+ B@(x,y))where $ i s c l a s s i c a l and
2.6.3.
c o n t a i n s only x and y f r e e , w e may introduce a r e c u r s i v e f u n c t i o n symbol f by t h e axiom 2.6.4.
@(x,fx)
The axioms f o r s e t theory.
2.7.
These a r e t h o s e of Z F , w r i t t e n
with l a r g e v a r i a b l e s , except t h a t t h e axiom of i n f i n i t y i s replaced by
E
X
2.7.1.
3 +-+ (&) (X=x).
W e can i n c l u d e any state-
ment which holds i n some f i x e d model of ZF, so long a s we w r i t e it using only l a r g e v a r i a b l e s ,
E,
and =.
I n p a r t i c u l a r , we s h a l l include choice.
Extensionality i s written
(VX)(X E Y
2.7.2. N o axiom under 2.7.
++
x E
Z)
-f
Y=Z
s h a l l i n c l u d e anything b u t l a r g e v a r i a b l e s ,
E,
and =
except 2.7.1 and t h e i n s t a n c e s of replacement.
W e s h a l l admit a l l and only
t h o s e i n s t a n c e s of replacement which c o n t a i n no
5.
2.8.
The d e f i n i t i o n of
(VX)(X
E
Y
+-+
-
@(X)) * Y = IXl@(X)I
(VY)-r(VX)(XE Y
@(XI)
-t
IXl@(X)I =
0
where b i s c l a s s i c a l .
3.
Rules.
Modus ponens; t h e q u a n t i f i e r r u l e
where a i s n o t f r e e i n 4; and t h e r u l e
Remark.
The r e a d e r w i l l have n o t i c e d t h a t t h e r e are t h r e e s e p a r a t e
r e s t r i c t i o n s i n IST:
(1) t h e r e s t r i c t i o n on s u b s t i t u t i o n of e q u a l s ( t h r e e
s p e c i a l c a s e s of it, 2.4.2 a=B A @ ( a )-+
@(B));
-
2.4.4
i n s t e a d of t h e g e n e r a l schema
( 2 ) t h e r e s t r i c t i o n on 2.6
(@ required t o b e c l a s s i c a l ) ;
and (3) t h e r e s t r i c t i o n on set formation (2.8) and t h e r e s t r i c t i o n on replacement (2.7)
p).
(note t h a t { X ( @ ( x ) )is n o t even well-formed i f @ c o n t a i n s
W e s h a l l show i n 17 below t h a t t h e f i r s t and t h i r d of t h e s e r e s t r i c -
t i o n s are e s s e n t i a l i f w e are t o have t h e d e s i r a b l e p r o p e r t i e s t h a t
Intensional Set Theory
k
( c l o s e d ) i m p l i e s k $ ( t ) f o r some t e r m t
(=)]?Q(X)
FQ
51
V FJ,
(2)
FQ o r k$.
(closed) implies
(3)
however, w e do n o t know, and so we
With regard t o t h e r e s t r i c t i o n on 2.6,
Does the s y s t e m o b t a i n e d f r o m I S T by
pose a s our f i r s t open q u e s t i o n :
d r o p p i n g f r o m 2.6 the r e q u i r e m e n t t h a t Q be c l a s s i c a l , s t i l l e n j o y the
-
p r o p e r t i e s (2) 4.
(3)?
We c o n j e c t u r e an a f f i r m a t i v e answer.
The T r u t h Definition.
Pick a modelm of Z F , and form IST* from
m
IST by a d j o i n i n g names of a l l elements of
as terms of t h e second kind,
b u t without any s p e c i a l axioms f o r t h e s e new t e r m s .
+
IST
+ IST p l u s -
2.6.2
+
0
i s IST
IST
0
minus a l l r e c u r s i v e f u n c t i o n symbols, and IST
;+I is
a l l d e f i n i n g axioms 2.6.4
2.6.3
c l e a r what IST
+
+
follows:
n
+
i s t h e union of a chain {IST.} of systems d e f i n e d i n d u c t i v e l y a s
mk
a r e provable i n IST'
f o r which t h e "supporting theorems"
+
b u t n o t i n IST
m
i s sound f o r c l a s s i c a l formulas, i . e . ,
%?+ 9.
f o r any m < n.
Q means f o r Q a c l a s s i c a l formula of 1s.:.
I t is
We claim t h a t
+
i f ISTOF-Qc l a s s i c a l , then
+
Il be a proof of Q i n ISTO, and form JIo from JI by
Indeed, l e t
e r a s i n g a l l occurrences of B.
no
Then
is, o r can t r i v i a l l y be made i n t o ,
a proof of Q i n ZF (more p r e c i s e l y , i n a conservative e x t e n s i o n of ZF obt a i n e d by adding t h e names of elements o f m and t h e obvious definingaxioms for
0, {I,U , 5,
{I}
and t h e lower c a s e v a r i a b l e s ) .
Suppose now t h a t w e have defined
+
shown t h a t IST
+
mb
FQand v k
So Z F
+ IST
$ f o r Q c l a s s i c a l and i n
$I.
and
i s sound f o r c l a s s i c a l formulas; w e s h a l l do t h e same f o r
mb
To d e f i n e Q f o r c l a s s i c a l $I E IST,+l it i s enough t o a s s i g n a ISTn+l. denotation t o a l l c l o s e d terms f t f o r which f i s i n b u t n o t i n IST+ and f o r which t h a s already been assigned a d e n o t a t i o n
+
f i n i t i o n of IST
n+l
unique h e r e d i t a r i l y f i n i t e X
+Q
By t h e de-
E
f o r which
for a l l classical Q
by t h e same argument used f o r
+ 1- N x ) (3 1 y ) J, ( x , y ) .
Ji (x,f x ) and IST,
px) (Ply)@(x,y); t h e r e f o r e ,
t h e i n d u c t i v e hypothesis
8
n'
Em.
t h e r e i s a formula @ ( x , y ) with j u s t x and y f r e e such
t h a t f i s introduced by t h e formula
meaning of
T
+ ISTO, v
E
IST;+l.
+ Ji (.c,X).
By
l e t f t denote t h e This d e f i n e s t h e
+
Now l e t ISTn+lF 9; then
Q; t h e r e f o r e
+ ISTn+l
and by induc-
t i o n t h e whole system IST+ i s sound f o r c l a s s i c a l formulas.
Remark 1.
Henceforth, unless t h e c o n t e x t c l e a r l y i n d i c a t e s otherwise,
whenever we speak of ZF we s h a l l mean a c o n s e r v a t i v e extension of Z F obt a i n e d by a d j o i n i n g t h e names of a l l elements o f m , t h e d e f i n i n g axioms f o r
-0,
{I,
etc.,
+.
and a l s o t h e d e f i n i n g equations of all f € IST
J. MYHILL
52
Remark 2 .
The r e a d e r may wonder why we have been s o d e t a i l e d i n t h e
proof of t h e soundness of IST'
f o r c l a s s i c a l formulas, and why we d i d n o t
s e t t l e it i n s h o r t o r d e r by appealing t o t h e well-known theorem on elimina b i l i t y of d e f i n i t e d e s c r i p t i o n s . p l y f a l s e f o r IST
+
The reason is t h a t t h i s theorem is sim-
(and IST), as i s seen from t h e following counterexample.
L e t Q be an undecidable c l o s e d formula. V (x =
{El
A
.
Suppose now t h a t iQ))
Then I S T
k
(3x) ( ( x =
0A
$)
we add t o IST a t e r m A of t h e f i r s t
kind by t h e d e f i n i n g axiom
( f o r which t h e supporting formula corresponding t o 2.6.2 b u t not t h e one corresponding t o 2.6.3).
t e m i s n o t a c o n s e r v a t i v e extension of IST. i n IST
+
i s forthcoming,
W e claim t h a t t h e r e s u l t i n g sys-
Using 2.5.4
we e a s i l y prove
(4)
tJ By ( 4 ) and t h e r u l e -
BJI
By ( 5 ) , (6) and 2.2.3
But by ( 3 ) t h i s is n o t a theorem of IST, and consequently t h e a d d i t i o n of ( 4 ) t o IST does n o t g i v e a c o n s e r v a t i v e extension. Our s t r a t e g y i s as follows.
(Cf. Smullyan [ 9 1 ) .
F i r s t w e extend t h e above t r u t h d e f i n i -
t i o n ( r e l a t i v e t o t h e modelm) t o n o n - c l a s s i c a l formulas, then i n t h e f o l -
+
lowing s e c t i o n we prove t h e soundness of IST
and a f o r t i o r i of IST, and
f i n a l l y i n 16 w e i n f e r ( 2 ) - ( 3 ) and o t h e r n i c e p r o p e r t i e s of IST from t h i s result. The extension t o n o n - c l a s s i c a l formulas i s a s follows:
mb $.
T1.
I f Q i s c l o s e d and atomic, $ i s t r u e i f f
T2.
I f $ and
T3.
I f $ i s c l o s e d , i $ i s t r u e i f f ($ i s n o t t r u e .
JI are c l o s e d , $ A JI i s t r u e i f f
$ and
tJ a r e each t r u e .
53
Intensional Set Theory T4.
terms t
.
T5.
If ( v X ) $ ( X ) i s c l o s e d , it i s t r u e i f f $ ( t ) i s t r u e f o r a l l c l o s e d
I f ( v x ) $ ( x ) i s c l o s e d , it i s t r u e i f f $ ( t ) i s t r u e f o r a l l simple
terms t , where t i s simple i f it i s c l o s e d and of t h e f i r s t kind and cont a i n s no r e c u r s i v e f u n c t i o n symbols. T6.
I f $ i s c l o s e d , @$ i s t r u e i f f I S T + k $ .
T7.
An open formula i s t r u e i f f i t s c l o s u r e i s .
It is immediate t h a t f o r c l a s s i c a l formulas
5.
The Soundness Theorem.
+
theorem of IST
0.
4, $ i s t r u e i f f ?7(
I f we t r y t o prove d i r e c t l y t h a t every
is t r u e (by induction on t h e l e n g t h of p r o o f ) , w e a r e
we t h e r e f o r e d e f i n e an a u x i l i a r y tech-
stopped as soon as we g e t t o 2 . 2 . 1 .
n i c a l notion t r u e * ( c f . Kleene [41) whose d e f i n i t i o n i s obtained from t h e d e f i n i t i o n of " t r u e " by r e p l a c i n g " t r u e " by "true*" throughout T1-T5 and T7, and s u b s t i t u t i n g t h e following f o r T6.
?$
T6*. I f $ i s c l o s e d ,
i s t r u e * i f f $ i s t r u e * and IST'F-0.
W e now prove by induction on t h e l e n g t h 1 of t h e proof of $ t h a t i f
IST+F$
then $ i s true*. Then $ i s an axiom which does not c o n t a i n any r e c u r s i v e
B a s i s 1 = 1.
f u n c t i o n symbol.
The only c a s e s which g i v e any t r o u b l e a r e 2.4.3
and
2.4.4. R e 2.4.3.
is $ , ( x )
A x = y
-f
$,(y),
where w e can assume without
loss of g e n e r a l i t y t h a t $ ( x ) c o n t a i n s no f r e e v a r i a b l e s o t h e r than x . 0
What has t o be shown i s t h a t
i s t r u e * f o r tl and t 2 simple.
$o.
W e proceed by induction on t h e s t r u c t u r e of
The b a s i s i s s t r a i g h t f o r w a r d , a s are t h e c a s e s where ,$I
c a t i o n o r t r u t h - f u n c t i o n a l combination of simpler formulas. consider t h e case where $ (t ) i s B $ ( t l ) . 0 1 Assume then t h a t ? $ ( t l ) and tl = t a r e true*. 2
is a quantifiI t remains t o
Then $(t,) and tl = t 2
are t r u e * , whence by t h e i n d u c t i v e hypothesis so i s $ ( t , ) .
Also $ ( t l ) i s
provable.
By i n d u c t i o n on t h e sum of t h e l e n g t h s of tl and t 2 , tl = t
provable.
By 2.4.3,
B $ ( t Z ) , q.e.d. R e 2.4.4.
$ is
$(t,)
is provable.
Since it i s a l s o t r u e * , so i s
2
is
J. MYHILL
54
$,(a)
A
B(a=B)
-f
$,(B
There a r e f o u r c a s e s , depending on whether a and 6: a r e l a r g e o r small v a r i ables.
We c o n s i d e r , f o r example, t h e c a s e where a i s l a r g e and B i s small.
Again t h e proof i s by induction on t h e s t r u c t u r e of $o, and again t h e only i n t e r e s t i n g c a s e i s when v a r i a b l e s except a.
c$
0
(a)has t h e form g $ ( a ) and c o n t a i n s no f r e e
What h a s t o be shown i s t h a t
is t r u e * f o r a l l closed terms tl and simple t e r m s t 2 . A s s u m e then t h a t g $ ( t l ) and g ( t =t ) a r e t r u e * . Then so i s j r ( t l ) 1 2 whence by t h e i n d u c t i v e hypothesis so is $ ( t 2 ) . Also $ ( t l ) and t =t 1 2 are provable, whence by t h e r u l e $/g$ so i s B ( t =t ) and by 2.4.4 so i s j r ( t 2 ) . 1 2 Since $ ( t ) i s a l s o t r u e * , so i s B $ ( t 2 ) , q.e.d. 2 We now r e t u r n t o t h e main induction on t h e l e n g t h 1 of proof. Suppose t h a t every theorem of I S T
+ IS . t r u e * i f its proof is
of l e n g t h
5
1 , and con-
Then $ is e i t h e r an axiom o r
s i d e r a theorem la with proof of l e n g t h lfl.
obtained from a preceding l i n e o r l i n e s by one of t h e t h r e e r u l e s .
ter case being t r i v i a l , we c o n s i d e r only
an axiom.
The l a t -
Two new c a s e s arise,
namely t h a t $ i s an i n s t a n c e of 2.3.1 and a t l e a s t one r e c u r s i v e f u n c t i o n
i s a d e f i n i n g axiom 2 . 6 . 4
symbol occurs i n t , and t h a t
symbol.
f o r such a f u n c t i o n
The l a t t e r case c r e a t e s no d i f f i c u l t i e s .
R e 2.3.1.
$ has t h e form
where t i s of t h e f i r s t kind and where without l o s s of g e n e r a l i t y w e can as-
sume t h a t $,(t)
is closed.
LEMMA.
W e f i r s t prove t h e
+ kt=tl.
There i s a simple term t' f o r which IST
We prove t h i s for all t e r m s t of t h e f i r s t kind o c c u r r i n g i n t h e proof
of which $ i s t h e l a s t l i n e by a s u b s i d i a r y induction on t h e l e n g t h n of t. If n = l , t i s
0
and t h e r e i s nothing t o prove.
must have one of t h e forms { t
1,
1
tlUtZ,
obvious, w e consider t of t h e form f t l ,
or ftl.
For t h e i n d u c t i v e s t e p , t The f i r s t two c a s e s being
where f i s introduced by 2.6.4 and
where f o r some formula $ ( x , y ) c o n t a i n i n g only t h e f r e e v a r i a b l e s i n d i c a t e d
Intensional Set Theory
have p r o o f s of l e n g t h
5
55
1 and a r e consequently t r u e * by t h e hypothesis of
By t h e hypothesis of t h e s u b s i d i a r y induction t h e r e
t h e main induction.
i s a simple t e r m t 2 f o r which tl=t2i s provable.
Since ( 7 ) i s t r u e * , t h e r e
i s a simple term t 3 f o r which $ ( t 2 , t 3 ) i s t r u e * .
Since ( 8 ) i s t r u e * ,
So J , ( t z , t 3 ) i s provable.
B$(tZ,t3) i s true*.
is J , ( t 2 , f t 2 ) .
Since ( 9 ) is provable, so
Since ( 7 ) i s provable, so i s (vyz) ($ ( t 2 , y ) A J, ( t , , z )
Putting a l l t h i s together, t3 = f t Hence IST+/-t=ftl=t3
and consequently t 3 = f t
2 and t 3 i s simple.
+
y=z).
i s provable.
1 This concludes t h e proof of t h e
lemma. Returning now t o t h e proof of t h e t r u t h * of 2.3.1, $o(t).
I f t i s simple, then $ i s t r i v i a l l y t r u e * .
l e t $ be ( V X ) $ ~ ( X ) +
I f n o t , use t h e lemma
t o f i n d a simple t ' f o r which ISTkt=t'. Since t=t' i s provable a n d c l a s s i c a l , it i s provable i n Z F and consequently t r u e and true*. ( V X ) $ ~ ( Xi s ) true*. t u r e of $,(t),
$o s i m i l a r
Then $ ( t ' ) i s t r u e * . 0
By an induction on t h e s t r u c -
t o t h a t used i n connection with 2.4.3 above, so i s
q.e.d.
+
This completes t h e proof t h a t every theorem of IST that
+ IST
i s sound*).
is true* ( b r i e f l y ,
By induction on t h e s t r u c t u r e of $, w e have t h a t $
i s t r u e i f f it i s t r u e * .
A s u s u a l , t h e only c a s e worth w r i t i n g down i s
when $ is c l o s e d and has t h e form Then $ i s t r u e * .
provable.
Assume t h a t
?$.
(Suppose $ i s t r u e .
Then $ i s t r u e * .
then J, i s provable and $ i s t r u e . )
Then J, i s
Conversely, i f $ i s t r u e *
Consequently, every theorem i s t r u e .
This i s t h e soundness theorem. Remark.
The meaning of t h e t r u t h - d e f i n i t i o n
0(t) is
w e regard (Vx)$ (x) a s t r u e j u s t i n case
i s c l e a r l y unchanged i f
t r u e f o r a l l closed terms
of t h e f i r s t kind, r a t h e r than j u s t t h e simple ones. t h e d e f i n i t i o n we had i n mind a l l along. use it a s our o f f i c i a l one till a f t e r 6.
Of course, t h i s w a s
But w e were not i n a p o s i t i o n t o
t h e soundness theorem was proved.
The Existence and Disjunction P r o p e r t i e s THEOREM A.
having only x f r e e ) :
The following t h r e e c o n d i t i o n s a r e e q u i v a l e n t
(0
56
J. MYHILL
Proof. If IST
k
(2)
IST
(3)
IST
t-
$ ( t ) f o r some simple t. $ ( t ) f o r some c l o s e d t of t h e f i r s t kind.
Since ( 2 ) + ( 3 ) and ( 3 ) + ( 1 ) a r e t r i v i a l , we only prove ( 1 ) + ( 2 ) .
+
( 3 x ) E $( X I , a f o r t i o r i IST
$(t).
@ x ) g $ ( x ) and by t h e soundness
Hence t h e r e i s a simple term t f o r which I S T + ~
theorem ( S x ) p $ ( x ) is t r u e .
This proof i s n o t n e c e s s a r i l y a proof i n I S T , b u t it can be made i n -
t o one by r e p l a c i n g every atomic t e r m which i s not a t e r m of I S T by
0.
L i k e w i s e w e prove THEOREM B.
IST
I-
IST
t- B$ V
c l o s e d t e r m t. THEOREM C.
k
(m)B$(X) (closed) i f f IST
?I$
(closed) i f f IST
I-
$ ( t ) f o r some
$ o r IST
b I$.
W e leave it a s an easy e x e r c i s e f o r t h e r e a d e r t o extend Theorem C t o
t h e case where $ o r I$ c o n t a i n s l a r g e (but of course n o t small) v a r i a b l e s f r e e , and a l s o t o prove THEOREM D.
(Uniformity theorem, c f .
(VX) (3y)fr$(x,y) ( c l o s e d ) , t h e n IST
(vx) (3Y)B$(X,Y) ( c l o s e d ) , a t most x f r e e .
then IST
k
[ 6 ] , p.
(3y) (vx)B$(x,y).
381). If IST
I f IST
k
(VX)$(X,t(x)) f o r some term
t(x)
with
The c r u c i a l observation i n both c a s e s is t h e l a s t sentence i n t h e proof of Theorem A. Theorem A immediately implies t h e following form of Church's t h e s i s . THEOREM E.
k
If IST
t o be a r e c u r s i v e f u n c t i o n of
(VX E
u)(3y
E
w l E J $ ( x , y ) then y can be taken
X.
W e a l s o have
THEOREM F.
( V x ) ( $ ( x ) -+ B $ ( x ) ) ( c f . 2.6.3)
I f IST
( c l o s e d ) then
( i . e . , C 1 over t h e admissible set F = HF, s e e [l], p . 4 7 ) . Conversely, i f y C F i s r.e., t h e r e i s a formula $ ( x ) d e f i n i n g y f o r which { x l $ ( x ) } i s r.e.
(vx)( $ ( X I
IST
Proof.
-+
B$(x)).
A s s u m e IST
(Vx) ( $ ( x ) + B $ ( x ) ) .
Then by soundness t h i s i s
t r u e and whenever a c l o s e d t e r m t of t h e f i r s t kind s a t i s f i e s $ ( t ) , IST $(t).
By soundness again,
IST
{ x l $ ( x ) } i s e q u a l t o t h e r.e.
$ ( t ) i m p l i e s t h a t $ ( t ) i s t r u e ; hence
set of t f o r which IST
$(t).
Conversely, i f y C F i s r.e., w e can introduce i n t o IST by 2.6.4
two re-
c u r s i v e f u n c t i o n symbols f l and f a , such t h a t f l d e f i n e s a p r i m i t i v e recurs i v e f u n c t i o n which enumerates t h e set of codes (Barwise, l o c . c i t . ) of elements of y and f 2 d e f i n e s a map which a s s i g n s t o every n E w t h e element of r" coded by n.
The formula @ ( x ) is now
Intensional Set Theory
By
E
w ) (x = f 2 f 1 y )
and it s a t i s f i e s t h e given condition because by 2.4.3
x = f f y A B(f2fly = f f y ) 2 1 2 1
+
B(x = f f
2 1y)
i s a theorem of IST. This t a n g e n t i a l connection between modal l o g i c and admissible set theory perhaps deserves f u r t h e r i n v e s t i g a t i o n . Open question.
I t i s easy t o prove (analogously t o Theorem F ) t h a t
every provably r e c u r s i v e f u n c t i o n ( i . e . ,
function f ( x ) = U(py)T(e,x,y) f o r
(Vx)( g y ) T ( e , x , y ) ) i s d e f i n e d by a r e c u r s i v e
which I S T o r e q u i v a l e n t l y ZF
f u n c t i o n symbol, and conversely t h a t every such symbol d e f i n e s a r e c u r s i v e f u n c t i o n (on t h e c o d e s ) . s i v e function? eluded me.
Does every such symbol d e f i n e a provably r e c u r -
I t would be a s t o n i s h i n g i f n o t , and y e t t h e proof has
I am even i n c l i n e d t o c o n j e c t u r e t h a t f o r each such symbol f ,
t h e r e i s a number e f o r which I S T F (Vx E w ) ( 3 y ) ( T ( e , x , y ) A f f 2 ( x ) = f 2 U ( y ) ) where f 2 i s t h e coding function used i n t h e proof of Theorem F.
7.
Concluding Remarks and Extensions.
There a r e c e r t a i n strengthen-
ings which one might want t o make i n t h e axioms 2.4 and 2 . 7 ;
namely, one
might want t o allow modal a b s t r a c t i o n terms, and t o have f u l l s u b s t i t u t i v i t y i n s t e a d of i t s s p e c i a l i n s t a n c e s 2.4.1
-
2.4.3.
E i t h e r of t h e s e c o u r s e s
would d e s t r o y our n i c e p r o p e r t i e s , as w e now show. I f w e allow f u l l comprehension
whenever ( Z )(VX)( $ ( X ) +
x E Z) ,
(by 2.4.1 and t h e r u l e f o r
B);
even with a modal 9, then i n p a r t i c u l a r
consequently
J. MYHILL
58
By 2 . 4 . 2
kX
= Z A X
E W + Z E W,
I-
x
=
z
-+ 2
SO
f
{ Y J B ( Y= X ) l
and by (10)
This conclusion l i k e w i s e follows from t h e Strengthened 2.4
i f we put B ( X = 2 ) f o r $ ( 2 ) .
But t h e conclusion (11) d e s t r o y s t h e d e s i r a b l e f e a t u r e s of our system, a s i s w e l l known.
Specifically, replace
x
i n (11) by
{x E
wl$},
where $ i s
any c l a s s i c a l sentence; then
and i f (11) w e r e provable we'd have a l s o
By t h e !-rule
we'd also have
so by 2.2.3
which with ( 1 3 ) y i e l d s
k
Ji
even f o r undecidable $, we'd
eJi f o r every have I- $: V 3
+-+
sentence $.
9
Since
$
v iJi
f o r such $ I s , c o n t r a d i c t i n g
Theorem C. Similar arguments can e a s i l y be given t o r e f u t e Theorems A and B f o r t h e strengthened systems c o n t a i n i n g e i t h e r (10) o r ( 1 2 ) ; so w e must l i v e w i t h t h e r e s t r i c t i o n s on both comprehension and s u b s t i t u t i v i t y . s t r i c t i o n s of course do n o t exclude any c l a s s i c a l arguments.
These re-
59
Intensional Set Theory W e do n o t know i f t h e r e s t r i c t i o n on 2 . 6 t h a t $ be c l a s s i c a l i s l i k e -
wise e s s e n t i a l , and we c o n j e c t u r e t h a t it i s n o t . Another Q u e s t i o n .
I s t h e r e a t r a n s l a t i o n of t h e i n t u i t i o n i s t i c connec-
t i v e s l i k e t h a t i n [ 3 ] which c o n s e r v a t i v e l y extends something l i k e i n t u i t i o n i s t i c ZF?
( I t cannot be a s t r a i g h t f o r w a r d t r a n s l a t i o n ; f o r any system
of i n t u i t i o n i s t i c set-theory of which w e might want t o prove t h a t I S T was a conservative extension would presumably contain ( t h e e q u i v a l e n t o f ) abs t r a c t s c o n t a i n i n g non-classical
connectives, and under t h e t r a n s l a t i o n
t h e s e would correspond t o modal a b s t r a c t s which a r e not p r e s e n t i n I S T and which, as w e have j u s t seen, cannot i n g e n e r a l be added t o I S T without destroying i t s n i c e p r o p e r t i e s .
However, it i s p o s s i b l e t h a t some modal
a b s t r a c t s might be added which would enable both Theorems A-C and a cons e r v a t i v e extension r e s u l t t o be proved.) F i n a l l y w e t u r n t o t h e question of extending IST by v a r i o u s t r u e formulas so a s t o r e t a i n t h e p o s s i b i l i t y of a soundness theorem and i t s c o r o l l a r i e s A-E.
Such formulas f a l l i n t o t w o c l a s s e s .
I n t h e f i r s t class
a r e t r u e formulas such a s
(Barcan's axiom f o r l a r g e v a r i a b l e s ) .
Such formulas w i l l be c a l l e d c l a s s i -
cally v a l i d meaning t h a t they are t r u e * and become theorems of ZF when a l l
B's are deleted.
There i s no d i f f i c u l t y i n c a r r y i n g over t h e proof of t h e
soundness theorem and i t s c o r o l l a r i e s t o e x t e n s i o n s of IST by c l a s s i c a l l y v a l i d formulas; t h e only p o i n t t h a t r e q u i r e s c a r e i s t h a t " t r u e " and "true*" a r e now defined with r e f e r e n c e t o t h e system c o n t a i n i n g t h e new axioms; i.e.,
"provable" i n T6 and T6* (954-5)
now r e f e r s t o t h e extended system.
More p r e c i s e l y , i f $ is any formula of I S T and C any extension of IST, we say t h a t $ i s t r u e w . r . t .
C i f 4 s a t i s f i e s a truth-definition
t h a t of 14 except f o r reading
Z+k@f o r
I S T + k @ i n T6;
f i n e t r u t h * , soundness and soundness* w . r . t .
C.
which i s l i k e
analogously w e de-
With t h i s terminology we
have, with no change i n t h e argument, t h e e x t e n d e d s o u n d n e s s theorem; i f Z
i s a n extension of IST by c l a s s i c a l l y v a l i d formulas, then C i s sound w . r . t . E , and i n consequence Theorems A-D hold f o r Z; do Theorems E and F.
i f i n a d d i t i o n I: i s r . e . , s o
The proof i s e x a c t l y t h e same as f o r IST.
The second c l a s s of t r u e formulas w e might want t o add a r e those which, while t r u e * , become f a l s e o r a t l e a s t unprovable i n ZF i f a l l leted.
Examples a r e
B's
a r e de-
60
J. MYHILL
K i s some p a r t i c u l a r c r e a t i v e s e t ) ;
( u n s o l v a b i l i t y of t h e h a l t i n g problem:
(negation of Barcan's axiom f o r small v a r i a b l e s ) ; and
(vx
E
( 3 y E w ) B $ (x,y) +
w)
(GE
w) (vx
E
E
w) (3y
0)
( T ( e , x , y ) A $ (x,U(y) 1 ) (17)
(Church's t h e s i s ) . C a r e l e s s t h i n k i n g w i l l r a p i d l y convince you t h a t a l l t h e s e a r e t r u e *
(w.r.t.
t h e system obtained by adding them t o I S T a s axioms).
In f a c t , i n
an e a r l i e r v e r s i o n of t h i s paper I claimed t h i s was "obvious" and spoke glowingly of f o r m a l i z i n g , e.g., problem i n IST
+
(15).
t h e proof of t h e u n s o l v a b i l i t y of t h e word-
However, Shapiro challenged m e t o prove t h o s e
"obvious" r e s u l t s and I could n o t ; e.g.,
t o prove (15) t r u e * w . r . t .
IST
+
(151, we'd f i r s t make s u r e t h a t t h e d e f i n i t i o n of K was such t h a t {nl I S T
n E K1
Kl were r e c u r s i v e l y i n s e p a r a b l e , then i n f e r t h a t
and (nl IST
f o r some n
IST
0'
( 1 5 ) was t r u e * .
+
(15)
ti
EG E
K and I S T
+
ti no
(15)
K and hence t h a t
Unfortunately t h i s i n f e r e n c e i s i n v a l i d u n l e s s we know be-
forehand t h a t IST
(15) i s c o n s i s t e n t , and w e d o n ' t know how t o do t h a t
f
without having shown before t h a t t h a t (15) i s t r u e * ; so t h e a l l e g e d l y "obvious" proof i s c i r c u l a r . Thus, t h e s t a t u s of p r i n c i p l e s l i k e (15)-(17) and indeed of t h e whole venture of using n o n - c l a s s i c a l e x t e n s i o n s of IST t o c a r r y o u t p r o o f s of und e c i d a b i l i t y r e s u l t s within t h e system remains d i s t r e s s i n g l y up i n t h e a i r . Some idea of t h e d e l i c a c y of t h e s i t u a t i o n may be granted from t h e f a c t t h a t
w e have constructed a n o n - c l a s s i c a l extension of I S T by a s i n g l e formula $ such t h a t (1) Q i s t r u e w . r . t .
r
r
= IST
+
@;
(2)
r
i s c o n s i s t e n t and y e t ( 3 )
does n o t s a t i s f y Theorem A ( e x i s t e n c e p r o p e r t y ) . W e t h e r e f o r e conclude by proposing t h e following open question: Let
I' be an extension of IST by (15)-(17) (or similar formulas, e . g . , EX) A (B(zEA)
x n y = O - + ( 3 u n c o u n t a b l y r n a n y A c ~ ( v z ) ( ( ? ( z€ A ) * z +-+
z
.
E y,
(1)
(w.r.t.
r)?
IS
r
consistent?
( 2 ) If so, are t h e new axioms t r u e
( 3 ) If so, are they t r u e * (SO t h a t
properties A-F) ?
r
enjoys t h e useful
Intensional Set Theory
61
REFERENCES Barwise, J., Admissible Sets and Structures, Springer, Berlin, 1975. Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967. GCldel, K., "Eine Interpretation des Intuitionistischen Aussagenkalkuls", Ergebnisse eines mathematischen Kolloquiums 4 (1932), 39-40. Kleene, S. C., "Disjunction and Existence under Implication in Elementary Intuitionistic Systems", Journal of Symbolic Logic 27 (1962), 11-18. Myhill, J., "Some Remarks on the Notion of Proof", Journal of Philosophy 57 (19601, 461-471.
, "Constructive Set Theory", Journal
of Symbolic Logic 40
(1975), 347-382. Shapiro, S., "Epistemic and Intuitionistic Arithmetic", this volume. Sierpinski, W., Algebre des Ensembles, Warsaw, 1951. Smullyan, A. E., "Modality and Description", Journal of Symbolic Logic 1 3 (1948), 31-37.
This Page Intentionally Left Blank
In tensional Mathematics S. Shapiro (Editor) @ Elsevier Science Publishers B. V . (North-Holland), 1985
63
A GENUINELY INTENSIONAL SET THEORY
Nicolas D. Goodman State University of New York at Buffalo Buffalo, New York U.S.A.
John Myhill, in the introduction to [ 3 3 , proposed to extend the ideas of Shapiro [41 to Zermelo-Fraenkel set theory (ZF) so as to produce a modal set theory conservatively extending ZF which would have the disjunction and existence properties and in which various principles of constructive mathematics could be expressed.
He found, however, that in order to have
the numerical existence property, he needed to distinguish between arbitrary sets and explicitly listed hereditarily finite sets.
The resulting theory,
called IST for "intensional set theory", is rather complex. Moreover, as a modal set theory, IST leaves much to be desired.
In particular, it
lacks a full axiom of comprehension. It is the purpose of this paper to show that Myhill's original program can be carried out in a straightforward way, without the introduction of more than one kind of set and without any restriction on the comprehension axiom, provided only that the set theory is made sufficiently intensional. Thus in our theory, called
ZFM
for "modal Zermelo-Fraenkel", the axiom of
extensionality is weakened to read that two sets which necessarily have the same members are identical. The elements of our domain of discourse ought therefore to be thought of as properties rather than as sets. This has the disadvantage that ZFM.
ZFC
(ZF with choice) is not directly a subtheory of
There is, however, a simple interpretation of
ZFC
in ZFM which
we show to be sound and faithful. The organization of the paper is as follows. theory ZFM.
In Section 1 we describe the
In Section 2 we show how to interpret ZFC in ZFM and
prove that the interpretation is sound and faithful. In Section 3 we adapt the method of Myhill [21 to ZFM in order to prove the existence and disjunction properties.
In Section 4 we show that ZFM also has the numeri-
cal existence property and discuss the treatment of Myhill's "recursive function symbols" in ZFM.
N.D.GOODMAN
64
1. Modal Zermelo-Fraenkel Set Theory.
The theory ZFM is formulated in quantified S4
without identity and
with a single binary predicate 'Q.Our notation differs from that of Myhill [31 in that we prefer to indicate the modal operator by than by
We use 'I) instead of
B.
in order to leave
E
E
0
rather
free to denote
an extensional membership relation to be introduced in Section 2 .
A
should, strictly speaking, be read 'x has the property y
formula x n y
More specifically, the language of the binary relation symbol 'I)
, and
the quantifier V the usual way.
,
ZFM has variables x,y,z,u,V,w,
the propositional connectives
the modal operator
0.
I .
...,
i and
-.L
,
Formulas are built up in
The other truth-functional connectives and the existential
quantifier are defined. The axioms and rules of inference of ZFM are as follows:
-
I.
All classical propositional tautologies.
11.
From $ and
111.
nz
-$.
U$
-Do$.
IV
.
V. VI
o $ A Q (16
.
VII
From
.
VIII. IX
.
$
Vx$(r)
From $
4
--
$) -.
$
infer $
.
U$.
infer U $ .
$(y)
,
is free for x
provided y
$(x) infer
$
-. V z $ ( x ) , provided
in
0(3)
.
Z is not free in
2.
(Modal Extensionality)
~ V z ( z* ~ xzrjy) A xrju * y y u . In the light of I X , we define the relation of intensional identity by letting x
2
y
be an abbreviation for the formula UVz(zllx
relation is stable in the sense that ZFM ZFMIX Z X If
$(z)
is any formula and y ZFM
We write v x l y $ X.
1X
zL y
.
zvy)
.
This
Moreover,
.
is free for x E y A$(T)
-. O(x 2 y)
*t
#(y)
in
$(x) , then
.
for Vx(z'I)y* $) , and we write azTlyp for 3z(xqyAg5)
(We11-foundedness)
vxt-rlyva:Fj(y)
-
$(z)l- v z
.
.
A Genuinely Intensional Set Theory
65
XI.
XII.
XIII.
XIV.
provided y XV.
.
is not free in the formula @ ( a )
(The Power Property) @ j ~ V Z [ Z q y .+
XVI.
vU((uqZ
-
Uq3C)l
.
(Infinity) * 0VI)yVv E y)].
XVII.
(Modal Replacement)
-
0VyTz
provided
L)
A Vu((Z(y,u)
b[@(Y,Z)
fi~VZ[Z7L)
*
XVIII. (Collection)
V Y Y ?az ~ ~@(y,z) provided u
u
=
211
3 y Y y T z A (Z(y,z))l ,
,
is not free in @(y,z)
-
-.
but
3u v
is free for z
u
YI-IX
in @(y,z)
.
aznu @(y,z) ,
is not free in @(y,z)
It is perhaps an inelegance to have both XVII and XVIII.
However, the
It would make no difference,
content of the two axioms is quite different.
of course, if XVIII were replaced by an unmodalized replacement axiom. XIX.
(Choice)
-
-
V z q z 3zJWQz) A V z q r V ~ ) v z[ b ( u ' Q z A u T ( W ) Vu((ulz
.+
uqL))I
3y V Z ~ 3S24 V ~ ) ( ( w q yA
This completes the description of the theory ZFM.
L ) ~ z
.+ L)
u)
.
Observe that there is
no restriction on the formulas occurring in the axiom schemes X, XIV, XVII, and XVIII other than that there be no collision of bound variables. We will need an auxiliary theory called ZFM'
.
This theory is obtained
66
N.D. GOODMAN
from ZFM by adding new terms in stages as follows. Suppose we have proved
-
3yUVx[xTly where x c @
is the only variable free in $
with the axiom
-
Vx[xTlc@
.
Then we add a new constant
.
$1
Of course we extend the axioms and rules of
particular, VII should read
-
VZ$ (o) $(t) where the term t is either a constant c+ @(XI
ZFM to the new language. In
I
or a variable free for x
in
.
It is easy to see that ZFM' is, if 2.
,
$1
is a formula of
$
The Interpretation of
is a conservative extension of
ZFM and ZFC
&
ZFM'
ZFM
Zermelo-Fraenkel set theory in ZFM.
.
1$,
then ZFM
ZFM.
t
$
.
That
We wish to interpret classical
The idea of the interpretation is
to define a relation = of extensional identity and then to take the membership relation x to some z
E
of
y
ZFC as meaning that z
which has the property y
.
is extensionally equal
In order to define extensional
identity we proceed somewhat informally in ZFM. For any properties x
and y , let { S l y ]
We may let (x]
Ovu((u~z- U f o U Iy ) . Then, for any z and y
(z,y)
is {{x]
be
(x,xj
, (r,y])
.
be the unique z
.
-D
x
uAy 5 u .
Now observe that, intuitively, we should have x = y to which the property x
to which the property
the ordered pair
We can prove in ZFM that
(z,y) 5 ( u , u )
u
,
such that
just in case for every
applies there is an extensionally equal v
y applies, and conversely. More generally, suppose
2
is any relation with the property that whenever
u
with U ~ Z there is some 0 with
Uqy
( 3 , ~ ')Q z
such that
, then for any
(u,v)
lz,
and
conversely. Then an informal application of the well-foundedness of the relation
9
shows that z
must be a subrelation of the relation of ex-
tensional identity. Conversely, if x
and y
are extensionally identical,
then by restricting the relation of extensional identity to a suitable
67
A Genuinely Intensional Set Theory
initial segment of the universe we can find a relation z property such that
(x,y) Q' z
More formally, let
B(Z)
with the above
.
be the following formula of
ZFM:
V u V u [ ( u , u ) 'Qz -. Vp'Qu3qTiv( ( p , q ) ' Q z ) A V p ' Q v 3 q ' Q u ( ( p , q ) Tz ) l
Then we take x
=
y
.
to be an abbreviation for the formula
~z[(z,Y)'Qz A e
mi
.
That this notion of extensional identity is adequate to the intuitive conception is shown by the following result. Theorem 1. ZFM Proof. -
r
=
s.
.+Vu'Qx
(x,y)'QzA
B(Z)
.
(7.4 = V )
.
Suppose, first, that x
=
Z V T y (u = V ) A Vu'Qy 3 V ' Q Z
We argue informally in ZFM.
so that
z
13: = y
For any
r
and
But from this and the definition of
s
,
if
y. Choose
( r , s ) ' Q z , then
it follows that
B(Z)
Vu.U?lx3V'Qy ( u = V ) AVu'Qy 8U'Qx (u =
V )
.
(1)
Conversely, assume (1). Then Vu'QxZziIV'i)g
(u,U)'Qz A 9 ( z ) I
By the collection axiom XVIII, let A
.
be such that
VU.U?~Z 3 z T A Z V U T t~(u,v) ~
By the comprehension axiom XIV, let B
A e(z)1
.
be such that
oVz[z.U?lB* z'QA A B ( z ) l , and, using XIII, let R be the union of B .
It is easy to see that BE?).
Thus
v u ' ~ 3 : 3 ~ q( (yU , W T R ) A B ( R )
. .
Similarly, construct S
so that vu'Q,u3V'Qx ( (u,U)7 S ) A B (S)
Forming the union of R
and
T
S
by
XI1 and XIII, we obtain a property
such that Vu.U?lx3vqy( ( u , V , ) T T ) A V u ' Q y 3 V ' Q (x ( u , V ) T T ) A 0 ( T )
But then, forming the union of T which verifies that
x
y,
with
{ ( x , y ) j , we find a property
completing the proof.
It follows from Theorem 1 by transfinite induction using ZFM
1x = x .
.
Moreover, we have immediately that
X
that
z
N.D.GOODMAN
68
t-
ZFM
X =
y M y = X.
The proof in ZFM that extensional identity is transitive is only slightly less trivial. Specifically, we show by transfinite induction on x , using
,
the symmetry of =
that
.
VuVz[3: = y h y = z - z = 21
Thus ZFM
/-
3: =
yAy = z - x =
to mean that 3zly
Next, in ZFM, define msy
2. (3: =
ZFM
1 3: = y A x 6 z
ycz,
ZFM
/-
zsy
z)
.
Then we have
and 3:
= yA z a
More generally, let us say that a formula fd
.
of the language of ZFM
extensional iff it is built up from formulas esy
and
3: =
is
by means of
y
(not the modal operator #(x) is extensional and y is free for z in @(z),
the propositional connectives and quantifiers Then if
ZFM Theorem 2. If fd
t
m = yA2(3:)
fd(y)
.
(Soundness of the interpretation).
is an extensional formula and ZFC
Proof. By induction on the proof of fd
12 ,
then ZFM
in ZFC.
1 fd .
In the light of what we
did above, it suffices to consider the non-logical axioms of Extensionality. We argue in ZFM.
1.
ulx
.
Vuly
8Vls
2.
Foundation.
Then u w , and so u e y .
.
(u = v )
Thus, by Theorem 1,
We must show that Vx@(x)
=
y
.
-
zcy)
.
Say
Similarly,
We have
.
follows that @ ( z )
-
#(y).
.
Suppose z a .
Thus vz6Zfd(z)
,
and so
vzc[vym fd(Y)
.
.
Then choose y v x
Since the formula fd
shown that Therefore vxfd(x)
$(2)1
. In ozder to apply the corresponding axiom
ZFM, suppose that vyl)Xfd(y)
.
x
.
Suppose the formula $(x) is extensional. Arguing in
vx[Vya#fy)
that z = y
ZFC.
Suppose that V z ( z a
Thus 3vTy (u = v )
ZFM, suppose that
of
0).
-
fd(m)
fdw1.
is extensional, it
.
That is, we have
X so
A Genuinely Intensional Set Theory
Empty Set.
3.
that
4.
By the corresponding axiom XI of ZFM, choose
.
O ~ Z ( l Z ~ y )Then
of ZFM choose z
.
be given. By the corresponding axiom XI1
so that
-
ovu(uq2
Say
UEZ.
vu
so
UQZ
.
Choose U q z
.
=y
= xvu
u
so that u = U .
Conversely, say u = x
Thus VU(UE2
5.
-
free.
Then U
u =
-
.
the extensionality of
.
, and
is extensional and does not By the corres-
z =
.
.
24.
is extensional, we also have $(u)
.
.
We have u q x A # ( u )
Choose u q x with
we have
gj,
y , and so
.
zqx A$(z)l
since the formula pr
Conversely, suppose z a A $ ( z )
Z
so that
Then choose u q y with
Clearly a = .
Z xvV
Then 3 U q z (u = a )
Let x be given.
We argue in ZFM.
ponding axiom XIV of ZFM, choose y
Suppose z s y
.
y
u = x v u = y)
0Vz"Z'Qy
.
$1
3
Separation. Suppose the formula $ ( z )
contain y
so
ZJ
certainly ~ z ( 1 z c y )
and y
Pairing. Let x
u = x
69
2
=u.
@(z)
.
Again using
Therefore u q y , and so zsy
.
Thus
'.. z ~ x A p r ( z ) l.
Vz[zey 6.
Unions.
choose y
Let
x
be given.
By the corresponding axiom XI11 of ZFM,
so that
.
OVz[z~y NOW suppose vex h z s u .
choose Wqu
z =
with
uqx
Choose 13.
.
3u((u~xAz~ull so that
Evidently 13qy
v Z [ 3 V ( U W A Z C U ;-m
7.
Power Set.
choose y
Let x
be given.
so that
-
OVZ[ZQf
Suppose a
u
is such that Vu(usa
so that
Obviously vu(u'Qv
-
uqx)
, and
= u.
Then
Therefore Z s y .
zeyl
Hence
2f.u.
Thus
.
By the corresponding axiom XV of
vu(uqz
-
OVu(uqU
.
V
uw)
.
-
uqxZ)1
.
.
By comprehension in ZFM , choose
uqxA UEZ)
so U q y
ZFM,
.
We claim that a = U .
First,
N.D. GOODMAN
70
say
p q z .
p = q.
,
.
pqV
p a .
and so
qcz,
But then
v e r s e l y , say
z = 2,
pcz,
Then
qqv.
and so
Then
p l 3 c A poz zoy
as w e claimed, and s o
VZ tV7L((UEZ 8.
Collection.
ZFM
that
axiom XVIII of
ZFM,
Wqx
Then choose
9.
.
Then
choose
y
fd
0
.
qq.2 ( p
Therefore
-.)
U a )
+
.
Con-
.
q)
=
Thus
2631 i s extensional.
fd(u,V)
.
Assume i n
V U ~ L Z ~ V @ ( M , V ) By t h e corresponding V24~3c3Ut)’iiyf d ( u , v )
so t h a t
vqy
W e can f i n d some
By t h e corresponding axiom XVI of
A
Now say u a
with
ZFM,
fd(L),U)
.
U
v
V
.
.
voy
Also
choose
[Lhjqu U V z ( z li y ) A Vgqu 3 . ~ 01Vo(z~qz ~ * ovqy choose
.
@(u,v)
.
BY t h e comprehension axiom X I V ,
with
W e have shown t h a t
is e x t e n s i o n a l , it follows t h a t
vum 3veyfd ( u , v ) Infinity.
u =W
with
Since t h e formula Thus
.
Vpqz3qq?lv ( p = q)
Thus
.
suppose t h e formula
vua3Vfd(u,v)
qqx
Thus we can choose
so t h a t
=_ y ) 1
.
so t h a t
We claim t h a t
V p V A Vq(qW-.) 0 4 T P ) For, suppose not.
plAA qqp
Say
A
iOqqp
.
(2)
*
B
Choose
oVr[rqBff r v A A i ( r 5 p ) l
so t h a t
.
Then
3 y q B O V z ( i z 7 y ) A Vy7BBzqBO Vv((v7zand so Now,
pqB
,
zqA
.
voz.
f i r s t case
Choose
qcy
pVA
with
,
y)
gel.
Moreover, say
,
Choose
y = p
,
so t h a t
uvv((uq2- o o q p v Say
V E
which i s a c o n t r a d i c t i o n .
KyyTIAVz((7zcy)
and then t a k e
OuqyV
qqz and so
with
Ucy
.
1,
= q .
2)
z pl
.
Either O q g p
I n t h e second case
or
0 =
q
p .
p , and so
In the U = y.
Thus
vv ( V O Z -. vcy v V For t h e converse, suppose f i r s t t h a t
vcy.
= y)
.
Then
Vcp.
Choose
qvp
A Genuinely Intensional Set Theory
with
V =
Then, by ( 2 ) ,
q.
other hand, suppose that
Hence q T z ,
Oq'Qp. =y
V
.
71
Then O = p .
and so
But
On the
VEZ.
p q z , and
so
UEZ.
Thus we have shown that Vy€A3zoA?$V(O€z 10. Choice.
Assume in ZFM vx6A3y((y€Z) h
v
Ooy
=
V
.
y)
that
VZSA vy€A(Z # y
We must construct a set C such that
TCEA@VZ(ZBCA zecc
h
l&(ZeC
+.
z = u)
(3)
ZOy)).
.
(4)
We cannot immediately apply XIX since it may be that the property A applies Thus the hypothesis of
to many sets all of which are extensionally equal.
XIX may not be satisfied. In order to get into a position where we can apply XIX to obtain C , therefore, we first apply XIX to choose one representative from each equivalence class of A under extensional identity. Let B be the collection of these equivalence classes. More formally,
-
using XV and XIV, construct B so that oVv(v'QB
3zTA
0
Vz(z'Qw
-
.
z =2 A z ~ A ) )
Evidently VvWg B 3y (y'Qw)
.
Moreover, it is not hard to check that V u ~ B V u T p [ & ( z TA ~ zTu)
Thus we can apply XIX to find D
Vz(zVw
-
Z'Qu)I
.
so that
.
VU'I)Bkb~fz(z'QLJ I\ zllW * z :u )
BY XIV we may assume that
It follows that Vz'e'llD3y (y'Qx). Next suppose x:"TIDA y v D (31,
x =y
that x :y
.
.
Say
z ' Q z h z'Qy
.
Then zecc A zoy
Therefore there is some Y'QB by our construction of 17.
such that
Z ~ L A )
so that, by y'Qw
,
so
Thus we have shown that
v x q D VyqD [ Z Z (zTZ I\ Zqy) It now follows that we can apply XJX to D
,
-B
x
3
yl
.
to obtain a set C such that
V x T D X u V z ( z l l C A z'Qx * z
2
u)
.
N.D.GOODMAN
12
Again we may as well assume that
.
Vz~C&qD(zTx) We must still check that C
so suppose that xeA.
holds.
-
is the desired choice Choose W s B
that is, that (4)
that
so
O V z ( z v ~+t z = X A z T A ) .
Thus
is the equivalence class of x
W
let y
be the representative of
Vz(zTDA z l w Then evidently y = x .
.
zlg- z
is the desired unique element of x
Conversely, suppose zeC
and
pqA Therefore, by ( 3 ) , class of x
q
=
z = u.
so
~ 7 . A4 46p p
in C
that u w
.
First
.
so that z = q .
There
Then
p = x , and s o p l w
as above. Hence
and
u ,
A
,
Choose q T C
zw.
such that q q p .
that
so
u).
5
it is clear that u q y , and hence, since x = y
must be some p Q D
D,
that
so
.
z :y )
+t
the construction of
That is, choose y
But, since y I D , we can choose u Vz(z'l)CA
Now we claim that u
. Then by
in D
W
A
,
~ G x .
where
is the equivalence
rJ
y , and so q'l)y
.
It follows that
Thus Vz(zeC A
zw
z = U)
.
That completes the proof of the theorem. In order to prove the converse of Theorem 2 we define a mapping the formulas of
ZFM
into those of
from
ZFC which, up to provable equiva-
lence in ZFC, fixes all the extensional formulas. The translation is obtained by simply erasing we take to be
Z0
Then
(2
-
to be xey
(xvy)'
Go ,
g)O
take
0
and taking
to mean
(Vx@)O to be Vxflo , and take ZFC to x
is provably equivalent in
every axiom or rule of inference of derivable rule of ZFC
.
( D@)O
=
y
.
(0 *
11)'
to be gjO
.
Moreover,
ZFM goes over into a theorem or
The definition of
cx,y]
definition of the doubleton (unordered pair) of x definition of
In more detail,
6 .
, take ( - 1 0 ) ~to be 70O , take
becomes the usual and y
,
and the
(x,y) becomes the usual definition of the Wiener-Kuratowski
ordered pair. The translation of Theorem 1, then, asserts that
13
A Genuinely Intensional Set Theory
(5 =
-
y)'
VUCC
A V U C ~~
~ V E ~ = ( U v)O
An argument by t r a n s f i n i t e induction on
1-
ZFC
@
in
ZFC
z =y
.
B
y)O
-
then shows t h a t
( ~ c y )i s~ e q u i v a l e n t i n
From t h i s it follows t h a t if
(5 =
V C C ( U= v ) O .
i s any e x t e n s i o n a l formula of
ZFM,
then
ZFC
1
ZFC
scy
to @
-
.
go.
Hence Thus
t h e following theorem has been e s t a b l i s h e d . Theorem 3 ( F a i t h f u l n e s s of t h e i n t e r p r e t a t i o n ) . formula and
ZFM
1- @ ,
then
1
ZFC
If
i s an e x t e n s i o n a l
@
@.
I n o t h e r words, w e may t h i n k of our i n t e n s i o n a l s e t theory conservative extension of c l a s s i c a l s e t theory 3.
A Model of
ZFM
ZFM
as a
ZFC. I t w a s observed by Shapiro
i n t h e Style of Friedman.
i n [ 4 1 t h a t Kleene's s l a s h could be adapted t o a modal a r i t h m e t i c formalized i n
Myhill extends S h a p i r o ' s v e r s i o n of Kleene's s l a s h t o h i s
S4.
modal s e t theory i n [ 3 1 .
Kleene's o r i g i n a l c o n s t r u c t i o n was considerably
g e n e r a l i z e d i n Friedman I11 and then a p p l i e d t o an i n t u i t i o n i s t i c s e t theory by Friedman and Myhill i n [21. [ 2 1 t o our modal s e t theory
adapted KLeene's c o n s t r u c t i o n .
W e now adapt t h e c o n s t r u c t i o n i n
i n t h e same way t h a t Shapiro o r i g i n a l l y
ZFM
Although t h e a d a p t a t i o n i s n o t completely
r o u t i n e even f o r o u r axioms, it should be n o t i c e d t h a t t h e proof we g i v e i s q u i t e s e n s i t i v e t o any change i n our axioms.
Indeed, t h e axioms of
ZFM
were a r r i v e d a t by t r y i n g t o f i n d axioms s t r o n g enough t o i n t e r p r e t
ZFC
f o r which a c o n s t r u c t i o n of t h e s o r t we now g i v e could be c a r r i e d through. Recall t h a t
terms
ZFM'
i s t h e theory obtained from
For each c o n s t a n t
c@.
introduce a p o s s i b l e c o n s t a n t
c c
of
@
@,X.
ZFM'
by adding a b s t r a c t i o n
ZFM
X , we
and f o r each s e t ( c Z I x ) - be
We let
) + be X . Then, f o r any p o s s i b l e c o n s t a n t (c @VX a c o n s t a n t of ZFM'
a ,
c
, and
@
let
t h e symbol a- i s
.
If
a
and
b
are p o s s i b l e c o n s t a n t s , we w r i t e ZFM'
If
X
is any s e t , w e say t h a t
are p o s s i b l e c o n s t a n t s with
a , let
Aa
a-b
X
of an
aeA
b i f f a+ = b'
i s e x t e n s i o n a l i f f whenever and
U LA B :p < a] . we mean
N
acX, Then l e t
and
.
Vz(zqa- * s q b - )
then
be t h e s e t of a l l p o s s i b l e c o n s t a n t s
e x t e n s i o n a l s u b s e t of By the
t
a
A
t h e l e a s t ordinal
beX.
a
and
b
For each o r d i n a l
a
such t h a t
be
U {AU:a
(Y
such t h a t
a+ is an
an o r d i n a l ] ada.
.
N.D.GOODMAN
14
We form a theory language of
ZFM* by adding every
aeA as a new constant to the
ZFM, by suitably extending all the axioms and rules of
ZFM
to the new language, and by adding all axioms of the form vxtxTic@-,x where
is a formula of
@
-
$1 ,
with only x
ZFM*
free, and where
is the
@-
formula of ZFM' obtained from @ by replacing each occurrence of a conThe language of ZFM* has a proper class of terms. stant a in @ by a-
.
If
@
is a formula of ZFM*, then ZFM* /-#
We now define truth for sentences of i)
aqb is true iff aeb'
ii)
-16 is true iff
iii)
@
iv)
n @ is true iff
is not true.
@
...,c
ZFM*
and
ZFM*
1 @. ...,y),
a- b.
Suppose @(x,yl,
all of whose free variables are displayed.
eA and @(a,cl, ...,c,)
Moreover, if Proof: -
is true and
@
Suppose a,bsA
formula of
ZFM*
is true, then
1. @(a,yl, ...,y,)
,
@(b,cl,
...,cn
1 @(b,yl,..
then ZFM*
is a If )
is true.
.,yn)
.
We prove the following simultaneously by induction on the structure
.
of the formula @(x,yl,. .,y,) For any
ZFM*
c,,
...,cneA,
...,cn ) 1. Z(a,yl ,...,y,)
only if ii)
ZFM*:
. .
Lemma 1.
i)
.
@-
-B $ is true iff either @ is not true or $ is true. Vx@(x) is true iff @(a) is true for every aeA
v)
cl,
if and only if ZFM'
@(b,cl,
:
the sentence
@(a,c l,...,cn)
is true if and
is true.
..,
@(b,y,
All the cases are straightforward.
,...,)y, . We make essential use of the axiom IX
of modal extensionality. We are now in a position to prove our main result.
...,x
Theorem 4. Suppose @(xl, variables are displayed. then the sentence #(a,,
_Proof: -
)
is a formula of
...,anen .
Suppose al,
...,an )
ZFM
If
all of whose free
ZFM
1 @ (xl,...,x )
We argue by induction on the proof of
@(xl,
...,x,)
logical axioms and rules I-VIII require no new ideas.
in
ZFM. The
We consider the
remaining axioms. Ix.
,
is true.
suppose that OVz(z7a- zqb) and
avc
are true.
We have that
A Genuinely Intensional Set Theory
+,
But if dca
Similarly b+ sional and X.
-
then since Vz(zqa
5 a+, i-
acc
.
+
and so a
z'llb) is true, it follows that deb'.
= b+.
Thus a-b.
Therefore bGc+, and
Suppose $(x) is a formula of
75
But
+
c
is exten-
bqc is true.
so
ZFM* with only z free.
Suppose that
the sentence
VzIVy7x $ ( y ) -. $(x)1 is true. Consider any a that $(a)
acA.
is true.
induction hypothesis, $(b) is true.
vyqa$(y) XI.
Let
a = c#,o '
extensional, acA
.
'tlz(iz7a) is true. XII.
where
#
is true.
ff
of
and so, by the
Then bca',
That completes the induction.
is the formula
Moreover, ZFM*
.
(x E z)
I
Let $
Since 0 is
1. Vz ( i z la) , and the
Thus the sentence Uvz(iz7a) be the formula x
sentence
is true.
f
a
v
x
I b.
Let
(Y
be
Let
X The set X
is true.
is true. Thus we have shown that the sentence
Hence $(a)
Suppose a,bsA.
such that a,beAU.
We show by induction on the rank
Say b7a
= iceAU: c-a
or
.
c-b]
is extensional. Take d = c+-,x. UVu(u1d- u
Z
Then the sentence
aVuEb)
is true. Suppose asAU.
xIII.
X The set X
Let $ = {csAU:
be the formula 3v(Vqa A x I V )
+ cab
for some b o a ' ]
$-,x -
.
Let
.
Then the sentence
is extensional. Take d = c
nVz[z7]dl* 3u(uVa A z y v ) l is true.
XIV. Let
Suppose acAU 9
and
$(x) is a formula of ZFM* with only x free.
be the formula xga A \ ( x )
X The set X
+
= [ceAU: csa
.
Let
and
$(c)
is extensional by Lemma 1. Take
is true] d
=
.
cg-,x.
Then the
N.D. GOODMAN
16
sentence o V x [ x l d - sYqa A q(3:)l
is true. Suppose
XV.
aoA (Y
.
$ be t h e formula V u ( u ~ s +u l a )
Let
X X i s extensional.
Then
= lcaA : c+ (Y
5 a+]
q-,x.
Take
.
Let
.
The sentence
d = c
oVz[zYqd- V u ( ( u l z + u 7 a ) l
is true.
XVI.
c$,o
Let
-
be t h e formula
2)
7 (3:
E
3:)
.
.
It i s t r u e t h a t O v Z ( i Z ~ O * )
6
Let
be
c
I3
Next l e t
aaA,
O*
be
be t h e formula
VuflUYquA VyYqu 3z?lu ~ V O ( V ~ Z0 ( v l y ) V v E y ) If
and l e t
dJ'
-
x9uI
.
aa be t h e formula
let
0
v
(3:Yqa-)
,
a-
3: 3
let
X
a' = c
and l e t
and
ZFM*
1- b l a ,
Then, f o r any
acA,
it i s t r u e t h a t
= [b: E i t h e r Q
a''a
baa'
Let
Y
0
(U'Qa) V v E a )
be t h e l e a s t s e t such t h a t i f
aaY
a, if
and
Using t h e axioms XVZ and XIV of
ZFM' Hence
b-a]
*
oVv((vYqa'-
t h a t , f o r any
or
bma'
,
.
bNO*
then
(5)
then
bay,
Then l e t
bey.
and such w*=c
e ,y
ZFM, we s e e t h a t
1 ~ p o 0 3 : ( z ~ q p8)- .
w*oA.
Clearly
is true.
00*lw*
Therefore it is t r u e t h a t
.
03yYqw* O V z ( i z l y ) Reasoning informally i n
ZFM*
,
that OVV(VYq2
Evidently
zTIw*
.
ZFM*
-
(2,lyI v
2,
(6)
yqw*
suppose t h a t
0
'
E 24)
.
Then choose
Z,
so
.
Thus w e have
vyVoj* 3Z71w*
O&J(V~z*
0
(UTy) v
2, 5
y)
.
(7)
I1
A Genuinely Intensional Set Theory
On t h e o t h e r hand, say
a d .
By (51, s i n c e
a'eY,
it i s t r u e t h a t
3zqw* o V V ( ( Y T Z * o ( ~ v a V) u F a ) . Combining t h i s with ( 7 ) shows t h a t t h e sentence
o v y ' y q o * a z q o * ~ V 2 , ( 2 , ~ z (vqy, -~
= yl
2,
P u t t i n g t h i s t o g e t h e r with (6) g i v e s t h e d e s i r e d r e s u l t .
is true.
XVII.
v
$(y,z)
Suppose
Suppose
i s a formula of
y
with only
ZFM*
and
z
free.
and suppose it i s t r u e t h a t
aeA,
o v y ' ~ a ~ [ + ( y , z ) A Y u ( + ( y , u -) + u
5
z)I
.
Then t ~ ~ y y ~ a ~ z [ ~~j ~( u y ,( z+ )( y , -, u )u :Z ) I
ZFM*
Therefore, using t h e axiom XVII. i n
ZFM*
,
ZFM* t . ; r i u o ~ z ~ z ~3 y u(*y q a A
Let
0
+(y,z))l
.
be t h e formula
.
3.y (.yOa- A +(y,x)-) Then
.
i s a t e r m of
ce
ZFM'
f o r some
X = (bcA:
.
Let
+,
cca
t h e sentence
$(c,b)
is true}.
This set e x i s t s s i n c e t h e sentence
v.yfynaazI$(y,ztAV~u(#(y,uf
+
z)l
u
i s t r u e , and the s e t is e x t e n s i o n a l by Lemma 1. Take
d = c JlPX
it i s t r u e t h a t o v z I z ' l d * 3y(y71aA $ ( y , z ) ) l XVIII.
f o r every
X _C A that
$(y,z)
Suppose
Suppose
i s a formula of
Suppose it i s t r u e t h a t
aeA.
t h e r e e x i s t s some
baa'
be a s e t such t h a t f o r every #(b,c)
ZFM*
with only
such t h a t
bsa+
Then
.
vyTa3zlj,(y,z)
COA
-
.
y
z
and
free.
Then w e know t h a t
+(b,c)
is true.
t h e r e e x i s t s some
COX
Let
such
i s t r u e , and l e t
Y
= {d6A: d-c
is an e x t e n s i o n a l set.
Then
Y
take
e = ce , y .
XIX.
Suppose
f o r some
0
Let
Then it i s t r u e t h a t A
i s an element of
A
ceX]
.
be t h e formula ~ v y q a 3 z ' q e #(y,z)
(
x 5)
,
and
.
such t h a t t h e antecedent i s t r u e .
N.D. GOODMAN
78
This asserts that if and if a+
and
subset of u[a acA+
+:
a
is non-empty, and that if a,bcA+ Hence let X
b+.
=
containing exactly one element from each
consist of all c where
@,Y '
+
then
are not disjoint, then a+
asA+]
and let Y
Let C = c b
b+
aCA+,
pr
such that c-b
.
is the formula l ( z E z)
Xn
be the unique element of
be a
a+ for
for some b
in X.
+.
Suppose acA
Let
Then it is true that
a+.
VZ(Z~CA zTa* z E bt
-
Thus the consequent is true. That completes the proof of Theorem 4. Corollary 1 (Disjunction Property).
1- 0 @ v
and ZFM
ZFM
Proof: Either Since g5
and
.
Suppose
Then either ZFM
or a$ is true.
OpI
#
O$
are sentences of
Corollary
$ are sentences of
1$ . 1 @ or
ZFM
Thus either ZFM*
2
(Existence Property).
Suppose @ ( z ) is a formula of
ZFM'
1 @(a)
such that ZFM'
1- 3 z U p r ( z ) .
.
Since the sentence a ~ o $ ( x ) is true, we may choose bcA
Og5(b)
is true.
4.
k@
Hence
ZFM*
t
#(b)
.
But let a = b-
1 @(a) . The Natural Numbers in Z F M .
ZFM with
Then there exists a term
Proof: ZFM'
ZFM*L$.
it follows that either ZFM
ZFM,
only z free, and suppose that ZFM of
or
1. $ .
or ZFM
a
and
@
1- @
.
so that
Then
The purpose of this section is to show
that we do not need to introduce Myhill's explicitly listed sets or his recursive function symbols into our theory in order to have the numerical existence property. We continue the notation from the verification of the axiom of infinity X V I in the proof of Theorem 4. Let if a is a term of ZFM'
,
w
let ffa
be the term
cB of
ZFM'
.
Moreover,
be the formula
o(z7a)Vx I a ,
.
and let a* = c setting
--* rn = n
Q
For each natural number n , let
Theorem 5 (Numerical Existence Property). ZFM
n
be defined by
a when m = n + l .
with only z
Suppose $ ( z ) is a formula of
free. Suppose that ZFM'
k
bTIGap(z).
Then there is
A Genuinely Intensional Set Theory
a natural number n
such that
ZFM'
t
#(
79
n).
Proof: Let ~ ( r be ) the formula Vu[3yTu o V z ( 1 ~ 1 9 A) V y l u 3~11.4~ V Z , ( ( V-,~ ~0Z( ~ 7 1 V ~ ) Z, :y ) +3~7ju]
.
We have ZFM
1 &[5(X)A
Hence, by Theorem 4, choose asA <(a)
.
0#(2)1
such that
<(a)Ang(a)
is true, it must be the case that a€Y.
a , we see that there is some natural number But
ZFM*
1 @(a)
,
and so ZFM'
t
$(a-)
.
is true. Since
By induction on the rank of such that ZFM'
n
Thus ZFM'
1-
#(;)
a- 1
,
n.
as
required. In order to treat Myhill's recursive functinn symbols, suppose that ZFM
t
where the formula $ ( I , z ) modal operator
0
.
-
~ r 3 y U 0 z ( g ( 2 , z ) z E y) is any formula of
,
ZFM, possibly containing the
Since the identity relation
S
is stable, it will
follow that ZFM
1 g(r,z)
0
#(s,z)
.
Then we can conservatively add a function symbol f
to ZFM with the
axiom
.
U V I f J (Z'f(Z))
Of course, it would not be conservative to add such a function symbol f
we only had the premise ZFM
-
~rXy/z(gl(x,z) z =
j)
if
.
REFERENCES Friedman, Harvey, "Some Applications of Kleene's Methods for Intuitionistic Systems", in A.R.D. Mathias and H. Rogers, eds., Cambridge Proceedings 1971, Springer, Berlin, 1973, pp. 113-170.
z , Myhill, John, "Some Properties of Intuitionistic Zermelo-Fraenkel Set Theory", in A.R.D. Mathias and H. Rogers, eds., Cambridge Summer School in Mathematical Logic, Proceedings 1971, Springer, Berlin, 1973, pp. 206-231. Myhill, John, "Intensional Set Theory", this volume. Shapiro, Stewart, "Epistemic and Intuitionistic Arithmetic", this volume.
This Page Intentionally Left Blank
intensional Mathematics S. Shopiro (Editor) @ Elsevier Science Publishers B. V. (North-Holland), 1985
EXTENDING &DELIS
81
MODAL INTERPRETATION M T Y P E THEORY AND SET THEORY
Andre j &edrov University of Pennsylvania P h i l a d e l p h i a , Pennsylvania U.S.A.
C&el
showed i n ( 1 1 t h a t Heytinq's p r o p o s i t i o n a l c a l c u l u s can be
f a i t h f u l l y embedded i n L e w i s ' modal p r o p o s i t i o n a l c a l c u l u s that
54
S4
, hence
c o n s e r v a t i v e l y contains both c l a s s i c a l and i n t u i t i o n i s t i c Using t o p o l o q i c a l Boolean a l q e b r a s , a similar
propositional calculi.
i n t e r p r e t a t i o n can be extended to the p r e d i c a t e c a l c u l u s without e q u a l i t y i n 161, chapter XI).
(e.g.,
p r o v a b i l i t y of
4 + 04
;
The u n r e s t r i c t e d use of e q u a l i t y leads to t h e
thus one p o s t u l a t e s
only for formulae without 0
, and
f o r a r b i t r a r y formulae. When attemptinq to build type theory or set theory based on has t o be c a r e f u l with comprehension, s i n c e the e x i s t e n c e of for
4
containinq
, toqether
0
5 = 1 + o(x=y) ( c f . 171 p r o v a b i l i t y of
1x1 needs
$(XI}
$ +
04
(21
S4
, one
#(x))
with ( 1 ) l e a d s to t h e p r o v a b i l i t y of
and i n t h e presence of e x t e n s i o n a l i t y to the (cf. [51).
only to t h e case where
(51 4 ( ~ ) ] with
4
Thus Myhill 151 restricts
4
does not contain
containinq
0
.
+(f)
in
However, one
0 t o be a b l e t o extend t h e
i n t e r p r e t a t i o n discussed i n [61 t o type theory and set theory i n a s t r a i q h t f o r w a r d way.
Thus one has t o dispose of e x t e n s i o n a l i t y or qive
a s p e c i a l s t a t u s to "modal" sets.
The f i r s t approach is discussed by
82
A.
Goodman i n [ 3 1 .
~FEDROV
I n the second h a l f of the paper ( $ 3 ) we show that
intuitionistic
can he i n t e r p r e t e d i n a s e t t h e o r y based on
ZF
,
54
somewhat s t r o n q e r t h a n Goodman's. I n t h e f i r s t h a l f of t h e paper, we d e f i n e a n e x t e n s i o n a l t y p e t h e o r y based on
,
54
show t h a t it h a s t h e e x i s t e n c e and d i s j u n c t i o n
p r o p e r t i e s ( $ 2 ) , and show t h a t t h e i n t i u t i o n i s t i c type t h e o r y can he f a i t h f u l l y i n t e r p r e t e d i n it ( $ 1 ) .
To m o t i v a t e this i n t e r p r e t a t i o n ,
c o n s i d e r ( a f t e r Rasiowa and S i k o r s k i ) t h e t o p o l o q i c a l i n t e r p r e t a t i o n of 54
l o q i c , w i t h o u t r e s t r i c t i n q o u r s e l v e s to t h e f i r s t o r d e r case.
Q = P(x) w i t h
t y p i c a l s i t u a t i o n is t h a t of a (complete) Boolean a l g e b r a an i n t e r i o r operation
n = @(x)
I:
I
and t h e (complete) Heytinq a l q e b r a
open i n the topoloqy on
of s u b s e t s of
i n t e r i o r operation
,
Q + Q
.
, as
q i v e n by the
S t a r t i n q f o r example with n a t u r a l numbers
the qround t y p e , we c a n d e f i n e h i q h e r t y p e s as
s e t s , e.q.
The
i n Takahashi [El.
-
N
as
Q-valued (or Q-valued)
For example, the t y p e
[
N1
of
i n t u i t i o n i s t i c ( u n a r y ) number p r e d i c a t e s c o n s i s t s of ( a l l ) f u n c t i o n s
P: IN P(;)
+
A
n
which s a t i s f y the e x t e n s i o n a l i t y c o n d i t i o n
= ;[El
<
P(m) ; the
type
(
EJ)
of classical ( u n a r y ) number
p r e d i c a t e s c o n s i s t s of ( a l l ) e x t e n s i o n a l maps
i s a s u b s e t of ( N) below, n o t of
((
d e f i n e d o n l y on in
((
N))
-
[[
.
n-valued
IN11
i s a s u b s e t of
([
+ Sl
N1)
.
, to
More p r e c i s e l y , f u n c t i o n a l s i n
Althouqh
h i q h e r type
T
if
,
9
N1
N11
are
( u n a r y ) number p r e d i c a t e s , whereas f u n c t i o n a l s On
a i s a n o b j e c t of (Q-valued) p: a + t9(=)j , although n o t e x t e n s i o n a l on
contains
t h e map
[
he c l a r i f i e d
N)) are d e f i n e d on a l l Q-valued ( u n a r y ) number p r e d i c a t e s .
the o t h e r hand,
T
,
R: N
0
and
, satisfies:
so we s a y t h a t
belongs t o the t y p e
-
(TI
, but
n o t n e c e s s a r i l y to
(TI.
I n t h e s e q u e l , we s h a l l c o n s i d e r t h e i n t e r a c t i o n between the h i e r a r c h i e s of h i q h e r o r d e r
Q-valued and
-
Q-valued f u n c t i o n a l s q i v e n by
a3
Extending Godel's Modal Interpretation
-
il
t h e f a c t t h a t operations i n
can be descrihed a s o p e r a t i o n s i n
il
as
follows :
and the o t h e r operations coincide.
Thus on the s y n t a c t i c a l l e v e l we
d e f i n e the followinq i n t e r p r e t a t i o n of i n t u i t i o n i s t i c type theory (formulated without comprehension terms):
where ( i n t u i t i o n i s t i c ) types a r e defined s t a r t i n q from the ground type and forminq
1.
[T
1
,...,'I-n I
Modal type theory
from a l r e a d y formed types
MT
T
1
,..., -n T
(1>
.
We now formulate a hiqher-order
54
c a l c u l u s containinq a l l
c l a s s i c a l , i n t u i t i o n i s t i c and modal hiqher-order p r e d i c a t e s .
1)
A
1
.
A. &%DROV
84
Types:
-A
i s a type.
( u ,...,u
7
either
I7 7
or
i -
1
and
)
n -
1
7
If
1
,..., -n 7
,..., -n 1 7
7
(if
i -
i -
-A is a c l a s s i c a l type. (z > 1) , then (7 1
-
A) 1
7 )
(2>
Lanquaqes: --
T
-
f r e e variables
-
closed variables
-
equality
=
7
T
5 , 7
, T
,1
5
u denotes i -
,
= A
U
-
i -
denotes
7
.
i -
a r e c l a s s i c a l types
n -
7
If
, then IT 1 ,...,7n I -
1)
-i
then
is a c l a s s i c a l t y p e .
n
A is an i n t u i t i o n i s t i c type. -
types
7
if
i
7 ,...,7
,...,-
,
1)
a r e t y p e s ; where
f
If
( 2>
a r e types
are i n t u i t i o n i s t i c
7
is an i n t u i t i o n i s t i c type.
... of , ...
.
7
each type
7
of each type
.
.
7
f o r each type
,..., -n
1
V a r i a b l e s and e q u a l i t i e s of i n t u i t i o n i s t i c type a r e assumed t o be the same
as i n a qiven s u i t a b l e f o r m u l a t i o n of i n t u i t i o n i s t i c type theory. Formation r u l e s : Terms:
-
f r e e v a r i a b l e s of type
7
-
f r e e v a r i a b l e s of type
[7 1
(pl,...,p
)
n -
-
7
denotes
-
i -
if
7. f
-
A
-
..,7
t h e bar i f
If
s
7
i -
and
formula.
t
If
n
,...,u u
each
i -
if
T
i -
.
)
n -
7
.
are also terms of t y p e
7
(Ol
) , where n -
,.
,..., -I
denotes
f r e e v a r i a b l e s of type (rl
Formulae:
"-i
, where
a r e terms of type
7
i -
, and
= A -
"i -
are a l s o t e r m s of type
denotes e i t h e r
-
7
i -
7
or
i -
(omit
= A ). -
a r e t e r m s of type
t
1
r e s p e c t i v l e y , and
t
t
,
s = t
i s an ( a t o m i c )
a r e t e r m s of t y p e s n is a t e r m of type 1u 1
,..., -n ,...,u-n (where
...,-
,
7
7
T
1
)
85
Extending Giidel’s Modal Interpretation denotes e i t h e r i (atomic formula.
IJ
If
-
i s a t e r m of t y p e
t
-
or
Ti
then
Ti
-
IT
1
...,t-n ) i s an
t(t,,
,...
,...., -n 1 , then
,t 1 n -
t(tl
T
a n (atomic) formula. If
4
and
J,
4
V $
4
&@(z) and vzT4
( )~
are formulae, so are
0
A $
I
+
1 4 , and I 3 4 If
+(fiT) i s a formula, so are
T
.
Axioms :
I.
Classical p r o p o s i t i o n a l c a l c u l u s .
3.
Q u a n t i f i e r axioms: T
vz OCZ) + 4 ( t )
is a t e r m of type
t
where
.
res t r i c t i o n s 4.
T
,
s u b j e c t t o the u s u a l
S p e c i a l t y p e axioms:
where
t
1
,...,t
n
r e s p e c t i v e l y , and
t
= t
’A
2
are terms of t y p e s u
T
is a t e r m of type
1
,..., T
[T
1
n
,..., -n 1 . T
$
is
86
A.
t
where
a
t
2
denotes e i t h e r
i -
5.
and
1
SCEDROV
IT
a r e terms of type
T i -
T
if
i -
T
or
f
1 and
n -
1
i -
if
T
5 ,
or
i -
.
= A -
Equality axioms:
where T = [T
u
and
I
..., -n 1 T
1
are terms of type
v
T
f o r some
T
for
T =
I
and
0
n -
1
i s an
a r b i t r a r y formula. u
= v r\ w(u i T i 1 - i- -
b.
each
i =1,
w
and
C.
where
u
(a
1
T
u
and
I
# ( u ) + $(v)
O(u = v) A
,Tilai+l,...la -i-1 - -
or
T
j
.
)
n -
for
T
i '
,
,
a r e t e r m s of type
v
n -
v are t e r m s of type i -
l . . . l ~
T
)
-
and
i -
denotes e i t h e r
j
where
I
)
-
1 -
is a t e r m of type
a
where
...,"
l . . . , ~ ~ ,n. .+. w(u , ~1 l . . . , v i , . . . , ~
T
Cp
I
is an a r b i t r a r y
formula. d.
U = U
T
u = v +
v = u
u = v
V = W
T
T
u
where 6.
I
A
T
T
v and w
+
U = W
T
,
a r e t e r m s of type
.
T
Comprehension schemata:
where no
in and
0 $
0 occurs i n
Cp
, and
types of a l l v a r i a b l e s occuring
a r e c l a s s i c a l (such a formula contains no f r e e occurrence of
Cp
is c a l l e d c l a s s i c a l ) , y
.
Extending GBdel's Modal Interpretation
-
-
1 T nov+ 1
(Tl,...,T
31
b.
IT
3y
c.
1
,...,
1
T
7
-
,...,x
+(lc,
7
1 Vx 1
is
,
...V-X C " _ I-"_ (y(x
9*(5,
x )
, . . . I
,...,7 x
,
))
* - t r a n s l a t i o n of a formula of the i n t u i t i o n i s t i c y
T
... 71n
Vx-(u(zl
u
where
and
T
b.
does not occur f r e e .
,...,71 x
)
-
v(5,
,...,71 x
a r e t e r m s of s u i t a b l e type
v
1 4 .
,v
-
2
a r e t e r m s of type
Axiom (5b) implies
...I
1 7
= [T
-"_
,..., -n I . T
1
u = v /I 9 ( u ) + $(v)
not j u s t f o r c l a s s i c a l
9 , b u t f o r a wider c l a s s of formulae $(u) with no
i n which
occurs only a s a " c l a s s i c a l coordinate" i n subformulae of r e s t r i c t i o n s i n 5b).
9
a theorem of
-
0
9
u
(cf.
axiom scheme 6a could be s t a t e d f o r those
Also,
not j u s t for c l a s s i c a l Obviously, i f
u = v
T
7 u
where
dbkoJ,(x)
))
2
Extensionality: a.
Remarks:
)
-2
T
EOV% 1
type theory i n which 7.
( y ( q, . . . I x
is an a r b i t r a r y formula.
4*
where
7
vxg
-2
Ip
where
...
87
;
0 ,
a l l our r e s u l t s hold f o r t h a t version a s well.
is a theorem of i n t u i t i o n i s t i c type theory, then
$*
is
MT ( f o r comprehension and e x t e n s i o n a l i t y , observe t h a t
U v ~ J I ( ~,)and
and provable i n
- o$*
).
Moreover, i f
0
is c l a s s i c a l
MT, then it is provable i n c l a s s i c a l type theory ( b u i l t
only on c l a s s i c a l t y p e s ) . t h e MT-proof of
+*
(p
, and
To see t h a t , e r a s e a l l
i n a l l formulae i n
a l s o change types of v a r i a b l e s r e p l a c i n g b r a c k e t s
with parentheses and e r a s i n q a l l bars above types. i n classical type theory.
0's
The r e s u l t is a proof
88
A.
~EEDROV * - i n t e r p r e t a t i o n is f a i t h f u l :
I t is not immediate t o s e e t h a t
the
r e s t of t h i s s e c t i o n i s devoted t o t h e proof of f a i t h f u l n e s s .
Definition.
Let
be a complete t o p o l o g i c a l Boolean a l q e h r a with an
51
i n t e r i o r operator
a l g e b r a of a l l f i x e d p o i n t s of
D
=
is a type)
T
T
ui
if
-
- -
i
En-
denotes
Bilyi
Eg-
7
- -
[T i . . . r T
-D
1
(Ill,
n -
ED
-
or. T
= A i -
T
where
1
i -
u
i -
i
A c o l l e c t i o n of nonempty S e t s
51-frame
for
‘ci -
for all
.
I
;
be the complete Heytinq
51
let
is c a l l e d
Bilyi
for all
-
, and
Q + Q
I:
1. =
l,...,~
Ti
-
i=
... -1 ,
1,.
*
; where
u i denotes
and i f
for
if:
..,Z
-
;
’
denotes
ISi
I
;
Yil - i- -
so t h a t
Ill
)I.
where
1 -
(a1
u 1
,...I
otherwise, and
denotes e i t h e r
7
i -
g or
- Eg
-
T
i -
denotes e i t h e r
.
( T l, . . . I
T
i -
7
-
if
.
89
Extending G6dePs Modal Interpretation
1-
I n the above, f u n c t i o n s
I*
Let
=
-A
-1: ax a
=
[ a = a1 A -
la
= a 1 2
la,
4 a2
1
Then d e f i n e
I*
7
g
are d e f i n e d as
+
T
u,v E g
*)
[a
= a ] 1
24
inductively:
and
denotes
Then i t is e a s y to see t h a t a l l
-
iff
1
10
=
0 1
7 = [Tl
,..., -n 1 7
.
i n h e r i t properties ( 1 ) - ( 3 ) .
( f o r i n t u i t i o n i s t i c type t h e o r y ) i s d e f i n e d s i m i l a r l y .
a-frame
In p a r t i c u l a r
be any mapping s a t i s f y i n g :
T
<
= 7
+ n
1
Remark.
x
-i
follows :
where
7
1: g
=
[ a = 61
II
= \ [ a = 61
a
,
where
a16
E g7
(7
intuitionistic
type).
Definition.
By an assignment
a s s i g n i n g elements of
7
0
(W.r.tl
a-frame
0
)
we mean a f u n c t i o n
to a l l f r e e v a r i a b l e s of any t y p e
Then by any t e r m of type
7
and any formula
4
T
.
we d e f i n e
t h e i r i n t e r p r e t a t i o n as p a r t i a l maps:
It!:
V
2
+
DT
(1 is
the set of a l l f r e e v a r i a b l e s )
1
A.
90
S~EDROV
defined on a l l assignments by: T
I t
= t
1 7
2
ICE)
f o r any f r e e v a r i a b l e s
=
\[t,](v)= [t,j(v)l t]
5
T
w
i s not to be confused with topoloqical-
I;!
on
2
)
i s t h e assignment given by:
-a
U
w ( 5 )
a
I n the above,
I-
v(c')
=
v
T
(where the i n t e r p r e t a t i o n
Boolean-valued e q u a l i t y
where
of any type
n
,
i f ( f r e e ) variable
,
i f it is
T
i s any assignment, but
depend only on the f r e e v a r i a b l e s occurring i n
5-
=' T
i s not
. t+](v) and Iti(v) 4
and
t
.
Remark.
A l l w e r e a l l y have to suppose about t h e t o p o l o g i c a l Boolean
algebra
0
is that a l l meets and j o i n t s i n t h e above d e f i n i t i o n a r e
d e f i n e d , and w e s h a l l do so i n t h e sequel.
0
does not have t o be
91
Extending a d e l ' s Modal Interpretation
complete.
We want t o work only with frames l a r g e enouqh to s a t i s f y
comprehension p r i n c i p l e s , b u t a t the same t i m e we d o . n o t want to restrict o u r s e l v e s only to " f u l l " frames ( a f o r every type
$2-frame
9
'
I
0
i s f u l l when
= F
T
T ) f o r the reason t h a t w e w i l l use a Heytinq-valued
completeness theorem for i n t u i t i o n i s t i c type theory ( i n t e r p r e t a t i o n i n
-
&frame is defined as above, droppinq the c l a u s e f o r C
But so f a r we
1.
do n o t even know t h a t the f u l l frame w i l l do f o r comprehension.
However,
t h e following lemmata a r e h e l p f u l :
Lemma 1 . l .
Let
2
[a=Bt A
Proof: Remark.
be any
Sl-frame.
tBq.ul < taql
a,8,y
Lemma 1.1 qenera i z e s obv ously t o
Let
2
be any
$2-frame, and l e t
formula of the modal type theory 1
for any
T :
Eb
Straightforward.
Lemma 1.2.
T
r
Then f o r any type
T
a ,...)a 1 -"_
.
MT
f o r any assignment
.
Then:
...,-" a
+(%T
T
)
be a
with the f r e e v a r i a b l e s T
Consider t h e map:
p a n y "predicates".
1
T
p: 2 x...xD' -
+
S l
given by:
92
A. ~ ~ E D R O V
'I
i
B, E D -
a
f o r any
r e s p e c t i v e l y . Here we allow t h e p o s s i b i l i t y t h a t i' 1 t h e values of some f r e e v a r i a b l e s are f i x e d .
Proof: -
Lemma 1.1
.
Otherwise ( w r i t i n q
that
~ ( 1 5= a
-
I$
If A
-
0
By induction on the complexity of
is
[ O2 ( a ) j is
If
O~ A
0,
;
A
toea)]
for
.
If
is atomic, then use
I#
\$(a')\(v)f o r
such a
considering f o r b r e v i t y only one v a r i a b l e a t a t i m e ) :
O, , we
have
tOl(a) A + , ( a ) l A
i t a = ~ =I
j o l ( a ) {A
1ta=8l < ( i n d . hyp.) < [ $ , ( 8 ) l A \ 0 2 ( B ) ] = t O l ( B ) A 0 2 ( 8 ) 1 , V
0, ,
the induction hypothesis gives f o r
tOiCaq A 1la=8] < tO,(8)]
-
-
< tO1(B)]
V
= 1,2
102(8)1 = ! O l ( B ) v b 2 ( B ) j
so the claim follows takinq the j o i n of t h e left-hand s i d e and usinq distributivity.
-
If
41 is
0,
+
0, ,
the i n d u c i t i o n hypothesis f o r
IJsing the induction hypothesis f o r
1
(I
2
gives:
i n t h e similar way, w e g e t :
,
Extending Gddel's Modal Interpretation what had t o be shown.
-
a
vz $ ( ~ , r f ),
$(E) i s
If
The case
=l$
then f o r any
93
is similar.
a y E D
, we
have:
so the claim e a s i l y follows by takinq the meet of the riqht-hand s i d e
.
a
over
2
-
@(a)i s
If
$(x,=), the
a 3~
/$(r,a)l
ila-el
c
induction hypothesis gives:
t$(y,5))
<
v
tq(6,~)j
6€Ea for
a
E
y
,
so the claim follows hy taking the j o i n of t h e left-hand
0
0
s i d e over
-
If
Q
and using d i s t r i h u t i v i t y .
is o $
hence applying
Remark.
, the
1
induction hypothesis qives:
t o t h a t inequality:
The i n s p e c t i o n of the preceedinq proof shows that i f
$(a)
s a t i s f i e s t h e r e s t r i c t i o n s discussed i n the remark on axiom 6a ( i n particular i f
Lemma 1.3.
Let
$
is c l a s s i c a l ) , then
s-2
P(a) A I
Ia=El <
be a t o p o l o q i c a l Raolean a l q e b r a ,
a l g e b r a of a l l f i x e d p o i n t s of the i n t e r i o r operator
P(B)
-
s-2
1:
.
the Heytinq s-2 + S 2
,
p -t-
-
En
94
A.
.
IET
for all
9-
Join
t
EEDROV exists.
pz
exists iff join
In
tET
a
that case:
L
exists, then meet
+g
If meet
r\" E._
exists and:
t€T
tpr -
Moreover, for any
-
z.2 E n
:
-
P :3 n
=
I(E
; 3)
n
and
1~
n
=
1 - r ~
I
and all other operations coincide. Proof:
Straightforward computation (cf. Rasiowa-Sikorski [ 6 1 ) .
Lemma 1.4.
Let
-
be any
intuitionistic type theory.
Then for any
for every intuitionistic type
T
, and
-
D
1 is any assignment w.r.t.
where Proof: -
+
n-frame and
,...,an))H - -
similarly for
g
T
for
and
-
.
g
such that
7
2
H
= :
extending 1 (as a function).
0
.
We again write T
7
1 I+(=, ,...,-"_ a "))(XI
1
O-frame
any assignment 1 w.r.t.
By induction on the complexity of
/@(a,
any formula of the
where
v(a
i
-)
-i,
= ai
-
, and
1
5
ExtendingG(jdePs adel's Modal ModalInterpretation Interpretation Extending
BBasis: asis:
10(a,
is aatomic, is t o m i c , tthen hen
$$
If ~f
...., . a
) =
t3B(al
,..., ... , a--nn )I ]f-H-H
\\E(al 0(al,
,..., - - .
-
2''
B(a == B(a 11
a
A
a,$ €
I£
an)lD
If aa,,@ @ E Z t T 1,, where where rT ~f
== ala=BIp
95 95
,...,a ,...,
a ) ==
t a d \ H=
then
-
1
@
is
1
= ( i n d . hyp.)
A
=
then
2 '
I@*{ );(
A 421!(~)
A (4
12- n
-
1
2 2
(I)
=
I@ I ( v ) hn 1;A \$
= \@*I );(
1
g- n
*
I@211(~)
1 (x)
2 g
vV and
and s i m i l a r l y for for
-
~f
0 is 4
+ +2
+
*2
then
.
i 1 is
4
+
(;I)
2 -
=
to($;
) =
@
I D -
4 2 11D(X)
=
1
below.
(v) h2i,(~) 1 3 - H
= 1 t@*l ( )
-
+
+
t r e a t e dd as as
j
* I@*1 I 2 (i) ( 4 2~1 ( x ) - fi
= ( i n d . hyp) =
= I
.
3
[a=!3ID
-
is aan then: is n iinnttuuiitti ioonni isst ti icc ttyyppee,, then:
I n d u c t i v e step:
-
1
( 4
+
n
+
*
-
t(2\D(~)
-
-
4 ) ID(X)
A. QEDROV
96
-
=
I
n" Y €DT
U
~ ( )5 , where
c'
if
5
U
is not
T
5
f o r any f r e e v a r i a b l e
w (5 ) = 7
T
of the i n t u i t i o n i s t i c type theory, and:
f o r any f r e e v a r i a b l e
5'
of the modal type theory.
Definition:
1
n-frame
D is
s a i d to be s a t u r a t e d if the mapping
T,
T
for any assignment 1 (where occuring in
4
)
element of
1
D
-
b)
a ,...,a' 1 -"_
are a l l f r e e v a r i a b l e s
is: (T r...rT
a)
T
1
element of
g
(T ,...,T 1
)
-
if
4
is c l a s s i c a l
if
4
possibly contains
)
-
0
91
Extending Godel's Modal Interpretation
"I element of
c)
D
r...r'T
1
1 -
if
Q
is
JI* f o r some formula
JI
of the i n t u i t i o n i s t i c type theory. ( h e r e we allow t h e p o s s i b i l i t y t h a t the values of some f r e e v a r i a b l e s a r e f i x e d .I
Proposition 1.1. provable i n
Proof: -
MT
.
Let
g
be any s a t u r a t e d
Then f o r any assignment
4
O-frame and
a formula
:
Straiqhtforward.
In t h e sequel, we make use of the following f a c t ( c f . e.g.,
Takahashi 181
f o r a somewhat d i f f e r e n t system):
P r o p o s i t i o n 1.2.
be t h e i n t u i t i o n i s t i c type theory (with
IT
Let
e x t e n s i o n a l i t y ; formulated without comprehension t e r m s ) , formula of
IT
Heyting algebra H -
n o t provable in C
, saturated
IT
.
C-frame
let
$ i
he a
Then t h e r e e x i s t s a complete
fi
w.r.t.
and an assignment
such t h a t
Remark.
Such a completeness is u s u a l l y proved for
comprehension terms).
Proof:
Let
A
But
IT'
is conservative over
he the Lindenbaum alqebra of
IT'
complete, a l l meets and j o i n s needed t o i n t e r p r e t a r e countable, a s is (e.q.,
A
itself.
i n 1611, there e x i s t s a s e t
a l g e b r a monomorphism
If:
A +
(i.e.,
IT'
.
IT
.
IT
Although i t is not
3 and
v
e x i s t and
By a well-known theorem of Sikorski
X
with
of i r r a t i o n a l s and a Heytinq-
@(x) p r e s e r v i n g a l l connectives
and
98
A.
~EEDROV
' 3
be the c o l l e c t i o n of a l l terms of can be made i n t o a
type)
t
function . . A
t(ul
t = t
by
,...,u-n
= h(Rt(u
)
,...,u . we
,...,
TO'
o f type
{ 3 ' I
=
is a
7
T
of t y p e
t
Let
defines a
A , otherwise:
7 =
1
IT'
.
C =
Thus t a k e
every t e r m
C-frame:
if
.
A
r e l e v a n t i n f i n i t e meets and j o i n s i n
) I ) , where
Rt(u
1
,...,u-
is the class of
)I
a
u in A have to check that each t is w e l l - d e f i n e d , n 1 by i n d u c t i o n on the type of t if 7 = A t is j u s t t Suppose t(u
,..., -,v ,..., -n ..., . -( a ,...,a ..a
u
u
1
n
i = I,
)I
t(v
2
C
-
= vi
v ) n A
we have
t(u
Now l e t
=
1
a
1 1
n
*
) =
{I7 I 7 71
A
L
1
n
A
*A
that
. E
be a l l f r e e v a r i a b l e s . BY e x t e n s i o n a l i t y ,
)
, so
)
Then
IT'
*
t
I-
t(u
,...,v-
,,...,- .. u )
.
A
in
)I
Therefore,
is a w e l l - d e f i n e d f u n c t i o n . 7
, E
T
f o r each type
1
= { t(
t
I t1
.
.
r= '21
=
h(Y t 1
= t 7
2
I)
f o r each t y p e
By the use of e x t e n s i o n a l i t y and e q u a l i t y axioms, one can check
is indeed a s a t u r a t e d
stitution.
I n particular, l e t
free variable
tti(1) =
a
2
Let
men it i s e a s y to check t h a t 7
j
, where
is a type)
.
2
of a s u i t a b l e type; s i n c e
u )I = n t ( v 1 n
1
)I =
n
-
2
,..., t ( v ,...,v -
lt(u
as f u n c t i o n s f o r
Ilu. (al,...,a
.. -( a ,...,a -j.
,...,u.-
i s a t e r m of type
= v
1.i
Thus by e q u a l i t y axioms,
, i.e.,
,..., L
.
u
a
f o r a l l terms
v
i
A
,...,11 ,
In particular, let
,...,a a
ui(%
t ui 1
A
in
i s a monomorphism.
IT'
= 1
n
1
IT'
are w e l l d e f i n e d and
Thus f o r a l l
lu.
1
L
v
1
.
L
,
s:
a'
, for 1
i
each t y p e
be the assignment 7
, and
f o r any comprehension t e r m
i s a sub-
Any assiqnment
C-frame.
t
T
i(a.1 2
AT = a
%
f o r any
d e f i n e the i n t e r p r e t a t i o n by and any assignment
1
.
Then
99
Extending a d e l ' s Modal Interpretation Now we are i n t h e p o s i t i o n t o prove our main r e s u l t :
Theorem 1.1.
Proof: -
4 he a formula of
Let
for some assiqnment
T
-
Proposition 1.2.
Now l e t
[61).
r
T
-
and l e t
2.
[#*),(:)
(i.e.,
Sl
2
1 w.r.t.
,
if
T
C-frame
IT
# T
I
a s i n the
S2
w.r.t.
I
i
,
S2
cf.
S2-frame defined by:
is i n t u i t i o n i s t i c type
otherwise,
be any assignment w.r.t.
-
is not provahle i n
the set of "opens" of
be a s a t u r a t e d
2' , F
-
Then
There e x i s t s a complete t o p o l o g i c a l Boolean algebra
is
C
such t h a t
.
If 4
One d i r e c t i o n is a l r e a d y proved.
t o ] H (-v ) #
1.4,
IT
, so +*
R e a l i z a b i l i t y for
2
.
extending
is not provable i n
MT
By lemmata 1.3-
(by Proposition 1 . 1 ) .
M T
Here we modify so-called Friedman r e a l i z a b i l i t y (developed f o r i n t u i t i o n i s t i c systems i n [ 2 1 and [ 4 1 ) t o prove t h e d i s j u n c t i o n property, i.e.,
where
0 and J1 are sentences of MT
where
k
T
3~ 0
3
, and 'I
MT'
MT
the e x i s t e n c e property:
@(t
is closed, and the system
35Ta#()'
assume t h a t i n
@(zT)
MT
for some closed term
MT'
is defined as follows:
t h e r e are countably many c o n s t a n t s 0
W e could make t h i s p r e c i s e by considering a system
t
MT
3
of type
A
obtained from
. MT
100
A.
EEDROV 0
by adding these c o n s t a n t s , and work with
throuqhout;
MT
but prefer MT
.
a
,
0
not t o complicate n o t a t i o n . 3~ 0
Then whenever
Note t h a t
is conservative over
MT
% ...* x ( y ( x l...l~ 1 .. +(x.,, ..., 1 ) -- -1 7!
5
is an
Y
,...,
11
i n s t a n c e of a scheme of qroup 6 and
@(q,...,a
v a r i a b l e s of
-"_
a a a r e t h e only f r e e 1 -2 introduce a new constant c of t y p e
)
0
and add the d e f i n i n g axiom:
L e t us c a l l the r e s u l t i n g system
extension of languaqe
obtained from
MT
a
(with c o n s t a n t s
MT
+
i -
n-tuples -
of type
MT'
c
comprehension c o n s t a n t s
X
c o n d i t i o n s on
for
c
1 by adding
i -
of type
where
T
c
0
1
(i.e.l
T
MT
+
or indexed
i -
1 of a p p r o p r i a t e type.
is a
of ordered
a
constants
t o be a t e r m of
0 111
a
i s a s e t of type
MT
.r
JI
0t
+
of p o s s i b l e t e r m s of
W e s h a l l a l s o need another
).
c
and
T
I t is a conservative
(with c o n s t a n t s
MT
"indexed" comprehension c o n s t a n t s c o n s t a n t of
.
MT'
We d e s c r i b e the
:
Definition.
a
Let
i
- ah
of type
f
i=j.
T = [T
and
T
c0
MT'
denote
1
cJI and
-
+
-
-
i
11
We say t h a t a set T
;
0
, then
0lX
-
T
e r r
c
(c
0 tx
of ordered
means
c
'I
~ I Z T qiy
111
of type
-
c
-
(c
are p o s s i b l e terms
JI ty
c
otherwise
0
c
and 111
means
x=x .
I+=&
;-tuples
of p o s s i b l e
T
(% 1 ,
terms
c
, . . . l ~I n
?=? I
( a , ) = (a.) = a
Let
If
-1) of qiven types -1
is e x t e n s i o n a l if
t l...lt I
-1
e i t h e r of the following holds: T =
T
i -
(a
1
or
,...,u -1 T
i '
Let
and for each
2
=
{L E
{l,
IE
{ 1,
...,2) I
...,3 u
i -
a
i -
denotes
denotes e i t h e r
-
T
1 ,
i -
101
Extending a d e l ' s Modal Interpretation
J =
1,.
..,;I\;
,...)71t
(%
)
t u 1 72 2
.
-
1
+ -ti .
MT'
= u
Let
(U , . . . r ~
...,-1 E ,...,1) . t 1
(t
)
-1
1
Let
E1,
for a l l
-A
T
Then
{lr...r;}
I
7 = f7 ' ...'T
-
E x ,
iE
for all
-
(one of them miqht be empty).
t ,. u -i 7 -i - i- -
and
E5
and
I-
MT'
t
1. E
for all
11
Then
...,nu-
(u -1
-
u
=
-1
-
.
E5
)
and
-2
i
+
MT
W e d e f i n e c l o s e d terms of
+
MT
Closed t e r m s of
d)
c
a l l constants
-
c
-
if
9 '11
1
= ( a l '...,a
7
n -
of t y p e
A
denotes e i t h e r
T
are a l l c o n s t a n t s T
of type
a
-i
of t y p e
, where
each
a
-
i
T
+
t
i -
of
MT
if
=
7
7
[T~,...'T
of .type
i -
n t -) n
of c l o s e d t e r m s
,... -
Q
For each formula
7
' t % of n
5
( w e s a y that
is an
occuring i n
of
+
MT
of
t
-i
t
,...,-n
+(tl 7
let
of type
MT+
t
;
7
i -
z-tuples
, and
(we s a y t h a t
41
9
-
be a formula of
MT'
.
Conventions on t y p e s made i n 0 1 . extend t o
4 be a s e n t e n c e of
+
MT
.
MT'
and
W e d e f i n e the n o t i o n " (0
R(+) ) i n d u c t i v e l y as f o l l o w s :
is a
).
4 by e r a s i n g s e t - i n d i c e s o f c l o s e d t e r m s of
o b t a i n e d from
write
i ' -
,... -
is an e x t e n s i o n a l s e t of o r d e r e d
(t ,t E i m p l i e s MT' n 1 s t a b l e e x t e n s i o n a l set of type
e)
7
7
T
..., -
1 (t, ,
5
I
n -
1 (tl
_?-tuples
or
i -
T 1
e x t e n s i o n a l set of type
-
+
, and
MT
7
X is an e x t e n s i o n a l s e t of o r d e r e d closed t e r m s
of
such that:
MT'
i s a c o n s t a n t of
9
inductively.
MT
+
.
+
MT
Let
is r e a l i z a b l e " ( w e
102
R!EDROV
A.
R(t
= t 1 u 2
t - t
3
1 u
2
The main r e s u l t of this p a r a q r a p h is:
Theorem 2.1.
R(O1
Let
Q
be a s e n t e n c e of
MT
.
Proof: -
.
Then
By i n d u c t i o n on the l e n g t h of t h e proof o f
MT
+
$
.
implies
W e only check
a l l g r o u p s of axioms which are n o t i m m e d i a t e l y r e a l i z a b l e , s i n c e the
r u l e s of i n f e r e n c e are p r e s e r v e d :
Group 4b.
w h e r e each
i.e.,
MT*
For the l a s t o n e , l e t
.
u
i -
and t be c l o s e d t e r m s of t y p e 1 2 Then b o t h of t h e m are a l s o of t y p e a = (u ..,u 1 n t
,.
denotes e i t h e r
t, = t T 2
+ and
t
1
T
i +
or
.
R(tl
= t )
= t . 2
other h a n d , the axiom i t s e l f qives
A
MT'
;t 2 )
t
+
+
is j u s t
- ;t2- .
tl
means
So,
t
1
= t
2
R(O(t,
-
1 T
.
t
2 '
on the
;t Z ) ).
103
Extending Gsdel's Modal Interpretation Group 5a.
,
T #
c l o s e d t e r m s of t y p e
,... .
.
+
By i n d u c t i o n on the complexity of
...,t-
,v
u
Let
be any
,
any c l o s e d terms of t y p e s n some of the f o l l o w i n g cases do n o t o c c u r . ) t
1
,T (If t = A , 1 n S i n c e we wurk i n a classical l o g i c , i n t h e i n d u c t i o n s t e p it s u f f i c e s to T
check o n l y
,
A
' I
,
Basis:
Case 1:
is
#(u)
( tl,...,t
n -
~
u
..., - .
t ) Then n +i f f ( tl,...,t ) ~ v n
u(tl,
, where
u = w 7
Then
R(u = w)
MT'
axiom i t s e l f , +(u)
Case 3:
a = (a ,...,a
)
n -
1
+
=
+
w
+
is
u = w
which i n c l u d e s u
+
#
metatheory, R(4 (v))
.
Case 2:
#
2
and qives
R(v=u)
is
MT'
#,
u ~
-
w
A +2
R(u=v)
is
-rl,
,a
+
+
.
+ u
and
, So v
+
= v
+
=
=
+ w +
.
so
.
R(u = v) 7
, and
w
7
.
by the
.
R(v = w)
-
denotes e i t h e r
i -
+
is some c l o s e d term o f type
w
.
;v
-
+i f f
.
and
Let
(using M).
R($(u))
= v
, i.e.,
a
u
v)
R(U = T
7
i -
implies
u
.
or
+
= v
+
Then
, so
w)
Induction step: Case 1:
= w
+
. w ( t , ,...,u ,...,t . -n+ a
n+ 1
u
, where
each
u
R(V =
,..., ,...,t-
(tl
v
, where
is
4(u)
- ;w-
MT'
, i.e.,
Case 4:
.+
- u ;w
and
'I
means j u s t
R ( U = w)
v
- u = v
implies
7
is some c l o s e d term of type
w
MT'
means that
7
means that
R(u = v )
-
is
#(u)
C a s e 2:
.
and 0
3
1
.
)
R(U = v) 7
w+
Since
means
u
7
v
,
is e x t e n s i o n a l , we have
...,v ,..., -n+ t
(tl,
.
E w+
1
By the i n d u c t i o n h y p o t h e s i s and l o q i c i n the R(# (u)) 1
R(u=v) So i f
contradiction.
and
and
R(#
2
(u))
R(SI,(u))
R(Q(v))
, then
R(#
.
1
(v))
Then n o t
and
R($(u))
the i n d u c t i o n h y p o t h e s i s
A.
SLTEDROV
Let
R(u=v)
104
Case 3:
$(u)
R($(W,V))
u
+
+
= v
w
f o r some c l o s e d t e r m
R($(w,u))
Case 4 :
.
P
3~ $(u,x)
is
8
$
,
SO
P R ( 3 z $(lfrVI)
is
a$ ,
Let
$ (u)
MT'
of
. R(u=v)
and
p
.
and
P
R(32$((U,x))
.
Thus
By the i n d u c t i o n h y p o t h e s i s ,
R(O$(u))
.
Thus
MT'
-
,
= v
u
R ( $ ( u ) ) , By t h e i n d u c t i o n h y p o t h e s i s ,
and
The r e a l i z a h i l i t y of axioms 5b and 5c is proved s i m i l a r l y .
C a s e 4 of t h e
h a s i s i n t h e above proof shows t h e s i g n i f i c a n c e of t h e r e s t r i c t i o n s made As remarked i n 51) 5b could be'extended
i n Sb.
classical
.
4
t o a t least a l l
The r e a l i z a b i l i t y of Sd is s t r a i g h t f o r w a r d and it w a s
a c t u a l l y p a r t l y proved above.
Group 6:
W e have t o show t h e r e a l i z a h i l i t y of
Comprehension schemata.
e v e r y i n s t a n c e of schemata 6a-6c.
a.
Let
of
,...,a
a 1
MT
terms), T
n+l -
+
be a l l f r e e v a r i a b l e s of a classical formula
2+E
(which might c o n t a i n c o n s t a n t s
a
i -
b u t no comprehension
t ) . . . , t of MT+ 1 m r e s p e c t i v e l y , we want to show:
For any c l o s e d terms
,..., -n+mT
Suppose t h a t a l l of
tl
,...]-m t
of types
are comprehension terms of
+
MT
;
from the subsequent d i s c u s s i o n it w i l l be clear what to do i f some of t h e m are c o n s t a n t s
a j
MT'
(T
t e r m of
of type
Thus w e c o n s i d e r :
1
. F i r s t of n o t e t h a t ,..., -n , s i n c e t ,... -m all,
T )
1
c
4
is n o t a
occur i n
0
.
105
Extending Girdel's Modal Interpretation 7
7
7
1 + ' ( a ,...,a 1 -1
: =
3%-
7
...-m_- -(+(a
n+ 1
n+m
Y
vq ...vrn
A
1
c
is
t
where
Qi
1
"7 ,...,
X
5
type
c
= {(u
.
7i}
-
,...,u ) I
Moreover,
u
i
-
v
-i Ti - i
= v
c
un,tl
).
J -
Using 5b and
,...,x- -
MT'
for a l l
i=
of
m
MT*
, we
i -
i s of
have t o
X
Since t h e types under c o n s i d e r a t i o n
..., - ,... -m
R($(u1,
u ,t ,t )) n 1 ( i n t h i s case this means just
l,...,~
i =I,...,;
u
and
t
is a term of
,
0'r
extensional.
for a l l
i
a.
"
a r e a l l c l a s s i c a l , t h i s j u s t means t h a t :
u
is a c l a s s i c a l
I
,..., - ,..., -1 )
R(+(u~
1 To show t h a t
5 is
-1
x (+'(x ) 1 n c i s a term Of
-
n -
1
,...,a r 1)
and t h e r e f o r e :
,...,? )
show t h a t
and
+
...,-n-
h1
MT'
,..., Jim 1) .
,c
-2 4J1
Let
.
( p o s s i b l y containing c o n s t a n t s
MT
$,(q
-1
f&
7a one e a s i l y shows t h a t
(7
,...,a r 1 -
(y (r 1-1
L = 1 ,...,m
for all
,Y ,...,y 7.l2
- -
formula of
type
,,...,a
) imply
R(+(vl,..., v n , t l
-
I . . . ,
t 1)
-
.
This is the consequence of the r e a l i z a b i l i t y of 5b, as remarked Since we have e s t a b l i s h e d ( 2 ) , to show ( 1 ) i t s u f f i c e s to
before.
show t h a t f o r all closed t e r m s T
1
... -
,
,7
n
. . . r ~
,..., u
n
of
+
MT
of types
respectively:
b u t this is obvious, s i n c e (U r 1
u1
1 E n -
5
.
R(c
(u
+',5
1
... -n
,
,u 1 )
means
A.
106
6b. -
~CEDROV 0'
We have t o be more c a r e f u l now i n d e f i n i n g
;
s i n c e i n this case
5b does n o t s u f f i c e , we u s e 5c i n s t e a d ( t o g e t h e r w i t h 7 a ) . T
T
+I(%
1
T
,...,a')
%-
: =
71
AOvr
n+l
...n
i
f
T
...11--(+(a, ,...,21,...,"_ Y
n+m
( y (r
1
i l
Then we have the analogue of ( 2 ) . E x t e n s i o n a l i t y of
5 is a
a ,Y
,...,I-
a
of
6c.
)
y )
.. +J1 ( r1 , . . . , ra
))
-1
-1
5 as
We define
before.
weaker requirement and c a n be s e e n
d i r e c t l y from the r e a l i z a b i l i t y of 5 c .
c
Therefore
4
A
Let
-
is a term 8
5
and it is a q a i n s t r a i q h t f o r w a r d to g e t the analogue of ( 1 ) .
MT
W e show the r e a l i z a b i l i t y of:
,...,X
where
is the
$*
T
-m
,
* - t r a n s l a t i o n of a formula of t h e i n t u i t i o n i s t i c
. So l e t
,...,z-1)
This is enough, s i n c e ( 3 ) and 6c are i n t e r d e d u c i b l e i n
type theory. 54
,z
-n- -1
t
,...,
T
n+l n+m i = 1 , . .,m -
. .
1
,...,t-m
be any c l o s e d terms of
res p ecti v el y , say
t
i -
is
MT
c
+
o f type
f o r each
W e want to show:
-
+*; (
,...I
x ,t 7 1
,...,tm 1 ) ) -
107
Extending Codel's Modal Interpretation
Define: T
1 +'(a 1
T
'c
,...,-"_ a 2)
n+ 1 315-
:
=
A
nvr
1
T
*
n+m ...2 y
...
I:
-n
--(+
(y (r 1 -1
I
,...,n-1 a , y ,...,y 2!
(a -1
,...,nr )
-1
--
J, ( r ,...rr )) 1 -1 -rl -1
-
nvq ...a z ( y (z ,...,z ) 3 -1 'h
A...A
U s i n g 5a and 7b ( n o t e t h a t a l l
J, (z m l
? ! -
are
J,
i -
.
k
$'
a
),
we get:
for some i n t u i t i o n i s t i c
$*
z
a
$
j
, possibly
)))
.
?!
*-translations
i n t u i t i o n i s t i c formulae, p o s s i b l y with constants MT
,...,a of
, and with constants
8_
Now l e t : X = -
)I
R(O$*(u
{(u r...ru 1 -
1
,...,-
u , t ,...,t 1 ) m n 1 -
and
c
To show t h a t
JI
*x is a
e x t e n s i o n a l and s t a b l e . r e a l i z a b i l i t y of 5a. Then MT'
k O +
-i
MT'
is of t y p e
, we
* (Ul
To show s t a b i l i t y , l e t
un,tl,
-1
. 5 is
5 f o l l o w s f r o m the
E x t e n s i o n a l i t y of
- r u ,t n 1 -
T ,}
have t o show that
,... ,..., -m - ,..., - - ..., - . -
t +*(u;
MT'
term o f
u
(u r...,u 1
)
-n
5
.
t-) hence also
t )
MT'
By ( 5 ) ,
-
k J,(Ul , . . . r U
)
-
.
The r e a l i z a b i l i t y of 6 c can be shown d i r e c t l y , u s i n g the f a c t that R(@*)
iff
R(o+*)
for i n t u i t i o n i s t i c
+
.
108
A.
SEEDROV
It i s r e a d i l y checked t h a t both versions of e x t e n s i o n a l i t y are
Group 7:
realizable.
The followinq c o r o l l a r y holds f o r
a)
Corollary 2.1.
t+
MT
b)
or
Disjunction property:
b J,
MT
Existence property:
term for
a)
If
.
t J,
Remark.
MT
But
if
t 04
if
such that
.
@(t)
MT'
-i
#A
v
OJ,
I
then
R(O+ V O J , ) MT
.
, so
One can a l s o add "indeterminates" = l,.,.,~
€or
=A ,
(If T
4
MT'
to
)
MT
c
1
. -'
or
T
,
..,c
n (with or without c o n s t a n t s
thereby g e t t i n g its conservative extension
MT+
.
1
Defininq
(where
a
j
from
MT+
-
= c i A'
MT'
and
MT*
C
were defined from
, and
MT
lettinq
C
C
c
as
),
and
MT:
C
MT:
only
a ). j -
T
T
then
I
a can be thouqht of as "indeterminates" i n t h e j
Constants
s t y l e of [ 4 1 .
OJ,
Xz'O+(x) ( c l o s e d ) , t h e r e i s a closed
is conservative over
MT'
b O+ v
MT
c
+=f
i
;
one then d e f i n e s r e l i z a b i l i t y analogously and proves
the soundness theorem.
Since
MT'
is conservative over
MT
I
we have
I
h t h e following:
Corollary 2.2.
Let
+
and
J,
be formulae of
c l o s e d , but with no parameters of type
A
1.
MT
If
MT
(not necessarily
04
V O$
I
:
i
a r e closed).
J,
MT
T
of type
MT'
with c o n s t a n t s
MT
roof: MT'
of
t
9 and
(both
a
with o r without c o n s t a n t s
MT
then
I09
Extending Gtidel's Modal Interpretation
+
MT
MT
or
6
.
n % 1 ,...,a -
Let
T
... -2
06) ,
V
x"(D+
then
T
T
Proof:
T
bv+ 1
MT
Hence, if
be a l i s t of a l l p a r a m e t e r s o c c u r i n q i n
0 and
-"_
6
T
.
Then a p p l y the above remark to
C o r o l l a r y 2.3. 'I #
A
MT
I- Vx 3~
Proof:
T
A
'/lr 3-~@(x,y)be
o + ( ~ , y,) then
Add i n d e t e r m i n a t e
2.1.
If
that
t
MTl
T
c
T
to
MT
V D$(C
A
.
T
1
1
a
j
,
...,c-n1' .
.
Let
If
.
T
ZI-WE +(x,y) MT
and u s e t h e analoque t o C o r o l l a r y
t
MTl
of
is some c o n s t a n t
c ; T
T
...,c-n-?I
,
a s e n t e n c e of
t
MT
1
with constants
+ ( c T , t ) f o r some term
c a n n o t depend on
of t y p e
a
j
.
A
But
, notice
+(cT,a ) -j
The r e s u l t s of $ 2 ( b u t n o t of $ 1 ) e a s i l y extend to the
Remark. system
A
MT
Consider t h e system
and l e t T
O+(cl
MTA
o b t a i n e d from
MT
with constants
axioms for the ground type El = A
.
a
by adding Peano j I n p a r t i c u l a r , t h e i n d u c t i o n scheme
i s g i v e n by:
where
A
i s any formula, and
MTA
c o n t a i n s a unary f u n c t i o n symbol
whose arguments and v a l u e s are of type
3.
Interpreting
ZFI
N
.
a 's
L
set theory
are numerals.
i n an i n t e n s i o n a l s e t t h e o r y
I n [ 3 ] , which is assumed to be known, N.D. Fraenkel-like
s
ZFM
based on
54
Goodman develops Zermelo-
logic, but lacking f u l l
110
A.
extensionality.
Rut
ZFM
VEDROV
d o e s c o n t a i n an analogue of " i n t u i t i o n i s t i c
e x t e n s i o n a l i t y " 7b i n t h e modal type t h e o r y I t w a s shown i n [31 that
ZFM
MT
c o n s e r v a t i v e l y extends classical
t h e o r y , by d e f i n i n g e x t e n s i o n a l e q u a l i t y i n
ZFM
c e r t a i n e q u i v a l e n c e r e l a t i o n , and d e f i n i n g t h e so as t o s a t i s f y f u l l e x t e n s i o n a l i t y .
i n t u i t i o n i s t i c set theory
developed i n 11.
set
by r e c u r s i o n as a &elation
simultaneously
One is tempted to do l i k e w i s e with
ZFI developed i n [ 4 ] ( i n t e r p r e t i n g , of
c o u r s e , i n t u i t i o n i s t i c sets as " h e r e d i t a r i l y s t a b l e " sets i n t h e n a p p l y i n g the
ZF
*-interpretation).
ZFM
and
However, a l t h o u q h i n t e n s i o n a l
i n t u i t i o n i s t i c C o l l e c t i o n s u f f i c e s t o prove i t s e x t e n s i o n a l c o u n t e r p a r t , t h i s is
g
This is j u s t a n o t h e r r e f l e c t i o n of the
t r u e f o r Replacement.
u n p l e a s a n t qap between Replacement and C o l l e c t i o n i n i n t u i t i o n i s t i c s e t t h e o r y , which h a s a l s o s u r f a c e d i n
ZFM ; i t seems t h a t addinq "modal
collection":
would s t r e n g t h e n the system ( a l t h o u g h XVII and XVIII are a l r e a d y i n ZFM), and i t is n o t known whether the r e s u l t i n g system s t i l l h a s for example the d i s j u n c t i o n property.
I n p a r t i c u l a r , it is n o t known whether modal
c o l l e c t i o n is t r u e i n the s e n s e of 131. Another d i f f i c u l t y is t h a t
n-induction
in
ZFM
does n o t s u f f i c e
(or a t least it is n o t obvious t h a t i t d o e s ) t o prove i n t u i t i o n i s t i c €-induction.
Thus, c o n t i n u i n g the numeration of axioms of
ZFM
i n 131,
w e add t h e new scheme:
with
Cp
possibly contining
0 ; and c a l l t h e r e s u l t i n g system
Adapting the n o t i o n of t r u t h from [31 t o
MZF
, we
MZF
.
have the f o l l o w i n g :
111
Extending Codel's Modal Interpretation P r o p o s i t i o n 3.1.
Proof:
Let
Every i n s t a n c e of
,...,2x ,..., - E
+(x 1
+(%,
Let
...,2 - . t ,x)
i.e.,
be a formula of
,x)
t t 1 -n 131, b u t r e f e r r i n g to
free.
is t r u e .
Xa.
MZF
with only
x
,...,
x ,x
-2-
-1
A'
A'
(where
i s d e f i n e d a n a l o g o u s l y to
rather t h a n t o
MZF
ZFM).
Let
o V & o V x ( ~ x+ $ ( X I )
suppose t h a t
+
A
in
$ ( z ) be $(z)) is t r u e ,
suppose t h a t :
Vzx(ov~(yIIx+ $(y))+
and t h a t
is true.
)
W e want t o show t h a t
o V ~ $ ( Z ) is t r u e , i , e , , :
vz$(z)is
and t h a t
true.
( 2 ) f o l l o w s from ( 1 ) by
p a r t i c u l a r , for any
tE
,
v ~ ( O v ~ ( y r +l x$(&!)I
$(h))w a s
ovx~yn:
+
VyCynt
+
+
$(y))+
A'
s(t) is
MZF*
true.
~l(t) is t r u e vzkt vx(xn5 + $(y)) + Jltx)) is $(I))
+
tE
A'
, completing
.
t Vy(ynlc +
But s i n c e
MZF*
tE
f o r any
A'
,
t V~(BJ +
of
ZF
, hence
n
t , as
$(y)),
i n [31) that
$(t) i s
true
the proof.
t o be
E
a l l r e s u l t s of [31 a b o u t
ZFM
in
, we
Xa
ZFM
is e a s i l y v e r i f i e d
n-induction"j
ZFI
Furthermore,
g e t the MZF
hold f o r
Before going on w i t h the i n t e r p r e t a t i o n of p o i n t o u t that a " u n i f i e d
,
A'
i.e.,
( m o d i f y i n g the arguments of [ 3 1 i n a s t r a i g h t f o r w a r d way). and t a k i n g
tE
Now it e a s i l y follows ( b y
true.
The t r u t h ( i n our s e n s e ) of a l l axioms of
by e r a s i n g
In
Since
assumed to be t r u e , f o r any
t r a n s f i n i t e i n d u c t i o n on the rank of f o r any
itself.
Xa.
in
€-induction
as w e l l .
MZF
, we
want to
A. S~EDROV
112
seems n o t to be t r u e , a l t h o u q h it i m p l i e s both
X
(of [ 3 1 )
and
Xa
.
Thus, as i n t h e case of r e p l a c e m e n t , we have t o work w i t h t w o d i f f e r e n t vers i o n s of an e s s e n t i a l s e t - t h e o r e t i c axiom:
one of t h e m is there to v e r i f y
( t h e t r a n s l a t i o n o f ) the i n t u i t i o n i s t i c v e r s i o n , the other v e r i f i e s ( t h e t r a n s l a t i o n o f ) the classical version. of
MZF
to i n t e r p r e t
ZFI
, not
just
However, in the s e q u e l w e u s e MZF
-
(X
+
all
XVIII).
W e i n t e r p r e t i n t u i t i o n i s t i c sets as " h e r e d i t a r i l y s t a b l e " sets of
MZF
.
x -
i s h e r e d i t a r i l y s t a b l e " is e x p r e s s e d i n MZF
HS(5) := 0 3 ~ ( f , u ), where
ml A F o r e v e r y formula
as
Z(z,g) is the formula:
.
(VmtJ (V&I~) (gu)A (vmu) ( v ~(~nx ) O(znv)) 0
of
ZFI
let
+*
be a formula of
f o l l o w s ( d e f i n i t i o n is by i n d u c t i o n on the complexity of
d e f i n e d as
MZF
4
.
):
Before s t a t i n g the main r e s u l t , we l i s t s e v e r a l t e c h n i c a l lemmata.
113
Extending Godel's Modal Interpretation Lemma 3.2.
,...,-2 x
Q(x 1 x
Let
,...,?! ,
are exactly
x 1
be a formula of
/-
MZF
Then
HS(x ) A -1
Proof: -
By i n d u c t i o n on t h e complexity of
Proof:
W e work i n f o r m a l l y i n
.
MZF
-
ov~(xn2 u n ~ A (3gy)E(=,XI E
W U {I}
.
x,
W e want t o show that
3:
If
iff
31 or zzy
Rut t h e n Let
5ng , we 31
,
-WOW _ ,
so
32
.
Usinq
BY union
let
-
.
If
z n !
,
xr\A) ,
W e have shown t h a t
(335)(3x)) , in
E(y,B)
w
Obviously,
y)B
then
.
.
B
be
is
Also,
for
F i r s t we show t h a t i n e i t h e r case, f o l l o w s by the second assumption and
, and
thus
(Vvyl$(32)
(3AA)(3gy)(E(=,A) A gA)
.
If
In p a r t i c u l a r ,
(3An!3(5nA)
.
is t r a n s i t i v e .
gx
and
(33?)(3A)
35
.
,
Hence
, so
i s t r a n s i t i v e , so 9 5 q i v e s
But that
( 3 3 9( 3 2 )
.
.
(32)E(y,B)
modalized, we a c t u a l l y have
.
o t h e r words
F i n a l l y , we show t h a t
(332)(E(~,A))
I n o t h e r words,
t h e proof.
.
(3&n!3(5nA)
then
(3&(331)(E(~,&) A
.
-
want to show that
so
(vzny)( 3 9 3 C(z,&).
Then
y , this
vx(yx .. n(yz)) .
then
wnA
5E
If
,
.
so that
ovz(znE- (3&nx)(znA)), i.e.,
Ovx(yB
- ~ ( y s.) )
.
.
(2,{I} 1 , i.e., nVx(vr\x p~v OV~(zn1gy)
be
vx(~q5
Ix
.
1
be given by
us , i.e.,
be
any
5
let
Let
.
+ * +a$*
A
n -
(vzny)(3AIIv)E(z,A)
comprehension (XIV), o b t a i n a set
Let
A HS(5
(vznl)(3A)E (?,A) , u s i n g
Since
numbering of axioms of [ 3 1 ) , we g e t
.
Q
...
whose f r e e v a r i a b l e s
c o l l e c t i o n ( X V I I I i n [ 3 ] ; h e r e and h e n c e f o r t h we r e f e r t o the
"non-modal"
(XIII),
ZFI
So
3 W , and
hence
.
But s i n c e t h e assumptions w e r e
O(3E)E(x,B), i.e.,
HS(y)
, which
completes
A. S~EDROV
114
Proof: -
Again we work informally i n
equivalent t o
o(2ny)
.
(z=!)*
i s (by Lemma 3 . 1 )
(=nz.~ n x ) l ( m o d a l i t i e s i n o(znz)
ovEtHS(z) +
can be dropped s i n c e
OVz(z?lx + HS(5))
have
MZF
HS(q)
HS(y) 1.
and
HS(z)).
O v ~ ( +~ n ~
and
and
Now by Lemma 3.3,
5
Thus
f
Y
.
we
W e now state and prove t h e main r e s u l t of t h i s paragraph:
Theorem 3.1. x 1
,...,-”-
Proof:
x
$(%
..
Then
ZFI
Let
free.
.
r
~
-”_
be a formula of
t $(x
r . . . r ~
By induction on the length of the
with e x a c t l y
implies
)
3
1
ZFI
ZFI-proof
of
o n l y n o n t r i v i a l cases and leave t h e rest t o the reader. informally i n
MZF
.
Elf-
ZFI
By Lemmata 3.1 and 3.3,
it s u f f i c e s t o prove
HS(x) - A HS(y) A HS(2)
(E=x)* A 3: +
antededent g i v e s
5H
A
is
VzCz
( a ) Extensionality i n
32
.
Assuming t h a t a l l parameters i n
it s u f f i c e s to show:
This is e q u i v a l e n t to:
Then
$
mx . p~ by
Ex1
4
.
We check
W e again work
+
B u t by Lemma 3.5,
Y E i
.
the
IX.
a r e h e r e d i t a r i l y s t a b l e , by Lemma 3.1
115
Extending GGdel's Modal Interpretation
and thus also to:
( c ) Power-set i n HS(x)
, by
is
ZFI
~ Z V ~ ( V ~ (E Uz + 2 E 5)
, we
have a set
y'
By t h e comprehension axiom (XIV), l e t
nv=( HS (=I
HS ( x )
, we
have
+ (nVu(ur(z +
-
oVu(unz
But a l s o
O V ~ ( =+ ~o(zr\y)) ~
n
(
Replacement i n ZFI
+ 3 5 v ~ E[ Y ~
unx)
-
-
y
g y )
- 3x(~
. .
~) ) )
is
.
zny'1
be given by:
.
OV=(=ny + nvu( unz + HS ( u ) )
0% ( z q y +
(d)
By t h e power set axioms i n
such t h a t :
o V ~ ( v $ ~ n+ ~ylz)
Then
Assuming
it s u f f i c e s to show t h a t :
Lemma 3.1,
by t h e argument i n the proof of Lemma 3.5.
MZF(XV)
- 2 E y) .
On the o t h e r hand, s i n c e
.
Also,
Thus by Lemma 3.4, By Lemma 3.4 again,
Vy(y E
E 5 A $J(Y,~ ) I) I
.
Hs(P)
, and
( y , ~A) Vl($ (ynu)
~f+
~f
+
thus (B)
u=z)1
5 and a l l parameters i n
a r e h e r e d i t a r i l y s t a b l e , it s u f f i c e s to show t h a t
.
o V ~ ( ~+ n yFLs (=) )
J,
.
A. ~ E D R O V
116
BY B y Lemma 3.3,
writing
++(y,z) (y,~)ffor or
H S ( ~ )A A HS(y)
HS(~) HS(5) A
q*(y,~) $*(y,z) , it
is
enouqh to show t h a t :
By Lemma 3.5, antecedent gives:
hence :
.
OV~l(yri~ + 3li+(y,=)A Vg(+(y,uf + (USE)) 1) MZF
(XVII) in By modal replacement (XVII)
and theref ore:
oV=[HS(~) + (zr(w BY Lemma 3.2,
~ h u sby ( c ) ,
-
, we
- 3x(pxA +(y,z)))l .
+(y,=) n+(y,=), and oVz(~n1-o(=n~)
3.4,
we have
(e)
Infinity i n
HSlw)
.
ZFI
.
2 sso o that:
have a sset et
since
HS(x)
,
-
y ~ x ~(YJIZ).
~ l s o , oV=t~ny+ Hs(=) 1
.
BY Lemma
is:
3g[3y(y E 2 A (V')l(? E y)) A Vy(y E 2 +
vv(v E z
3=(5 E 2 A
-1
EY V
v=x)) ) 1
~t s u f f i c e s to show t h a t :
for f o r then we w e can r e s t r i c t s t a b l e sets, s e t s , and thus r e p l a c e
,
, vx ~ Z Z by
s e tt is c e r t a i n l y h e r e d i t a r i l y s t a b l e , so so
i n t h i s formula to h e r e d i t a r i l y
.
(x=y)* (?z)*
The " n e c e s s a r i l y empty"
uvE-r(Enyl uV~-~(zrlyl can be c e r t a i n l y
Extending Godel's Modal Interpretation by u v ~ ( H S ( z )+ c]~(EI@) , and
replaced
... .
~ Z ( H S (A~ )
stable
A
...
restricted to
Thus we j u s t have t o show t h a t f o r some h e r e d i t a r i l y
)
:
w r i t t e n O@(5) i n s h o r t .
-u -A
By
be a s e t such t h a t
By t h e axiom of i n f i n i t y i n
@
(u) .
MZF ( X V I )
By t h e comprehension axiom ( X I V )
, let
, let
be such t h a t :
-
S i n c e the " n e c e s s a r i l y empty" set is u n i q u e l y determined ( u p t o t h e same h o l d s f o r s u c c e s s o r s , i t is e a s y to s e e t h a t Furthermore, w e c l a i m t h a t
, because
v~[&y~ + H S ( ) I
2
(XIV), look a t t h e s e t
If
set
5
w
It s u f f i c e s to show
g i v e n by
be such t h a t
vu(-
+ - t ( ~ n z))
.
+ ~ ( g z ) , ) and t h e r e f o r e
There i s no
Such a
by axiom (X).
s o that:
zni
obviously
V'(_wl~)
and
@(yw
a contradiction.
By t h e above,
3~g i v e
-
$(Y)
.
~ ( g x ) )
Let
.
2
HS(z)
.
Thus
0VV (!Jx
So by Lemma 3.4,
+
(1) )
we would have
But H S ( ~ ),
be g i v e n by:
Since
3:
, 32 ,a
Applying Lemma 3.4 a g a i n , we s e e t h a t
contradiction.
HS(A)
, completing
.
~ ( p n ~ ) )
e x i s t s by axiom (X), f o r o t h e r w i s e we would have
k ( V z ( z n w + 7(=nz))
Indeed,
and
By t h e comprehension axiom
(D) is modalized.
yz
i s nonempty, l e t
.
o V ~ [ p r+ l ~HS(2)I
*E(~A
),
the proof.
A. SfEDROV
118
Acknowledgments.
The r e s u l t s of t h i s p a p e r were o b t a i n e d i n May 1979
d u r i n g the a u t h o r ’ s g r a d u a t e s t u d i e s a t S t a t e U n i v e r s i t y of New York a t Buffalo.
I
am g r a t e f u l t o P r o f e s s o r John Myhill f o r many h e l p f u l
s u g g e s t i o n s and criticism of e a r l i e r v e r s i o n s . P r o f e s s o r N.D.
S p e c i a l t h a n k s are due to
Goodman f o r i l l u m i n a t i n g d i s c u s s i o n s on the s u b j e c t of
f 3 , and f o r the e l e g a n t d e f i n i t i o n of h e r e d i t a r i l y s t a b l e sets
w i t h o u t the development of r a n k s .
119
Extending Ctidel's Modal Interpretation
References
K.
&el,
"Eine I n t e r p r e t a t i o n des I n t u i t i o n i s t i c h e n Aussagenkalkuls", E r g e b n i s s e e i n e s mathematischen Kolloquiums (19321, 39-40.
[ 21
H. Friedman, "Some A p p l i c a t i o n s of Kleene's Methods for I n t u i t i o n i s t i c Systems" i n A . R . D Mathias and H. Rogers, eds.: Proceedings Cambridge Summer School i n Mathematical &, 1971, S p r i n g e r LNM 337, B e r l i n , 1973, 113-170.
I31
N.D.
f41
J. Myhill,
Goodman,
" A Genuinely I n t e n s i o n a l S e t Theory",
this volume.
"Some P r o p e r t i e s of I n t u i t i o n i s t i c Zermelo-Fraenkel S e t Theory", i n A.R.D. Mathias and H. Rogers, eds.: Cambridge Summer School i n M a t h e m a t i c a l e , Proceedings 1971, S p r i n g e r LNM 337, B e r l i n , 1973, 206-231.
--[51
J. Myhill,
f61
H. Rasiowa, R. S i k o r s k i , Warsaw, 1963.
I71
A.E.
I81
M. Takahashi , "Cut-Elimination Theorem and Brouwerian-Valued Models for I n t u i t i o n i s t i c Type Theory", Comment. Math. Univ. P a u l i 19 (1970), 55-72. --
" I n t e n s i o n a l S e t Theory", this Voltme.
Smullyan,
The Mathematics g
Metamathematics,
"Modality and D e s c r i p t i o n " , JSL
11. (19481, 31-37.
s.
This Page Intentionally Left Blank
Intensional Mathematics S.Shapiro (Editor) @ Elsevier Science Publishers 8.V . (North-HoNandJ, 1985
121
CHURCH'S THESIS IS CONSISTENT WITH EPISTEMIC ARITHMETIC Robert C. Flagg State University of New York at Buffalo Buffalo, New York U.S.A.
The idea of using epistemic notions to provide a setting for integrating classical and intuitionistic mathematics is developed in the work [161 of Shapiro (in the present volume).
He sets up a formal system of epistemic
arithmetic (EA) conservatively extending classical arithmetic and possessing certain properties reflecting features of intuitionistic arithmetic. Goodman [4] showed that
Godel's Translation (EA) also conservatively
extends intuitionistic arithmetic. The methods of Goodman and Shapiro consist, in part, of adapting to (EA) standard tools from the metamathematics of classical and intuitionistic systems. An important tool in intuitionistic metamathematics is the recursive realizability interpretation. In this paper we show how to adapt this interpretation to an epistemic setting. realizability model for ( E A ) .
We thus obtain a recursive
As an application it is shown that Myhill's
"Epistemic Church's Thesis" can be consistently added to ( E A ) . The paper consists of two parts.
In Part I, we prove a version of
Funayama's Theorem for the recursive realizability model.
The main tool in
the arguments is the notion of adjoints for preordered sets. All the basic properties of these are developed from scratch, so the paper is completely self-contained. In Part 11, we use our version of Funayama's Theorem and ideas from the algebraic approach to recursive realizability, developed by Hyland, Johnstone and Pitts, to construct a recursive realizability model for epistemic arithmetic. It is shown that Epistemic Church's Thesis is valid in the model and consequently that this schema can be consistently added to (EA). The results presented here are based on Chapters I and I1 of the author's Ph.D. Thesis written under the direction of N. D. Goodman, to whom we wish to express our indebtness. We would also like to thank Bill Lawvere for many helpful discussions on this material.
R.C.F L A X
122
PART I: FUNAYAMA'S THEOREM FOR RECURSIVE REALIZABILITY The main objective of this first part is to "integrate" recursive realizability and the classical notion of truth. This is accomplished by adapting the construction used in the proof of Funayama's Theorem.
Funayama's
Theorem asserts that any complete Heyting algebra can be embedded in a complete Boolean algebra by a map that preserves finite infimums and arbitrary supremums.
For this result to be relevant here, the recursive
realizability model must be described as a "generalized" complete Heyting algebra.
Such a description has been given by Hyland, Johnstone and Pitts
in [6]. Their generalization of a complete Heyting algebra is the notion of a "family of Heyting algebras indexed by the category of sets". Roughly, such an indexed family consists of the following data. (I)
For each set X , there is associated a Heyting prealgebra R(X) ;
f:X-Y, there is associated an
(11) For each function
order preserving map f * : R ( Y ) -,R(X), called "substitution along f
'I,
and two order
preserving maps
,v : R ( X l f f
3
R(Y)
-a
1)
which are left-adjoint and right-adjoint to f , respectively. There are certain conditions that these data have to satisfy and there is some additional structure required for higher-order properties; but, we w i l l not need to mention these explicitly.
The idea, then, for embedding recursive realizability into a Boolean setting is to apply Funayama's Theorem to each R ( X ) prealgebra B(X)
obtaining a Boolean
and a mapping
j*:R(XI
4
B(X) .
The prealgebra R(X) need not be complete; however, the higher-order structure provides "enough" completeness for the argument to work nonetheless. for R
Also
for any map
f:X
4
Y
of sets the appropriate substitution map
gives rise to a substitution map
f#:B(YI
4
B(X)
Finally, because the construction of B(X)
. is "uniform in
x",
we can
Epistemic Church’s Thesis
“lift” the adjoints Xf
4 f* 4 vf
123
to obtain adjoints 3;
4 f # 4 V! .
To fill in the details of the above sketch, we will have to review some elementary results from lattice theory.
Since adjoints will play a central
role in our final arguments, it seems natural to make a systematic use of categorical methods right from the start. This amounts to the study of 2-valued category theory. Thus, the first few sections below consist of standard results from category theory simplified to the 2-valued setting.
1. Preordered Sets. A
preordered
set is a pair
P
= ( P I d ) consisting of a set
P and a bi-
nary relation d on P which is reflexive: for all p
E
p , p
transitive: for all p , q , r
E
C p ; and P , if p L q
We have not required that the relation
C
C P.
be
P , if p
antisymmetric: for all p , q
and q 5 r , then p
6
q
and q
If this extra condition is satisfied, we will call P
C
p ,
then p = q .
a poset.
Given any preordered set P I there is a naturally associated poset, denoted by
P I E , obtained from P
equivalence relation
f
by “moding out” with respect to the
on P defined by if p C q
p - q
and q 5 p .
When working constructively, it is not always possible to replace P with
P/E
.
Moreover, some of our constructions below are much more transparent
when carried out on P
itself.
The next example describes the preordered set of principal interest to us in the sequel.
1.1.
Example.
Let N denote the set of natural numbers.
nary relation d on
-
R
the power set of
Define a bi-
N ,
by the prescription
p
C q
if there is a partial recursive function f
for all t
p , f
is defined at t and f ( t )
6
such that
q .
It is easy to see that this relation is reflexive and transitive. We therefore obtain a preorder, which is also denoted by
1.2.
Definition.
Let P and Q
is called a morphism
from
P
be preordered sets.
Q if it
R. A
map F : P - Q
R.C.FLAGG
124
preserves As
order:
for all p,p'
E
usual we will write F:P
to indicate that F 1.3.
p,p'
S
P , F(p) L F(p')
implies p
implies f ( p )
p'
6
f(p')
.
Q
to Q .
is a morphism from P
Definition. A morphism F:P C
-
P ,p
-a
Q is called conservative if for all
S
p'
.
The collection of morphisms from P to Q inherits a natural preorder from Q : if for all p
F L G
E
morphisms
from
P I F(p) L G(p)
.
Q p and call it the preorder of
We will denote this preordered set by
P to Q .
Adjoints for Preorders.
2.
The description of adjoints in the present context is very simple, since in a preordered set "all diagrams commute". 2.1.
P
and Q be preordered sets.
Definition. Let P
to Q
is a pair (F,G) : p
where F
and
-Q
An
adjunction from
I
are morphisms
G
F + p+ Q G
and the following condition is satisfied for all p F(p) L q If
(F,G):P-
we will say F
$,
.
adjoint to F _ __-
if and only if p
5;
E
p
G(q)
.
-1
q
6
Q:
G and G is right-
is left-adjoint to
We will a l s o use the notation F
and
in this situation.
G
As a convenient characterization of adjunctions, we have 2.2.
Theorem. Suppose F : P
orders P
and
Q
.
-t
Q and G:Q
-
P
are morphisms of the pre-
Then F is left-adjoint to G
if and only if the
following two conditions are satisfied: (I)
For all p
E
p , p L GF@)
(11)
For all q
E
Q , FG(q) S q .
Proof. Suppose F
4 G.
Then
F(p) L q
for all
11
C
P
;
if and only if p
and all q E Q .
6
G(q)
Taking q = F ( p 1
.
gives condition (I) and
Epistemic Church's Thesis
taking p
125
gives condition (11).
= G(q)
Conversely, assume (I) and (11) hold. F(p)
L
Then
implies G F ( p )
q
implies p
6
5
G(q)
.
G(q)
The opposite implication is similar. // We will refer to the inequalities (I) and (11) of Theorem 2.2 as the Triangle Inequalities. The next two results are easy consequences of the Triangle Inequalities. Composition of Adjoints.
2.3.
Suppose
are adjunctions. Then
(F,G):A
-
B
-
(H,K):B
and
C
--.C
(HF,GK) : A is also an adjunction. Uniqueness of Adjoints.
2.4.
orders Q
and P ,
adjoint to G
.
Theorem.
2.5.
P is a morphism of the preF,F':P -B 4 are both leftin the preorder of morphisms Q p . Suppose G : Q
Suppose the morphisms
Then F Suppose
=
F'
(F,G):P
-Q
is an adjunction.
F is conservative if and only if for every p
(I)
E
P
p ;
GF(p) (11)
Then
is conservative if and only if for every q E Q ,
G
FG(q)
=
q
.
Proof. From the Triangle Inequalities for F
-1
G , for any p
E
P we
have FGF(p)
5;
F(p)
and
p s GFtp)
.
If F is conservative, from the first of these we get Thus G F ( p ) E p
GF(p1
.
6
p .
Conversely, suppose G F ( p ) :p for all p E P .
If F ( p ) L F@')
.
then
p So
p
rS
p'
.
L
GF(p)
L
GF(p')
S
This completes the proof of (I).
p'
. The proof of (11) is dual.//
R.C. FLACG
126
Supremums and infimums are defined for preorders just as for posets.
We
will use the standard notation
p = to indicate that p
lP,Ii
E
I
-
V
icI
and p =
pi
*
Pi
i €1
is a supremum, respectively infimum, of the family
The supremum and infimum are unique up to
t
.
Evidently, the empty supremum, if it exists, is a least element. this by
Dually, the empty infimum, if it exists, is a greatest
1.
we denote this by
element.
We denote
T.
Our next result establishes one of the most useful properties of adjoints. Theorem.
2.6.
Suppose ( F , G ) :P 2 Q
is an adjunction. Then (I)
Proof.
If p =
,
pi
V
i €1
suppose p =
V
then F ( p ) =
pi.
Then directly from the definitions we have
i€1
F(p)
I; q
F(pi);
V
i €1
if and only if p L G ( g ) if and only if for all i E I , p , S G ( q ) if and only if for all i I
so F ( p )
=
V
i€1
F(pi)
.
This gives (I)
,F ( p i )
69
.
.
The argument for (11) is dual. // Thus "left-adjoints preserve supremums" and "right-adjoints preserve
.
infimums"
3 . Equivalence of Preorders
There is a relation between preorders, weaker than isomorphism, which still has the property that two preorders related in this way can be "identified" in many situations. 3.1.
This relation is based on a weaker notion of inverse.
morphism F:P * Q
G:Q - P
is called a quasi-inverse for the if the following two conditions are satisfied:
Definition. A morphism
for a l l
p
E
P, G F ( p )
Z
p ,
127
Epistemic Church's Thesis
and for all q Notice that if G
E
Q, F G ( q )
3
q.
is a quasi-inverse for F , then G
is both a left-
adjoint and a right-adjoint and, moreover, F is a quasi-inverse for G 3.2.
Definition.
A
morphism F : P * Q
.
is called an equivalence if it has
a quasi-inverse. 3.3.
and Q
Definition. Two preorders P
there is an equivalence F : P
-t
Q.
4. Monads for Preorders
Suppose F : P
-I
are said to be equivalent if
Q is left-adjoint to G:Q
-
Then the composition
P.
T = GF:P * P is an endomorphism of P
.
Using the Triangle Inequalities, we easily get
(~1) for all p EP, p
L
T(p)
i
and (M2) for all p EP, T P ( p ) L T ( p ) . We will call an endomorphism P : P * P on P A
.
satisfying (Ml) and (M2) a
natural question to ask is whether every monad can be defined by a
suitable pair of adjoint morphisms. We will show that this is indeed the case. 4.1.
be a monad on the preorder P
Definition. Let T
the subpreordering of P
.
Then P"
is
with elements given by PT= [PEPIT@)
6
p]
.
By (Ml), it follows that P T consists of exactly the "fixed-points" of T
:
(Ml) and (M2) together give for: all
We can therefore regard T
p
6
P, T T ( p ) E T f p )
as a morphism from P
this morphism by FT:P-P T Also,
we will denote the inclusion of p T
GT: P T - t P .
. to P T
. into P
by
.
We will denote
R.C.FLAGG
128
Then a simple calculation gives: F
T Thus FT-/ G 4.2.
.
T( p ) L q
for all p , e P and all q , B P
T
if and only if p L G ( q )
T
,
.
Summarizing, we have
Theorem. Let ?'
be a monad on the preorder P .
an adjunction
-
(FT, GT): P T T such that T = G F
.
Then there is
PT
Now suppose we begin with an adjunction (F,G):P
-
62,
construct the associated monad X = GF on P , and then the preorder of T fixed points P How do Q and P T compare? Well, from the Triangle
.
Inequalities, any q ' Q
satisfies
TG(q) = GFG(q) L G(q) * G
can therefore be regarded as a morphism from Q
denote #is
morphisrn hy
to P T
.
We will
PT,
K:Q
and call it the comparison morphism for the adjunction
4.3.
(F,G):P
Q
(F,G):P
-Q
Theorem. Let
be an adjunction, and let
.
T = GF be the associated monad on P .
If G
is conservative, then the cornparison morphism K:Q
-
P
T
is an equivalence.
Proof: Let J : P T - r Q be the restrict of F to P T servative, by ( 2 . 5 ) , for any q q
Moreover, if p
PT
5
F G ( q ) = JK(q)
Since G
is con-
.
, then U ( p ) = GF(p)
Thus J
.
EQ
is a quasi-inverse for K
Dually, a comonad on a preorder
Q
p
and so K
. is an equivalence. //
is an endomorphism X:Q -. Q
( ~ 1 ~for ) all q
E
Q,
H(q) L q ,
and (MZ')
for all ~ E Q , ~ ( q L) H H ( ~ ) .
satisfying
Epistemic Church's Thesis
129
Our discussion of monads dualizes in the obvious way to comonads. We will write
Q H = Lq.EQIq 6 H ( q ) 1 ,
,
G H (q) = H ( q ) and
FH(q) = q
.
Then
Q
(FHtGH):QH is an adjunction and H 5.
=
FH GH'
The Adjoint Lifting Theorem
Suppose T
and S
are monads on the preorders P and
We will say a morphism
in the preorder of morphisms
Q
and
is proper for T
U:P-rQ
P
UT
.
=
Q, respectively. S
if
SU
For any such proper morphism, we
obtain a morphism
8:PT* Qs by restriction.
5.1.
-
Let T and S be monads on P and Q , Q be a proper morphism for T and S , and be the restriction of U Then
Adjoint Lifting Theorem.
respectively. Let U:P let z : P p *
Qs
(I) If U then
.
has a right-adjoint R:Q
-
P
,
U has a right-adjoint E:Qs-
PT
given by the prescription
-
R(q) = R(q)
(11) If
U has
a
for U E Q
S
.
S
.
left-adjoint
L:Q * P , then
has a left-adjoint E:Qs*
PT
given by the prescription
L(q) Proof.
= TL(q)
for
q EQ
For (I), it will suffice to show R
restricted to
Qs
takes
R.C. FLAGG
130 values in PT
.
But, using the Triangle Inequalities for the adjunction (U,R) :p
for any q
E
Q s, we
-a ,
have T E ( q ) s RUTR(q) s RSUR(q) 6
Consequently, for q
E
as,
T
~ ( qE )P
RS(q) R(q)
.
.
For (II), we will directly verify the Triangle Inequalities. L e t S
q
.
Then UL(q)
q
s SUL(q1 f
UTL(q1
-
.
Ur,(q)
T Let p € P
.
Then
Heyting Prealgebras
6.
A lattice is a preorder which has supremums and infimums of finite families. Suppose L
is a lattice. We will write
and
(xo A x l ) For each element x by
z,
for
i6
A
xi.
(0,1}
of L , there is a natural endomorphism of L induced
namely
x A-:L-L. lattice H
6.1.
Definition.
xSH
the endomorphism
A
is a Heytinq Prealqebra if for every
x has a right-adjoint, denoted by
A -:H
x
-- .
H
Since left-adjoints preserve supremums, any Heyting algebra satisfies the
Epistemic Church's Thesis
131
V-distributive law":
"A,
for any family of elements
(yiIi c I
exists.
is a complete preorder which satisfies the
A,
Conversely, if H
such that the supremum
V-distributive law, then, by the Adjoint Functor Theorem, H
is a
Heyting prealgebra. "Negation" in Heyting prealgebras can be defined by 1I =
Lemma. Let
6.2.
(I) (11)
x
(111)
1I 1 1 7 3 2 .
(I) (11)
(X
*
I).
be an element of the Heyting prealgebra H .
Then
x A T I ~ L ; 5
71s
;
=
Lemma.
6.3.
x
Let x , y
be elements of the Heyti..: prealgebra H .
x L y implies ~y L i (x V y ) A i y
6.4. Definition.
Then
T Z ;
A Boolean prealgebra is a Heyting prealgebra satisfying
the condition: for all
I,T= I(
V TX)
.
As a convenient criterion for a Heyting prealgebra to be a Boolean pre-
algebra, we have
e. A Heyting prealgebra
6.5.
H
is a Boolean prealgebra if and only
if for all z E H
1 7 x LX. 7.
Geometric Morphisms
The natural notion of morphism between Heyting prealgebras is that of a map preserving all of the structure mentioned in the definition. However, for the applications we have in mind, another notion turns out to be more use-
ful. 7.1.
-
-
Definition. Let C and D be Heyting prealgebras. A geometric
morphism f:C such that f*
-1
D consists of a pair of morphisms f*:D
4
C and f,:C
D
f* and f* preserves finite infimums. We call f* the
R.C.FLAW
132
f and f* the direct image of f
inverse image of
~~
.
The composition of two geometric morphisms
f
9
and D + E
C-sD
is the pair
g *f:E
BY ( 2 . 3 ) ,
-
* *
g
'f = (f 9 4*f*)-
* *
C is an adjunction, and trivially, f g
preserves
finite infimums. Thus g
f:C
-I
E
is a geometric morphism. 1.2.
Definition.
We call a geometric morphism conservative if its inverse
image morphism is conservative. 8. Modal Operators
Suppose f = (f*,f,):C and D
prealgebras C
. . (
.
D is a geometric morphism between the Heyting By our work in sectio,i 4, we know that the compo-
sition
0 = f*f*:D
is a monad on D
.
-. D
Since f* preserves finite infimums, 0
satisfies the
extra condition (El
0
for r , y ~ ~ (, z A Y )
= 0
AO
y
.
D on a Heyting prealgebra D satisfying the condition (E) a diamond-operator on D . Such operators are considered in
We will call a monad O : D
4
Fourman and Scott [ 2 1 , where they are called J-operators. 8.1. Theorem.
algebra D
.
Let
0:D
-L
D
be a diamond-operator on the Heyting pre-
Then the preorder of fixed points of
0,
D o , is a
Heyting prealgebra with operations given by the following prescriptions:
0
P =o,;
0
(IV)
TD = T ;
(V)
(Z+)
0
=sag.
Epistemic Church's Thesis
Proof.
By Theorem 4.2 there is an adjunction (
where
133
-0 0
0,i):D
D ,
i is the inclusion morphism of D
into D .
Conditions (I) and
(11) now follow, because left-adjoints preserve supremum.
By axiom (Ml) f o r monads, V T 2 T
0
(IV) follows. Because
Thus v ( x A y )
S
r Ay
0.
Suppose, x
D
Thus x A - : D
4
,
,
0T
so
T
and
0.
T E D
Condition
preserves order, we have
and (111) follows.
Then for any y a D .
0.
D is a proper map for
By the Adjoint Lifting Theorem,
the map
x A-:D
0
-t
D
0
has a right-adjoint given by
'0
x s D y = x * y , as required. // 8.2.
D.
Corollary.
Let
9
be a diamond operator on the Heyting prealgebra
Then there is a geometric morphism
0
f:D such that 8.3.
0 = f , f*.
Example.
Let d E D , where D
endomorphism Vd:D
-
D
is a Heyting prealgebra. Define an
-
D
by putting
Od(x) = (x =) d ) It is easy to see
0,
3
d.
is a diamond-operator on D .
Boolean quotient with respect
d .
We call it the
RC.FLAGG
134
Theorem.
8.4.
D.
d be an element of t h e Heytinq prealgebra
Let
Dod,
t h e preorder of f i x e d p o i n t s
of t h e Boolean q u o t i e n t
0, ,
Then
forms a
Boolean prealgebra. Proof. ___
Since
From ( 8 . 1 ) , w e have
a
for
Dv
D,
i s t h e same a s t h a t f o r
0
x = (x =a d )
-ii
Consequently, t o show
0
D
it follows t h a t
.
i s Boolean w e need t o show t h a t f o r a l l x
E
0,
D
(sad) * d r z .
0.
D
But t h i s follows from t h e d e f i n i t i o n of
//
We w i l l a l s o need t o consider comonads t h a t p r e s e r v e f i n i t e infimums. 8.5.
Definition.
E
Let
be a Heyting p r e a l q e b r a .
E
i s c a l l e d a box-operator on
A comonad
0:E
+
E
i f t h e following two c o n d i t i o n s a r e
satisfied: (I)
OTZT;
(11)
for a l l
r,y
Dual t o (8.1), w e have 8.6.
E.
Theorem.
Let
0:E
E , u (x A 9 )
E
-
E
2
ox
Any.
be a box-operator on t h e Heyting p r e a l g e b r a
Then t h e preorder of f i x e d p o i n t s of
0,
E,,
i s a Heyting p r e a l g e b r a
with o p e r a t i o n s given by t h e p e r s c r i p t i o n s :
8.7.
Corollary.
Let
0
be a box-operator on t h e Heyting prealgebra. Then
t h e r e is a geometric morphism
f:E
-
E,
135
Epistemic Church's Thesis such that
0
C.
.
f*f,
If we begin with a conservative geometric morphism
f: E + D , then, by Theorem 4.3, the comparison morphism,
K
-D
E,
is an equivalence of preorders. quasi-inverse for K follows that K I, V, A
,T
is both a left- and a right-adjoint for K , it
and its quasi-inverse will preserve all of the operations
and
*
.
Consequently, in all "logical" contexts, we may
with E,.
identify
J
Now, by ( 3 . 2 ) , (2.3) and the fact that a
This sort of identification will be made on
several occasions below. 9. Recursive Realizability
Recall the preordered set R
p
of R
of example 1.1.
as a proposition and elements z
If we think of an element
of p
as realizers of p ,
then we can mimic the recursive realizability interpretation of the logical operators to show that R
is a Heyting prealgebra.
To see this, let
(. , .) be some standard premitive recursive pairing function for N 9.1.
Theorem.
The preordered set R
.
is a Heyting prealgebra with
operations defined as follows: (I)
I = the empty set;
(11)
(pvq) = 5(0.2)12~PI u C L d l x E q I i ( p A q ) = {(x,y>I.:ep and y " q 3 ;
(111) (IW
T-
(V)
(p
N
* q)
=
5el for all z , e p , { e ] ( x ) i and { e l
(2)
eq]
.
We are using the standard notation
Cel (2) to denote the value of the e'th unary partial recursive function at the argument x ,
if it is defined.
We will also use the
the standard interpretation. Thus if g i ( U l , involving the variables V 1 ,
...,U n ,
hUL.. .u,.
...,U n )
then $(U1,..
.,Vn)
"A-notation'' with
is some expression
R.C. FLACG
136
...,an
will denote the function whose value at the n-tuple al,
is
....,an) .
@(a,, Finally, we will write U
for Ul,. . . v
n'
if this does not cause confusion. To extend this algebraic formulation of the propositional logic of recursive realizability to account for quantifiers, we will use adjoints of "substitution". 9.2.
Definition.
For any set x , we define a binary relation Sx
on
the set R(X) = the set of all functions from X
to R,
by the prescription. # S x & if there is a unary partial recursive function f #(x) into @(z) for all z E X .
mapping
Notice that # S x *
is equivalent to the assertion
We will call an element of the above intersection a witness of the inequality #Sxrji 9.3.
.
Theorem. The pair
(R(X),S )
X
is a Heyting prealgebra with
operations defined pointwise. 9.4. Definition.
-~ map induced a f
Let f
be a mapping from
is the mapping
f*W for @'R(Y)
=
f*:R(Y)
x
to Y .
* R(X)
Ax. #(f(x))
The substitution
given by
I
.
-
The next three lemmas follow at once from the definitions. 9.5.
-.
For any mapping f:X
is order preserving. 9.6.
Lemma. (I)
The operation
-
( )
*
*
Y , the substitution map f : R ( Y ) *R(X)
is "functorial":
If I:X -. X is the identity map on X , then * I :R(X) R(X) is the identity map on R(X) ;
Epistemic Church's Thesis
(11) If f:X -8 Y and
-
g:Y
9.7.
e. For any m
pping f : X
preserves all the operations
Z , then
f)*
($7 0
A,
137
-8
=
f* 0 $?*.
Y , the A , T
V,
su
and
It can be shown that each substitution map
*
stitution map
*
of R ( Y )
*
f
:R
.
has both a left- and a
f
right-adjoint. We will describe these adjoints in the case when
f is
surjective. The description in this case is slightly simpler than the general one, and we need the result only for surjective maps. 9.8.
Theorem.
For any surjective map
f:X
-8
Y , the induced map
f*:R(Y) * R(X) has a left-adjoint 3f:R(X)
-8
R ( Y ),
given by
3f (z) .u{G(x)If(r)
=
yl
z c R(X) ,
for
and a right-adjoint
vf ($1
).(fI
= )ry.nCz(z)
=
yl
z~R(x).
for
Using the surjectivity of f , the conditions
__ Proof.
Zf(53) L y $ if and only if
$L
X f*($)
and
f*($)L x p '
if and only if
$6
v
Yf
(2)
,
can be established by a simple calculation. // The next result establishes a key relationship between the operations and 8 (
.
We call it "Beck's Condition", but, in fact, it is a
8( ) very special case of this stronger result. 9.9.
where
~
Beck's Condition for R .
n
and
n'
( )
Consider the commutative diagram
are the natural projections and g g(z,s) = (z,f(s))
for z
EZ,
ZCX.
satisfies
*
,
138
R.C.FLAGG
proof. For -
(I), we need to show the following diagram commutes:
The argument for (11) is dual. 10.
//
Funayama's Theorem
We are finally ready to address our main problem of embedding recur iv realizability into a Boolean setting. As indicated in the introduction this will be done by modifying Funayama's Theorem.
In our preferred
terminology Funayama's Theorem can be formulated as follows. 10.1.
Theorem.
-
For any complete Heyting algebra Q , there is a complete
Boolean algebra B
and a conservative geometric morphism j : B
Q.
Johnstone [81 has given a proof of Funayama's Theorem which we will adapt to our present setting. Let X
be any set. Applying (9.8) to the projection
p: X
x
R - X ,
we obtain an adjoint situation
*
( p ,V
P
:R(X)
-A
In fact, this is a geometric morphism
R(X x R)
.
139
Epistemic Church's Thesis
AX:R(X since, by (9.7),
preserves
p*
R)
x
T
-
,
R(X)
and A .
There is a "preferred element" of R(X
, namely the projection map
R)
x
s:X x R - R .
Let =
0 :R(X
R ) -. R(X x R )
x
be the Boolean quotient with respect to
be the preorder of "fixed points" for prealgebra and
ox
with
(
ox.
Also let
By (8.4),
B(X)
is a Boolean
is the inverse image of a geometric morphism
whose direct image is the inclusion of A,
.
s
,
B(X) into R(X
X
R).
Composing
ox, ix) gives a geometric morphism
-
jX:B(X) with
* i,W
kp.
=
R(X) ,
($(Z)
p) =p
=)
and
We claim this geometric morphism is conservative. Again by (2.5), it will suffice to show:
(I) Let k
for all @,ER(X),jx*jx (Id)
5;
$
.
be a code for the identity recursive function. Let t 6 N for all e . E N , it](.)
r
satisfy
{ e ] ( k ).
Then t witnesses the inequality
n
t
(m)p )
=)
pi
$(XI
,
P as can be seen by taking p = equality is just
(j,j
.* ( $ ) I
(ce)
.
@(z)
.
But the left-hand side of this in-
The inequality (1) follows.
R.C.FLAGG
140
We obtain 10.2.
Theorem.
-
For any set X , there is a Boolean prealgebra
a conservative geometric morphism jX:B(X)
B(X)
and
R(X) .
By our remarks at the end of section 8, Theorem 10.2 permits us to "identify" R(X)
with the set of fixed points of the box-operator
ax = j x * o ji:B(X) -. B(X)
.
Moreover, by (8.61, under this identification, the operations of R(X) are given by
We now turn to the problem of constructing "quantifiers" for B , appropriately related to,the "quantifiers" for R
.
which are
Again these will be
adjoints to substitution. Let f:X
-
Y be an arbitrary map. Define F:XxR-YxR
F(r,p) = ( f ( z ) . p ) 10.3.
for
5 EX,
p.ER.
e. The substitution map F* :R(Y x R )
is proper for Proof. -
0,
and
by
ox.
-
R(X x R)
We have to show that F*
Let @,ER(YxR).
0,
5
ofl* .
Then (F*
Vy)@= F*(hyp. (y,p)
=)
= k p . ( @ ( P ( X ) .p)
*
=
k p . ((F
=
COT*)l a .
@)( z , p )
One consequence of this lemma is that F*
values in B ( X ) .
We
pl * p) =)
p)
*p)
3
p)
=)
p
/! restricted to B ( Y )
will denote this restriction of F*
by
takes
f# and call
Epistemic Church’s Thesis
it the substitution map induced by
141
f for B .
The next two lemmas follow at once from the corresponding results (9.6) and (9.5) for
.
0”
f#:B(Y)
Lemma. For any mapping f:X * Y , the substitution map -IB(X) is order preserving.
10.5.
Lemma.
10.4.
The operation
-
( )#
is “functorial”:
X is the identity map on X , then I# :B(X) is the identity map on B(X) j
(I)
If I:X
(11)
If f:X
Y and g:Y * 2,
Lemma.
For any map
f:X
preserves all the operations
B(X)
then
( g o f ) # = f # o g# 10.6.
4
.
the substitution map f# :B(Y)
Y, I ,v, A ,
1- and *
of
B(Y)
.
-
B(X)
Proof. This follows at once from Theorem 8.1, Lemma 10.3 and the fact, (9.71, $4
that F*
preserves the operations of R(Y
x
R)
.
For example, if
$I are elements of B(Y) , then
and
Combining (9.8),
(10.3) and the Adjoint Lifting Theorem, we obtain
10.7.
For any surjective map
Theorem.
-
f:X
f#:B(Y) B(X) has a left-adjoint
given by
and a right-adjoint
-
Y , the induced map
R.C.FLAGG
142
given by
10.8.
vf;.Ib= # VF;.Ib, 0 ' B ( X )
Beck's Condition for B .
zx
where
B
n'
and
-
Consider the commutative diagram
X
T
I
X
are the natural projections and g g(z,z) = (z,f(z)),
ZEZ,
satisfies
s e x .
Then (I)
3",g#
= f#3", ;
Proof. First consider the diagram ZxXxR-Xx
where P and P'
P
R
are the natural projections,
and
This diagram is of the form described in Beck's Condition for R ,
*
*
(1)
3 G Z F 3
(2)
Y G E F V
P
and
P'
*
*
P'
P
*
For (II), we need to show that the following diagram commutes:
So let $ ' B ( Z x Y )
.
Using ( 2 ) and the definitions, we have
so
Epistemic Church's Thesis # # (V,g 1 pr
# = VJG
143
* $1
*
p
VpG @
a F*Vp #
F * ( V : , ,d1
(f# v# p . To establish (I), we can argue in almost the Same way; but now, in addition
to (l), we will use the fact that
v$* 10.9.
Lemma.
For any map
is proper for __ Proof.
Oy
and
U
f:X
-
=
F*O Y
.
//
Y, the substitution map
X'
Recalling the definitions, we have for any
Z eB(Y)
and
Consequently, for any
@ 6
B(Y) ,
Combining this lemma with the Adjoint Lifting Theorem and the Uniqueness of Adjoints, we obtain the following relationships between the quantifiers for R and those for B : for any
r'
and
pr
E
R(X)
R.C.FLAGG
144
PART 11: RECURSIVE REALIZABILITY FOR EPISTEMIC ARITHMETIC The recursive realizability interpretation provides a useful method for obtaining consistency results for intuitionistic systems. We want to adapt this interpretation so that it can be used to give analogous consistency results for epistemic systems. Most of the work has already been done in section 1.10. To see how to complete the model construction, we will carefully present the algebraic approach to recursive realizability due to Hyland, Johnstone and Pitts. After this it will become quite clear how to proceed in the epistemic case. The model we obtain will be used to show the consistency of the epistemic version of Church's Thesis.
To show this schema is valid in our model, we
will use the GSdel Interpretation of intuitionistic systems into classical modal systems and the validity of the standard Church's Thesis in the Recursive Realizability Model.
1. Formal Systems for Arithmetic We will be considering two formal systems for first-order arithmetic: intuitionistic and epistemic. These systems are simple extensions of the corresponding logical systems obtained by adding axioms for the elementary arithmetical operations and a rule to express the principle of induction. The primitive symbols of intuitionistic arithmetic consist of a countable list
...
V11V2'
I
of variables, parentheses and also the following symbols: (I)
the constant symbol: 0 ;
(11)
the unaiy function symbol:
(111) the binary function symbols: (IV)
the logical symbols:
=
,
A,
'; +,
( )
.;
v, * , V,
3.
The primitive symbols of epistemic arithmetic are those listed above together with
(v)
the provability symbol:
0.
The terms of arithmetic are defined in the usual way.
Also the formulas of
intuitionistic arithmetic are defined in the standard way.
For epistemic
arithmetic, in the definition of formulas we must add the clause
Epistemic Church's Thesis if
cp
i s a formula, so i s
145
nrp
.
We w i l l w r i t e I
=
(O=O')
;
and
The notions of
free
and bound v a r i a b l e s and t h e operation of s u b s t i t u t i n g
a term
t
simply
c p ( t ) , w i l l be taken f o r granted.
for the variable
i n t h e formula
2,
cp(v)
,
denoted
The systems of a r i t h m e t i c are formulated a s sequent c a l c u l i .
We use Greek c a p i t a l l e t t e r s
i s standard.
sets of formulas.
I-, A ,
n, ...
cp(t/v)
or
Our n o t a t i o n
t o denote f i n i t e
W e w i l l write
r,
A
for
rua,
and
r, 1.1.
Definition.
cp
for
r u [(PI .
sequent is a s t r i n g of symbols of t h e form
A
r t . A p where
and
A
are f i n i t e sets of formulas.
and
A
are called the
antecedent and t h e succedent, r e s p e c t i v e l y , o f t h i s sequent.
The Axioms and Rules of I n t u i t i o n i s t i c Arithmetic R
cpl-t;
=o
tt=t;
=
s=t
-
-*
r=s, s = t
=3
v 1= U l ,
A1
t ' = O
A2
s'=t'
1
A4
11
A5
t
A3
A6
t
t=s;
/-
r=t;
..., v n = u n ttcg = t ( U ) 1.; 1S=t;
S + o =
S + t '
9 ;
=
( S + t ) ' ;
s * O = O ; s o t '
=s't+S;
;
R.C. FLAGG
146
provided x is not free in
r;
The Axioms and Rule of Epistemic Arithmetic are obtained from those of intuitionistic arithmetic as follows: The rules
I--
and
&
are modified to allow more than one formula in
the succedents.
r
.*.
- A , v ~ cp(z)
provided x does not occur free in
r,
A.
Also the following two rules for the provability operator are added:
Epistemic Church's Thesis
147
We will write (IA) and (EA) to denote the systems of intuitionistic and epistemic arithmetic, respectively. The notion of derivation for the two systems is defined in the usual way as an appropriately formed tree of sequents. We will write
where S
is one of the systems (In) or (EA), if there is an S-derivation
whose last sequent is
r 1A .
We will write
if
r 1 a
and
~ 1 - .r
S
2.
S
The Ggdel Interpretation
Our proof of the consistency of Church's Thesis with epistemic arithmetic makes use of the Gb;del Interpretation of intuitionistic systems into classical modal systems.
In this section we describe the interpretation
and prove a simple preservation result. 2.1.
Definition. For each formula
'p
of (IA) we associate a formula cp
of (EA), by induction on the complexity of 0
(I)
For
(11)
(cp v $ID 5 ('pa v qJo,
(1x1)
(9 A $lo
(IV)
('p
(V)
-
q'
(3Zcp)
(VI)
atomic, cp
'p
(VZcp)
('p'
0
n 0
5
.(cpn
2
qJ
as follows:
9;
;
A qJo) -#
'p,
;
U
) ;
o
= 33rp ; 0 = OVZCp .
We want to show that simple formulas of (IA) are "preserved" under the operation
( )
0
.
For this purpose it is convenient to make use of the
following notion. 2.2.
Definition. A formula (p of (EA) is called stable if
R.C.FLAGG
148
The proofs of the next three propositions are simple exercises in formal manipulation and so will be omitted. 2.3.
Proposition. Let (I)
The formula
(11)
If 'p
2.4.
2.5.
,$ ,
.
$(z) be formulas of (EA)
Then
is stable;
O'p
$ are stable, then so are
and
(111) If $(z)
'p
is stable, then so is &$(z)
('pV$)
and ('pA#)
;
.
Proposition. The following sequents are derivable in (EA):
41-(OCpA u#) 4t. ~ V 09. z
(1)
o('pA#)
(11)
OVZ'p
;
Proposition. The following rule is permissible in (EA):
=+{$-
provided each y
6
r
is a stable formula of (EA)
.
We will write 2 6
y
for 3z(y = z
+
z) ,
and z < y 2.6.
for x S y A z P y .
Definition. The class of Ao-formulas is defined inductively as
follows:
(I)
Each atomic formula is a A,-formula;
(11)
If 'p, $
are Ao-formulas, then so are ((p A $) , ('p V ' $ )
and
$1
('p-
(111) If $
;
is a AO-formula, then so are
VZC
va
and
3x<<$. 2.7.
Definition. The class of &formulas
is defined inductively as
follows: (I)
Each AO-formula is a Cformula;
(11)
If cp,
(111) If
jd
2)
are x-formulas, then so are
is a Cformula then so are W x
) we will write T ( ~ (e,g,y)
and U(y,z)
(cp A $) and
$
('p
$);
and h $ .
to denote the natural &formulas
149
Epistemic Church's Thesis
of
representing Kleene ' s "T-predicate" and the result extracting
(CA)
function.
It will also be convenient to write 'p(r,u(y))
for 3 z ( u ( y , z ) A c p(x ,z ))
We want now to show that all >formulas are stable.
Lemma. For each atomic formula
2.2.
I-
EA
'p
of (EA) ,
'p-ncp *
and
t-f-
OTY
EA proof. -
It suffices to consider the case
'p
for variables
(u=V),
But this case follows by a simple inductive argument.
z. If (p is a stable formula of
2.9.
h < v'p Proof. 2.10.
.
(EA), then the formula
is also stable. This follows by induction on
Theorem.
Proof.
u , V
//
If
'p
#.
//
is a x-formula of (EA), then
'p
is stable.
First note that any Ao-formula is equivalent to a formula obtained
from atomic and negated atomic formulas by the operations of conjunction, disjunction and bounded quatification. But such formulas are stable by ( 2 . 8 ) and ( 2 . 9 ) .
(2.3),
this, by ( 2 . 3 ) and ( 2 . 9 ) ,
Consequently all Ao-formulas are stable. From it follows that all %formulas
are stable. //
Using (2.10), by a simple induction on the complexity of &formulas, we obtain: 2.11.
3.
Theorem.
For any Cformula
'p
of (IAI,
A Model for (IA)
In this section, using our work in Part I, we present the algebraic approach to recursive realizability developed by Hyland, Johnstone and Pitts [61. The basic idea of this algebraic approach is to regard the elements of
B = P(N) as "truth values". The underlying set of the model is the set N
of standard natural numbers. We make equality for natural numbers
R.C.FLAGG
150
R-valued i n t h e following d e f i n i t i o n . 3.1.
Definition.
The e q u a l i t y map
!,=,IR:
NXN -R
i s defined by
[rn
=
1.3
n]
=
i
if
m=n
otherwise.
$
Using t h e standard a r i t h m e t i c a l o p e r a t i o n s , w e can i n t e r p r e t t h e t e r m s of ( I A ) i n t h e obvious way.
3.2.
Definition.
t
For any term
of ( I A ) and any sequence
d i s t i n c t v a r i a b l e s , which incLudes a l l t h e v a r i a b l e s of
Ill,...,
Un
of
t , we define a
mapping
]t;uI: t
by induction on t h e complexity of (I)
lui;ul
(11)
lo;
(111)
(t';u1
= hu
(IV)
jt+s;ui
=
(V)
It
.I
=
hg.
=
hu. 0;
as follows:
8;;
lt;vl+1;
h g * It;'/+
s;vl = hv
s-3,
lt;LlI
Is;v] ; 9
I.;gl
i
There i s an abuse of n o t a t i o n i n the above d e f i n i t i o n t h a t r e q u i r e s some comment.
It;vl
W e have used t h e n o t a t i o n
ambiguously t o denote a
function
]t;v]:rJn
-N
,
and t o denote the value of t h i s f u n c t i o n a t t h e p o i n t
...,Un )
(U1,
of Nn
.
The context should be s u f f i c i e n t t o allow t h e r e a d e r t o d i s t i n g u i s h between t h e two d e n o t a t i o n s . 3.3.
Definition.
For any formula
'p
of (In) and any sequence
of d i s t i n c t v a r i a b l e s , which includes a l l t h e f r e e v a r i a b l e s of d e f i n e an element 'p
a s follows:
['p;?IR
of
Ul, 'p,
...,U n we
R(Nn), by induction on t h e complexity of
151
Epistemic Church's Thesis
where f i s the projection
Nn induced by the inclusion of If
(VI)
;c
'2
&
in
mn-1
v.
does not occur among ul,
...,u n ,
1vx;ccP;vg - = YT(ex=z;x,g3= 8rp;x,g3, If x
does occur among
Ul,
...,
then
.
then with the same notation as in
V,,
clause (V).
f*("Lzcp;g*1)
[Vxcp;g] = The symbols
v,
A
.
and a used above are intended to denote the operations
of the Heyting prealgebra R(Nn)
.
One should think of the function [cp;E] as the operation which sends an n-tuple of numbers kl, kn to the set of "realizers" of the sentence
-
cp(k,,
-
...,k,) .
...,
Under this interpretation, the clauses (II)-(IV) for the
propositional operators are transparent, since the operations of R(Nn) are defined pointwise.
To appreciate the clauses for the quantifiers,
consider the case of a sentence &cp(x)
[ 3 m p ( x )1 = =
Thus an element of [&rp(x)] realizes
cp(z) .
.
Then
[ z = ~ ; z ]A [cp(x) ;x!)
u
x,E N
Cb,e>le.e[ t p ;dI ~
is a pair
.
( x , e ) where x,C N
For a sentence v x q ( x ) , we have
and e
R.C.FLACC
152
Thus an element of
[ % k ( p ( 5 ) ] is
a code e
{el (2)
such that for all z C N ,
of a total recursive function
cp(z) .
realizes
We now turn to the task of showing the soundness of this interpretation for (IA). The argument is standard; however, because of the algebraic guise,
some points may appear unfamiliar.
Lemma.
3.4.
among
Let t be a term of (IA) all the variables of which occur
...,U n .
Vl,
containing
Vl,
Let w
...,V n .
be another sequence of variables
Then I t ; . [
where
n:N
r -t Nn
= I . ; t
0 lr,
is the projection induced by the inclusion of 2 into
w.
Proof. This result follows at once by induction on the complexity of
terms.
//
Lemma. Let occur among v l ,
3.5.
containing Vl,
where
n:N
Proof.
'p
be a formula of (IA) all the free variables of which
...,V n . ...,Vn .
Let
be another sequence of variables
W l,...,lJr
Then
r -D Nn is the projection induced by the inclusion of 2 into
We argue by induction on the complexity of
(p
For
(p
atomic, the result follows from (3.4).
For
c+S
(ql v I/J2),
(ql A I/J2)
or
($
-t
I/J2) ,
.
the result follows
from the induction hypothesis and the fact, (1.9.71, that preserves Suppose
'p
2
v,
b q 9 .
A
and
*
*
n
.
First assume z
does not occur among g .
p':N
X
Nn
-t
N"
and
p:N
X
Np-DNp
be the obvious projections, Then the diagram
Let
W.
153
Epidemic Church's Thesis
where T' =
-
h -(z,n(gl) ,
if of the form required in Beck's Condition. Consequently
3 TI' P
*
*
n3+
=
Using this and the induction hypothesis, we get Bq3
]I
ez=x;z,gll
= 3
P
A
II@;z,g3,
= 9 n'*((z=z;z,gl A
P
= 7* 3
P'
=
The case when
3:
([a:=x;z,gA j e$;a:,v]l)
g
the basis of the functoriality of The argument for cp we omit it. 3.6.
Lemma.
f
.
n*(eq%gh
does occur in
[@;z,g])
reduces to the case just considered on ( )
*
.
vx$ is completely dual to that for cp
E
&$,
so
// Let
which occur among
s ( V l,...,V
n)
.
ZI~,..., u
n
variables of which occur among IS(tl,
...,t,);lJI
=
be a term of (IA), all the variables of
...,t be . Then
Let tl, lJ1,...,
terms of (IA) all the
Wr
hw_. Is;yI cltpgl ,..., It,;&
.
Proof. This follows by an easy induction on the complexity of terms. 3.7.
Substitution Lemma.
Let
cp(V1,
the free variables of which are among
...,U n ) V1,
be a formula of (IA) , all
...,2,n .
terms of (IA), all the variables of which are among
where
f:Nr-
Nn
...,tn
Let tl,
be
lJl,...,lJr.
Then
is the function
.
hw_ (It1;gl,.. ;Itn;gl Proof. -
//
We argue by induction on the complexity of
cp
.
R.C. FLAGC
154
For
'p
For
'p
atomic, the result follows from (3.6).
(Q1 A
(lbl v
or
$J2)
($1 -t $,),
the result
follows from the induction hypothesis and the fact that f* preserves V , A and =)
.
Suppose
'p
3x@.
First assume x
does not occur among
w.
Let
p ' : N X Nn * N n and
p:NX N be the obvious projections.
where
f'
=
P
-t
ZIP
Then the diagram
*-
(3C,ltl;x,WI,...,ltn;x,wl, ,
is of the form required in Beck's Condition. Consequently,
3J'
The case when x does occur in
*
w
the basis of the functoriality of
=
f
*
3P,
reduces to the case just considered on ()
*
.
We again omit the argument for the case when to that f o r 3.8.
cp
8x1)
e. Let
among V1,...,Vn.
. //
-
cp
p
vx$,
since it is dual
t be a term of (IA), all the variables of which occur Then
155
Epistemic Church's Thesis
Proof. Clearly, the,function
f = L?. [t;zJj be a code for the recursive function g
is recursive. Let e
g ( V 1 = s((V)o,-..,(V)n-l) Then e
satisfying
.
is a witness of the inequality
u,;c]A . . . A [ U n = Vn;g]L [t= t;xl . // 3 . 9 . Lemma. Let yl,. . . 61,. ..,6 be a sequent of (IA) and let 4 V 1 , ...,V n and W 1 , ...,W r be two sequences of distinct variables, both [v,
=
containing all the free variables occurring in the sequent yl, . . .,yp h1,...,6 Then the following two conditions are equivalent:
q
Proof. -
n
a
D
We establish this result in the case when all the variables
...,
Vl,
.
Vn
occur in the sequence
2
.
The general cases follows by con-
sidering a sequence containing all the variables Vl,
..., .
Wl,
Let
1J
...,U n
and
Wr
.
that
n:Nr*
Nn
be the projection induced by the inclusion of
into
The implication from (I) to (11) follows from (3.5) and the fact n*
preserves order and A
.
Using
0's
to "fillout" an n-tuple to
an r-tuple, we can construct a function
f : Nn -I Nr such that f(Ul,.
where for each v
i
..,On)
...,r ,
i=l,
ti a v
under the inclusion of g
(11) holds. order and A
I
... ,I tp;?l
= ( tl 721 ,
3
into
if w.
g;
)
,
corresponds to the variable
otherwise ti
2
0
.
Now assume
Using the Substitution Lemma and the fact that f* preserves I
it follows that
R.C.FLAGG
156
But from this, (I) follows by (3.8).
,...,yp
3.10.
Definition. Let y,
say y,
,...,yp 1- 61 ,... ,64
if for any list
t
61
,...,64
be a sequent of (IA). We
is valid in R , denoted by
(Ylf...IYP
R
//
/-
6 1 t - - . r $1
,
vl, ...,vn of distinct variables, containing all the free
variables occurring in the sequent yl,.
.. , yP 1 61,. .., & q J
This definition makes sense on the basis of (5.9). Let cp(s) be a formula of (IA) and let t be a term of
3.11.
Then the following twa sequents are valid in R:
(IA).
t
(1)
cp(t)
(11)
VsCp(s)
Proof. -
3;c(p(r) ;
.
cp(t)
Let v l , . . . , v
n be a list of distinct variables containing all the
free variables of the formula c p ( t ) f:Nn-
.
N
Let x Nn
be defined by
f(vl Finally, let 'TI
of
n:N
X
I . . . ,
an* NX
is the identity on
IiX
...,vn ) .
= (It;2l,v1
vn)
be the natural projection.
.
it follows that
is the identity on
Thus, using the Substitution Lemma, it follows that et
=
tiV1 A ecpw;l.'g
From this and ( 3 . 8 ) , we get (I). The argument for (11) is dual.
Then clearly,
From this, by the functoriality of
//
earcp;v1
.
( )
*,
157
Epistemic Church’s Thesis Soundness Theorem f o r R
3.12.
.
Let
I- k A be a sequent of
(IA).
If
r 1. A ,
(I)
IA
then (11)
Proof. -
R
1 (r
.
We w i l l show t h a t (11) holds f o r each axiom of ( I A ) and t h a t i f it
holds f o r t h e prernisses of some r u l e of ( I A ) , then it a l s o holds f o r t h e conclusion.
The Theorem then follows by induction on t h e length of a
d e r i v a t i o n of (I).
Most of t h e axioms a r e q u i t e e a s i l y handled, so we consider only a few cases. =2
-
r=s, s = t
t
r = t . For any n-tuple
kl,...,kn,
an element of
and
But c l e a r l y then
a = Ir;vl(k) = ltiEl(&) , and
a E [r = tic] (6)
.
Consequently t h e r e c u r s i v e f u n c t i o n
kc- W o witnesses t h e i n e q u a l i t y
[ r = s ;v] A [s= From this our r e s u l t follows.
We need t o show n
n
tic] S
[r=
tic] .
R.C.FLAGG
158
w
where Vl,
is a sequence of distinct variables containing the variables
...,V n
...'un.
and ul,
Clearly the function
f
= h w . It;gl
be a code for the recursive function g
is recursive. Let e
g(W) = f((")o,.*.'(W)r-l)
Now for any
satisfying
*
' U p , if the left hand side of the above inequality is
W1,..
not empty, then
and so It(g;gI = lt(g;gI = It follows that e
..., r ) .
f(W1,
W
witnesses the inequality
r
A7
cp(O),
vz(cp(z)
Suppose cp(0)
-
cp(z'))
has only
1 VzcpPz) . free. Let
2,
@ =
€cp;z,ul.
From the Substitution Lemma, we have €rp(O)
-
and
~ v ~ ; c ~cp(zl));~l (~) Let
;vl
= hV
-
@(O,V)
n (izj a
= AV.
(g(z,u) a
.
@ ( z + ~ , V ) ) )
ZEN
be the function on the left-hand side of the inequality (1)
lcp(0) ;vI
A CVX(qJP(2)
Then for any element a ~ @ ( O , v )amd
e
@(z,V)
N
I
an element of
€VXqJ(Z) ; P I
$(V)
.
is a pair
(a,e>
is the code of a total recursive function f
fying for all Z E N mapping
V
-. cp(r')) ;Vl s
, where
satis-
f ( z ) is the code of a partial recursive function into @ ( x + l , t Y ) . Consequently, the function g defined I
by g(0)
=
g(z+l) =
a
J
lf(z,j k?(z)),
Epistemic Church’s Thesis
159
is a total recursive function satisfying f o r all z Any code for
will therefore be an element of
g
E
N , g(z) t$(z,u)
[vzcp;v](v)
.
.
This
informal description of how to pass from an element of
(icp(0) ;ul
A [V~(cp(z)
to an element of
-
cp(x’)1 ; v l ) ( u )
evxcp;ug ( V ) is easily converted into a witness for the inequality (1). The validity of
?gfollows.
The validity of the rules for the propositional operators follows because each R(N
n
)
is a Heyting prealgebra.
We give a couple of illustrations.
We suppress the parameters, since they are just carried along in the argument. Assume
and
Then
and (11) for the conclusion follows.
To simplify notation, we again suppress the parameters and moreover we assume
r
and
A contain only one formula. Our assumptions become
[d
lul
6
V
evl
A
h!~l6 161
and
.
R.C. F L A W
160
Then
[yl
h-$I ev3 A ( E 6 3 c evl A ( e 6 1 5 (Ivl A
A
Ed) A (kd-.!$l) V !$I) V ("fl A !$I A
161 The validity of the rules for the quantifiers depends upon the adjunctions
a , -/ ,* 4Y n , the
Substitution Lemma and Lemma 8.11. We give the details
&
for the rules
and
u,
the arguments for
&
and
are
dual.
We consider the case when
r
is empty, A
and I is the only variable free in tp(z)
consists of a single sentence
.
The general case is easily
obtained by a notational modification of the argument given below. Suppose the only variable free in
!y=y;yl
t is y
A !v(t);yl
. 6
Assume
16;yI
.
By Lemma 5.11,
BIy=y;yl A evIT;yI s ICp(t);yl. It follows that
k-!!
.--
r
V Z (11~
provided I does not occur free in
r .
Assume
n
Since R(N x N
)
is a Heyting prealgebra, it follows that
Using the Substitution Lemma, the left hand side of this inequality is equal to
Epistemic Church's Thesis
where
TENX N
n 4
N
i s t h e n a t u r a l projection.
161
Consequently, s i n c e
v ~ ,
n*-i
a s required. The v a l i d i t y of t h e r u l e
zJ
is quite clear.
F i n a l l y , consider t h e r u l e
Again suppose
r,
f r e e v a r i a b l e s of
A
c o n s i s t of a s i n g l e formula each.
T, A , c p
a r e among
U1,...,
up
Assume a l l the
and assume
and
Then
4.
A Model f o r (EA)
Our c o n s t r u c t i o n of a " r e c u r s i v e r e a l i z a b i l i t y model" f o r epistemic a r i t h m e t i c depends on t h e r e s u l t s of s e c t i o n 1.10 and very c l o s e l y p a r a l l e l s t h e c o n s t r u c t i o n given i n t h e l a s t s e c t i o n . g r e d i e n t s are the f u n c t o r i a l i t y of t h e o p e r a t i o n
( )#
Again t h e key in-
, the
adjoint
R.C.FLAGG
162
-/
situations 3;
#
n
-1 v:
and the Beck Condition for B
.
A s before, the underlying set of the model is the set of standard natural
numbers.
A l s o the terms of arithmetic are interpreted just as in
Definition 3.2. It will be convenient below to "identify" the Heyting prealgebra R(X1
,
with the preorder of fixed points, B(X)o
.
any set X
X
B. For any map f: X
-
(1
*
.
Y and any
f*W 4.2.
for
Under this identification, the next lemma follows from
Lemma 1.10.3 and the functoriality of 4.1.
of the box operator Ox
E
Definition. For any formula cp
f#(@,
@
ER(Y) ,
.
of (EA) and any sequence U 1 ,
..., n V
of distinct variables, which includes all the free variables of cp , we define an element [ c p ; ~ ] ~ of B(Nn) , by induction on the complexity of cp
, as follows: B
R
(1)
etl= t2;gl
=
€tl=tygl
(11)
ecpv $;gl
=
ecp;$l
(111)
!cpA+;gl
=
[tp;vJ A
(IV)
Icp-+;vl
=
Icp;gl = h g 3 ;
(v)
If x
v eijJ;g1;
e+;vl;
does not occur among Ul,
€arcp;gl where
;
n:N x Nn- N
...,V n ,
3n([z=ziz,24 A
then
€(p;z,trI),
is the natural projection.
If s
does occur among
ul,...,u n , let 2' denote the sequence obtained from 2 by deleting Then e3xcp;g =
where f
is the projection N
induced by the inclusion of (VI)
f#(earcp;g'I),
2'
72-
in
Nn-l
2
If z does not occur among
Cvzq7;gl
=
. V1,...,
Vn(Cz=z;z,~1 =)
u n , then
€cp;z,vI).
z.
Epistemic Church's Thesis
If x
...,V n
does occur among V1,
163
, then with the same notation as in
clause (V),
Cvxcp;vI (VII)
lay. ;vl
v,
The symbols
I\,
=
4cp;vl
*
and
=
f# (iVxcp;g'l) ;
.
n used above are intended to denote the
operations defined in section 1.10 f o r the preorder B(N,) 4.3.
z. Let cp
occur among containing
r n:N
where
be a formula of (EA) all the free variables of which
...,vn . Let vl, ...,0, . Then V1,
-
.
LI1,..., up
be another sequence of variables
Ccp;gl = n#tCcp;gl,, n N is the projection induced by the inclusion of v into
LI. -
Proof.
We argue by induction on the complexity of
(p
.
If cp is atomic, by (4.1) and (3.51, we have R C W g = .!cp;WI R 5 ?iC[cpiv]
Ir#Ccp;g]B
2
For compound formulas, the argument is completely parallel I5 that given for (3.5). 4.4.
//
Substitution Lemma. Let
tp(V1,
...,V n )
free variables of which are among V1, all the variables of which are among
where
f:NrM
Nn
be a formula of (EA), all the
...,v, .
W1,
...,Wr .
Then
is the function
hw* (Itl;gI
I . .
.,It,;gI) .
Proof. We argue by induction on the complexity of
If cp
...,t
Let tl,
is atomic, by (4.1) and (3.7), we have
rq
.
be terms,
R.C.FLAGG
164
For compound formulas the argument is completely parallel to that given for ( 3 . 7 ) .
//
The next result is proved in the same way as Lemma 3.9. 4.5 V,,
Lemma. Let yl,. ..,yP
...,
Vn
and
WI,
...,W p
1
61,.
..,bn
be a sequent of (EA) and let
be two sequences of distinct variables, both
containing all the free variables of the above sequent. Then the following two conditions are equivalent:
r
a
V
Let y, ,...,yp 6, ,... ,6 ,...,yP 1 6 , ,...,69 is valid in B , 4 B k ( Y , , . . . ~ Y ~ 16,,....6q)
4.6. Definition.
say y,
...,
be a sequent of (EA).
We
denoted by
,
if for any list v l , of distinct variables, containing all the free Vn variables occuring in the sequent y,, ,yp 6., ,6q ,
...
1.
...
Again, the next xesult is proved just as the corresponding result, (3.11) , was proved. 4.7.
=.
Let cp(x) be a formula of (EA) and let t
be a term.
Then
the following two sequents are valid in B :
4.8.
cp(t)
(11)
VxqW
3mp(z) ;
t
cp(t)
.
Soundness Theorem for B (I)
r t EA
then
t
(1)
A,
.
Let
r
A
be a sequent of ( E A ) .
If
165
Epistemic Church’s Thesis (11) ~
(r
B
t
A)
.
Proof. We will show that (11) holds for each axiom of
(EA)
and that if it
holds for the premises of some rule of (EA), then it also holds for the conclusion. The Theorem then follows by induction on the length of a derivation of (I). We will only consider cases when the arguments are not straightforward modifications of the arguments for the corresponding cases in the proof of the Soundness Theorern.for R
A7 Suppose
cp(O), v X ( c p ( X ) cp(0)
has only
. 1- v X c p ( X ) .
-I cp(Z’))
free. Recall that
V
B ( N x N ) ~ R ( N xN x R )
and the preordering of B(N
N
x
agrees with that of R(N X
)
B
@ = [cp;s,V]
E R ( N x Nx R)
N x
R)
.
Let
.
From the Substitution Lemma, we have
hup
CcpPcO) ; V 3
and
[vX( c p ( X )
-
cp(xl
) ) ;VJ
=
h ~ p .n
X C N
*
Z(O,V,P) ,
(GI
(Z( X , V , P I
a
z (X + 1, V , P I 1 )
Let $ be the function on the left-hand side of the inequality (1) [ c p ( O ) ; ~ ]
/t
“e‘;c(cp(~)*cp(~‘));t)] e V ~ q ( ~ ) ; t r l .
Then for any elements V . 6 N pair
and p , k R , an element of
( a , e ) where a € $ ( O , O , p )
and e
$(U,p)
“is” a
is the code of a total recursive
function satisfying: for all x , t N , f ( r ) is the code of a partial recursive function mapping function g
@(s,V,p)
into @ ( x + l , U , p )
.
Consequently, the
defined by g(0)
= a
g ( 5 + 1 ) = Cf(z)I(g(z)) ,
is a total recursive function satisfying : for all Any code will therefore be an element of
3: 6
[hp;V](U,p)
(1) follows, and from this we get the validity of A7.
N
, g(z),c@ ( X , U , p )
.
The inequality
.
R.C.FLAGG
166 Assume
Since the box-operator on
B(Nn )
o!cp;gI
satisfies
Bcp;gI
Y
we at once get
as required.
Assume
Since
we get
Also, since
we get
Consequently since
as required.
//
0
preserves order and
A
,
Epistemic Church's Thesis
where
is any formula of ( E A ) .
cp
167
This is the natural epistemic analog of
Church's Thesis for intuitionistic arithmetic, which is the schema:
CT(cp) where
cp
vx3ycp(Sry)-3ekz%(T(e,x,y)
A cp(Z,U(y)))
,
.
is any formula of (IA)
In this section we will show that ECT is valid in the model B .
From this
our main theorem, stating that Epistemic Church's Thesis can be consistently added to (EA), follows. Suppose P is some q-ary relation symbol.
Let L(P)
denote the language
of arithmetic extended by adding P as a new relation symbol.
5.1.
Definition. For any
@ c R ( N q ),
defined on all formulas of
to an interpretation [ * ] ( R " ) ,
e.1 R
we extend the interpretation L(P)
,
by
adding the clause
C m , , . ..rt4 ) ;gl ( R , @ ) =
ja(lt,;&...,lt
.Ul)
4'-
to Definition 3 . 3 . We extend the notion of validity in R for formulas of the language L(P)
,
to a notion of validity in
(R,ja)
in the obvious way.
The usual argument showing that Church's Thesis is valid under the recursive reallzability interpretation gives 5.2.
Theorem. For any formula cp
of
and any function @
L(P)
in
R ( N ~ ) ,
(I?,@) If u(vl, then
...,vq
)
=!
-
E('p)
is a formula of (EA), with at most
V l'.
.. ,v 4
free,
[ oo;vlB is an element of R ( N 4 ) .
We can take this for g5
above.
In this case
we will write
tcpig1 ( R * o ) for
['p;w]
( R , @ ),
and (R,Q) 5.3.
Ul,.
l=
'p
Definition. For any formula
..,V 9
for
w,@)
l=
cp
of (EA), with at most
U ( V l,...,Uq)
free, we extend the interpretation
.
(
9
0
of (IA) into (EA) to
R.C.FLAGG
168
an interpretation
( )(
"),
P(t,,
..
defined on all formulas of
L(P)
, by
adding
the clause . I
t
P
4
Ga(tl,
...,t
4
to Definition 2.1. 5.4.
Change of Basis Theorem.
Let P
be a
let U be a formula of (EA) with at most ?I1, formula cp
wl,
...,w p
Proof.
of
L(P)
q-ary relation symbol and
...,u4
free. Then for any
all the free variables of which occur among
J
We argue by induction on the complexity of
cp
.
For cp an atomic formula of (IA), the result follows directly from the definitions. Suppose cp
P(tl,
...,t4 ) .
Then
and
Our result then follows by the Substitution Lemma for B
.
For compound formulas we will use the relationships between the logical operations of R
and those of
and Lemma 1.10.9). Suppose
and
so,
'p
('po
3
(cf. the remarks after Theorem 1.10.2
We give two illustrations.
* 'pl)
.
Then
using the induction hypothesis,
as required.
169
Epistemic Church's Thesis
Suppose cp
vx$.
Then 'Po'
OVX$
0
and so, using the induction hypothesis,
['p;glR= vJez=X;z,gl
=?
I
oV#([x=z;x,wl
=
1 ovmjJ;g]
BQJ;x,g4R ) B$+ 0 ;z,gl 3)
as required. // 5.5.
Corollary. Let P and
formula
'p
of
L(P)
(R,a) 5.6.
be as in the Theorem. Then for any
Q
,
/=
cp
cp("
if and only if B
')
Theorem. Let 'p be a formula of (EA). Then
From this, 5.2 and 5.4 it follows that B
5.7.
0
ECT
7
Corollary. The system (EA)
Proof. -
('p)
r
+
Y
.
//
(ECT) is consistent.
Suppose on the contrary, that (EA)
Then for some formulas
'pl,...t'pn
+ (G)1 I
A
-
.
R.C.FLAGG
170
By the Soundness Theorem for 3 , we get
n B C-IU
A
~
,
~
(
'
i=1 But this clearly contradicts the Theorem.
.p
~
)
//
[*IB .
We close by giving an explicit definition of the interpretation
This will provide a version of recursive realizability for epistemic arithmetic. The basic notion is: "the number e set
realizes the sentence
relative to the
'p
X of realizers of absurdity."
This is denoted by erx'p
1
and defined by induction on the complexity of (1) for
'p
as follows:
atomic,
er 'p if and only if X
('p
(2) erx(y*#)
Vu((arX'p a te3 ( a )4
'p
is true) or
( e EX)
;
if and only if &
{el (a)rx$)
;
(3)
erxV?Jcp(v) if and only if Vu((e] (a)4
(4)
erXOcp
YU (E
( ~ ~ yX = (br J cp)
&
( e ] (a1rxcp(a))
;
if and only if
Y
=)
{a] ( b )c
&
{ a ] ( b ) c X) 1(el ( a )4
The clauses for the logical operators
v,
A
and
3
those above by way of their definitions in terms of
&
( e ] ( a ) X)
.
can be obtained from I,
-.
and
v.
The results of this and the last section can be developed directly from the above definition. avoided.
In this way all the category-theoretic work can be
However with this approach, the basic motivation for the model
construction is lost; moreover, the arguments tend to become longer and obscured by numerous tedious calculations.
Epistemic Church's Thesis
171
REFERENCES [l]
Barry, M., "Toposes Without Points," J. Pure and Applied Algebra 5 (19741, 265-280.
[21
Fourman, M. P. and Scott, D. S., "Sheaves and Logic", Applications of Sheaves, ed. Fourman, Mulvey and Scott, Lecture Notes in Mathematics 753 (Springer 1979), 302-401.
[31
G6de1, K., "Eine Interpretation des Intuitionischen Aussagenkalkuls," Ergebnisse eines mathmatischen Kolloquiums. 4 (1932), 39-40.
[41
Goodman, N. D., "Epistemic Arithmetic is a Conservative Extension 49 (1984). of Intuitionistic Arithmetic", Journal of Symbolic&L
[5]
Hyland, J. M. E., Notes on Realizability Toposes, University of Cambridge, 1979.
[6]
Hyland, J. M. E., Johnstone, P. T. and Pitts, A. M., "Tripos Theory", Math. Proc. Camb. Phil. SOC. 88 (1980), 205-232.
[7]
Johnstone, P . T., "Adjoint Lifting Theorems for Categories of Algebras", Bull. Lond. Math. SOC. 7 (1975), 294-297.
[8]
Johnstone, P. T.,
[9]
Lauchli, H., "An Abstract Notion of Realizability for which Intuitionistic Predicate Calculus is Complete," Intuitionism and Proof Theory, ed. A. Kino, J. Myhill and R. E. Vesley (North Holland, Amsterdam, 1970), 227-234.
[lo]
Lawvere, F. W., "Adjointness in Foundations," Dialectica 23 (1969), 281-296.
[ll]
Lifschitz, V . , "Constructive Assertions in an Extension of Classical Mathematics," Journal of Symbolic Logic 47 (1982), 359-387.
[121
Maclane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics no. 5 (Springer-Verlag 1971).
[131
Makkai, M. and Reyes, G. E., First-Order Categorical Logic, Lecture Notes in Mathematics 611 (Springer-Verlag 1971).
[14]
Myhill, J., "Intensional Set Theory," this volume.
[15]
Rasiowa, H. and Sikorski, R., (Warsaw 1963).
1161
shapiro, S., "Epistemic and Intuitionistic Arithmetic," this volume,
Topos
Theory, (Academic Press, London, 1977).
The
Mathematics
of Metamathematics,
172
R.C. FLAGG
[171
Takeuti, G.,
Proof Theory (North Holland, Amsterdam, 1975).
I181
Troelstra, A. S . , Metarnathematical Investigations of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics 344 (Springer, Berlin, 1975).
Intensional Mathematics S. Shapiro (Editor) @ Elsevier Science Publishers B. V. (North-Holland), I985
173
CALCULABLE NATURAL NUMBERS
Vladimir L i f s c h i t z Department of Mathematics U n i v e r s i t y of Texas a t ~1 Paso E l Paso, Texas U.S.A.
INTRODUCTION
L e t A b e a mathematical a s s e r t i o n which h a s been n e i t h e r proved n o r r e f u t e d , and l e t f : w -+ o be d e f i n e d by
f(n) =
1 i f A is true, 0 otherwise.
Since f ( n ) d o e s n o t depend on n , f i s r e c u r s i v e (even p r i m i t i v e recursive).
However, one c a n n o t compute f ( 0 )as l o n g as A remains u n s o l v e d ,
and it s e e m s t h a t one h a s t o s o l v e A t o a r g u e t h a t f i s i n t u i t i v e l y computable. Examples of t h i s k i n d have been known f o r a l o n g t i m e ;
P&er
[ll]
and Heyting [ 2 1 claim t h a t such examples r e f u t e t h e c o n v e r s e o f C h u r c h ' s thesis. I n f a c t , f i s r e c u r s i v e i f t h e r e e x i s t s a number m such t h a t
f
=
cpm (where ( ( P m ) m E w
i s a s t a n d a r d GBdel numbering of t h e unary r e c u r s i v e
f u n c t i o n s ) ; b u t , t o compute f ( n ) , one h a s t o know n o t merely t h a t m e x i s t s , b u t a l s o what m is.
The t h e o r y of r e c u r s i v e f u n c t i o n s d o e s n o t p r o v i d e any
means t o e x p r e s s t h i s d i s t i n c t i o n .
According t o Heyting [ 2 1 , t h e t h e o r y of
r e c u r s i v e f u n c t i o n s h a s made t h e n o t i o n of a c a l c u l a b l e f u n c t i o n more p r e c i s e , b u t n o t t h a t of a c a l c u l a b l e number. Heyting views t h i s d i f f i c u l t y as a n argument i n f a v o r of t h e cons t r u c t i v i s t program.
As a matter of f a c t , i n t u i t i o n i s t i c number t h e o r y can
be viewed as a t h e o r y o f c a l c u l a b l e n a t u r a l numbers:
it shows how t o oper-
a t e w i t h numbers under t h e assumption t h a t t h e y a l l a r e c a l c u l a b l e . the intuitionist,
(1) simply d o e s n o t d e f i n e a t o t a l f u n c t i o n .
For
Intuition-
i s t i c mathematics makes t h e n o t i o n of c a l c u l a b l e number more precise a t t h e p r i c e of f o r b i d d i n g n a t u r a l numbers i n t h e wide s e n s e , which l e a d s t o t h e n e c e s s i t y of r e v i s i n g t h a v e r y fundamentals of mathematics.
V. LIFSCHITZ
174
In t h i s paper another theory of c a l c u l a b l e numbers i s proposed.
We
extend c l a s s i c a l a r i t h m e t i c by adding t h e unary p r e d i c a t e K which means "is calculable".
The l o g i c of t h e extended theory i s more r e s t r i c t e d than
c l a s s i c a l , b u t t h e r e s t r i c t i o n s do n o t a f f e c t t h e K-free fragment of t h e theory, so t h a t t h i s fragment e x a c t l y c o i n c i d e s with c l a s s i c a l a r i t h m e t i c . Hence t h i s theory of c a l c u l a b l e numbers does n o t involve any r e v i s i o n of classical results. On t h e o t h e r hand, q u a n t i f i e r s r e s t r i c t e d t o c a l c u l a b l e numbers have
t h e meaning c l o s e t o t h a t of i n t u i t i o n i s t i c q u a n t i f i e r s .
Formulae of in-
t u i t i o n i s t i c a r i t h m e t i c can be t r a n s l a t e d i n t o t h e extended language by r e s t r i c t i n g a l l q u a n t i f i e r s t o K , and, i n t h i s sense, t h e extended system a l s o contains i n t u i t i o n i s t i c arithmetic.
Other t h e o r i e s i n which c l a s s i c a l
and i n t u i t i o n i s t i c systems c o e x i s t a r e developed by Shapiro [121 and Myhill
[lo]; i n s e c t i o n 6 we compare t h e i r approach with ours. Some r e s u l t s of t h i s paper are announced i n [6J. In [81 we show how t h i s theory of c a l c u l a b l e numbers can be developed on t h e b a s i s of an informal axiomatization, and a l s o d i s c u s s e x t e n s i o n s of t h e p r e s e n t system.
Language.
1.
T e r m s and formulae of i n t u i t i o n i s t i c f i r s t - o r d e r
_-
a r i t h m e t i c HA and c l a s s i c a l f i r s t - o r d e r a r i t h m e t i c
HIC
a r e b u i l t up i n t h e
standard way, using v a r i a b l e s f o r n a t u r a l numbers, c o n s t a n t
0,
symbols f o r
p r i m i t i v e r e c u r s i v e f u n c t i o n s , p r e d i c a t e =, and l o g i c a l symbols &,+,V,B. The numeral r e p r e s e n t i n g n i s denoted by
n.
Symbols
1,V ,
tf
a r e used a s
a b b r e v i a t i o n s w i t h t h e usual d e f i n i t i o n s :
Now w e extend t h e language of HAc by adding atomic formulae of t h e _I
form K ( r ) f o r a l l t e r m s r.
K ( r ) reads:
"r i s c a l c u l a b l e " , o r "r can be
found". Here i s t h e intended meaning of some types of formulae of t h e extended language. For any formula A ( x ) ,
Calculable Natural Numbers 3x(K(x)
175
& A(x)),
or, shorter,
means: one can f i n d an x such t h a t A ( x ) .
For i n s t a n c e , f o r any unary func-
t i o n symbol f ,
expresses t h e c a l c u l a b i l i t y of an upper bound of t h e set of s o l u t i o n s of f(y) =
0.
On t h e o t h e r hand,
or, shorter,
means: A ( x ) whenever t h e value of x i s known.
For i n s t a n c e ,
expresses t h e a s s e r t i o n t h a t one can f i n d a r b i t r a r i l y l a r g e numbers with the property A ( y ) .
Notice t h a t
expresses a s t r o n g e r condition: t h e p o s s i b i l i t y of f i n d i n g y A ( y ) , even when x i s unknown, i . e . ,
impossible.
uniformly f o r a l l x.
2
x such t h a t
This i s c l e a r l y
One can express t h e s t r o n g e r form of r e c u r s i v e n e s s discussed
i n t h e I n t r o d u c t i o n by r e s t r i c t i n g t h e e x i s t e n t i a l q u a n t i f i e r (over Gddel numbers) i n t h e d e f i n i t i o n of r e c u r s i v e n e s s t o K.
On t h e o t h e r hand, t h e r e
i s a much simpler condition expressing t h e n o t i o n of computability:
V. LIFSCHITZ
116
i f x i s known then one can f i n d f ( x ) .
which means:
Thus w e c h a r a c t e r i z e Using t h e precise
computable f u n c t i o n s a s t h o s e p r e s e r v i n g t h e property K.
d e s c r i p t i o n of t h e semantics below one can show t h a t t h e s e two c o n d i t i o n s a r e equivalent. We extend t h e d e f i n i t i o n s ( 2 ) , ( 4 ) of negation and equivalence t o t h e extended language; p o s s i b l e d e f i n i t i o n s of d i s j u n c t i o n i n t h e extended language a r e discussed i n s e c t i o n 5.
2.
Kolmogorov [41 showed t h a t i n t u i t i o n i s t i c proposi-
Semantics.
t i o n a l l o g i c can be i n t e r p r e t e d a s t h e l o g i c of c o n s t r u c t i v e problems. semantics of our extended language i s based on t h e same idea.
The
The p o s s i -
b i l i t y of g e n e r a l i z i n g Kolmogorov's i n t e r p r e t a t i o n t o e x t e n s i o n s of c l a s s i c a l systems i l l u s t r a t e s t h e f a c t (pointed o u t by Heyting [l, 55.21)that it provides i n t u i t i o n i s t i c l o g i c with a meaning independent of t h e i n t u i t i o n i s t i c conception of mathematics. A constructive problem is defined t o be simply an a r b i t r a r y set IT of
n a t u r a l numbers.
solves
IT.
If e
E
IT,w e
say t h a t e i s a solution of
or that e
IT,
The d e f i n i t i o n below a s s i g n s a c o n s t r u c t i v e problem t o every
closed formula of t h e extended language; t h u s c o n s t r u c t i v e problems are s i m i l a r t o t r u t h v a l u e s i n t h e t r a d i t i o n a l semantics of c l a s s i c a l mathematics.
By asserting a closed formula A , one claims t h a t
a solution of the
constructive problem assigned t o A has been found. The r e l a t i o n " e s o l v e s ( t h e c o n s t r u c t i v e problem assigned t o ) A" i s defined by induction on t h e number of l o g i c a l symbols i n t h e closed formula A , thus:
1.
e solves r = s i f r
2.
e solves K ( r ) i f e e q u a l s t h e value of r.
3.
e solves A
&
standard enumeration of w 4.
e solves A
-f
= s
is t r u e .
B i f e i s t h e number of a p a i r < e l , e2> i n t h e 2 , such t h a t el solves A and e 2 solves B . B i f , f o r any n which solves A , cp ( n ) i s d e f i n e d
and solves B . 5.
e solves VXA(X) i f e solves ~ ( i i f) o r every numeral
6.
e solves 3xA(x) i f e solves A ( i j ) f o r some numeral
ii.
i.
There i s a c l e a r s i m i l a r i t y between t h i s d e f i n i t i o n and Kleene's r e a l i z a b i l i t y [ 3 , 5 8 2 1 ; more about t h a t a t t h e end of t h i s s e c t i o n .
Clauses 5,
6 a r e simpler than t h e corresponding c l a u s e s i n Kleene's d e f i n i t i o n (they
177
Calculable Natural Numbers
resemble t h e c l a u s e s of t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n introduced by Kreisel and T r o e l s t r a [ 5 1 t o prove t h e f i r s t - o r d e r e x p l i c i t d e f i n a b i l i t y
property f o r t h e theory of s p e c i e s ) . Notice t h a t we do n o t c l a s s i f y c l o s e d formulae of t h e extended language as t r u e o r f a l s e ; i n s t e a d , our semantics d e f i n e s under what condit i o n a c l o s e d formula c a n be a s s e r t e d .
I t does not make sense t o ask
whether a formula i s t r u e ; one may only ask f o r t h e s o l u t i o n s of t h e cons t r u c t i v e problem corresponding t o t h a t formula. To i l l u s t r a t e t h e above d e f i n i t i o n , we s h a l l apply it f i r s t t o ( 5 ) . According t o t h e d e f i n i t i o n , e s o l v e s ( 5 ) i f , f o r every numeral f;, e s o l v e s ~ ( f ; )+ K(f
(ii)), o r , e q u i v a l e n t l y , i f , f o r every m s o l v i n g ~ ( f ; )q,e ( m ) s o l v e s
K(f(i)).
But n i s t h e only number s o l v i n g K ( n ) , and f ( n ) i s t h e only num-
ber s o l v i n g K ( f ( i i ) ) .
Hence e s o l v e s ( 5 ) i f f f o r every n , Ve(n) = f ( n ) .
Thus t h e c o n s t r u c t i v e problem assigned t o ( 5 ) i s t h e set of Gddel numbers of f .
By a s s e r t i n g (5), one claims t h a t a m d e l number of f has been
found, i.e.,
t h a t f was found t o be r e c u r s i v e by a c t u a l l y c a l c u l a t i n g one
of i t s Gddel numbers.
This a g r e e s with t h e intended meaning of ( 5 ) .
Our n e x t example is v x K(x).
A number e s o l v e s t h i s formula i f it
s o l v e s every formula of t h e form K ( n ) .
n.
But t h e only number s o l v i n g K(f;) i s
Hence t h e c o n s t r u c t i v e problem assigned t o v x K(x) h a s no s o l u t i o n s .
A
s i m i l a r argument shows t h a t even
has no s o l u t i o n s . Consider now t h e induction scheme
I f A(x) i s allowed t o c o n t a i n K,
(7) g e n e r a l l y cannot be solved.
As a m a t -
t e r of f a c t , a number s o l v i n g t h e antecedent of ( 7 ) g i v e s information about, f i r s t l y , how one can s o l v e A ( E ) , and secondly, how one can solve A
(i')
provided a s o l u t i o n of A
t o solve A(:)
(G) is
given.
This i s c e r t a i n l y s u f f i c i e n t
f o r any given number n ; but t o solve V x A ( x ) w e have t o spec-
i f y a common s o l u t i o n of a l l formulas A ( ; ) . i f A ( x ) i s K(x)
.
Actually, ( 7 ) h a s no s o l u t i o n s
The s i t u a t i o n changes i f t h e second e x p l i c i t l y shown
q u a n t i f i e r i n (7) i s r e s t r i c t e d t o K.
Then we can assert t h e i m p l i c a t i o n ,
even i f t h e q u a n t i f i e r i n t h e antecedent i s r e s t r i c t e d t o K too.
This
V. LIFSCHITZ
178
modification of induction i s included i n t o t h e l i s t of p o s t u l a t e s of t h e formal system described i n t h e n e x t s e c t i o n . The d e f i n i t i o n of ” s o l v e s ” can be reformulated a s follows.
To every
formula A of t h e extended language w e s h a l l a s s i g n a K-free formula x and then d e f i n e : e s o l v e s a c l o s e d formula A i f
eg
? i s true.
t i v e d e f i n i t i o n of s u s e s t h e following n o t a t i o n : j,,
A,
The induc-
j, a r e t h e p r i m i t i v e
r e c u r s i v e function symbols r e p r e s e n t i n g t h e i n v e r s e s of t h e standard enumeration of u L ; T , U r e p r e s e n t Kleene‘s T-predicate and r e s u l t e x t r a c t i n g f u n c t i o n ( i . e . , T ( x , y , z ) means t h a t z i s t h e Gddel number of t h e computation of (px(y), and U(z) i s t h e r e s u l t of t h e computation).
W e de-
fine :
x s r = s
-=def
r = s,
where Q i s a q u a n t i f i e r .
The equivalence of t h e s e two d e f i n i t i o n s is imme-
diate. There e x i s t s a simple connection between c o n s t r u c t i v e problems assigned t o K-free formulae and t h e i r t r u t h values. formula A w e a s s i g n a number e
A
er = s
“A & B
“A + B
To every closed K-free
i n t h e following way:
is 0 ,
i s t h e number o f t h e p a i r < e A , e >, B
i s a Gddel number of t h e c o n s t a n t f u n c t i o n Axe
B’
eQxA is e A . 2.1.
I f a closed K-free formula A is t r u e then eA s o l v e s t h e con-
s t r u c t i v e problem assigned t o A .
Calculable Natural Numbers 2.2.
179
I f a c l o s e d K-free formula A is f a l s e then t h e c o n s t r u c t i v e
problem assigned t o A has no s o l u t i o n s . These a s s e r t i o n s can be e a s i l y proved simultaneously by induction on t h e number of l o g i c a l symbols i n A .
They show t h a t our semantics, when ap-
p l i e d t o K-free formulae, i s e q u i v a l e n t t o t h e c l a s s i c a l semantics. Now we s h a l l d e s c r i b e t h e r e l a t i o n between t h e semantics of t h e extended language and Kleene's r e a l i z a b i l i t y x f A , defined f o r every K-free A a s follows:
x r r = s
= -def
r = s,
For every K-free A , l e t A
f
s t a n d f o r t h e formula obtained from A by
r e s t r i c t i n g every q u a n t i f i e r t o K.
2.3.
For every K - f r e e
A,
Proof by induction on A. with a q u a n t i f i e r ; i n
The only n o n - t r i v i a l c a s e is when A s t a r t s
V. LIFSCHITZ
180
Proposition 2.3 gives a decomposition of Kleene's realizability into two simpler operations.
It also gives a precise meaning to the assertion
that quantifiers restricted to K are similar to intuitionistic quantifiers. It should be observed, however, that realizability is not considered a correct interpretation of arithmetic by those constructivists who reject Church's thesis.
Moreover, one can claim that realizability renders the
constructive meaning of a formula correctly only if x constructively.
r
A
is understood
(This distinction is crucial, e.g., for the question of
whether Markov's scheme is realizable).
In the present theory we accept
exactly the more liberal approach; the definition of "solves" is intended to be used in the context of classical mathematics. These remarks show that the relation of our semantics to constructive mathematics is not so direct.
There seems to be a more immediate connection between formaliza-
tions of constructive mathematics and formal systems with postulates for K. 3.
gK
Formal system.
is the formal system in the extended
language with the following postulates. A.
Logical postulates:
B.
Al.
Intuitionistic predicate calculus with equality.
A2.
-?T
A3.
*A
A4.
-(XI.
= y + x = y. +
m.
Arithmetical postulates: B1. 7 ~=' 0. B2. 83.
x ' = y'
-+ x = Y. The definitions of primitive recursive function symbols.
Postulates for K:
C.
C1.
Thus
K(5).
C2.
K(x) + K(x').
C3. K
A(:)
&
vx
EK
[A(x)
+
A(x')]
+
vx E K A ( x ) .
is based on a logic intermediate between classical and (the
disjunction-free fragment of) intuitionistic. For any K-free A ,
-ns
-+A
(induction on A , using A 1
closed under classical logic. K-free A
-
HAK
A3); thus the K-free fragment of
!I K
Furthermore, using A4, we get for every
.
1s
Calculable Natural Numbers
181
B4
implies t h e induction scheme ( 7 ) f o r every K-free A ( x ) . Hence a l l p o s t u l a t e s of gca r e d e r i v a b l e i n HiK , and HI K 1. s an extension I t follows t h a t
of HAC. _-
On t h e o t h e r hand, t a k e any proof i n HAK and r e p l a c e every subI-
formula of t h e form K ( r ) i n it by r = r ; t h e r e s u l t i n g sequence of formulae can be e a s i l y made i n t o a proof i n HAC.
I t f o l l o w s t h a t a K-free formula
- 1
is provable i n
3.1.
HAK
HAK
only i f i t i s provable i n
Hic.
Thus w e have:
HA".
i s a conservative extension of
Since t h e i n t u i t i o n i s t i c meaning of a closed a r i t h m e t i c a l formula A i s c l o s e t o t h e meaning of A + , it seems p l a u s i b l e t h a t t h e p r o v a b i l i t y of A
+
i n a reasonable formal system i n t h e extended language should be equiva-
l e n t t o t h e p r o v a b i l i t y of A i n a f o r m a l i z a t i o n of i n t u i t i o n i s t i c arithmetic.
K
We do n o t know what f o r m a l i z a t i o n corresponds i n t h i s sense t o
g.
but believe t h a t t h i s is precisely
,
I t i s easy t o prove t h i s c o n j e c t u r e
i n one d i r e c t i o n :
3.2.
Hi
For every c l o s e d K-free formula A , i f
A then
HeK
A'.
The converse of 3.2 i s discussed i n s e c t i o n 7. Now we s h a l l sketch t h e proof of t h e soundness of t h e semantics described i n s e c t i o n 2. a closed formula A i n
K
HAK
with r e s p e c t t o
We should show how f o r any proof of
one can e f f e c t i v e l y f i n d a number which s o l v e s A .
Hence it s u f f i c e s t o give an e f f e c t i v e proof of t h e following p r o p o s i t i o n :
3.3.
I f HAK - I
A then f o r some numeral
HAc
ii,
1 ii
A.
Notice t h a t A i s n o t r e q u i r e d t o be c l o s e d ; t h e reason is t h a t 5 , unl i k e most o t h e r r e a l i z a b i l i t y i n t e r p r e t a t i o n s , commutes with q u a n t i f i e r s . The proof i s s i m i l a r t o t h e usual proof of t h e soundness theorem f o r Kleene's r e a l i z a b i l i t y [9, and A4.
13.2.41.
It i s easy t o check t h a t
We s h a l l consider two p o s t u l a t e s : A3
HAc
proves
Recall t h a t f o r any metamathematical expression
r
representing a recursive
f u n c t i o n of a v a r i a b l e x, A x . T s t a n d s f o r a Gddel number of t h i s f u n c t i o n . Arguing i n
Ezxfz
2
HAc,
assume y ?-i%xA.
A ) , which, by (81, i m p l i e s
ny.0 5 - t i ~ x+~a m . (81, 0 s - r J c ( x ) -
Then, by (81,
3r(o ? - A ) ,
Furthermore, x
s
3Z(Z
i.e.,
s_
0
3 x A ) , i.e.,
s_ E m . Hence
~ ( x i)m p l i e s 3 y ( y
s
~ ( x ) ) and, , by
V. LIFSCHITZ
182
4.
In this section we discuss proof-
Explicit definability.
theoretic closure properties of ing modification
s'
x s ' K(r) x s' A
&
Their proofs are based on the follow-
of s , similar to Kleene's q-realizability: def r
x s ' r = s
HAK .
= s,
adef x = r, B
sdef jl(x)
s' A
&
j2(x) s ' B,
The proof of 3 . 3 extends to this modification and gives: 4.1.
If
HAK k
A
then for some numeral
yeK
?1 5' A .
From this proposition we easily derive the following theorem similar to the so-called existence property in intuitionistic logic: 4.2.
If
HAK k
3x
E K A ( x ) and then for some numeral :!35K
A
(GI.
From 4.2 and 3.1 we immediately conclude: 4.3.
K If $
k
3x
E K A(x) and
A ( x ) is K-free then for some numeral
H --. A ck A ( : ) . Using 4.1, we can also prove the closure under Church's rule:
4.4. If some numeral i?
yeK
HAc
v x E K By
E K
v x Rv(T(:,x,v)
A(x,y) and A(x,y) is K-free then for & A(X,
U(v)
)).
Notice that in all these theorems, like in 3 . 3 parameters are allowed. Now we shall use 4.4 to show how
@"
reflects the difference between
two kinds of definitions of recursive functions discussed in the introduction.
Since the language of
"AK does not
contain symbols for functions deHAc and K by adding
fined by formulae like (11, we conservatively extend
a unary function symbol f to their languages, and its definition
Calculable Natural Numbers
183
to their lists of postulates, where A(x,y) is a K-free formula with all parameters explicitly shown such that
K
.
The resulting systems are denoted respectively by S and S We would like K to know when 5 proves condition (51, expressing the computability of the newly introduced function f.
The answer is given by the following proposi-
tion.
4.5.
sK proves
( 5 ) iff for some numeral i
s proves
K Thus S proves
Formula (9) expresses that n is a G8del number of f. the Computability of f if for some n, f = cp arithmetic. To prove 4.5,
is provable in classical
K
assume first that S proves ( 5 ) .
K Then HA -.. proves
and we apply 4 . 4 and the definition of f. Assume now that
5
proves (9).
Then HAc - _ proves
It follows that this formula is provable already in K by 3.2, HA proves
[9, 13.8.63.
--
Now (5) follows from ( 9 ) ,
(lo), and the uniqueness condition for T,
If f is defined by the formula expressing (I), i.e.,
Then,
V. LIFXHITZ
184
where A i s a c l o s e d K-free formula, then ( 5 ) i s provable i n
-.-. 1-
o r HAc
5.
y1.
K .
iff
HIC 1-
A
This f a c t e a s i l y follows from 4 . 3 .
Disjunction.
There a r e two d i f f i c u l t i e s connected with t h e use
of d i s j u n c t i o n i n t h e language of
K
.
Firstly, the rule A + C , B + C A V B - t C
t u r n s o u t t o be i n c o r r e c t .
(111
Recall t h a t d i s j u n c t i o n i s a defined connec-
t i v e , and A V B has been d e f i n e d only f o r K-free A , B . A , B a r e K-free,
b u t l e t C be allowed t o contain K.
A s s u m e then t h a t
I t may happen then t h a t
both premises can be solved i n t h e sense of s e c t i o n 2 , and t h e conclusion cannot.
Example:
( s e e t h e d i s c u s s i o n of (6) i n s e c t i o n 2 ) .
This may seem puzzling, s i n c e
(11) i s c o r r e c t both i n c l a s s i c a l and i n t u i t i o n i s t i c l o g i c .
Notice, how-
e v e r , t h a t d i s j u n c t i o n i n (11) i s ” c l a s s i c a l ” , and t h e semantics w e use i s “constructive“.
This discrepancy, e x i s t i n g n e i t h e r i n purely c l a s s i c a l nor
i n purely c o n s t r u c t i v e systems, i s r e s p o n s i b l e f o r t h e i n v a l i d i t y of (11). One can j u s t i f y (11) with K - f r e e A , B i f C i s r e q u i r e d t o be ( e q u i v a l e n t t o )
a negated formula. Secondly, w e do n o t see any reasonable g e n e r a l i z a t i o n of d i s j u n c t i o n defined by ( 3 ) t o a r b i t r a r y formulae of t h e extended language.
There i s
c e r t a i n l y no hope t o g e t (11) c o r r e c t ; but i f we d e f i n e A V B by ( 3 ) f o r a r b i t r a r y A,B then something worse happens: w e cannot j u s t i f y even A V A A.
(Counter-example: t a k e K ( x ) t o be A ) .
-+
The same happens i f w e d e f i n e
AVBby
( t h e same counterexample).
There a l s o seems t o be no way t o treat d i s j u n c -
t i o n a s an undefined connective and add a c l a u s e f o r d i s j u n c t i o n t o t h e
185
Calculable Natural Numbers d e f i n i t i o n of "solves" i n such a way t h a t A
v
A
-f
A would be s o l v a b l e and,
a t t h e same time, t h e meaning of K-free d i s j u n c t i o n s would n o t change. On t h e o t h e r hand, t h e r e i s a u s e f u l d e f i n i t i o n of d i s j u n c t i o n f o r a r b i t r a r y formulae which i s
not
equivalent t o ( 3 ) f o r K-free formulae.
It
can be c a l l e d " i n t u i t i o n i s t i c d i s j u n c t i o n " :
The following p r o p e r t i e s of VI
A + A V
I
a r e derivable i n
HaK:
B,
B + A V
A + C , B + C A V B + C I
I
B,
'
and, f o r K-free A , B ,
A v
I
B - t A V B .
Proposition 4 . 2 implies:
gK
A VI B then
HAK
1-
If
6.
A l t e r n a t i v e approaches t o t h e s y n t h e s i s of c l a s s i c a l and con-
s t r u c t i v e s y s t e m of arithmetic.
A or
HAK
5.1.
B.
As mentioned i n t h e I n t r o d u c t i o n , t h i s
theory of c a l c u l a b l e numbers may be viewed a s a s y n t h e s i s o f c l a s s i c a l and c o n s t r u c t i v e systems of a r i t h m e t i c .
I t i s o f t e n s a i d t h a t , when l o g i c a l
symbols a r e i n t e r p r e t e d c o n s t r u c t i v e l y , t h e combination n 8 r e p r e s e n t s This i d e a i s suggested by t h e f a c t t h a t t h e r e s u l t of
c l a s s i c a l existence. inserting -before
every 3 i n an a r i t h m e t i c a l formula A i s a theorem of
_-
HA i f A is a theorem of
HAc
(provided d i s j u n c t i o n is e l i m i n a t e d ) .
One
might t h i n k then t h a t a s s e r t i o n s about n a t u r a l numbers involving both
-_
c l a s s i c a l and c o n s t r u c t i v e q u a n t i f i e r s can be expressed i n HA by u s i n g 1 1 3 f o r t h e former and factory.
3
for the l a t t e r .
This method seems, however, u n s a t i s -
Take an a s s e r t i o n of t h e form: " t h e r e e x i s t s (non-constructively)
an x such t h a t one can c a l c u l a t e a y such t h a t A ( x , y ) " .
The t r a n s l a t i o n
V. LIFSCHITZ
186
suggested above does n o t work: it makes both q u a n t i f i e r s non-constructive, n o t only t h e f i r s t one.
This d i f f i c u l t y can be seen even more c l e a r l y i f
we consider a s l i g h t l y more complicated a s s e r t i o n : " t h e r e e x i s t s (nonc o n s t r u c t i v e l y ) an x such t h a t , f o r every w , one can c a l c u l a t e a y such t h a t A(x,w,y)
".
The t r a n s l a t i o n
does not work; n e i t h e r , it seems, does anything e l s e , u n l e s s Church's t h e s i s i s accepted and an e x p l i c i t r e f e r e n c e t o r e c u r s i v e f u n c t i o n s i s made. K In HA we would w r i t e simply
--
From a proof of t h i s formula one can e x t r a c t a r e c u r s i v e f u n c t i o n f s u c h t h a t f o r some x
b u t n o t n e c e s s a r i l y t h e v a l u e of t h a t x . The systems of Shapiro [121 and Myhill [lo] a r e i n some r e s p e c t s " s i m i l a r t o o u r s . Like $IA, they c o n s e r v a t i v e l y extend f a m i l i a r formal sys-
t e m s of c l a s s i c a l mathematics, and, a t t h e same t i m e , have t h e e x i s t e n c e p r o p e r t y , and are capable of expressing many s p e c i f i c a l l y i n t u i t i o n i s t i c distinctions.
Here a r e some f e a t u r e s of t h o s e systems which o u r s does not
share. (i)
They use a l o g i c a l operator which means "provable", o r "know-
a b l e " , i n s t e a d of our p r e d i c a t e K.
Thus t h e i r languages a r e s i m i l a r t o
t h o s e of modal l o g i c r a t h e r than t o t h e language of f i r s t o r d e r p r e d i c a t e logic. (ii) They a r e c l o s e d under c l a s s i c a l l o g i c , while
K
is not.
(iii)They do n o t prove t h e s u b s t i t u t i v i t y of i d e n t i c a l s and of equi-
v a l e n t subf ormulae, while HAR does. I-
I t seems, furthermore, t h a t t h e approach of Shapiro and Myhill i s
s u b j e c t t o a c r i t i c i s m s i m i l a r t o t h e one brought forward a t t h e beginning of t h i s s e c t i o n i n connection w i t h t h e i d e a of using 4 f o r classical existence.
In fact, t h e i r constructive existential quantifier
XI
isdefined
Calculable Natural Numbers
where
g
is t h e new o p e r a t o r .
187
The theorems
? A + A ,
--
B A + BB A , 3x
g
A
-t
5
3x A
imply
i.e.,
Ex 3
I
EIx 31y A .
y A
i-+
y A
cf
The obvious equivalence
3x 3
I
3 y BIx A
a l s o seems u n d e s i r a b l e .
7.
Conjecture on
2"and 53.
I n s e c t i o n 3 w e conjectured t h a t t h e
converse of 3 . 2 holds, so t h a t t h e p r o v a b i l i t y of a closed a r i t h m e t i c a l formula A i n Heyting's a r i t h m e t i c can be c h a r a c t e r i z e d a s follows:
bA
i f and only i f
HAK
A+.
This c o n j e c t u r e , i f c o r r e c t , g i v e s a new meaning ( a f t e r 2 . 2 ) t o t h e idea t h a t r e s t r i c t i n g q u a n t i f i e r s t o K i s a t r a n s l a t i o n from i n t u i t i o n i s t i c a r i t h m e t i c t o t h e theory of c a l c u l a b l e numbers.
The d i f f i c u l t " i f " p a r t
s e e m s very p l a u s i b l e because of t h e following p a r t i a l r e s u l t which e s t a -
b l i s h e s t h a t t h e p r o v a b i l i t y of A each of t h r e e systems extending
+ i.n
HA
HAK
implies t h e p r o v a b i l i t y of A i n
in quite different directions.
d i f f i c u l t t o imagine what kind of p r i n c i p l e unprovable i n i n each of t h o s e t h r e e e x t e n s i o n s .
I t seems
can be proved
V. LIFSCHITZ
188
Notation :
MpR is the primitive-recursive Markov scheme
+R(x)
+
3xR(x),
where R(x) is a primitive recursive predicate; ECT
is the extended Church
thesis
vxB ( X where A ( x , y ) , B ( x ) are arithmetical formulae, and B contains 9 before primitive recursive formulae only; CA- is the following restricted form of comprehension:
where A ( x ) is a formula of the theory of species HAS [9, 11.9.51; HAS- is
HAS with CA replaced by CA-.
(ECTo is of interest in connection with
Kleene's realizability 19, 13.2.141; 6.1. provable in
CA- is defined in 171).
For any closed arithmetical formula A , if (1) HAC,
(ii)
HA +
+
M
ECTo, (iii)
PR
HAK
A+ then A is
€fez-.
In [ 7 1 we asked whether HAS _ - - is conservative over HA.
A positive
answer would imply the validity of the conjecture in question. Re (i).
As remarked in connection with 3.1, the result of replacing
each occurrence of K(r) in a proof in in
gic.
"AK
by r = r can be made into a proof
When applied to A + , this operation gives a formula equivalent to
A.
Re (ii).
ii 2
A+.
HAK I- A+ implies that, for some numeral i, HAc HAc n r A . In Hi + MpR, ii 5 A is equivalent to
By 3.3,
Then, by 2.3,
negative formula; since
HAc
is conservative over
5
tive formulae, it follows that this formula implies A in Re (iii).
HA +
M
PR
HA
A is provable already in I3E
+
a
with respect to nega-
+
MpR, and
ECTo 19, §3.2.18(i)I.
In [71 we assigned to every arithmetical formula A a
formula A* of the theory of species in the following way.
To construct A * ,
one replaces each variable in A by a species variable, restricts each quantifier to the condition CN(X), "X is a classical natural number", defined by
Calculable Natural Numbers
189
and r e p l a c e s each f u n c t i o n symbol f by t h e f u n c t i o n symbol f * defined i n
HAS-
by
f*(X1,
(0 i s
..., XK) = x y v x 1 E X1 ... xk E
Xk
..., Xk)
(f(Xl,
considered a 0-ary f u n c t i o n symbol, so t h a t O* = { O ) ) .
t h i s t r a n s l a t i o n t o t h e language of
HAK
= y).
Now we extend
a s follows: K ( x ) should be t r a n s -
l a t e d by
3 x (X = { X I ) .
W e s h a l l show t h a t , f o r every c l o s e d theorem A of HAS-;
HeK , A*
i s a theorem of
then t h e a s s e r t i o n t o be proved follows from t h e equivalence of
i
( A ) * and A i n
HAS-.
It s u f f i c e s t o show t h a t
each of t h e axioms A2-C3. theorem A of
K, i.e.,
HA.
HAS-
A*
f o r t h e u n i v e r s a l c l o s u r e A of
According to [71,
HAS-
A* for every c l o s e d
Hence we should only c o n s i d e r t h e p o s t u l a t e s involving
A3, A4, C 1 , C2, C3.
The only n o n t r i v i a l case i s A3.
To s i m p l i f y
n o t a t i o n , assume t h a t y i s t h e only parameter of A o t h e r than x; w e should show then t h a t
HAS-
proves
According t o [ 7 ] (Lemma 2 and t h e concluding remark),
f o r each F beginning with negation. ACKNOWLEDGMENTS.
HAS-
proves
Apply t h i s t o 7 d * .
I would l i k e t o thank Solomon Feferman, Michael
Gelford, Nicolas Goodman, Yuri Gurevich, Georg Kreisel, Grigory Mints, John Myhill, Stewart Shapiro, Richard Vesley, and P e t e r Winkler f o r h e l p f u l comments.
V.LIFSCHITZ
190
REFERENCES [
11
A. Heyting, M a t h e m a t i s c h e G r u n d l a g e n f o r s c h u n g .
Intuitionismus.
B e w e i s t h e o r i e E r g e b n i s s e d e r M a t h e m a t i k und ihrer G r e n z g e b i e t e 3 ,
no. 4, Springer, Berlin, 1934. [
,
21
"After Thirty Years," L o g i c , M e t h o d o l o g y and P h i l o s o p h y of
Science. P r o c . of the 1960 I n t e r n a t i o n a l C o n g r e s s , ed. by E. Naqel, P. Suppes, A. Tarski, Standford University Press, Standford, 1962,
194-197. S. C. Kleene, I n t r o d u c t i o n t o M e t a m a t h e m a t i c s , North Holland, Am-
sterdam, P. Noordhoff, Groningen, D. van Nostrand, New York, 1952. A. N. Kolmoqorov, "ZUr Deutung der intuitionistischeu Logik," Math Z . 35 (1932)' 58-65. G. Kreisel and A. Troelstra, "Formal Systems for Some Branches of In-
tuitionistic Analysis," A n n u a l s of Math. L o g i c 1 , 229-387. V. Lifschitz, "A Conservative Extension of
HiC with the E-property,
"
Notices Amer. Math. SOC. 25 (19781, A-362.
,
"An
Intuitionistic Definition of Classical Natural Num-
bers," P r o c . Amer. Math. SOC. 77 (19791, no. 3, 385-388.
, "Constructive Assertions
i n an Extension of Classical Mathe-
matics," J. Symb. L o g i c 4 7 , 359-387. Metama thema t ica1 I n v e s t i g a t i o n of In t u i ti oni s t ic A r i t h e t ic and A n a l y s i s , L e c t u r e Notes i n Math. 3 4 4 , ed. by A. Troelstra, springer,
Berlin/Heidelberg/New York, 1973. J. Myhill, "Intensional Set Theory," this volume.
R. Peter, "Recursivitat und Konstruktivitlt," C o n s t r u c t i v i t y i n Mathem a t i c s , ed. by A. Heyting, North Holland, Amsterdam, 1959, 226-233. S. Shapiro, "Epistemic and Intuitionistic Arithmetic," this volume.
In tensional Mathematics S. Shapiro (Editor) @ Elsevier Science Publishers 8.V. (North-Holland), 1985
191
1
NODALITY AND SELF-REFERENCE
Raymond M.
Smullyan
Department of Mathematics Lehman College, City University of N e w York Bronx, N e w York U.S.A. INTRODUCTION AND SUMMARY
W e consider some self-appliedprotosyntacticalsystems r e l a t e d t o v a r i o u s modal systems.
In each of our systems w e have b u t one p r e d i c a t e
v a r i a b l e P ranging over p r o p e r t i e s ( o r sets) of t h e expressions of t h e system.
As soon a s an i n t e r p r e t a t i o n i s given t o t h e one symbol P , each
sentence becomes t r u e o r f a l s e .
Now, suppose w e select an a r b i t r a r y s e t
of sentences a s axioms and an a r b i t r a r y set of i n f e r e n c e r u l e s .
W e can
then i n t e r p r e t P t o mean p r o v a b i l i t y within t h e very axiom system--we t h i s t h e self-referential i n t e r p r e t a t i o n of t h e axiom system.
call
In g e n e r a l
it would be a s u r p r i s i n g coincidence i f a l l t h e provable sentences of t h e system turned o u t t o be t r u e under t h i s s e l f - r e f e r e n t i a l i n t e r p r e t a t i o n . W e are i n t e r e s t e d i n those systems i n which j u s t t h i s c u r i o u s phenomenon
occurs--such
systems w e c a l l self-referentially correct.
look a t t h e m a t t e r from a d i f f e r e n t angle:
W e might a l s o
Suppose we f i r s t g i v e an i n t e r -
p r e t a t i o n t o P and then c o n s t r u c t an axiom system i n which a l l t h e prova b l e sentences are t r u e under t h e i n t e r p r e t a t i o n .
It could happen t h a t
t h e property assigned t o P under t h e i n t e r p r e t a t i o n t u r n s o u t t o be coext e n s i v e with p r o v a b i l i t y i n t h e axiom system.
Again, t h i s would be a s u r -
p r i s i n g coincidence. Our main system 8* is an analogue of t h e modal system K 4 ( c f .
Boolos [ l l ) with c e r t a i n s u b s t i t u t i o n axioms added which provide enough fixed-points f o r t h e arguments of GBdel's second incompleteness theorem and M b ' s theorem t o go through--indeed system G ( c f . Boolos [ l ] ) .
s* h a s
t h e f u l l power of t h e modal
I t i s a very simple system whose p r o v a b i l i t y
p r e d i c a t e s a t i s f i e s t h e Hilbert-Bernays d e r i v a b i l i t y c o n d i t i o n s and a l s o y i e l d s t h e diagonal lemma needed f o r providing necessary fixed-points.
The
system i s provably c o n s i s t e n t on p u r e l y f i n i t a r y grounds, y e t i t s consistency, though e x p r e s s i b l e i n t h e system, i s n o t provable i n t h e system.
R.M. SMULLYAN
192
Although the system has certain features in common with Peano Arithmetic, it also exhibits some curious differences (cf. Theorem 3 ) . We also consider some self-referential systems related to modal systems other than K4--the systems 54, G and G* (cf. Boolos).
The counterpart
of G turns out to be self-referentially correct, but not the counterparts of S4 or
G*
when substitution axioms are added.
Informally speaking, Gbdel 121 proved the incompleteness theorem by constructing a sentence G which expressed its own non-provability in the system. Now, in Smullyan [ 5 1 we considered a "dual" form of the argument (cf. Theorem 2, Chapter 111) which yields a sentence J which expresses, not its own unprovability, but rather its own refutability in the system. (By a refutable sentence we mean one whose negation is provable.)
These "duals"
of Gddel sentences were later studied by Jerislow 131 who used them to show
certain redundancies in the Hilbert-Bernays derivability conditions, and are sometimes referred to as "Jerislow sentences."
Now, one can similarly
"dualize" the argument of LUb and thus obtain an alternative proof of Ldb's theorem (LUb 141).
This "dual" argument is not to be confused with Kripke's
2
method of obtaining LUb's theorem as a consequence of Gbdel's second incompleteness theorem.
Kripke's argument (which can also be "dualized") we
consider and modify in our sequel (this volume). We conclude this paper with a discussion of the "dual" of Lab's argument and a brief indication of some useful ways our systems can be extended and modified.
1.
Formalism and Semantics of 8,8*.
nine symbols: P D y
'
3 I ()
We shall use the following
X.
Informally, the symbol P is a predicate variable ranging over sets of expressions built from these 9 symbols; D is a name of the diagonal function (to be defined); the symbols y and
are used to construct names of
all the expressions ( y ' is the name of the first symbol P, y " is the name of the second symbol D, etc., and concatenation of names denotes concatena-
tion of the expressions named); the symbol tion;
I
3
stands for material implica-
stands for logical falsehood; the parentheses are used in the usual
manner for punctuation; and the symbol x is a variable ranging over the set 3
of expressions of the systems.
In our metalanguage we use expressions of the object language (i.e., expressions built from the 9 symbols) to denote themselves.
(E.g., we
write "P is a predicate variable," meaning that "P" is a predicate
Modality and Self-Reference In t h e o b j e c t language each expression X h a s a name 'X'
variable.)
.. ., x by
3,
y',
which
is t h e r e s u l t of r e p l a c i n g each occurrence i n
is very d i f f e r e n t from X - - i t
X of P, D,
193
..., y9,
y",
respectively.
( W e w r i t e y9 t o
mean t h e expression y followed by 9 primes; more g e n e r a l l y w e use yn t o mean y followed by n primes.)
Thus, f o r example, t h e name of PDx i s
y ' y l l y l I' l l 1 1 I I
W e use " A " , " B " , "C" a s v a r i a b l e s i n our metalanguage ranging over
expressions of t h e o b j e c t language, and a s a l r e a d y i n d i c a t e d , f o r any exp r e s s i o n A , by !A1
we mean t h e name of A .
Terms, Formulas, Sentences
by t h e r u l e s :
-
We
i n d u c t i v e l y d e f i n e t h e set of t e r m s
(1) For any expression A , i t s name 'A'
v a r i a b l e x is a term; ( 3 ) I f t i s a t e r m , so i s D t .
i s a term; ( 2 ) The 4
By an atomic formula we mean e i t h e r I or any expression P t , where t is a t e r m .
The set of formulas i s defined i n d u c t i v e l y by t h e r u l e s :
Every atomic formula i s a formula; ( 2 ) i f A , B are formulas, so is ( A
(1) 3 B).
By a sentence w e mean a formula i n which t h e v a r i a b l e x does n o t occur, and by a constant term w e mean a term i n which x does n o t occur. f o r (X
W e use t h e u s u a l abbreviations-X
(X A Y ) f o r 4 d v 4'); (X
5
31);
Y ) f o r ( ( X 3 Y ) A (Y
3
(X v Y ) f o r
X)).
(-X 3 f);
W e d e l e t e paren-
t h e s e s when no ambiguity can r e s u l t . When we p u t quasi-quotes around an a b b r e v i a t i o n , w e mean t o d e s i g n a t e t h e name, n o t of t h e expression which appears w i t h i n t h e quasi-quotes,
of t h e expression which it abbreviates--for name of t h e expression (PDX
example, by r"f'DX'
but
we mean t h e
3 I).
S u b s t i t u t i o n and Diagonalization
-
For any expression A and any t e r m
t , by A ( t ) w e mean t h e r e s u l t of s u b s t i t u t i n g t f o r x i n A .
It is t r i v i a l
t o v e r i f y t h a t f o r any formula A and any term t , A ( t ) i s a formula, and i f
t is a c o n s t a n t term, then A ( t ) is a sentence. For any e x p r e s s i o n s A , B by A ( r B 2 ) w e mean, of course, t h e r e s u l t of s u b s t i t u t i n g t h e term 'B1 f o r x i n A .
Since rBl is a c o n s t a n t t e r m , then
i f A i s a formula, A ( ~ B ' ) i s a sentence.
BY A r B l we mean ~ ( r ~ l ) .
By t h e d i a g o n a l i z a t i o n of A w e mean A r A 1 .
Again, i f A i s a formula,
then A ' A ~ i s a sentence.
Designation
-
W e i n d u c t i v e l y d e f i n e t h e designation r e l a t i o n between
c o n s t a n t terms and expressions by t h e r u l e s :
(i)r A 1 d e s i g n a t e s A;
(ii)if
R.M. SMULLYAN
194
t d e s i g n a t e s X , then D t d e s i g n a t e s t h e d i a g o n a l i z a t i o n X r X 1 of X.
I n t e r p r e t a t i o n s and Truth
-
For any s e t M of e x p r e s s i o n s , we induc-
t i v e l y d e f i n e a sentence t o be M-true b e r s h i p i n M ) by t h e r u l e s : l i e s i n M(thus, e.g., A rA 1
( t r u e when P i s i n t e r p r e t e d as mem-
(1)P t i s t r u e i f f t h e t e r m designated by t
P ~ A T i s M-true
l i e s i n M); ( 2 ) I i s n o t M-true;
i f f A l i e s i n M; P D ~ A ' i s M-true i f f ( 3 ) ( A 3 B ) i s M-true i f f e i t h e r A i s
n o t M - t r u e o r B i s M-true.
Independent Truth
-
W e c a l l a sentence S independently t r u e i f f it i s
t r u e under every i n t e r p r e t a t i o n of P--i.e.,
i f f S i s M-true f o r every s e t M
of expressions. Every t a u t o l o g y , f o r example, i s independently t r u e .
A l s o , f o r any
t e r m s t , t which d e s i g n a t e t h e same expression, t h e sentence P t l 1
2
independently t r u e .
are independently t r u e . 1
L e m 1
3
Pt
(For example, a l l sentences of t h e form PD'A'zP'A
2
is
rA'l
More g e n e r a l l y :
- For any formula 0 and any c o n s t a n t t e r m s tl,t2 which desiq-
n a t e t h e same expression, t h e sentence @ ()t e @ ( t) i s independently t r u e . 1 2 One e a s i l y proves t h e above lemma by induction on t h e complexity of 0.
The above lemma i m p l i e s , i n p a r t i c u l a r , t h a t f o r any formula 0, and any expression A , t h e sentence 0 (DrAl ) z
Fixed P o i n t s
-
t h e sentence S G arS1
@ (
rA7-1)
i s independently t r u e .
W e c a l l a sentence S a fixed-point of a formula @ i f f
i s independently t r u e .
We might remark t h a t t h i s notion of "fixed-point''
i s a p u r e l y seman-
t i c one, and does n o t involve t h e notion of p r o v a b i l i t y i n any axiom system. W e s h a l l l a t e r have occasion t o speak of a sentence S being a fixed-point
of 0 f o r an axiom s y s t e m in
a.
a,
meaning t h a t t h e sentence S
QrS7
i s provable
But f o r t h e t i m e being, we a r e u s i n g " f i x e d p o i n t " i n a p u r e l y se-
mantical sense. The following b a s i c p r i n c i p l e d e r i v e s u l t i m a t e l y from GBdel [ 2 1 :
Theorem A
-
(Semantic Fixed-Point Theorem):
Every formula @ ( x )h a s a f i x e d
point. To prove Theorem A, w e introduce t h e following d e f i n i t i o n s and no-
tions:
Modality and Self-Reference
For any formula @,
we l e t Q D be t h e formula O(Dx)--i.e.,
of s u b s t i t u t i n g t h e t e r m DX f o r x i n &-and
@. (For PDx
J
example i f 0 i s t h e formula Px
PDDx.)
195
the r e s u l t
w e c a l l OD t h e diagonalizer of
>PDx, then O
i s t h e formula
D
I t i s e a s i l y seen t h a t f o r any expression A ,
@,lid’
is t h e
same sentence as @(DrA1)--that i s , i f we f i r s t s u b s t i t u t e Dx f o r x i n 0 f o r x i n t h e r e s u l t , w e would g e t t h e same sentence by sub-
and then rA1 s t i t u t i n g D‘A~
for x in
@.
Then by lema 1, f o r any expression A , t h e sentence O(DrA’)=
OrArAll
i s independently t r u e , and t h u s ( i n t h e new n o t a t i o n ) t h e sentence @ TAl
D
5
@ r A r A l l i s independently t r u e .
Then ODrOD1
W e now t a k e f o r A t h e formula OD.
D
rO
’’ i s inde-
D
So, l e t t i n g Pi* be t h e sentence o D r O D l , we see t h a t
pendently t r u e .
@*
Or@
E
zOrO*l i s independently t r u e , hence
@*i s
a f i x e d p o i n t of Q.
This
proves Theorem A. We s h a l l henceforth r e f e r t o t h e sentence O*--i.e., QDrQDl--ast h e canonical f i x e d p o i n t of
a.
t h e sentence
This sentence O* i s O(D‘@(Dx)l) .
For example, t h e canonical fixed-point of Px i s PDrPDxl: t h e c a n o n i c a l f i x e d p o i n t of -Px i s -FDr-PDxl--i.e.,
(PDr(PDx
t h e canonical f i x e d p o i n t of (Px
J
2.
3 Lj7
Self-Referential Interpretations.
formulas and t e r m s of our language.
3 I).
Y ) i s (PDrPDx
3
Y1
We l e t
For any sentence Y, 3
Y)
.
2 be t h e set of
L e t u s now consider an a r b i t r a r y axiom
system ff whose provable formulas a r e a l l sentences i n %--such an axiom sys-
t e m we term an axiom system i n t h e language 2. W e c a l l a sentence S true for Q i f f
s is
M-true where M i s t h e set of sentences provable i n
ff.
(In-
formally w e paraphrase t h i s by saying t h a t S i s t r u e when P is i n t e r p r e t e d a s p r o v a b i l i t y i n ff.)
We say t h a t ff i s self-referentially correct i f f
every sentence provable i n ff i s t r u e f o r ff. The notion of s e l f - r e f e r e n t i a l c o r r e c t n e s s i s perhaps t h e fundamental n o t i o n of t h i s paper. Self-referential
correctness has some curious properties:
It can hap-
pen t h a t an axiom system Q i s s e l f - r e f e r e n t i a l l y c o r r e c t , y e t i f w e d e l e t e one of t h e axioms, t h e r e s u l t i n g system may no longer be s e l f - r e f e r e n t i a l l y correct!
For example, we might have one axiom A1 which s a y s t h a t another
axiom A2 i s provable i n t h e system ( t h a t i s , we might have A1 being t h e sentence p r A z l ) .
Since A2 i s an axiom of t h e system, it i s c e r t a i n l y provable
i n t h e system, hence A
1
i s t r u e f o r ff. But i f we d e l e t e A2 as an axiom, it
may no longer be provable i n s u l t i n g system.
a, and hence
A1
w i l l become f a l s e f o r t h e re-
196
R.M. SMULLYAN
We may also have a self-referentially correct system a a n d a sentence X which is true for a, but if call this extension
a + {X]--the
x
is adjoined to
a as a new axiom--we
resulting system may fail to be self-
referentially correct. 2.1.
A
UniformIncompleteness Theorem. We call an axiom sys-
tem acomplete iff for every sentence S, either s or rS is provable in We call aconsistent iff for no sentence S is it the case that s,
-6
a.
are
both provable in&. The following theorem is a "uniform" version of Gbdel's first incompleteness theorem: Theorem 1
-
( a ) There is a sentence G which is true for all self-
referentially correct systems, but is provable in none.
Stated otherwise,
there is a sentence G such that given any self-referentially correct system a,
G
is true for
a but
not provable in
a.
(b) There exists no self-
referentially correct axiom system which is also complete. More specifically, if
a is self-referentially correct, then neither G
nor its
negation is provable in 0. Proof
-
(a) We take for G the sentence nPDr-?Dxl,
fixed-point of the formula R p x .
By Theorem
A,
which is the canonical the sentence Ga -prGl is
independently true, hence G is true for those and only those axiom systems for which -PrG7 is true. This means that
G
is true for those and onlythose
axiom systems in which G is not provable (for -erGT is true for C? iff G is not provable in
a).
If noti
a
is self-referentially correct, it cannot be
that G is not true for a but provable in a , hence G must be true for
not provable in a.
(b) Suppose a is self-referentially correct. then true for a but not provable ina.
a but
The sentence G is
Then the sentence W ) is false
(not true) for a, hence also not provable i n a (sincea is self-referentially correct).
Thus a is incomplete.
Remark; Without appeal to Theorem A, we can see directly that G is true for an axiom system0 iff G is not provable in it: mPDr,PDxl
in ff.
The sentence
asserts (for a) that the diagonalization of NPDX is not provable
But the diagonalization of-PDx is the sentence
G.
197
Modality and Self-Reference Throughout t h i s paper w e l e t G be t h e sentence NPDr-PDx?
and we r e f e r
t o it a s t h e GBdel sentence.
The Axiom S y s t e m s
3.
8*.
s*
- We
now t u r n t o a s p e c i f i c axiom s y s t e m
In s t a t i n g the axiom schemata and i n f e r e n c e r u l e s , X , Y , Z a r e any
sentences, 0 i s any formula, and t l , t 2 a r e any c o n s t a n t t e r m s .
- Ao:
Axiom Schemata P'X
A2:
prX'-T3 PfPrX7'
A3:
A l l tautologies
Y-I> (P'X'
A1:
3
I3 Pry')
@ ( t ,9) @ ( t 2providing ), tl,t2 d e s i g n a t e t h e same expression.
Inference R u l e s
--
R1:
(Modus Ponens) - From X ,
R2:
(Necessitation)
-
( X 3 Y ) t o infer Y
From X t o i n f e r P'X'.
Remark - I n s t e a d of axiom scheme A3 it would have s u f f i c e d t o t a k e P t Z , where tl,t2 a r e terms d e s i g n a t i n g t h e same
t h e simpler scheme: P t l
A mechanical i n d u c t i v e argument would show t h a t a l l sentences
expression.
of A3 would then be provable.
We sometimes w r i t e "A A3.
We note t h a t A
3
3
"
t o mean t h e set of sentences of axiom scheme
c o n t a i n s a l l sentences of t h e form I(D'A')
and i n p a r t i c u l a r a l l sentences
@ *L
@r@*7.
E @
('ArA''),
These "fixed-point sentences"
a r e i n f a c t t h e only axioms of A 3 which w i l l p l a y any r o l e i n t h i s paper. W e l e t 8 be t h e system 8* without axiom scheme A
Remark:
The system
different notation.
8 is l i t t l e
3'
more than t h e modal system K4 i n a
However, i n t h e n o t a t i o n of modal l o g i c , t h e r e i s no
immediately obvious way of g a i n i n g t h e power of A3.
There i s i n f a c t such
a way, and w e p l a n t o p r e s e n t it elsewhere.
3.1.
The S e l f - R e f e r e n t i a l Correctness of 8 , 8 * .
The f i r s t in-
t e r e s t i n g t h i n g t o prove about t h e s y s t e m s 8 , 8* i s t h a t they are s e l f referentially correct.
But f i r s t w e must t u r n t o some more general
considerations. W e a r e l e t t i n g 2 be t h e s e t of terms and formulas of 8 " .
Until furth-
e r n o t i c e , axiom system w i l l mean axiom system whose provable formulas a r e a l l sentences and a l l i n 2.
For any axiom system C? and any s e t M of
R.M. SMULLYAN
198
sentences, by ff
+M
we mean the system resulting from ff by adjoining all
elements of M as axioms. For a single sentence X, by result of adjoining X as an axiom to
a.
a + {x) we mean
extension of ff iff the theorems (provable sentences) of
theorems of ff.
a +{XI--i.e.,
the
We call an axiom system ff', an
a',include the
By a simple extension of a w e mean one of the type, any extension resulting from ff by adjoining a single sentence
as an axiom. It is possible for a system a t o be self-referentially correct, a sentence X might be true for tially correct.
a+
{XI, but ff
+
{XI might not be self-referen-
This motivates the following definition:
We shall say that ff is essentially self-referentially correct iff for every set M, if each element of M is true for ff
+ M,
then ff + M is self-
referentially correct. Suppose d is essentially self-referentially correct. Letting M be the empty set, it is vacuously true that all elements of M are true for ff
6'.
+ M,
hence ff + M is self-referentially correct.
But0
+M
is the system
Thus any system which is essentially self-referentially correct must
also be self-referentially correct. We now need:
-
Lema 2
If 0 is essentially self-referentially correct, and N is
any set of sentences all of which are independently true, then ff + N is essentially self-referentially correct. Proof
-
Assume hypothesis.
M is true for
Now take any set M such that every element of
(a + N ) + M; we
are to show that
( a +N ) + M
is self-referen-
tially correct. Now,
( a +N ) + M is the + (M U N), and
are true for 0
system ff
+
(M U N).
Thus all elements of M
so are all elements of N (since they are in-
dependently true) so all elements of M U N are true for 0 + (M U N). Therefore,
+
(M
u
N ) is self-referentially correct.
NOW we prove:
Theorem 2: Proof
The systems 8 , 8* are essentially self-referentially correct.
- We first show that 8 is essentially self-referentially correct. Let M be any set of sentences all of which are true for 8 + M.
first show that all axioms of 8
+M
are true for 8
+ M.
We
We are given that
199
Modality and Self-Reference
all elements of M are true for 8 of 8 are true for 8
+
+
M, so it remains to show that all axioms
M.
Re A*, all tautologies are independently true, hence true for 8 Re A1, to say that Pr(X say that if Pr(X
3
Y)l
3
( P r X 1 3 Pry1) is true for 8
3
Y)’, P r X 1 are both true for 8
equivalent to saying that if X
+
M, so is Pry1.
Y,X are both provable in 8
3
But this is so, since modus ponens is an inference rule of 8 Re A2, to say that P’X1 X is provable in 8
ence rule of 8
+
+
3
is true for 8
P‘P‘X’’
M, so is PpX’.
+
M
+
+
+
M.
M is to
This is
M, so is Y.
+
M.
is to say that if
But this is s o , since R2 is an infer-
M.
This proves that all axioms of 8
+
M are true for 8
+
M.
Next we must proceed by induction on the length of a proof, but we must be careful, since modus ponens preserves truth for 8
+
M, but rule R2
doesn‘t! So the property on which we must perform the induction is not
+
+
truth for 8
+
M, but truth for 8
axioms of 8
+
M are true for 8 + M, and are a l s o provable in 8
must have this compound property.
M and provability in
g
Since all
M.
+
M, they
Also modus ponens clearly preserves this
property (since it preserves truth for 8
+M
and also provability in 8
+
M,
Rule R 2 also prebecause modus ponens j.s an inference rule of 8 + M). serves this property, because if a sentence X has this property, then it is provable in 8
8
+
+ M,
so P‘X’is
true for 8
+
M and also provable in
M (by R2).
This completes the induction, hence every sentence provable in 8 is true for 8
+
M, hence 8
+
+
M
M is self-referentially correct. This proves
that 8 is essentially self-referentially correct. It then follows from Lemma 2 that8* is also essentially selfreferentially correct because 8* is 8
+
A3, and all elements of A 3 are in-
dependently true.
2 -
Corollary 1
Then system 8* is consistent.
Corollary
is true for g* but not provThe sentence G, viz. -PD~-PDx’,
able in 8 * . Corollary 3 Corollary 4
-
The system 8* is not complete. For any sentence X, if PrXl is provable in g*, so is X.
The
same is true for the system 8 . Corollary 1 is immediate, since an inconsistent system cannot be selfreferentially correct.
Corollaries 2, 3 follow from Theorem 2 and Theorem
R.M. SMULLYAN As f o r Corollary 4, suppose P r X 1 i s provable i n g(g*). Then p r p i s
1.
t r u e f o r g&*) s i n c e 8(g*) i s s e l f - r e f e r e n t i a l l y c o r r e c t .
4.
Adjunction of G , N G .
Our system
which t h e deduction theorem holds.
i n one important
Therefore, s i n c e t h e Gadel sentence f o r
(Peano Arithmetic) is undecidable i n P.A., one can a d j o i n e i t h e r it o r
i t s negation t o P.A.
and t h e r e s u l t i n g system i s c o n s i s t e n t .
tem g*, t h i s i s n o t so: G
S* d i f f e r s
Peano Arithmetic i s a f i r s t - o r d e r theory i n
r e s p e c t from Peano Arithmetic:
P.A.
This means y. i s
g(g*).
provable i n
With our sys-
As w e s h a l l see, if w e a d j o i n t h e Gddel sentence
(which i s t r u e f o r g * ) , t h e r e s u l t i n g system i s i n c o n s i s t e n t , b u t if w e
a d j o i n N G (which i s f a l s e f o r 8 * ) t h e r e s u l t i n g system i s c o n s i s t e n t , b u t not s e l f - r e f e r e n t i a l l y c o r r e c t . W e s h a l l use t h e t e r m “normal“ i n a weaker sense than it is used f o r
modal l o g i c :
W e c a l l an axiom system ff a normal system i f f i t s set of
provable sentences c o n t a i n s a l l t a u t o l o g i e s and i s c l o s e d under modus ponens and n e c e s s i t a t i o n .
-
Lemma 3
The sentence G i s not provable i n any c o n s i s t e n t normal
system i n which a l l sentences of axiom schema A3 a r e provable. Suppose ff i s normal and extends t h e Set A3 and t h a t G i s provable
Proof:
0. The sentence G e -PrG’ i s i n A3, so i s provable i n ff. HenceNPrGT i s provable i n 0. But s i n c e G is provable i n ff and a i s normal, PrG’ i s provable i n 0. So G,-Gare b o t h provable i n ff, hence is inconsistent. in
Th eore m 3
Proof
-
8*
- S* 3. {GI +
{GI extends A3, i s normal and G i s provable i n it.
+
above lemma, g* Remark
is inconsistent.
-
So by
{G} i s i n c o n s i s t e n t .
More s t r o n g l y , i f w e l e t go* be t h e system
schemes A1,A2, t h e system go*
g* without axiom
+ {GI i s i n c o n s i s t e n t .
The s i t u a t i o n a r i s i n g from a d j o i n i n g -Gto
gf s t r i k e s u s a s more
curious:
Lemma 4
-
For any system Owhich i s e s s e n t i a l l y s e l f - r e f e r e n t i a l l y
Modality and Self-Reference
20 1
correct and for any sentence X: If X is false for every simple extension of Q which is self-
(1)
+ {XI.
referentially correct, then X is false for 0
If X is true for every self-referentially correct simple exten-
(2)
sion of
-
Proof
a+
then
x
is true for
(1) Let
x
be a sentence which is false for all self-referentially
a,
correct simple extensions of ff.
{*}.
If
x
Suppose x is true
the following contradiction:
+ {x], we would get for 0 + {XI. Then ff + {x}
were true for a
is self-referentially correct (because a is essentially self-referentially
a + {x} is
a which is self-referentially So it is contracorrect. But this implies that x is false for ff + {XI! dictory to assume that x is true for a + {x}, hence x must be false for a + {x}. ( 2 ) Suppose x is true for every simple extension of ff which is selfreferentially correct. Then -x is false for every simple extension of ff correct), so
a simple extension of
which is self-referentially correct.
{*I,
hence
Theorem 4
-
x
is true for
a+
Then by (l), ( 4 )is false for ff +
{wx}.
L e t m be any essentially self-referentially correct axiom sys-
tem in which modus ponens is a rule of inference and in which all tautologies are provable.
Then
+
{,c}
is consistent but not self-referentially
correct. Proof
-
Let% satisfy the hypothesis.
Since modus ponens is an inference
rule of V(, it is also an inference rule of 131 any sentence X).
+
+
{I\G}
(in fact of V(
A l s o all tautologies are provable i n m , hence provable in
{ 4 } .Therefore if there is any sentence not provable i n R
then V(
+ I41
+ {x} for
+
{G} ,
is consistent.
Now, G is true for all self-referentially correct systems, hence certainly true for all self-referentially correct systems which are simple extensions of %.
Then by Lemma 4 (statement
( 2 )), G
is true for V(
+
[wG}.
+ {GI (because G is true for just is not provable). Since G is not provable in 711 +
This means that G is not provable i n m those systems in which G {&),
the system??( + [ & } Also,
must be consistent.
( 4 )is provable in@
true for it), hence%
+ [-GI
+
{4} but
false f o r m + { 4 }(since G is
is not self-referentially correct.
{4} is consistent but not self-referentially correct.
So%
+
202
R.M. SMULLYAN
From Theorem 4 and Theorem 2 (which tells us that
a*
is essentially
self-referentially correct) we immediately get: Theorem 5
-
S*
+
1-G) is consistent, but not self-referentially correct.
The Henkin Sentence H.
4.1.
of the formula Px.
We let H be the canonical fixed-point
It is the sentence PD'PDx'.
This is the Henkin sen-
tence in our formalism. By Theorem A , the sentence Ha PrH'
is independent-
ly true, hence H is true for just those axiom systems in which it is provSO for any axiom system
able.
a,H
is true for U
+ {HI.
If U i s essential-
ly self-referentially correct, then 0 + {HI is self-referentially correct. What about ff
+ {+I?
Well, if modus-ponens is an inference rule of
and all tautologies are provable in ff, then H is true for 0 + only if
a+ {d}is inconsistent, hence -H
{ w } is consistent. If furthermore correct, then
+
a+ {d}, which
is the case iff
Theorem 6
-
+
is true for
a+
is essentially self-referentially
{d} is self-referentially correct iff
+
{ifiand f}
{d} iff
Id}is consistent.
-81
is true for
We thus have
(a) If a i s essentially self-referentially correct, then ff
{H) is self-referentially correct.
provable in
+
(b) If furthermore all tautologies are
and modus-ponens is an inference rule of
a,then +
is
(-3
self-referentially correct iff it is consistent. Remarks
-
As w i i i
provable in g*.
ne seen, LOb's theorem holds for g* and hence H is
However, H is not provable in 8 .
Nevertheless, 8
+
{ H } is
self-referentially correct, by the above theorem. It might be amusing to note that the system {H}, whose only provable sentence is H, is self-referentially correct.
(It has just one provable
sentence sentence which says of itself that it is provable, and so it is!). LBb's Theorem for 8*.
5.
Boolos [l, Chapter 31 speaks of realiza-
tions and translations of modal formulas into sentences of Peano Arithmetic; here we do the analogous thing for sentences of 31. By a realization we shall mean a function 4 which maps every propoFor any'sentence A of modal
sitional variable p to a sentence ' ( p ) of 31.
logic, we inductively define its translation ' A ' I
= A;
( 2 ) p'
(A+ 3 B ' ) ;
=
in
by the rules: (1)
4 ( p ) (for any propositional variable p ) : ( 3 ) Ls 3 B)' = BY a translation of A we mean a translation
(4) (oa)' = prAQ7.
Modality and Self-Reference
@.The
under some r e a l i z a t i o n
203
following can be proved a s i n Boolos (Th. 1,
Ch. 3 ) .
-
Lemma 5
I f A i s provable i n t h e modal s y s t e m K4 t h e n every t r a n s l a -
t i o n of A is provable i n 8. Now, t h e sentence o ( p 5
( u p 3 q ) ) J ( ~ ( o gJ q )
i s provable i n
J OQ)
K4 ( c f . Th. 2 , Ch. 3 , Boolos), hence by t h e above lemma, f o r any sentences X , Y , t h e following sentence is provable i n
g*:
W e have so f a r defined a f i x e d - p o i n t of a formula 0 i n a semantical
as a sentence s such t h a t t h e sentence s
sense--i.e.,
07.97 i s independent-
ly true.
we s h a l l c a l l s a fixed-point of 0 i n an axiom system tence
s
E
a
i f f t h e sen-
i s provable i n ff.
0%‘-
From (L) above immediately follows: Theorem 7
-
[ A f t e r Lelb]
-
For any normal extension 0 of 8 and f o r any sen-
t e n c e Y , i f t h e r e is a fixed-point i n 0 f o r t h e formula PX P’(P~Y’
Proof
3Y)’
is provable i n
3 pryY
- Assume
able i n
Then t h e r e i s a sentence
hypothesis.
( P ~ X ’ 3 y ) is provable i n
0. Then from
a.
Since
3
Y , then
a.
ff i s
x
normal, p r x z
( L ) , by modus ponens,
such t h a t
(P‘x’
x
8
> y ) l i s prov-
w e g e t o u r d e s i r e d conclusion.
Since f o r every formula 0 t h e r e is a fixed-point i n $*, then we have Theorem 7.1
-
[Lelb‘s Theorem f o r 8*--Strong
t h e sentence p r ( p r y ’
Remark
is:
-
3y)l
Form1
J P r Y T i s provable i n
-
For every sentence y ,
8*.
One can e a s i l y see what t h e canonical fixed-point of P x
I t i s t h e sentence PD’PDx
3Y1
3Y
>Y-
From Theorem 7 and Lemma 5 e a s i l y follows:
Theorem 7.2
-
I f A i s provable i n t h e modal system G , then a l l t r a n s l a t i o n s
of A are provable i n
8”.
Theorem 7.1 y i e l d s t h e weak form of Lab’s theorem by v i r t u e of t h e following lemma, which we w i l l a l s o need elsewhere.
RM. SMULLYAN
204
Lemma 6 and pr(prY1
Proof
-
3
-
For any normal system
Y)'
3
any sentence Y , if Pry7
a,then
Pry1 are both provable i n
a is
Suppose
pr(pry7 3 ' ) y
3
a and
normal and t h a t (1) pry7
3
Y
so i s Y.
Y i s provable i n C?;
(2)
a.
pry1 i s provable i n
By (1) and normality, t h e sentence pr(pry7
3
Then by ( 2 ) and modus ponens, pry1 i s provable i n ponens, Y i s provable i n
3
a.
Y)'
i s provable i n
a.
a.
Then by (I) and modus
Theorem 7 . 1 and Lemma 5 y i e l d :
-
Theorem 7.3
[After Mb]
t e n c e Y , i f pry1
5.1.
3
-
For every normal extension
Y is provable i n
The Sentence Consis.
any axiom system
ff,
a of
8* and any sen-
so i s Y.
W e l e t Consis. be t h e sentence
e C ~ l .For
a i f f I i s n o t provable i n &-if a and t h e set of theorems of a Consis. i s t r u e f o r 0 i f f a i s consis-
0,Consis. is t r u e f o r
furthermore a l l t a u t o l o g i e s a r e provable i n
is closed under modus ponens, then tent.
Consis. is t h e sentence pr~'
NOW,
I f o r y,
3
I , hence by Theorem
w e see t h a t f o r any normal e x t e n s i o n of
provable i n
Theorem 9
-
a,so [A
is
1.
a of
7.3, t a k i n g
8 * , i f Consis. i s
This proves:
Uniform Version of Gddel's Second Incompleteness Theorern]
-
The sentence c o n s i s . i s n o t provable i n any c o n s i s t e n t normal e x t e n s i o n of
s*. Theorem 9.1
-
( a ) Consis. i s n o t provable in
g*; (b) 8* + {Consis.} is in-
consistent.
Remarks
-
Actually, t h e sentence Consis. z G i s provable i n
g*, and
so we could have derived Theorem 9 a s a consequence of Theorem 3 , which
would have followed t h e l i n e s of Glldel's o r i g i n a l proof of t h e second incompleteness theorem.
The above proof of Theorem 9 followed t h e l i n e s of
George Kreisel, who showed t h a t Gbdel's second incompleteness theorem can be looked a t a s a s p e c i a l c a s e of L8b's theorem. I n c i d e n t a l l y , by Theorem 7, t a k i n g ^PrConsis.-)
i s provable i n 8 * .
I
f o r Y , t h e sentence (Consis. z
Modality and %&-Reference
205
Although the proof of Th. 10 below is somewhat like that of Th. 3 it is also somewhat different, and we see no way to derive either theorem as a consequence of the other.
(The fact that G
Consis. is provable in g*
seems to have no bearing.) Theorem 10
+
-
For any axiom systemn satisfying the hypotheses of Theorem 4 ,
{.-Consis.) is consistent, but not self-referentially correct.
Proof
-
Letv' be any simple extension ofm. Then the set of theorems ofv'
contains all tautologies and is closed under modus-ponens. Therefore, is ('consistent. Consis. is true forq' iff ?I
If nowv' is self-referential-
ly correct, then fl' is consistent, hence Consis. is true for it. proves that Consis. is true for every simple extension of referentially correct.
Since
This
which is self-
is assumed essentially self-referentially
correct, then by Lemma 4 , Consis. is true for
+
{-Consis.},
and so %
+
(hConsis.1 is consistent (and hence also not self-referentially correct, since -Consis. is false for it but provable in it). Corollary
-
S* + {-Consis.} is consistent but not self-referentially cor-
rect. Some other systems.
6.
axiom scheme added:
(G)
Let & be the system
- All sentences Pr(P'Y?
3
&is the counterpart of the modal system G.
Y)'
with the following >Pry1.
Of course, & is a sub-
system of 8* (by Theorem 7), but a subsystem of a self-referentially correct system is not necessarily self-referentially correct. Nevertheless, we have: Theorem 11 Proof
-
-
The system & is self-referentially correct.
Since 8 is essentially self-referentially correct, it suffices to
show that all axioms of scheme Pr(PrY7
3
Y)'
3
in &, then Y is provable in &. yr
3p'Yl
6.1.
(G)
are true for &.
Now, to say that
Pry7 is true for & is to say that if Pry3
3
Y is provable
But this is so by Lemma 5 (since P r P r Y T 3
is provable in &I. The Systems S*, &*.
We let2 (the counterpart of Lewis' modal
system 54) be the system obtained from 8 by deleting A3 and adding a l l
206
R.M. SMULLYAN
sentences P ' X
3
X a s axioms.
Let&,
( t h e c o u n t e r p a r t of t h e modal system
c a l l e d G* i n Boolos) be t h e system whose axioms a r e a l l t h e provable sent e n c e s of & t o g e t h e r with a l l sentences PrX7 3 X and whose only r u l e of i n ference i s modus ponens.
Let
i n g axiom scheme A3 t o $,
&,.
$*,
&,* be t h e r e s p e c t i v e r e s u l t s of adjoin-
W e do n o t know whether e i t h e r $ o r &,
s e l f - r e f e r e n t i a l l y c o r r e c t , b u t n e i t h e r $* nor
I,* is
is
self-referentially
This i s a consequence of t h e following f a c t :
correct.
Theorem 12 - There e x i s t s no s e l f - r e f e r e n t i a l l y c o r r e c t system ff with t h e following f o u r p r o p e r t i e s :
(1) A l l t a u t o l o g i e s a r e provable i n ff. ( 2 ) The
set of theorems of ff i s c l o s e d under modus ponens. A
3
a r e provable i n ff. ( 4 ) A l l sentences ' X P
2
( 3 ) A l l sentences of
X a r e provable i n ff.
Proof - L e t ff be any system having t h e above f o u r p r o p e r t i e s , w e show t h a t it i s n o t s e l f - r e f e r e n t i a l l y c o r r e c t . W e l e t G be t h e GBdel sentence -PD'-PDx'.
By (3) t h e sentence G
-PrG1 i s provable i n ff, and by ( 4 ) t h e sentence P'G-'
3
But G i s t a u t o l o g i c a l l y implied by t h e s e two sentences, hence by ( 1 1 , G i s provable i n
7.
ff.
5
G i s provable i n ff. (2),
Then by Theorem 2 , ff is n o t s e l f - r e f e r e n t i a l l y c o r r e c t .
Some Extensions of Our Language.
W e now d i s c u s s how our main
s y s t e m 8* can be extended i n some u s e f u l ways. W e b r i e f l y mentioned i n t h e i n t r o d u c t i o n t h e "duals" of Gadel sen-
t e n c e s and s t a t e d t h a t t h e r e i s a corresponding "dual" method of proving LBb's theorem.
Now, w e cannot c a r r y t h i s o u t i n S*, because although we
have negation i n t h e system w e have no t e r m f o r t h e negation function ( i . e . , t h e function which a s s i g n s t o each expression i t s n e g a t i o n ) .
And so we ex-
tend our formalism a s follows: W e add a t e n t h symbol N and t a k e y10 f o r i t s name.
Then we augment
t h e formation and designation r u l e s f o r t h e t e r m s by t h e r u l e s :
t e r m , so i s N t , and i f t d e s i g n a t e s A then N t d e s i g n a t e s 4.
If t i s a
We let
P1 be
t h e s e t of t e r m s and formulas of t h i s extended formalism and w e l e t 8,, 8,* be t h e systems 8 , 8 * , r e s p e c t i v e l y , only c o n s t r u i n g "term", "formula", "sentence" a s being i n t h e l a r g e r c l a s s %
1' A l l theorems and lemmas so f a r proved about
8 , S* a l s o hold f o r gl,
S1*, and w e s h a l l f r e e l y use them. L e t u s d e f i n e 5 t o be an anti-fixed-point of a formula 0 i f f t h e sen-
tence s
=
@ r w s - i s independently t r u e .
W e n o t e t h a t f o r any expression A ,
207
Modality and Self-Reference d e s i g n a t e t h e s a m e expression (namely, - A ) ,
t h e terms N ~ A - and ‘..A’
f o r any formula 0 ( x ) , by Lemma 1 (applied t o J !k,
hence,
r a t h e r than !Jk) t h e sentence
0r-a’ = Q (“A’) is independently t r u e . ( A l s o it i s an axiom of 8 *.) 1 Therefore, f o r any sentence S, t h e following p r o p o s i t i o n s a r e e q u i v a l e n t : (1)
S
(2)
S
QrwS1 i s independently t r u e . 3
Q(NrS1) i s independently t r u e .
Proposition (1) says t h a t s i s an anti-fixed-point (21 s a y s t h a t S is a fixed-point
of t h e formula Q(Nx).
Lemma - S i s an anti-fixed-point
of
of 0 ; p r o p o s i t i o n So w e have:
@ ( X I i f f S i s a fixed-point of
Q(Nx). The above lemma with Theorem A ( a p p l i e d t o X I ) g i v e s :
Theorem A‘ course).
-
Every formula Q i n !Jk has an a n t i - f i x e d p o i n t ( i n 2h1, of 1 In p a r t i c u l a r , t h e canonical fixed-point of Q(Nx) (which i s t h e
sentence Q (DN’Q (DNX)’
i s an a n t i - f i x e d p o i n t of @.
)
We henceforth l e t
5 be
canonical a n t i - f i x e d - p o i n t Q ( N x ) . ) The sentence an axiom of which i s
S,*:
5,then
Q(NDrQ(NDx)l )I
%
t h e sentence Q(DNrQ(DNx)’); we c a l l it t h e of Q.
( I t i s a l s o t h e canonical f i x e d p o i n t of
r ’ l i s n o t only independently t r u e , b u t it i s
Q 4
Since DrQ(NDx)l.d e s i g n a t e s t h e d i a g o n a l i z a t i o n of Q ( N D x ) ,
s. Also c4
-1
WDrQ(NDx)’ d e s i g n a t e s
a‘s’
t h i s sentence is Q
3
i s an axiom of
Qrs’.
A
3
(applied t o
designates
PI,of
s,hence
course), but
W e d e f i n e a sentence S t o be an a n t i - f i x e d p o i n t of a formula 0 for r an axiom s y s t e m s y s t e m a i f f t h e sentence S 0 - S is provable i n Thus
=
t h e canonical a n t i - f i x e d p o i n t
5 of
a.
Q is a l s o an a n t i - f i x e d p o i n t of Q f o r
t h e system 8*.
7.1.
The Jerislow and Rogers Sentences.
W e l e t J be t h e canonical
a n t i - f i x e d p o i n t of t h e formula Px; it i s t h e sentence PNDrPNDx‘. t h e J e r i s l o w sentence i n our formalism. pendently t r u e .
i s provable.
The sentence J
This i s
P r 4 - i s inde-
Hence J i s t r u e f o r j u s t t h o s e axiom systems i n which ( 4 )
Hence ( d ) ( l i k e G) i s t r u e f o r j u s t t h o s e systems i n which it
i s n o t provable.
Lemma 3 and Theorem 3 hold, r e p l a c i n g ”G“ by “4“.
Theorems 4 and 5 hold r e p l a c i n g ‘‘4 by““J” ( b u t t o prove Theorem 4, one needs statement (1) of Lemma 4 r a t h e r than t h e weaker statement ( 2 ) ) .
R.M. SMULLYAN
208
The Roger's sentence R i s obtained by t a k i n g t h e canonical a n t i f i x e d p o i n t of t h e formula -Px--it tence R
i s t h e sentence - P N D r ~ P N D x 1 .
The sen-
i s independently t r u e , hence R i s t r u e f o r j u s t those
-Pr-R'
%
axiom systems i n which (-R)
Hence (-I?)
i s not provable.
( l i k e t h e Henkin
sentence H) i s t r u e f o r j u s t t h o s e s y s t e m s i n which it i s provable. Theorem 6 holds, r e p l a c i n g "H" by "-R"
i n statement ( a ) , and "-H" by "R"
i n statement ( b ) . A Dual Form o f L6b's Argument
87.2.
t h e sentence P r P r Y 1 2 Y7
3
P T .
-
For any sentence Y , l e t Y+ be
Also i n modal l o g i c , l e t q'
be t h e sen-
t e n c e c(cq 3 q ) 13 oq. W e c a l l t h e next theorem a "dual" of Theorem 7 - - i t
r e p l a c e "S" by "8
-
Theorem '7
1'
For any normal extension
is an a n t i - f i x e d - p o i n t
-
Proof
a(-p
E
a l s o holds i f we
"
a of
8 and any sentence
i n Q f o r - ( P x 3 Y ) , then Y
Since t h e formula O ( p
+ ( W p 3 q ) ) 3 q , and
5
+ 1. s
+ 1. s
(Up 3 q ) ) 3 q
hence so is a ( p
Y , i f there
provable i n
provable i n K4, so i s
-(*p
+.
3 q)) 3 q
Lemma 5 , f o r any sentences X , Y t h e following sentence i s provable 0 + (L )P'(X = - ( P r 4 7 3 Y ) l 2 Y
.
Suppose now t h e r e i s an a n t i - f i x e d p o i n t X i n means t h a t X provable i n
5
ff.
a for - ( P x
-(Pr-Xl 3 Y ) is provable i n ff. Then P'X'"
Then by
i n 8: This
3 Y ) .
-(Pr-X1
3
Y)
is
a,
0 hence by ( L ) and modus ponens, so i s Y + .
Theorem 7'
i n l i e u of Theorem 7 p r o v i d e s an a l t e r n a t i v e proof of
LBb's theorem f o r t h e system
S1*,s i n c e
f o r S,*
w e do have an a n t i - f i x e d -
p o i n t f o r - ( P x 3 Y ) ( a s w e do f o r every formula).
This a l t e r n a t i v e con-
s t r u c t i o n i s of course a l s o adaptable t o Peano Arithmetic. Thus LUb's o r i g i n a l argument b o i l s down t o t a k i n g a fixed-point of Px 3 Y .
Our "dual" c o n s t r u c t i o n involves t a k i n g an a n t i - f i x e d p o i n t f o r
-(Px 3 Y ) .
For t h e s y s t e m
fur -(aJx
Yf--we
3
-(PND'(-PNDX
8.
3 Y)'
al*, t h i s
is e q u i v a l e n t t o t a k i n g a fixed-point
can indeed t a k e t h e canonical one, which is t h e sentence 3 Y)
.
Concluding Remarks.
formalism % can be extended.
There a r e s e v e r a l o t h e r u s e f u l ways our
For example, we could a d j o i n a symbol C and
t h e comma, and add t h e r u l e s t h a t f o r any terms tl,t2, t h e expression C(tl,t2)
i s a l s o a t e r m , and t h a t i f t 1' t 2 a r e c o n s t a n t terms d e s i g n a t i n g
Modality and Self-Reference X,Y, respectively, then C(t,,t,)
designates (X 3 Y).
209
(The symbol N then
becomes superfluous, and can be deleted, since we can then construe Nt as C(t,rll).)
If we base the systemss, 8 * in this extended formalism, a
host of other Ldbian type constructions become available which we consider in our sequel. One could also add a symbol B and the rules: If t is a term, so is Bt, and if t designates X, then Bt designates PrX7.
Then the
useful Corollary of Th. 4, Ch. 4, Boolos [11 applies. Finally, I wish to mention the rather obvious fact that although all the systems of this paper were presented as self-applied protosyntactical systems, they could as well have been presented as arithmetical ones: stead of "y" take "0",and let O,O',O", tural numbers 0,1,2
....
In-
..., be the usual names of the na-
Then take some GLldel numbering and redefine r X 7 as
the name of the Gddel number of A'.
A l l results of this paper apply equally
well to these arithmetical systems. We believe it desirable to have an abstract approach in which one can simultaneously talk about the systems of this paper--both in their protosyntactical and arithmetical versions--as well as usual first-order arithmetic theories like Peano Arithmetic, as well as systems of modal logic. do this in our sequel (this volume).
We
210
R.M. SMULLYAN NOTES
1.
This i s t h e t e x t of an i n v i t e d address presented t o t h e Association f o r Symbolic Logic a t t h e Annual Meeting, December 29, 1979, i n New York.
I t w a s presented under t h e t i t l e , "Self-Referential
Interpret-
a t i o n s of Modal Logic." 2.
This has been r e f e r r e d t o a s t h e " f o l k l o r e argument," s i n c e i t w a s never published, though an account of it i s given i n Boolos [lI and a l s o i n an a b s t r a c t form i n our sequel ( t h i s volume).
3.
The r o l e of t h e v a r i a b l e "x"
i s a b i t c u r i o u s , s i n c e w e have no
q u a n t i f i e r i n t h e system and i n f a c t t h e only axiom systems w e cons i d e r prove only sentences, and "x" does n o t occur i n any sentence. The symbol " x " i s used only t o f a c i l i t a t e d i a g o n a l i z a t i o n .
There i s
an a l t e r n a t i v e way of formulating our systems i n which we do n o t n e e d t h e v a r i a b l e x and i n which t h e diagonal f u n c t i o n can be replaced by t h e simpler norm f u n c t i o n of Smullyan 171.
W e plan t o d i s c u s s t h i s
elsewhere. 4.
For purposes of t h i s paper, w e could g e t by with a simpler set of terms--namely some name r A l . etc.,
j u s t those of t h e f o r m a o r
m , where a
is either x or
Thus we do not r e a l l y need such t e r m s as Dm, DDDCX,
s i n c e w e have no need t o i t e r a t e t h e diagonal f u n c t i o n .
But
then Theorem A of t h i s paper m u s t be modified: it only holds f o r formulas i n which D does not occur.
21 1
Modality and Self-Reference REFERENCES
[ll
Boolos, G.
[21
GUdel, K.
The Unprovability of Consistency, Cambridge U n i v e r s i t y
P r e s s , 1979. Itfiber f o r m a l u n e n t s c h e i d b a r e SBtze d e r P r i n c i p i a mathe-
matica und v e r w a n d t e r Systeme I , " Monatschefte fur Mathematik und
Physik 38 ( 1 9 3 1 ) , 173-198, E n g l i s h t r a n s l a t i o n i n From Freqe to G B d e l , e d i t e d by Jean van H e i j e n o o r t , 131
J e r o s l o w , R.G.,
Cambridge, 1967, pp.
596-616.
"Redundancies i n the H i l b e r t - B e r n a y s D e r i v a b i l i t y
C o n d i t i o n s f o r GUdel's Second I n c o m p l e t e n e s s Theorem," Journal of
Symbolic Logic 38 ( 1 9 7 3 ) , 359-67. [4]
LBb, M.H.
" S o l u t i o n of a Problem o f Leon Henkin," Journal of Sym-
bolic Logic 20 (1955), p p . 115-18. [5]
Smullyan, R . S t u d i e s #47,
[6]
Smullyan, R.
Theory of Formal Systems.
Annals of Mathematics
P r i n c e t o n U n i v e r s i t y P r e s s , 1959. "Some P r i n c i p l e s R e l a t e d t o LUb's Theorem"
-
this
volume. [7]
Smullyan, R.
"Languages i n Which S e l f - R e f e r e n c e i s P o s s i b l e , "
Journal of symbolic Logic 22 (19571, 55-67.
This Page Intentionally Left Blank
Intensional Mathematics S. Shapiro (Editor) @ Elsevier Scfence PubIishers B. V. (North-Holland), 1985
SOME PRINCIPLES
RELATED TO
Raymond M.
213
~ 8 ~THEOREM ' s
Smullyan
Department of Mathematics Lehman College, City University of N e w York Bronx, N e w York U. S.A.
INTRODUCTION AND SUMMARY
It h a s been shown by K r e i s e l t h a t Gddel's second incompleteness
theorem is e a s i l y d e r i v a b l e from Ldb's theorem.
I t has been shown by
Saul Kripke t h a t Ldb's theorem can be obtained from GUdel's second incomp l e t e n e s s theorem ( f o r e x t e n s i o n s of Peano Arithmetic). L e t u s make a d i s t i n c t i o n between what might be c a l l e d t h e weak and
s t r o n g v e r s i o n s of GUdel's second theorem and Ldb's theorem:
By t h e weak
v e r s i o n of Gddel's second theorem w e mean t h e statement t h a t i f Peano Arithmetic i s c o n s i s t e n t then t h e sentence Consis. i s n o t provable i n P.A.
(Peano A r i t h m e t i c ) .
By t h e s t r o n g version of Gddel's second theorem
w e mean t h e statement t h a t t h e sentence Consis.
3
NBew 'Consis?
(Bewtx) is t h e p r o v a b i l i t y p r e d i c a t e of P.A.;
a b l e i n P.A.
standard sentence expressing t h e consistency of P.A.;
X , r X 7 i s t h e name of t h e GUdel number of X . )
i s prov-
Consis. is t h e
f o r any sentence
By t h e weak v e r s i o n of
Ltlb's theorem w e mean t h e s t a t e m e n t ' t h a t f o r any sentence Y of P.A., Bew(rY1)
3 Y
is provable i n P.A.
so i s Y.
if
By t h e s t r o n g v e r s i o n of Ldb's
theorem, we mean t h e statement t h a t f o r any Y , t h e sentence Bew('Bew('Y') 3 Y')
3
Bewry'
i s provable i n P.A.
The s t a r t i n g p o i n t of t h i s paper w a s t h e r e a l i z a t i o n t h a t one can do t h e same t h i n g f o r t h e s t r o n g v e r s i o n s of Gddel's second theorem and Ldb's theorem a s Kripke d i d f o r t h e weak versions.
(Indeed, Kripke's argument
can be formalized i n t h e modal system K4, a s w e w i l l show.)
This l e a d s u s
t o something i n t e r e s t i n g : To o b t a i n Gddel's second incompleteness theorem by p u r e l y proof-
t h e o r e t i c methods, one somewhere along t h e l i n e f i n d s a fixed-point f o r
some formula.
(BY a fixed-point
of a formula Q ( x ) i s meant a sentence S
such t h a t t h e sentence S Z Q ( ' S ' ) i s provable i n t h e system under consideration.)
G8del d i d t h i s by t a k i n g t h e famous Gddel sentence G which i s a
R.M. SMULLYAN
214
fixed-point of t h e formula -Bew(x).
A "dual" scheme which works
Smullyan 141 and Jerislow [ 2 1 ) , i s t o t a k e a fixed-point
(cf.
f o r t h e formula
(For convenience we assume our theory has terms f o r a l l primi-
Bew(neg x ) .
t i v e r e c u r s i v e f u n c t i o n s ; neg(x) i s a term such t h a t f o r every sentence Y, t h e sentence neg7Y7= T;YY1 i s provable.)
Also, f o r Ldb's theorem, given a
sentence Y , t o show t h a t t h e sentence Bew('BewrY'3 henceforth a b b r e v i a t e by P.A.)
+ Y --to
+ Y
show t h a t
Y')
3
BewrY1--which w e
i s provable i n t h e system (say
involves f i n d i n g a f i x e d p o i n t f o r some r e l e v a n t formula.
l a used by Ldb 131 was Bew(x) 3 Y.
151 a "dual" c o n s t r u c t i o n a l s o works: -(Bew(neg x )
3 Y)
The formu-
A s w e showed i n our preceeding paper A fixed-point
f o r t h e formula
a l s o does t h e t r i c k .
Let u s look a t t h e m a t t e r t h i s way:
Suppose i n s t e a d of Peano Arithme-
t i c we have a f i r s t - o r d e r a r i t h m e t i c theory with a p r o v a b i l i t y p r e d i c a t e Bew(x) s a t i s f y i n g t h e Hilbert-Bernays d e r i v a b i l i t y c o n d i t i o n s , b u t which does not n e c e s s a r i l y s a t i s f y t h e condition t h a t all formulas 9 ( x ) have
fixed points.
Then it may be t h a t f o r some sentence Y, Y+ i s provable i n
t h e theory, and f o r o t h e r sentences n o t .
Then i n s t e a d of looking a t t h e
matter "globally", we might look a t it " l o c a l l y " and ask f o r a given Y,
+
what f i x e d p o i n t w i l l ensure t h e p r o v a b i l i t y of Y ? answers--one
W e a l r e a d y have two
t h a t t h e r e be a f i x e d p o i n t f o r Bew x 3 Y and t h e o t h e r t h a t
t h e r e be a f i x e d p o i n t f o r -(Bew(neg x) 3 Y ) . answers which we have found:
But t h e r e a r e s e v e r a l o t h e r
The two most p e r t i n e n t ones were found by
" l o c a l i z i n g " Kripke's argument, then s t r e n g t h e n i n g both t h e hypotheses and conclusion of t h e " l o c a l i z e d " argument, and t h i s l e d t o t h e f a c t t h a t a s u f f i c i e n t condition f o r Y sentence
s
+
t o be provable i n t h e system i s t h a t t h e r e be a
such t h a t t h e sentence ( Y V
(S
5
-Bewr(S
v
Y)'))
be provable.
A - f o r t i o r i , a s u f f i c i e n t condition f o r Y+ t o be provable i s t h a t t h e r e be a sentence S such t h a t t h e sentence s
C
-Bew('S
+
means t h a t a s u f f i c i e n t c o n d i t i o n f o r Y
V Y')
be provable.
t h a t t h e r e be a fixed-point f o r t h e formula -Bew(disj. d i s j . ( x , y ) i s a term such t h a t f o r any sentences X , Y , d i s j . (rX','-Y1)
= 'X
v
Y'
i s provable.
This
t o be provable ( i n t h e t h e o r y ) i s (x, 'Y')),
where
t h e sentence
T h i s , of course, provides an a l t e r n a -
t i v e proof of Ldb's theorem f o r P.A. The o t h e r p e r t i n e n t answer i s a "dual" t o t h e preceding one: L e t t i n g cond.
( x , y ) be a term such t h a t f o r a l l X , Y , t h e sentence cond.
r.y 3 Y '
i s provable.
A s u f f i c i e n t condition f o r Y
+ to
t h e r e be a f i x e d p o i n t f o r t h e formula Bew(cond. (x,'Y'))--in t h a t t h e r e be a sentence S such t h a t S
Bew'S
3 Y'
(rX1trY')
=
be provable is t h a t o t h e r words
be provable.
(Compare
Some Principles Related to Ub's Theorem
t h i s with LUb:
S I (BewrS'3
215
Y ) is provable.)
We have found s e v e r a l o t h e r answers t o o , which we might summarize i n t h e following theorem:
Theorern A
-
Given a theory ( T ) with a p r o v a b i l i t y p r e d i c a t e Bew(x) s a t i s f y -
ing t h e Hilbert-Bernays d e r i v a b i l i t y c o n d i t i o n s , f o r any sentence Y a suff i c i e n t condition t h a t Bew%ewrY'
3 Y'
3 Bew
:Y7 be provable i n t h e system
i s t h a t t h e r e be a sentence s such t h a t any of t h e following e i g h t sent e n c e s a r e provable:
(LBb [ 3 1 ) (Smullyan [ 5 1 ) (Proved h e r e ) (Proved h e r e ) (Proved h e r e ) (Proved h e r e ) (Proved h e r e ) (Proved h e r e )
The above f a c t s have t h e i r c o u n t e r p a r t s i n t h e modal system K4, t o which a good p a r t of t h i s paper i s devoted:
( f o r K4) we
By an L-formula
mean a modal formula A ( p , q ) i n which every occurrence of p
l i e s withinthe
scope of some occurrence of O(such formulas a r e s a i d t o be modalized i n p ) such t h a t t h e formula O(p mean t h e formula o(0q 3 q )
A (p,q)) 3 q
3nq).
4-
i s provable i n K4 (where by q
The well-known L-formula i s
up3
+
we
q . And
w e showed i n our preceeding paper, t h i s volume, t h a t i t s "dua1"c. W p
3 q)
i s an L-formula.
(2)'
The two main new L-formulas
(corresponding t o (21,
of Theorem A) of t h i s paper a r e O ( p 3 q ) and ,o(p are: U P
3
oqi -(OCP
3
0 9 )i O ( p
our o v e r a l l p l a n i s t h i s :
v
q) 3
v 4 ) . The remaining f o u r
oq and -(UP
3 q ) 13 0 4 .
In t h e f i r s t s e c t i o n w e c o n s i d e r some
Gddelian and LBbian p r i n c i p l e s i n an a b s t r a c t s e t t i n g which enables u s t o simultaneously t a l k about f i r s t - o r d e r t h e o r i e s , modal l o g i c s , and t h e s e l f r e f e r e n t i a l systems t r e a t e d i n our preceding paper.
I n t h i s a b s t r a c t set-
t i n g w e cannot speak of a p r o v a b i l i t y p r e d i c a t e Bew(x) ( s i n c e we have no such t h i n g a s v a r i a b l e s o r open formulas i n t h i s g e n e r a l s e t t i n g ) , so we introduce t h e notion of a p r o v a b i l i t y function--a
mapping from s e n t e n c e s t o
sentences s a t i s f y i n g t h e analogue of t h e Hilbert-Bernays c o n d i t i o n s .
I t is
i n t h i s general s e t t i n g t h a t w e f i r s t consider Kripke's argument and i t s
R.M. SMULLYAN
216
modifications. We then t u r n t o t h e modal system K 4 .
Our discovery of s e v e r a l f a c t s
about K4 was g r e a t l y f a c i l i t a t e d by use of a q u i t e b a s i c p r i n c i p l e - - t h e t r a n s l a t i o n theorem--which d a l system G ) .
w e s t a t e and prove f o r K4 (and a l s o f o r t h e mo-
This allows u s t o g e t new p r i n c i p l e s of K4 from known ones,
which when t r a n s l a t e d i n t o metamathematical t e r m s , y i e l d s new LBbian-type c o n s t r u c t i o n s from known Gbdelian-type c o n s t r u c t i o n s .
W e then g i v e meta-
mathematical a p p l i c a t i o n s of t h e s e r e s u l t s about K4.
1.
P r o v a b i l i t y F u n c t i o n s i n a General S e t t i n g .
W e consider an
axiom s y s t e m 4 whose language c o n t a i n s a t l e a s t a l l t h e p r o p o s i t i o n a l conIt w i l l be convenient t o t a k e L ( t h e symbol f o r l o g i c a l f a l s e -
nectives.
hood) a s a p r o p o s i t i o n a l c o n s t a n t and t o d e f i n e -X
Ell).
a is a contains
We s h a l l say t h a t
t e n c e s provable i n
as (X 3 r ) ( a s i n Boolos
t a u t o l o g i c a l l y c o m p l e t e i f f t h e s e t of sena l l t a u t o l o g i e s and i s c l o s e d under modus
ponens. We now c o n s i d e r a f u n c t i o n B which maps every sentence sentence B ( X ) of 0.
W e sometimes w r i t e BX f o r B ( X ) .
provability function for a i f f sentences X , Y of
B ~ : B ~ : B ~ :
If
x
a is
x
of
ato
a
W e shall call B a
t a u t o l o g i c a l l y complete and f o r a l l
ff t h e following c o n s i d e r a t i o n s hold: i s provable i n
a,
so i s B X .
B ( X 3 Y ) 3 ( B X 3 B Y ) i s provable i n
a.
BX 3 BBX i s provable i n ff.
I f B i s a p r o v a b i l i t y f u n c t i o n f o r ff, then we s h a l l say t h a t t h e p a i r q , B > i s an a c c e p t a b l e p a i r .
Example 1
-
Here are some examples:
For Peano Arithmetic, d e f i n e B ( X ) t o be t h e sentence Bew(rX1).
Since Bew(x) s a t i s f i e s t h e Hilbert-Bernays d e r i v a b i l i t y c o n d i t i o n s , B i s a p r o v a b i l i t y f u n c t i o n f o r P.A.
Example 2
-
I n t h e s e l f - r e f e r e n t i a l p r o t o s y n t a c t i c a l system S of our pre-
ceding paper, d e f i n e B ( X ) t o be t h e sentence PrX7, where of t h e expression X . systems, 8 ,
Example 3
-
rxl
i s t h e name
C l e a r l y B is a p r o v a b i l i t y f u n c t i o n for each of t h e
a*, al. 8,*. Consider t h e language of modal l o g i c .
For each modal sentence
Some Principles Related to Lab's Theorem A,
217
Given a modal axiom system M, t o say
d e f i n e B ( A ) t o be t h e sentence O A .
t h a t B i s a p r o v a b i l i t y f u n c t i o n f o r M i s e q u i v a l e n t t o saying t h a t M i s a normal extension of t h e modal system K4 ("normal" i n t h e sense of Boolos
[l]).
In p a r t i c u l a r , B i s a p r o v a b i l i t y f u n c t i o n f o r K 4 .
1.1.
L e t B be a p r o v a b i l i t y function f o r f f . W e s h a l l say t h a t a sen-
tence X i s weakly LUbian f o r f f with r e s p e c t t o *-more weakly LUbian f o r t h e p a i r d , B > - - i f f p l i e s t h e p r o v a b i l i t y of
x
LUbian f o r < f f , B > i f f t h e sentence
x)
(BX 3
x)
is
i n ff i m -
i n ff ( i n o t h e r words i f f e i t h e r ( B X x~ ) i s n o t
provable i n a o r X is provable i n ff). W e s h a l l say t h a t
B(BX 3
x
briefly that
t h e p r o v a b i l i t y of
. x + is
provable i n
Q(x'
x
is s t r o n g l y
i s t h e sentence
2 BX).
I f X i s s t r o n g l y LUbian f o r d r B > it i s c e r t a i n l y weakly LUbian f o r < a , B > , f o r suppose B ( B X 3
provable i n X
a,
x)
3 BX
so i s B ( B X 3 X )
is weakly LUbian f o r c f f , B > .
,
i s provable i n
a:
Then if B X = I
x is
and hence by modus ponens, so i s B X , and so
The converse i s not n e c e s s a r i l y t r u e ,
though it is t r u e t h a t i f a l l sentences X a r e weakly LUbian f o r
then a l l sentences a r e s t r o n g l y LUbian f o r d,B>--indeed, tence X , i f X
+ is .
f o r any p a r t i c u l a r sen-
weakly LUbian f o r ff, then X i s s t r o n g l y LUbian f o r ff
( t h i s can be proved i n t h e manner of Th. 4 of L e t Consis. be t h e sentence N B I .
11, Chapter 3 1 ) .
We s h a l l say t h a t e , B > weakly
obeys GUdel's second (incompleteness) theorem i f f t h e p r o v a b i l i t y i n Consis. implies t h a t
0 is
of
i n c o n s i s t e n t ( i n o t h e r words t h a t e i t h e r Consis.
is n o t provable i n f f or f f i s i n c o n s i s t e n t ) .
W e say t h a t d , B > s t r o n g l y
obeys GUdel's second theorem i f f t h e sentence B(Consis.1 3 4 o n s i s . provable i n ff. Now t h i s sentence i s B ( B I provably e q u i v a l e n t i n
f f
a to
3 I) 2 ( ( B L 2 I) 3 I ) ,
t h e sentence B ( B I 3 I) 2 B I .
is
which i s
Thus q , B >
s t r o n g l y obeys GUdel's second theorem i f f I i s s t r o n g l y LUbian f o r CQ,B>. It a l s o t r u e ( a s can e a s i l y be checked) t h a t e , B > weakly obeys GCldel's
second theorem i f f I i s weakly LUbian f o r d , B > . Now, ICripke's argument f o r showing t h a t every sentence X of P . A .
weakly LUbian ( f o r P . A . ) i.e.,
involves c o n s i d e r i n g t h e extension P . A .
t h e system P . A . with NX adjoined a s an axiom.
work i n our more a b s t r a c t s e t t i n g , s i n c e deduction theorem--i.e., in f f
+
ff
is
{-XI--
This p l a n w i l l not
does n o t n e c e s s a r i l y s a t i s f y t h e
w e may have sentences X , Y such t h a t Y i s provable
{ X I , y e t X 3 Y may n o t be provable i n f f .
follows:
+
So w e modify t h e p l a n a s
218
R.M. SMULLYAN For any sentence X w e l e t ff
be t h a t axiom system whose axioms a r e
X
a l l sentences Y such t h a t ( X 3 Y ) i s provable i n
i s provable i n ff.
which has no provX
( I n those s p e c i a l c a s e s i n which
t i o n theorem, then p r o v a b i l i t y i n ff
a+
a,and
Thus Y i s provable i n ff
a b l e sentences o t h e r than t h e axioms.
x
iff
a s a t i s f i e s the
3 Y
deduc-
is equivalent t o provability i n
X { x } , b u t we a r e i n t e r e s t e d i n t h e more g e n e r a l case.)
W e now d e f i n e B
X
a s t h a t f u n c t i o n from sentences t o sentences which
a s s i g n s t o each sentence Y t h e sentence B ( X 2 Y ) ( W e w i l l later see t h a t B
X
.
Thus B ( Y ) = B ( X 2 Y ) X
is a l s o a provability function f o r
aX'
.
and a l s o
f o r ff, b u t w e d o n ' t need t h i s y e t . ) W e now consider t h e ordered p a i r
-
Theorem 1
( a ) (After Kripke)
-
If
'G&,
BNX> and prove:
B , ~ > weakly obeys Godel's second
theorem, then X i s weakly LBbian f o r . l y LBbian f o r < U , B > , then ,,
Proof
-
( b ) Conversely, i f
BNX> weakly obeys GBdel's second theorem.
, a
in
implies t h e p r o v a b i l i t y of
A
in
&.
N o w , "B-1
sentence 18 (45) I), hence t h i s sentence i s provable i n 3
-~( 3 I4 ) is provable i n
ing provable i n
a,which
a,which
a,which
a-X
i s equivalent t o
B(-x
3
11
3
x
be-
a. Also
i s equivalent t o t h e p r o v a b i l i t y of d 3 I i n
i n t u r n i s equivalenk t o t h e p r o v a b i l i t y of X i n
Bq>
is the
iff
i s e q u i v a l e n t t o BX 2 X being provable i n
t h e p r o v a b i l i t y of L i n Qrr
is weak-
< a N x , BWX> weakly obeys GBdel's second theorem i f f t h e p r o v a b i l i t y
of ^ B 4 i
Y, . ,
x
0.
Therefore
weakly obeys GBdel's second theorem i f f t h e p r o v a b i l i t y i n ff of
BX 3 X implies t h e p r o v a b i l i t y i n ff of X--in
o t h e r words i f f X i s weakly
LBbian f o r W , B > .
Discussion
-
P a r t ( a ) of Theorem 1 i s e s s e n t i a l l y Kripke's argument:
Taking Peano Arithmetic f o r ff, t h e f a c t i s t h a t a l l e x t e n s i o n s of P.A.
do
weakly ( i n f a c t , s t r o n g l y ) obey GBdel's second theorem, and t h e r e f o r e f o r any sentence X , t h e p a i r theorem.
B-x
> does weakly obey GBdel's second
Therefore every X is weakly LBbian f o r P.A.,
s o LBb's theorem
holds f o r P.A. A s i n d i c a t e d i n t h e i n t r o d u c t i o n , one motivation of t h i s paper was
t h e attempt t o show t h a t t h e above theorem holds i f we r e p l a c e "weakly" by "strongly. "
1.2.
S t r o n g Properties.
The next theorem is b a s i c .
Same Principles Related to Ub's Theorem
Theorem 2
- For
219
any acceptable p a i r d , B > and f o r any sentence X ; ( a ) Bx
i s a p r o v a b i l i t y f u n c t i o n f o r ff. (b) BX i s a p r o v a b i l i t y f u n c t i o n f o r
Ox.
To prove Theorem 2 , we f i r s t l i s t some p r o p e r t i e s of d , B > which a r e
w e l l known i n t h e c o n t e x t of t h e modal system K 4 . provable i n
0,and w e w r i t e
By "provable" we mean
F X t o mean t h a t X i s provable i n ff. Then f o r
any sentences X,Y:
F B ( X 3 (Y 2 2 ) ) 3 (BX 3 (BY 3 BZ))
L1:
I f X 3 Y i s provable i n ff, so i s BX 3 BY
L2:
I f BX 13 Y i s provable i n
L3:
ff, so
i s BX 3 BY
F B ( X 3 Y ) 2 B(BX 3 B Y )
L4:
L is proved by two a p p l i c a t i o n s of condition B2 of t h e d e f i n i t i o n o f 1 a p r o v a b i l i t y function. A s f o r L a , i f X 3 Y i s provable, so i s B(X 3 Y ) ,
and hence so i s Bx 3 BY.
A s f o r L3, i f BX 3 Y i s provable,
so i s
B ( B X 3 Y ) , hence so i s BBX 2 BY, but a l s o BX 3 BBX i s provable, and hence
( s i n c e ff i s t a u t o l o g i c a l l y complete) so i s BX
B(X 3 Y ) i s provable, so i s BX Proof of Theorem 2
- We
3
3
BY.
A s f o r L4,
if
BY, and hence so i s B(BX 3 BY).
f i r s t prove ( a ) .
We must show f o r any sentences
X,Y,Z: (1) I f Y is provable i n ff, so is BX Y ; ( 2 ) Bx(Y 3 Z ) 3 (BxY 3 B x Z ) i s provable i n ff; ( 3 ) B Y 3 B g Y i s provable i n X X R e (1), Y 3 (X 3 Y ) i s a t a u t o l o g y , hence provable ( i n Hence
a;
BY
3 B(X 3 y)
BY.
i s provable (by L 2 ) .
a).
Now suppose Y i s provable.
Hence (by modus ponens) so i s B(X
3 Y)--i.e.,
Then so i s
so i s B Y. X
R e ( 2 ) : The sentence (X 3 (Y 3 Z ) ) 3 ( (X 3 Y ) 3 (X 2 Z ) ) i s a t a u t o -
logy, hence i s provable i n 0.
(B(X 3 Y ) 3 B ( X 3 2 ) ) - - i . e . ,
Then, by L1,
s o i s B ( X 3 ( Y 3 2 )) 3
Bx(Y 3 Z) 3 ( B Y 3 BxZ) i s provable i n X
ax.
Re ( 3 ) : B(X 3 Y) 3 ( X I B ( x 3 Y ) ) i s a tautology, hence provable i n
0. Then by
L ~ B(X , 3 Y ) 3 B(X 3 B(X 3 Y )1 i s provable i n
a, i . e . ,
B ~ 2 Y
BXBXY is provable i n ff. This proves t h a t Bx i s a p r o v a b i l i t y f u n c t i o n f o r
a. (b)
As f o r
ax,w e
a b l e i n ff i s provable i n
f i r s t n o t e t h e t r i v i a l f a c t t h a t anything prov-
ffx (because
Re (1) Suppose Y i s provable i n
i f Y i s provable i n ff, so i s X 3 Y).
ax.
Then X 3 y i s provable i n ff.
Then B ( X 3 Y) i s provable i n ff, which means t h a t BxY i s provable i n hence i n
Ox.
a,and
220
R.M. SMULLYAN
Re (2),
(3):
Since B (Y 3 Z) 2 (BxY 3 B x Z ) and Bx Y 3 B2 X
provable i n ff, they a r e provable i n
ax.
xY
a r e both
This completes t h e proof of
Theorem 2 . For any sentence X I w e temporarily l e t X* be t h e sentence B
W e now know t h a t Be
B-A.
say t h a t < f f , B
^x
M4M1
ff-.
is a p r o v a b i l i t y f u n c t i o n f o r a a n d f o r
= To
> s t r o n g l y obeys GUdel's second theorem i s t o say t h a t X* i s
provable i n ff; t o say t h a t
s t r o n g l y obeys GOdel's second theorem
i s t o say t h a t X* i s provable i n ffwxl o r e q u i v a l e n t l y , t h a t t h e sentence (-X
3X*)--or equivalently X V X*--is
provable i n ff.However, X* is prov-
a (and hence a l s o i n a,) because X* i s t h e sen=, -B(^X = J.)) = B("X 3 I ) , which i s provably e q u i v a l e n t t o
ably equivalent t o X+ i n tence B("X
B ( 4 f 3 -BX) I B X which i s provably e q u i v a l e n t t o B (BX 3 X ) 3 B X , which i s
t h e sentence X + .
Theorem 3
-
provable i n
-+
W e t h u s have
(a) d,B,>
a.
s t r o n g l y obeys GBdel's second theorem i f f X
( b )
(X V X ) i s provable i n
+ 1. s
s t r o n g l y obeys GOdel's second theorem i f f
a.
By a GOdel sentence f o r < f f , B > w e mean a sentence S such t h a t t h e sen-
tence S
= wB(S)
i s provable i n ff.W e can a l s o speak of a GBdel sentence
f o r d , B >, meaning a sentence S such t h a t S X
.-B
X
(S) i s provable i n 0, o r
of a GUdel sentence f o r d x , B x > , meaning a sentence S such t h a t S
= hBX(S)
i s provable i n Ox. We s h a l l s t a t e t h e n e x t theorem without proof ( s i n c e it i s w e l l known f o r f i r s t - o r d e r
t h e o r i e s , and a l s o follows from t h e well known f a c t
e) 3 (0- i 3 01) i s provable
t h a t t h e modal sentence o ( p
Theorem 4 [After GOdel]
- For
c o n d i t i o n s a r e equivalent:
i n K4).
any acceptable p a i r d , B > t h e following two
(1) There is a GBdel sentence f o r < a , B > ;
(2)
d , B > s t r o n g l y obeys GOdel's second theorem.
W e now apply Theorem 4 t o t h e p a i r d , B - x . X > , Theorem 5
-
and g e t
A s u f f i c i e n t (and a l s o necessary) condition f o r X t o be strong-
l y LObian f o r < a , B >
i s t h a t t h e r e i s a sentence S such t h a t S z d ( S V X )
is provable i n ff. Proof
- The
tence S
sentence S
2 -B_,(S).
- B ( S V X ) i s provably e q u i v a l e n t i n
Therefore t h e p r o v a b i l i t y of S
e q u i v a l e n t t o S being a GOdel sentence f o r d , B x > .
= H B (S V
ff t o t h e sen-
X ) i n ff is
But by Theorem 4
Some Principles Related to Lab’s Theorem (applied t o t h e p a i r true i f f X
+ is .
-X
221
> ) t h e r e i s a GOdel sentence f o r
nx> i f and
s t r o n g l y obeys GOdel’s second theorem, which i n t u r n i s provable i n
a
(by ( a ) of Theorem 3 ) .
of a sentence S such t h a t S
Therefore t h e e x i s t e n c e
4 ( S V X ) i s provable i n
a is
equivalent t o
X being s t r o n g l y LObian f o r ff.
W e have now proved (3) of Theorem A.
We have not y e t proved t h a t
Theorem 1 holds, r e p l a c i n g “weakly” by “ s t r o n g l y “ , but we w i l l .
I t is per-
haps e a s i e s t t o f i r s t do t h i s i n t h e c o n t e x t of t h e modal s y s t e m K 4 , t o which we now t u r n .
2.
Some P r o p e r t i e s of K4.
We formulate t h e modal system K4 a s i n
[l], except t h a t i n s t e a d of having a r u l e of s u b s t i t u t i o n , w e take t h e axioms a s axiom schemata.
S p e c i f i c a l l y we t a k e t h e following axiom sche-
mata :
A ~ :
A2: A3:
A l l tautologies A l l sentences C ( X 3 Y ) 2 (RX 3 O Y )
All sentences C X 2 O C X
The only i n f e r e n c e r u l e s are modus ponens and n e c e s s i t a t i o n (from X t o infer O X ) . W e t a k e one p a r t i c u l a r p r o p o s i t i o n a l v a r i a b l e q.
tence X we i n d u c t i v e l y d e f i n e t h e q - t r a n s l a t e of X
F o r any modal sen-
- which we
w r i t e q{X}--
by t h e following r u l e s :
[For example, i f X i s t h e formula o p a C ( O r o(q 2 p ) 3 o(q 2 ( o ( q 2 r ) 2 I)1.1 L e t us define
c
X t o be O ( q 3 X ) 4
.
3 I),
then q { X ) i s t h e formula
Then (roughly speaking) q { X } i s
obtained from X by r e p l a c i n g each subformula CY by 0 Y. Q The t r a n s l a t i o n theorem--which we a r e about t o prove--is
that for
any X provable i n K4, i t s q - t r a n s l a t e q { X } i s a l s o provable i n K4. Let O b e t h e axiom systen: K4 and B t h e f u n c t i o n which a s s i g n s t o each modal sentence X t h e sentence O X .
Since B i s a p r o v a b i l i t y f u n c t i o n
222
R.M. SMULLYAN
f o r K4, then Theorem 2 a p p l i e s t o
B
i s then t h e function which as4 So by Theorem 2 we have t h e following
s i g n s t o each X t h e sentence 0 X. 4 facts:
I f X i s provable i n K4, so i s 0 X. 4 0 (X 3 Y ) 2 (0 X 3 0 Y ) i s provable i n K4. 4 4 4 n X 3 c n X i s provable i n K4. 4 4 4
F1:
F2: F3:
Let us a l s o note t h e t r i v i a l f a c t :
we a l s o need t h e f a c t :
F ~ :
If
x is
a t a u t o l o g y , so i s qIX3.
One way t o prove F 5 i s t h i s :
...,
t i o n a l combination of X1,
..., X,)}
q{Y(Xl,
qIx 1--i.e.,
I f Y(X1,
..., X
) i s some t r u t h func-
then an easy induction shows t h a t
Xnl
...,
i s t h a t same t r u t h - f u n c t i o n a l combination of q{X1},
qIY(X1,
..., x,)
tautology, then l e t t i n g
xl,
1
...,
= Y(q{xl},
..., x
q{xn}).
I f now
be t h e subformulas of
x
x
is a
which a r e e i t h -
e r p r o p o s i t i o n a l l e t t e r s , I, o r expressions of t h e form o Y , then X is a
..., xn
t r u t h f u n c t i o n a l combination Y(X1,
...,
q{x
Y(Y1,
Y,,
1)
i s a tautology.
Theorem 6
Proof
..., Y
-
-
)
i s a tautology. SO
Y (q{xl},
[ T r a n s l a t i o n Theorem]
-
...,
) such t h a t f o r
any sentences yl,
so i n p a r t i c u l a r , \y(q{xl}, q{x,})
...,
i s a tautology.
I f X i s provable i n K4 so i s q{X).
I t s u f f i c e s t o show t h a t f o r any modal sentences X I Y , t h e follow-
ing conditions hold:
A:
I f X i s an axiom of K4, q{X) i s provable i n K4.
B:
I f q{X} and q{X 3 Y } a r e provable i n K4, so i s q { Y ) .
C:
I f SIX} i s provable i n K4, so i s qIOX}.
I t w i l l then follow by induction t h a t f o r any provable sentences X of
K4, q{X} i s provable i n K4.
223
Some Principles Related to Lab's Theorem
(1) I f X i s a tautology (axiom of scheme 1) then q t X }
To prove A:
i s a tautology (by F5), hence provable i n K4. t h e form C ( X 3 Y ) 3 (CX 3 CY).
Consider any axiom of
(2)
Now q { C ( X 3 U )
3
q{Cx 3 C Y } = q{c(x 3 Y ) 1 3 (SICXI) 3 (qtccy}) =
c
(aU 3 CY)} = q{C(X 2 Y )
dx
3 Y} 3
4 C q q { Y } ) (by Fact 4 ) , and hence provable i n K4 by F1
and "q{Y}" f o r " Y " . )
q{Cx 3 C C X } = q@X)
(Cq41X} 2
(taking " q { X } " f o r "X"
( 3 ) Take an axiom of t h e formCX 13OcX. 3
q{CCX} =
c9q{X}
3
c q{CX} 9
=
i s provable i n K4 by F4 ( t a k i n g "C X" f o r "X"). 4
3
c q{X} 9
3
Now,
CqcqqfX}, which
This t a k e s c a r e of t h e
axioms and proves A. To prove B:
Suppose g{X} and q { X 3 Y } a r e both provable i n K4. Since
q{X 3 Y } = q{X} 3 q{Y}, then q{X} and q { x } 3 q { Y } a r e provable hence q{Y} a l s o i s , by modus ponens. Suppose q{X} i s provable i n K4.
To prove C:
F1), b u t 0 q{X}
=
4
q{CX}, s o q { C x ) i s provable i n K4.
Then so i s
C q{X)(by 4
This completes t h e
proof. We now d e f i n e q'{X}--which
we c a l l t h e q'-translate of
x
by t h e f o l -
lowing r u l e s :
A t r i v i a l induction shows t h a t q'{X} i s provably e q u i v a l e n t i n K4 t o
(-q){X}, hence by Theorem 6: Th eore m '6
-
Remark
I f X is provable i n K4, so i s q'{X}.
- Theorem 60 w i l l
be used i n t h i s paper more than Theorem 6 ,
and w e could have proved it d i r e c t l y i n s t e a d of appealing t o t h e e q u i v a l e n c e of -p 3 q t o p
2.1.
v
q.
T r a n s l a t i o n T heorem for G.
This w i l l p l a y no r o l e i n t h i s
paper, b u t should be of i n t e r e s t t o those working i n t h e modal system G. The s y s t e m G i s obtained from K4 by adding a s axioms a l l sentences of t h e form C(0X I> X ) 3 C X .
Th eore m 6.1
- If
X i s provable i n G, so i s q { X } .
R.M. SMULLYAN
224
Proof
-
It suffices to show that the q-translate of every sentence
c(@X3 x ) 3 C X is provable in G. 3 a q 3
This q-translate is c ( q
3x1)
NOW, ( q 3 ( c ( q 3 x 1 3 ~ )3 )( c ( q T X ) 3 ( 4 c(q 3 (c(q3 x 1
3x1)
But c ( C ( q 3 X)
3 (q 3
3 C(C(g
3x1
3x1
3 (q
x)) I)c(q 3 X )
3x1
3X))
3x)
is provable in
53
G
(it is
C( Y 3
(C(q 3 x 1 3 X ) )
G.
Y) 3 I)
G.
We define an L-formula ("L" after Lob) as a
L-Formulas.
2.2.
is a tautology,hence
is provable in K4, hence in
is an axiom of
O Y , where Y is ( q X X ) ) , hence (by syllogism) C ( q c(q
3 (C(q
X).
) in p such that the sentence c ( p formula ~ ( p , q modalized
provable in K4 (we are using q+ to abbreviate C ( 9 q
A ( p , q ) ) 3 q + is
By a K-
3 q ) 3IJq).
formula we shall mean a formula A ( p , q ) modalized in p such that the formula A ( p , q ) ) ) I , q + is provable in K4.
c(q V ( p
Lemma 1
Proof
-
- For any
hence CX 2
Every K-formula is also an L-formula. sentences X, Y the sentence X
3 C(X V Y )
is provable in K4.
is provable in K4, so is OX
3 2.
q+) is provable in K4, so is C ( p
3
(X V Y ) is a tautology,
3
Therefore for any Z, if C(X V Y )
3
In particular, if C ( q V ( p f d ( p 8 q ) ) 2 + Thus every K-formula is
d ( p , q ) ) 53 q
.
also an L-formula.
G-formulas
-
By a G-formula
("G"
after Godel) we mean a formula @ ( p )
modalized in p (and with no other propositional variables) such that the sentence O ( p
I
@(PI)
3 I+
(which is o ( p
@(PI)
3
( C a 53 cl)) is provable
in K4. The following theorem is our main application of the translation theorem. Theorem 7
-
The q'-translate of any G-formula is a K-formula (hence also an
L-formula). Proof
-
Let @ ( p )be a G-formula.
Then by Theorem ,'6
Then O(p
3
@ ( p ) )3 I+ is provable in K4.
its q'-translate is also provable in K4.
translate is o(q v ( p
q ' { @ ( p ) ) ) 1) q v { L + 1 .
This q ' -
NOW, ~ ' { I + isIq
'ba
3 ~ 1 1 ,
which is n(q V w O ( q V I)) 2 O ( q V I) which is provably equivalent in K4 to
Some Principles Related to Ldb's Theorem
a )2 cq,
C(q V
(p
O(q V
q'C0
a K-formula.
and hence to
(p)1 )
3
+ q
C(Cq 3 q ) 2
nq, which is
225
Therefore
q+.
is provable in K4, which means that q ' { Q(p)} is
It is also an L-formula by Lemma 1.
Applications of Theorem 7
-
It is well known that -cp andC-p are
G-formulas (cf. also remarks following Th. 12, this paper). Theorem 7, their q'-translates are K-formulas.
Then by
The q'-translate of
-n(q V p), which is provably equivalent in K4 to wO(p V q ) .
-Cp
is
The 9'-trans-
late of O(-p) is c(q V -p), which is provably equivalent in K4 to O(p 3 4 ) . We thus have: Theorem 8
- The formulas -C(p
V q)
and C(p
3 q)
are K-formulas--and also
L-formulas. Applications to Provability Functions.
3.
we stated in 51 that
Theorem 1 holds if we replace "weakly" by "strongly" and we are now in a good position to prove it. We again consider an axiom systema and a provability function B for
a.
Boolos 11, Chapter 31 speaks of realizations and translations of modal
sentences into Peano Arithmetic.
We must do the analogous thing for trans-
lations of modal sentences to sentences of
a.
By a realization (in < a , B > ) we shall mean a function to each propositional variable p a sentence d(p) of of a modal sentence X under
a.
6 which assigns
The translation X d
is defined inductively by the rules:
(2) pd = z(p) (for each propositional variable p);
' I
= I;
xd
B 2Y ;
(ox)+= B ( Xz ).
(4)
(3)
(1)
( X ='YYld
=
BY a translation of x (in ) we mean a
translation under some realization d.
The following fact can be proved as
in Boolos (Th. 1, Ch. 3 ) .
-
Lemma 2
For any acceptable pair
of K4 are provable in
, all
translations of all theorems
a.
Now, by Theorem 8 , the formula -U(p V q ) is a K-formula, which means d
O(q V ( p
s, 2
x
in
a, we
+
is provable in K4. Then given any sentences take any realization which maps p to s and q to X, and Lemma p V g)) ) 3 g
gives:
-+ For any sentences S I X of G? (a) B ( X v (S S - B ( S V X ) ) ) =' X is -+ provable in (b) If x V (S 4 (S V X ) ) is provable in so is X Lemma
3
-
a,
a,
.
226
R.M. SMULLYAN We note t h a t ( b ) follows from ( a ) , because i f X V (S
provable i n
ff, so i s
B (X
v
tence f o r d W x I B
-X
Lemma 4 in
-
(5') i s provable i n
-B
a,,
I f t h e r e i s a GMdel sentence f o r
ff.
-X
3
(S
-B(-X
3 S))
i f f S i s a GMdel sen-
,-X And so (b) of Lemma 3 gives:
>!
is
-B (S v X ) ) 1.
(S
Now, X V (S f -B(S V X ) ) i s provable i n ff i f f -X is provable i n ff, i f f S
-B (S V X ) )
>, then X + i s provable
Now we e a s i l y prove:
Theorem 9
-
s t r o n g l y obeys G8del's second theorem i f and only i f
X i s s t r o n g l y LMbian f o r q , B > .
Proof
-
( a ) I f 'ff-xlB-x>
s t r o n g l y obeys G8del's second theorem, then by
Theorem 4 t h e r e i s a GMdel sentence f o r 'a-xlB-x>I
is s t r o n g l y LMbian f o r - d , B > . X+ i s provable i n
a,
so is X V
and hence by Lemma 4 X
( b ) The converse i s r e l a t i v e l y t r i v i a l :
+ X .
If
Then by ( b ) of Theorem 3 ,
s t r o n g l y obeys GBdel's second theorem. The above theorem i s t h e "strong" v e r s i o n of Theorem 1, and h a s t h e following c o r o l l a r i e s .
Corollary 1 d
-
I f d-xlB-x>
-
I f t h e r e i s a GMdel sentence f o r
s t r o n g l y obeys GMdel's second theorem, so does
r BMX' *
Corollary 2
then t h e r e is a
GMdel sentence f o r q7,BWx>.
Corollary 3
-
If
x
V
x+
is provable i n ff, so i s.'X
Corollary 1 follows, because i f d w x I B cond theorem, then X'
WX
> s t r o n g l y obeys GMdel's se-
i s provable i n ff, which by (a) of Theorem 3 implies Corollary 2 follows
t h a t d , B m X > s t r o n g l y obeys GMdel's second theorem. from Corollary 1 by Theorem 4 . provable i n
a.
second theorem. theorem.
As t o Corollary 3 , suppose (X V X+) i s
Then by (b) of Theorem 3 , -d ,B
-x w >
Then by Corollary 1, g,B,,>
Then by (a) of Theorem 3 , X
+
s t r o n g l y obeys G6del's
s t r o n g l y obeys GMdel's second
i s provable i n
a.
Corollary 3 could have been a l t e r n a t e l y proved from t h e following f a c t about K4:
Some Principles Related to Wb's Theorem T h e o r e m 10 Proof
-
The sentence O ( q V q
- The formula q 3 (04
3
+)
3 q+
227
is provable in K4.
q ) is a tautology, hence O ( q 2 (Oq
3
4 ) ) is
provable in K4, and hence so is Oq I> C(oq 3 q ) . From this it easily folU q , and hence to uq
lows that q+ is provably equivalent to O(0q I> q )
n(cq 3 q ) . From this it follows that ( q V 4') is provably equivalent to q V (nq 5 c(0q 3 q ) ) , and hence that the following is provable in K4: (1) c(q v 4+) 2 0 ( q v (0q
5 O(C4 3
Now, by Theorem 8, C(p 4 ) ) ) 3 q+ is provable in K4.
ing is provable in K4:
3 q)
9))).
is a K-formula--i.e.,
O(q V
(p
=
C(p 2
Then, substituting "tq" for " p " , the follow-
(2) C ( q V (Cq 3 U ( 0 q 3 ql
) ) 3 4.'
Then by (1), ( 2 ) and syllogism, the formula
C(q
v
4')
+ IS .
prov-
3 q
able in K4. Remarks
- From Theorem 10 and Lemma 2 it follows that for any
sen-
. provable in a. Then if x of a, the sentence B(x V x+) 3 X + is (x V x+) is provable in a, so is B(X V X+), and hence by modus ponens, s o is x+. This gives an alternative proof of Corollary 3 of Theorem 9. And
tence
I might mention that this corollary with Theorem 3 immediately gives Theorem 9 , so we could have given an alternative proof of Theorem 9 along these lines. More LL5bian P r i n c i p l e s for K4.
4.
The following two theorems
will be useful in obtaining more L-formulas and G-formulas. T h e o r e m 11
-
(a) If A(p,q) is an L-formula,
so
is 4(-p,q).
(b) If @(p)
is a G-formula, so is -@ ( - p ) .
-
Proof
(a) suppose A(p,q) is an L-formula--i.e.,
provable in K4.
means thatNA(-p,q)
ciple.
2
Substituting -p for p , the formula O(-p
is provable in K4, hence C(p
Remarks
O(p
E
,4(-prq)) 3 q
is an L-formula.
+ 1. s
+ .
A(p,q)) 3 q E A(-p,q))
-
3q
+
provable in K4, which
(b) Proof is similar to (a).
- We are tempted to call the above theorem a
"duality" prin-
Part (a) holds for K-formulas as well as L-formulas.
T h e o r e m 12
1s
If A(p,q) is an L-formula, then A ( p , i ) is a G-formula.
228
R.M. SMULLYAN
+
Proof
- Suppose U ( p = A ( p , q ) )
U(p
A ( p , l ) ) 3 I+ is provable in K4, which means that A ( p , i ) is a G-
I
=I q
is provable in K4.
Taking I for q ,
formula. Remarks
-
Theorem 1 2 is the basis of Kreisel's observation that
Gadel's second theorem can be looked at as a special case of Lob's theorem. In particular, since up G-formula.
Also, C ( p
is a G-formula.
is an L-formula. C p
3 q
3 q)
I--which is + o p - - i s
a
(Alternatively, since -qp is a G-formula, then by Theorem
is a G-formula, hence so is
11,
3
is an L-formula, hence C ( p I3 i)--which ism+--
4p.)
Our next theorem is somewhat in the character of a lemma. Theorem 1 3
-
(C(cq 3 q ) 3
Proof
-
-
in K4:
Then C ( p
5
Substituting Oq for q , the sentence C ( p But (04)'
A ( p , C q ) ) 3 9'
3 q+
+ .
A(p,q)) 2 q
is prov-
A ( p , . @ ) ) 3 (04)'
is provable in K4 by Theorem 13.
is
Therefore
is provable in K4, which means that A ( p , O q ) is an L-
formula. Since U p 3 q and -(D-p Theorem 15
3
If A ( p , q ) is an L-formula, So is A ( p 5 q ) .
Suppose A ( p , q ) is an L-formula.
able in K4.
9
(C(cLq 2 C q ) 3cnq)
nq) is provable in K4.
provable in K4. O(p
is provable in K4--i.e.,
- The following formulas are successively provable
Theorem 14 Proof
( C q ) + 3 q+
3
q ) are L-formulas, Theorem 14 gives
- U p 3 oq and -[nr, 3 w) are L-formulas.
By Theorem 15 and Theorem 12,
Some PrinciplesRelated to ulb's Theorem Theorem 16
- O p 3 OL and -(o.lp
3
229
CL)are G-formulas.
Then by Theorem 16 and Theorem 7 , we have: Theorem 17
- O(p V
q ) 3 Dq
and - ( O ( p
3
q ) 3 t l q ) are L-formulas--in
f a c t K-
formulas. This gives a l l the L-formulas needed for the proof of Theorem A stated i n the introduction.
R.M. SMULLYAN
230
REFERENCES
[l]
The Unprovability of Consistency, Cambridge University
Boolos, G.
Press, 1979. [2]
Jeroslow, R.G.,
"Redundancies in the Hilbert-Bernays Derivability
Conditions for Gddel's Second Incompleteness Theorem," Journal of Symbolic Logic 38 (1973), 359-67. [3]
Ldb, M.H.,
"Solution of a Problem of Leon Henkin," Journal of Sym-
bolic L o g i c 141
Smullyan, R.
20 (1955), pp. 115-118. Theory of Formal Systems, Annals of Mathematics
Studies #47, Princeton University Press, 1959. [5]
Smullyan, R.
"Modality and Self-Reference," this volume.
