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Introduction-Intensional Mathematics and Constructive Mathematics
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 world--its
e x i s t e n c e independent of p e r c e p t i o n .
1
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:
5
Intensional Mathematics and Constructive Mathematics
...
...
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
"
is o b t a i n e d hy addinq
32
fact
'I.
For example, on the 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
S. SHAPIRO
6
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. 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
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
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"
.
7
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
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
-
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))
I f it d i d , 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
K
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) &
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 differences.
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
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
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
K
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".
-
i s simply taken as
BR(&)
"
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 ,
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 world--its
e x i s t e n c e independent of p e r c e p t i o n .
1
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:
5
Intensional Mathematics and Constructive Mathematics
...
...
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
"
is o b t a i n e d hy addinq
32
fact
'I.
For example, on the 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
S. SHAPIRO
6
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. 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
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
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"
.
7
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
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
-
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))
I f it d i d , 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
K
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) &
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 differences.
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
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
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
K
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".
-
i s simply taken as
BR(&)
"
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 ,