Introduction-Intensional Mathematics and Constructive Mathematics

Introduction-Intensional Mathematics and Constructive Mathematics

Intensional Mathematics S. Shapfro (Editor] @ Elsevier Science Publishers B. V. (North-Holland), 1985 1 INTRODUCTION--INTENSIONAL MATHEMATICS AND CO...

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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 ,