Logic Colloquium'76: Proceedings of a conference held in Oxford in July 1976 (Studies in Logic and the Foundations of Mathematics, 87)

If cp(x, y) is prenex and z does not occur in cp(x,

F (Px > r)
(pz

F (Px?: r) cp(x, y)

(pz?:

>

y) ,

y) , r) cp(z, y) . r) cp(z,

If x does not occur in cp , then

F F cpv (Px?:

r)lIJ

(Px > r)(
r) lIJ

(Px?: r)(cpvllJ)

F CPA(Px >

r)lIJ -

(Px> r)(CPAlIJ)

r

< 1 .

then

HYPERFINITE MODEL THEORY

F 'P1\(Px

== r)tjJ -

(Px

23

== r)('Pl\tjJ) ,

>

r

0

.

1= l (Px > 1) 'I' ,

(6)

1= (Px == 0) cP •

The completeness problem for Lw p is the problem of characterizing the valid sentences by means of valid axioms and rules of inference.

The

reader can easily discover schemes of valid formulas in addition to those listed in the above proof.

Here are some examples which depend on

properties of the probability measure. If

F (Px

r > s ,

> r) cP -

(Px > s) 'I' .

F [(Px > r)(cpl\tjJ)I\(Px > s)(cpl\ltjJ)] -

F (Px>

r + s) 'P -

If rs

(Px> r+ s) cP ,

[(Px> r)(cpl\tjJ)v(Px > s)('P1\ ltjJ)

+ r' a' == 1, then

F l~PX > r)(Py > s) cP

1\

(Px > r')(Py > S')l 'P]

We shall now tum to the infinitary logic L P'

wI

L

wl

has all the

formation rules of L plus an additional infinitary rule: wp If

(2.11)

are formulas, then

x is an n-tuple of variables and

1\ cP (x) and

m m

'l'm(X), m

E: W ,

V'P (X) are formulas. m m

Notice that the probability quantifiers may be applied to formulas of L P as well as L P wI w

In general the formulas of L P belong to V("'R)

wI

but if the signature (I, ()')

r

belongs to V(R) , then the formulas of

belong to V(R) • Each formula of L has at most finitely many free variables. wlP

Since

countable intersections and unions of measurable sets are measurable, there is no difficulty in defining the satisfaction relation

for formulas of L

wl

We state the analogue of Proposition 2.6.

24

H.KEISLER 2.12.

PROPOSITION.

For each formula ;a(x) of L P and

wI

hyperfinite model m, the relation {a: m1=
on A is a Borel set.

We next prove a normal form theorem which is a uniform version of the monotone class theorem from measure theory.

Like the Prenex Normal Form

Theorem, it will be useful later on. By a monotone conjunction we shall mean an infinite conjunction

1\

wl

such that for each m ,

Bya monotone disjunction we mean a disjunction

Vt\Jm(X) such that

m

for each m ,

Thus monotone conjunctions correspond to intersections of decreasing chains, and monotone disjunctions to unions of increasing chains. The set of monotone formulas of L is the least set such that: "lIP (2.13) Every prenex formula of L

wp

is monotone.

(2.14)

A monotone conjunction of monotone formulas is monotone.

(2.15)

A monotone disjunction of monotone formulas is monotone.

The monotone complexity of a monotone formula is defined as follows. Prenex formulas have monotone complexity zero. of

I\
The monotone complexity

(x) is the least ordinal greater than the complexities of
m m

m < co .

The monotone complexity of V 2.16.

of L

wl


(x)

m m

Proof:


(x),

is defined similarly.

MONOTONE FORMUlA THEOREM.

is equivalent to a monotone formula

m

Every formula


(X)

Let M be the set of all formulas of L P which are equivaWI

lent to monotone formulas with the same free variables. formula is prenex and hence belongs to M.

Every atomic

Any infinite conjunction

HYPERFINITE MODEL THEORY

25

is equivalent to the monotone conjunction

/\ 'I'm

m

/\l\J m, m

Therefore to prove the theorem it suffices to prove that M is closed under negations, finite conjunctions, monotone conjunctions, and the quantifiers (Px> r] . It is easily seen by induction on monotone complexity that if 'I' is

monotone then (lq» Suppose 'I' m(X) conjunction.

If follows that if 'I'

E:

M.

E:

M for each m < wand

E:

M then (lq»

E:

M .

/\ 'I' (X) is a monotone

m m

For each m , choose a monotone formula l\J (X) equivalent m

'I'm(X) .

Then ~l\Jm(X) is a monotone formula equivalent to ~q>m(X), so

/\'1' (X)

E:

m m

M .

To show that M is closed under the quantifiers [Px > r) we first observe that for a monotone disjunction V cp (X) or conjunction /\ 'I' (X), m m m m (1)

1= (px> r)Vq>

(2)

1= (Px> r)/\q>

V (Px

m m m m

m

-

V

O
>

r) 'I'

m

,

1 /\m(Px> r+ r)({Jm

Since 1= 'I' -- l\J

implies

the disjunction V (Px > m

1= (px> r)q> --

(Px> r)ljJ ,

on the right of (1) is monotone.

conjunction /\(Px > r + m

on the right of (2) is monotone.

Also, each Moreover,

since r > s

implies

1=

(Px > r) ({J

--

(Px > s) cp

r

the disjunction

on the right side of (2) is monotone.

Obviously, if 'I' is a prenex formula

then (Px > r) 'I' is prenex and hence monotone.

Using (1) and (2), we see

H.KEISLER

26

by induction on monotone complexity that if ep is monotone then (px > r) ep EM.

Therefore if ep E M then (Px > r) ep EM.

To prove that M is closed under finite conjunction, we borrow a trick from classical measure theory (see Halmos [12]).

Fix the n-tuple x .

For each ep(X) in L p let wI

Then K(ep) is closed under equivalence. II'

E K(ljJ).

tions,

By symmetry, if tjJ E K(cp) then

Since M is closed under monotone conjunctions and disjunc-

K(ep) is closed under monotone conjunctions and disjunctions.

By

the Prenex Normal Form Theorem every formula of Lwp belongs to M. Suppose epo(X) is in Lwp '

Then ljJ E K(epO) for all tjJ(X) in Lwp '

It

follows that tjJ(X) E K(cpO) for all monotone ljJ(X) , and hence tjJ(X) E K(epO) for all tjJ(X) EM.

Now suppose ep(X) EM.

For any tjJ(X) in Lwp ' we

have ep E K(ljJ) and hence ljJ E K(ep). As before it follows that whenever ljJ(X) EM, tjJ E K(ep).

Therefore if ep(X) , ljJ(X) EM, then ep(X) 1\ ljJ(X) EM.

This completes our proof. By leaving the quantifiers out of the above proof we obtain an analogous result. 2.17.

MONOTONE FORMUIA THEOREM WITHOUT QUANTIFIERS.

Every formula ep(X) of L P without quantifiers is equivalent to a monowI tone formula ep' (X) of L p without· quantifiers. wI We remark that every formula of Lwp without quantifiers is prenex and hence monotone. We shall now introduce an infinitary logic which is richer than L p' WI the Suslin logic L)O(p'

The analogous Suslin logic

~w

' with universal

and existential quantifiers instead of probability quantifiers, was studied by Ellentuck [8]. In addition tothe symbols of L P' WI

L"p has a new infinitary connec-

--xx.

HYPERFINITE MODEL THEORY tive XX, called the Suslin connective. sequences of elements of w.

Let

w$

27

be the set of all finite

The logic Lxxp has all the formation rules

(2.1), (2.2), (2.3), (2.11) of L P plus the following new rule: wI (2.18)

x is an

If

n-tuple of variables and 'Ps(X) , s

€

w,S;;-, are

formulas, then XX '" (X) is a formula. s

wl '

Thus LXXpiS an extension of L

Notice that probability quantifiers

'

and connectives can be applied to formulas of L XXp as well as L

wl

and each formula of L XXp has at most finitely many free variables. The satisfaction relation

mF 'P[a] for L XXp is defined by induction

on the complexity of 'P(X) as usual.

The clause of the definition giving

the interpretation of XX is: (2.19)

mF XX"'s[a]

if and only if there exists f

Intuitively the finite sequences s

€

€ WW

such that

w,S;;- should be thought of as nodes

~

in a countable tree, and XX 'Ps(X) means that for some branch f

€

w w

of

the tree, the conjunction ~ 'l'f tm(X) of the nodes in the branch holds. prove that the satisfaction definition

mF 'P[a]

'P(X) of L XXP , we must show that the relation

To

is meaningful for formulas

{a:

m F 'P[a]}

is always

Loeb measurable so that the probability quantifiers can be interpreted. We need a classical lemma (for a proof see Kuratowski and Mostowski [15J, pp. 430-431). 2.20. space.

If X

s

LEMMA.

Let (B, 1') be a complete probability measure

is measurable for each s

€

JS- ,

then

is

measurable. 2. 21. PROPOSITION.

For every formula ",(x) of L XXp and

H.KEISLER

28

and every hyperfinite model ~, the relation {a: \U 1=
defined by


By induction on the complexity of
Suslin connective

)Q(

uses Lemma 2. 20.

that the relation {(a, b) : \U

1=

<jJ[a, bj}

The step for the

For the quantifier step we assume

is Loeb measurable.

It follows

from the Corollary 1. 13 that the relations

and

are Loeb measurable. and (Py

2;

These are the relations defined by (Py > r)<jJ(x, y)

r) <jJ(X, y) respectively.

The other steps of the induction are

easy. There are several reasonable variants of probability logic which we have not investigated.

One possibility, which was suggested by P.

Bankston, is to use the more general quantifiers (Px Borel subset of [0. lJ.

€

B) where B is a

Another possibility is to introduce probability

quantifiers on ri-tuples of variables, so that

(Pxy > r)
that the relation

{ (x, y)


iH

has measure greater than r , The results in this section hold for a much wider class of models, namely for all models with a family of IT-additive probability measures v n satisfying the Fubini Theorem.

We postpone our discussion of such models

until Section G, where we show that any sentence of L P which has a

WI

model of the general type has a hyperfinite model.

In the next two sections

we obtain results which apply only to hyperfinite models.

HYPERFINITE MODEL THEORY III.

29

THE MANY-VALUED LOGIC

In this section we shall introduce a second logic,

5L,

for the study

of hyperfinite models.

SL

is a many-valued logic with truth values in the hyperreal line '~R .

For this reason it is convenient to identify a hyperfinite model

with the two-valued model

where F : AtT(j) _ j

{O, l} is the characteristic function of Sj'

Our

results will apply to a broader class of many-valued hyperfinite models. A many-valued hyperfinite model is a structure

where (A, fJ.)

is a hyperfinite probability space, each a

€ A, and each i F. is an internal function mapping AtT{j) into *[-1,1] . The F. are

J

J

called the functions, or random variables of lU. The signature (I, c ) is defined as before. finite model.

In this section, model will mean many-valued hyper-

These models are of interest in their own right since they can

often be used to represent physical processes. ple of Brownian motion in Section 5.

The logic

We shall discuss the exam-

5L

has the following

symbols: individual variables equality symbol an n-ary connective C for each continuous real function C:Rn_R,

n2::0

three quantifiers

(max xj , (min xj ,

constants

€

I

5... dx

30

H.KEISLER

function symbols We use the notation is the quantifier Because formula.

SL

SL

for this logic because its most interesting feature

S... dx . is many-valued, we shall use the word term instead of

By an atomic term we mean an expression of one of the forms

where "i: ... , ulT(j) are constants or individual variables. ~ of

SL

The set of

is the smallest set which satisfies the following formation

rules. (3.1)

Every atomic term is a term.

(3.2)

If C is an n-ary connective and T

C (Tl, ... ,Tn) is a term.

... , Tn are terms, then l, (In the case n = 0, C is a real

number and C is a term. ) If T is a term and x is a variable,

(3.3)

(max x)T,

(min x)T,

STdx are terms.

Free and bound variables are defined as usual, with the quanttfiers binding the variables.

x.

T(X) denotes a term with at most the free variables

A term without free variables is called a constant term.

Notice that

a variable or constant alone is not a term, since terms are built up from atomic terms as defined above. The truth value Tlll[a]

of a term T(x) in a model III at an assignment

a is defined by induction on the complexity of terms. (3.4)

(3.5)

eq(x,y)Q![a,b] = {lo if a=b if a f b - ~Q!~ Fj (X) [a] = Fj(a)

Atomic terms with constants are interpreted in the natural way.

HYPERFINITE MODEL THEORY (3.7)

(3.8)

31

(max y)T(X,y)'!l[a]

maX{T'!l[a,b]: b

(min Y)T(X, y)'!l [a]

min {T'!l[a,b] : b lOA}

(jT(x'Y)dy)'!l[a] =

10 A}

L: T'!l[a,b] f1 (b) bEA

It can be shown that the truth value of T(x) depends only on

the assignment a of X.

'!l and

ST(X, y)dy gives the expected

The quantifier

value, or average value of T(x, y) as y varies. We see by induction on the complexity of T(X) that for each model '!l, the function f: An _ each term T(i) of say even more.

SL

= T'!l[ a]

*R defined by f(a)

Suppose the signature of L is internal.

Sl, and for each T the function f: SlX

>I'

n

N

-

We can

Then the set of

*N is an internal set,

~

'~R defined by

'!l

~

~

f('!l, a) = T [a]

Recall that, on the other hand, the interpretation of a formula


'!l is in general Borel but not internal.

3.9.

LEMMA.

For each term T(X) of

real number IITII such that for every model '!l and a

Proof:

Thus

is interpreted in '!l by an internal function.

all hyperfinite models '!l with universe sets A

is internal.

is internal.

S1 10

there is a least

An ,

By induction on the complexity of T .

Linear combinations of terms are terms because linear functions are connectives.

The function IITII

is a serninorm on the terms of

SL,

that

is, liT + UII ::s IITII + IIUII,

lIaTIi

A set D of terms is said to be dense in

SL

and every real

E

=

S1

> 0 there is a term TO (X)

lailiTH if for every term T(x) of

10

D such that

liT - Toll < e .

Two terms T(i) and U(i) are said to be equivalent if liT - UII = 0 . This holds if and only if for all '!land a,

T'!l[a]"" U'!l[a]

.

32

H.KEISLER n For each n and each real closed interval [r, s] , the set C (R ) of

continuous functions C: Rn __ R is a linear space with the seminorm

The classical Stone-Weierstrass theorem shows that there is countable set Co of continuous real functions which is dense in the following sense: For each n , [r, s], C Co

Co

€

n C(Rn)

€

n) C(R , and real

such that IIC - Coil <

E

E

> 0 there is a

on [r, s t . In particular, the

Set of polynomials with rational coefficients is dense. continuous real functions, let SL(e

o)

Given a set Co of

be the set of all terms T(x) such

that every connective in T(x) belongs to CO' 3.10.

STONE-WEIERSTRASS THEOREM FOR TERMS.

If Co is

a dense set of continuous real functions, then SL( CO) is dense in SL. Proof:

We show by induction on the complexity of terms T(x)

that for all real e > 0 there exists To(X)

SL(C

€

Every atomic term already belongs to SL(C ) ' O

o)

such that liT - Toll <

The quantifier steps in the

induction are easy, because

and

Assume the result holds for terms Tl(X), ... , Tm(X) , and let

Let

E

> 0 be real. Let

Then we always have

of SL E.

HYPERFlNlTE MODEL THEORY !If..~ ~ (TlLa], ... ,Tm[a])€

*[-r,r] m

There is a real Ii > 0 such that whenever max{ls£ - t£

I:

£ < m choose Co € Co

n

£ -s m} < Ii, we have T£O € SL(C O)

m) C(R

such that

33

~

~

s,t € [-r,r]

IC(s) - C(t) I < IIT£ - T.eo ll <

such that for all s € [_r,r]m,

m

and

e/2.

s .

For each

Take

IC(S) - Co(s)1 <

e/2,

and let

Then for all

~

and a,

< e/2 + e/2

=

e .

Therefore liT - TO II < e . This complete s the proof. Our use of the Stone-Weierstrass Theorem will depend on the fact that if L has a countable, signature then SL(C

o)

is a countable set of terms.

We shall frequently use the following consequence of the Stone-Weierstrass Theorem. Two models

~, ~

are said to be infinitely close in symbols

""

~

~

,

W

if T "" T~ for every constant term T . 3.11. tions.

If

LEMMA.

Let Co be a dense set of continuous func-

T~ "" T'.J> for all constant terms T € SL(C o)' then

Proof:

Consider an arbitrary constant term U.

W ""

~.

For each real

e > 0 , choose a term T € SL(C o) such that liT - UII < e. Then

34

H.KEISLER

Therefore U!ll "" U~ . The hyperreal universe (*[V(R)], E) is said to be IC-saturated, where IC

> w,

if for every set X of fewer than IC internal sets such that every

finite subset of X has a nonempty intersection,

nX f e ,

By the Internal

Definition Principle, this is equivalent to the property that for each n , every set of fewer than IC bounded formulas rp{x) with constants from "'(V(R)] which is finitely satisfiable in *[Vn(R» is satisfiable in ~'[Vn(R)] .

f is a near isomorphism from !lI to

~,

in symbols f:!lI:::'

is a bijection of A onto B and for every term T(X) of SL and

Equivalently,

('U, ala E A "" (~. fa)a EA'

~

, if f

a E An

r

Obviously, if !lI and 'J3 are

nearly isomorphic then they are infinitely close. 3. 12.

cardinal of *N.

ISOMORPHISM THEOREM.

Let IC be the (standard)

Suppose (*[ V(R)) ,E) is IC-saturated and the signature

of L is of cardinality less than IC. If !lI "" 'J3 then 'U and 'J3 are nearly isomorphic. Proof: Let

SLa

Assume 'U ""

~.

We use a back and forth construction.

be SL with a extra constants, and let A = {a

a

: a < IC}

r

B

= [ba :

a < x} .

We construct c E A and dEB such that a a A

= {ca :

a < Ie} ,

B

= {d0' :

a < z} ,

and for all constant terms T E SL/C(C ) ' T!lI "" T~. O T

(!lI,c/3)/3
0'

(We abbreviate

!lI by T .) Suppose c/3' ~ have been chosen for /3 < a.

is even, 1. e.,

0'

= X.

+ 2n where

Consider finitely many terms

X.

is a limit ordinal, put

CO'

If

= a1l. + n .

HYPERFINITE MODEL THEORY

35

where Co is a countable dense set of connectives. Let 0 < n < Choose functions C l, ... , C p

x

= T£~[C)

•

such that C£ (x) is near 1 at

Co

€

w

T£~[CJ . Precisely,

and near zero when x is away from

Then in the model (~, 93)13 < a ' the term

has value ('13.

~)f3

(1 - 1..)P. n

2:

< O!

.

Therefore this term has value

It follows that there is an element d

€

2:

(1 - 1..)p in n

B such that for

£=l, ... ,p, 1

-<: -n

There are fewer than

IC

terms in SLa(C ) ' O

of *[ VIR)] , there is an element d

O!

For odd

0'

= X. + 2n + 1

and then construct ca'

€

Therefore by x-saturatron

B such that for all terms

the argument is similar but we take dO' Then

= bx.+ n

T~ I'::! T'13 for all constant terms T

€

SLIC( Co)

By the Stone-Weierstrass Lemma 3.11 we have

Let f: A -

B be defined by f(c ) 0'

so dO' = df3 . Bya dual argument,

= d0'

,

0'

<

IC •

f is a function because

f is one to one.

f maps A onto B

•

H.KEISLER

36

because the ci s and d's exhaust A and B.

It follows that f is a near

isomorphism. The following analogue of the Isomorphism Theorem follows from the results in the Continuous Model Theory book [7].

We shall discuss the

relationship further in Section 6. 3.12.

ISOMORPHISM THEOREM WITHOUT INTEGRAL QUANTIFIERS.

Assume the hypothesis of 3.12.

Let M be the set of all terms T(X) of

S ... cIx

in which the quantifier

does not occur.

If

SL

T~ '" T'P for all constant

terms T € M, then there is a bijection f of A onto B such that for all T(X) € M and

a € An,

3.13.

T~[a] '" T'P[fa]

HYFERRATIONAL APPROXIMATION THEOREM.

model m there is a model 'P such that A and all the values of v Proof:

Let a

O

= B,

For every

(m, ala € A '" ('P, ala e A '

and G. are hyperrational numbers. J

be the element of A with the smallest non- zero

measure fJ.(a ) • Let H be an infinite hyperinteger large enough so that O

H

-1

€

o(fJ.(ao»

For a € A let mea)

=

[HfL(a)]

.

Define v

by

veal = m(a)/K

Thus v(a)/fL(a) '" 1 whenever fJ.(a)

fo

0 , and veal

=0

whenever fJ.(a)

'P has the universe A, the measure v , and the same constants a

i

=0

as m.

The relations of 'P are G. where J

By a straightforward induction on the complexity of terms one can show that for all T(x) and a € An, T(X) =

SU(X, y)dy

Tm[a] '" T'P[a] . The key step from U(X', y) to

is as follows.

T~ [a] = ~ u~ [a, b] fJ.(b) '" ~ U'P[a, b] fJ.(b) b€A

b€A

.

HYPERFINITE MODEL THEORY "" LU'J>[a,b]v(b)

=

37

T'J>[a)

bEA Our next theorem is an analogue of the Ehrenfeucht- Mostowski theorem for classical logic.

The classical theorem may be stated as follows (see

[ 6]). EHRENFEUCHr-MOSTOWSKI THEOREM.

Every infinite w l saturated model 21 for a countable first order logic L has an infinite ww set (X, <) of indiscernibles. That is, X ~ A and for any increasing n-tuples xl<· .. <xn' Yl<"'
Our analogous result will use Ramsey's theorem for finite sets. 3.14.

FINITE RAMSEY THEOREM.

For all positive integers

b , c , d there exists a positive integer a such that d

a ...... (b)c . That is, if A has power a, then for every partition of the set [A]d of unordered d-tuples from A into c parts, there is a subset B

~

A of

power b such that [B]d is included in one partition class. Note: If d a ...... (b)c ,the relation remains true if a is increased and b. c , dare decreased. Consider an internal model 21. mean an internal set X

~

By a set of indiscernibles in m we

A and an internal linear ordering < of X such

that for all n < wand any two increasing n-tuples a, (21 ,a) "" (m

,b). m is said to be infinite

3.15. signature.

from (X, <) ,

if A is infinite (but hyperfinite).

INDISCERNIBILlTY THEOREM.

Then every infinite model

b

Let SL have a countable

mcontains an infinite set of indis-

cemibles. proof:

Let Co be a countable dense set of connectives, and let <

38

H.KEISLER

be an internal linear ordering of A.

of terms in SL(C numbers. not T,e ~

o)

Let p(

raj '"

and a finite sequence q

T, q)

q,e , where

the Finite Ramsey Theorem,

IAI ....

d

(b)c

=

(ql"'" qm)

of rational

be the partition of [A]n determined by whether or

partition has Zm classes.

Therefore

Consider a finite sequence

a is increasing with respect to Let JAI

<. This

be the internal cardinality of A. d

By

for all finite b , c , and d.

[AI .... (b)c

for some infinite hyperintegers b , c , d.

Therefore

for each p( T, q) there is an internal set X ~ A of cardinal b such that [X]n is included in one partition class of p( T, many pairs (T, q).

q).

There are only countably

Thus by w 1- saturation there is a single internal set

X ~ A of cardinal b such that for all (T, q), [X]n is included in one class of peT,

(Tz , qz)

q) .

(Here we need the fact that two pairs (T

can be combined.)

l,

It follows that for any term T(X)

and any two increasing n-tuples

a, b

qll and €

SL(C

o)

from (X, <), T IU [a] ~ T~ [b]

Finally, by the Stone-Weierstrass Lemma 3.11,

for all finite increasing

a, b

from (X,

-c ) .

Given positive integers b, c , dIet r(b, c , d) be the least positive d

integer a such that a .... (b)c 3.16.

. The letter r stands for Ramsey.

INDISCERNIBILITY THEOREM WITH EXTRA CONSTANTS.

Let SL have a countable signature.

Let ~ be an infinite model and S an

internal subset of A such that for all positive b , m , d m r(b,Z ISl ,d):s Then the expanded model (~, ala Proof:

€

€

W

,

IAI .

S has an infinite set of indiscemibles.

There exist infinite b , m , d

€

~'N such that

HYPERFINITE MODEL THEORY m r(b,zISI ,d) :s JA[

39

.

Consider k-tuples of terms

of fL(Co) and rational numbers q = (ql"'" qk)'

Let q have length P.

This time peT ,q) is the partition of [A]n such that

"ii, b are in the same

class if and only if for all -;

Zk.JSJP

There are at most

E

sP and i:s k ,

< zlS[m partition classes. The proof of 3.15

can now be carried through to get an infinite set of indiscernibles for

Here is another indiscemibility result. 3. 17.

m'"

(A, Il.' Fj ) j

internal set.

E

Let

INDISCERNIBILITY THEOREM FOR FUNCTIONS.

J

be a model whose functions (F j ) j

E

Let

J form an infinite

< be an internal linear ordering of J. Then there is an

infinite internal subset K C J such that for any two increasing finite sequences

from (K, <) ,

Proof:

We may assume without loss of generality that all the F. 's J

have the same number m of argument places. dense.

Let Co be countable

By a constant term scheme T (8) we mean an expression which

becomes a constant term when symbols.

G is

replaced by a sequence of function

There are only countably many constant term schemes from

40

H.KEISLER

SL(C ) . O

Consider pairs (T(G),

q)

where T(G) is a k-tuple of constant

term schemes with G of length n , and p( T(G).

q)

q

is a k -tuple of rationals.

Let

be the partition of [JJn determined by whether or not the

value of T,E (Fi ' ...• Fi ) in I n

~ is ~

q,E' ,E

= 1,

... , k.

k There are 2

partition classes. As in the preceding proof. there is an infinite internal

q) ,

set K ~ J such that for all (f(G). partition class for peT (G),

q).

[K]n is included in a single

Using Lemma 3.11 again, we see that K

has the required property.

-j

We conclude this section with two examples of preservation theorems in SL.

The first example is quite elementary, while the second example

is more difficult and uses an analogue of the compactness theorem. By a homomorphism of

~

onto '.Il, in symbols f:

~

-- '.Il , we mean

an internal function of f of A onto B such that: (3.18)

For al I b

(3.19)

For all i

(3.20)

For all j

The image '.Il

B. v (b )

=

€

I • f(a

= bi

€

J and b

€

= f( ~)

i)

E

fl.(f

-1

(b)

.

Bcr(j) ,

may be thought of as the average of

~

under f .

A constant term T is said to be preserved under homomorphic images if whenever f I

~

--

'.Il , T~ "" T'.Il.

morphic images if whenever f:

~ --

(Recall that '.Il is atomless if each b

T is preserved under atomles s homo'.Il and '.Il is atomless, E

T~

"" T'.Il

B has infinitesimal measure

v(b) "" 0 .)

Given a function symbol F with m places. we define constant term

Sf

to be the

HYPERF1N1TE MODEL THEORY

Let T be a constant term.

3. 21. THEOREM. (i)

41

T is preserved under homomorphic images if and only if there is a

term U of the form

such that Tlll

!':l

U~ for all ~ .

T is preserved under atomless homomorphic images if and only if

(ii)

there is a term U of the above form such that T

mRJ

U

lll

for all atomless

m.

Every term of the form U is preserved under homomorphic

Proof:

images because each

SFj

is.

We prove (ii). Assume T is preserved under atomless homomorphic images.

Let

FI , ..• , Fn

be the function symbols occurring in T.

To

simplify notation we assume that these are all the function symbols of L. Consider two atomless models

~

, 'Jl such that

Let 1ll x 'Jl be the model

m X 'Jl

= (AXB,fJ.Xv,(ai,bi),H.). I '< JIE,J-n

where

Let "ITA: A X a -- A, "ITa: A x B -- B be the projection mappings. "ITA:

mx 'Jl --

each a

1ll

and

"IT

mx 'Jl -- 'Jl

B: such that f.l(a) i 0 ,

are homomorphisms, because for

~ Hj(a, b)(f.l x vXa, b)/IJo(a) b

Then

H.KEISLER

42

[Fj(a) -

(j"fj )~] ~ v(b) + ~Gj(b)v(b) b

b

and similarly whenever v (b) of- o ,

~Hj(a,b)(flXV)(a,b)/V(b) "" Gj(b) a Therefore

T~ "" T~ x '+l "" T'jl . Hence we may define a real function C(st( where

~

models set, and

c:

[0. l]n _

J-Fl) ~ , ... , st( J-Fn) ~ ) = stT ~ .

ranges over all atomless models. ~

such that for all a ~

R by

E

A , flea)

is atomless if and only if

Let M(a) be the set of all

< a . Then

~ E

M(a) is an internal

M(a) for some

a "" 0 . For

each real e > 0 and infinitesimal a > 0 , we have:

Therefore for each real

E

> 0 there is a real

It follows that C is continuous.

for all atomless

a> 0

such that (1) holds.

Thus

~.

The proof of (i) is similar but without the restriction to atomless models. There is also a simpler proof of (i) using the one-element homomorphic image of

~

instead of

~ X

'+l •

43

HYPERFINITE MODEL THEORY 3.22.

PROPOSITION.

The sentence

JJeq( x, y)dxdy

is preserved under atomless homomorphic images but not arbitrary homomorphic images.

In fact,

~

is atomless if and only if

[JJeq(x, y)dXdY]!U """ 0

.

Our next result will characterize terms which can be approximated by terms without equality. We shall need the following analogue of the compactness theorem.

Bv a statement for

S1

we mean a relation of the

form T E X where T is a constant term and X E ':'P(R).

A set g of

statements is satisfied in ~ if T!U E X for each (T E X) E g . 3. 23. is

1C- saturated

COMPACTNESS THEOREM.

Suppose

51

and g is a set of statements for

(*[ VCR)] ,

E )

of power less than

IC

If every finite subset of g is satisfiable, then g is satisfiable. Proof: (

'~[V(R)],

There is a bounded formula W( x, y) in the language

E)

a statement for

which holds if and only if x is a hyperfinite model,

5L,

and y is satisfied in x.

y is

For all Yl"'" YnEg ,

By lC-saturation there is a model x such that

(*[ VCR)] , E) t= A {
is said to be a refinement of 'j3 if there is a homomorphism f:

which preserves function values, that is, for all

A special type of refinement is a duplication. finite probability space U

='j3 • tJ

~

= (D, rr)

at do is defined by

a in

~

-- 'j3

A and j E I .

Consider an atomless hyper-

, and let dO ED.

The duplication

H.KEISLER

44 A

Then

~

B

x D •

f!(b,

= b.

is a refinement of 'j:\ with feb, d)

by splitting each b

€

B into a copy of

= v(b) IT(d)

d)

~

Intuitively,

~

is formed

.

m is

A constant term T is preserved under refinements if whenever

T~ "" T'll .

a refinement of 'll. 3.24.

PROPOSITION.

If T does not contain the equality

symbol then T is preserved under refinements. Proof:

Show by induction on the complexity of T{x) that if

refinement of 'll with f: ~ -

3. 25.

EXAMPLE.

'll , then for all

a

€

~

is a

An •

The term

JJe{ x, y)dxdy is not preserved under refinements. We shall prove a converse of 3.24. 3.26.

LEMMA.

First we need a lemma.

For every constant term T there is a

constant term U without the equality symbol such that for every duplication

~ = 'll. ~, T~ "" U~. Proof:

The intuitive idea is that since

IT

is an atomless measure

on D. each a e A has a set of duplicates of measure large compared to f!{a).

Formally the construction of U is a complicated recursion on terms

T{X) . For each n-tuple of variables x, consider partitions p of x .

For

each partition class c of p, introduce a new variable vc ' forming a new sequence o~ variables partition p of

x we

x., = (v

C : C E

p).

shall define a term T

p

For each term T{X) and

(Xp ). First. given a partition

HYPERFINITE MODEL THEORY

45

q on y =: (x, y), q - y is the induced partition on x and Tq - y is the term obtained from Tq by replacing vc where y

E

c by vc- {y} • Here

are the details. [Fj (X)]p is obtained by replacing each

'1c

by vc where

'1c

E

c .

[eq(x,y)]p =: 1 if x p y , 0 otherwise. 1 m - 1 m [C(T , ... , T )]p =: C(T , ... , T )

P

[(maxy)T(X,y)]

=:

p

P

max({(maxv)(T -y):p=:q-y}) ¢

q

[(min Y)T(X, y)]p is similar.

The idea is that for integrals we ignore the points where y equals something in X, while for max and min we take each case into account. It can be shown by induction that

of

x determined by a.

T

p

T~ [a) ~ Tp~[a)

if p is the partition

does not contain the eq symbol, so for a

constant term T, U =: T¢ has the required property.

3. 27.

THEOREM.

The following are equivalent for a

constant term T. (i)

T is preserved under refinements.

(ii)

For every real that liT Proof:

UII <

E

> 0 there is a constant term U without eq such E

•

(ii) ~(i) by Proposition 3. 24.

Assume (i).

Let

SLo

be the

sublanguage of SL with the constants and relation symbols occurring in T . Let Co be the set of polynomials with rational coefficients, and let u be the set of all constant terms U in U.

E

SLo(C

u is not empty because 0, 1

For each U

€

E

such that eq does not occur o) 1.1. Moreover, u is countable.

1.1 and rational number r , we have r - U

We first prove:

€

'tl and U - r

€

1.1 •

46 (1)

H.KEISLER For all rational r < sand

8

> 0 there is a term U

r,s,

rational Ii (r, s , 8) > 0 such that for all

oS

E

II and a

~,

To prove (l), let

By compactness, for each U (\I~)T

~

E

II there is a rational r.s

E

[r, s]

~U

~

2:

o(U) > 0 such that

o(U) .

Let '13 be such that (\lU E lr,s l ) st U'13 > 0 ,

and let ,'} = {U Ell: st U'13 :s o} .

Let g be the countable set of statements g= {U:S.!..:O
The set

s

U{T

E

"r r, s]}

is finitely satisfiable, because if

is not satisfiable and C( r

l,

... , r

m)

E

Co approximates

within 1/Zn, then we have the contradiction

By compactness the countable set

max] r

l,

... , r

m)

HYPERFINITE MODEL THEORY g U{T

is satisfied in some model For each U

€

47

':'[r, s ] }

~.

U, we have

€

st U'tl "'" 0 => st U~ -s 0 , and it follows that for all U

SL

Lemma 3.11 (for

without eq ), every U without the eq symbol

satisfies U~ "'" u'tl. respectively.

U, U~ "'" U'tl . By the Stone-Weierstrass

€

Let ~, , tI' be duplications of ~, tI

Taking U as in Lemma 3.26, we have

so T m"", T'tl . Therefore st T'tl

[r , s]

€

Thus any tI which satisfies g

satisfies st Ttl satisfies g real

E:

r, s

€

{U

r, s

[r, s]

2::

5(U)/2: U

€

[r -

> 0 there is a term Ur, s ,

minimum of finitely many U

liT - UII <

E:

as follows.

E:

intervals [rk, rk + 1]

€

€

g

E:

> 0 , every 'tl which

E:, S

+ E:]

•

E: €

U

formed by approximating the

r

By compactness, for each

r , s ,and a real 5 > 0 such that (1) holds.

> 0 we may construct a term, U Take IITII < M

of length

U

Pick C (u) k

ur , s}

Hence for each real

satisfies T

For each rational

€

Co such that

E:

U with

wand partition [-M, M]

0 "'" k "'" K .

E: ,

rk , rk + i:

€

€

Let

, 5 k = 5( r:k , r k + l' E:) •

into

48

H.KEISLER

Finally, let

where C E Co and C(u O " ' " uk) is within jukl s M+ E • Then UE e 1..1

and

E

of max(u O " ' " Uk) for

liT - UE!I < 3 E

A constant term is preserved under refinements

:E atomless

for every atomless '.Jl and every refinement !II of '.Jl, T!II "" T'.Jl.

models if Here is

an atomless version of Theorem 3.27. 3. 28.

THEOREM.

The following are equivalent for a

constant term T. (i)

T is preserved under refinements of atomless models.

(ii)

For every real

E

> 0 there is a constant term U without eq such

that for all atomless '.jl, Proof:

IIT'Jl - u'Jl lI <

E

•

The argument is like the proof of 3. 27 but restricted to

atomless models.

The compactness theorem holds for atomless models

because '.jl is atomless if and only if '.jl satisfies the countable set of statements

{f f eqj x, y)dxdy < n : 0 < n< w} 3.29.

EXAMPLE.

•

The term

(max x)(max y)[ F (x, y) - eq( x, y)1 is not preserved under refinements of atomless models. A model !II is uniform if fLea)

= l/IAI

for all a EA.

Our last result

of this section shows that for every model !II there is a uniform model

~

HYPERFINITE MODEL THEORY such that T~

TIS: for every constant term T without the eq symbol.

""l

3. 30.

refinement '13 of ('J5.b)bEB

""l

49

PROPOSITION. ~

For every model

and a uniform model

(~,b)bEB

. If

~

~

there is a

~

such that B

=C

and

is two-valued. we may take 'J5 and IS:

to be two-valued. Let a

Proof:

measure f1(a ) ' O

O

be the element of

m with the

smallest non-zero

Let H be a hyperinteger large enough so that to each

a E A we may assign a hyperinteger 0 < mea) such that Lm(a)

aEA and

L {rnra) : f1(a)

= o} /m(a o)

""l

0 .

Let the universe B have internal cardinality H.

Form 'J5 by letting f

-1

split each a E A into rma) elements of measure floral/mea) , with any compatible interpretation for the constants, and let

~

has the same universe, constants, and relations as 'J5, but each b E B

has measure l/H in ~.

IV.

ELEMENTARY SUBMODELS

This section contains a. downward Lgwenheim-Skolem theorem for SL which is closely related to the weak law of large numbers.

It also contains

a theorem about elementary chains which is related to the strong law of large numbers. Bya model we shall again mean a many-valued hyperfinite model. shall concentrate on infinite uniform models (where flora)

= l//AI

""l

We

0 for

50

H.KEISLER

all a

E

A ) and atomless models (where fL(a)

Rl

0 for all a

E

Our notion of a submodel differs from the classical notion.

A ). The idea

is to keep the universe A and the functions and constants the same, but restrict the hyperfinite measure fL to an internal subset of

~.

We shall work with the many-valued logic 5L. We ~ throughout this section that L has

~

countable signature.

Consider a model !lI and an internal subset B S A of positive (but possibly infinitesimal) internal measure fL(B) > O.

The conditional

measure fLlB is the measure on A defined by (4.1)

(fLIB)(D)

=

jJ.(Bf1D)jjJ.(B)

for internal subsets DCA.

Thus for an element a

E

A,

if a
E

B

By the sub model \\lIB of !lI restricted to B we mean the hyperfinite model !lIIB

= (A'fLIB,ai,F.). J 1EI,

JE J

If \\l is a two-valued model, then so is !lIIB. We say that

~

IB is an elementary submodel of \\lIB

< \\l

~

, in symbols

,

if for every term T(x) of 5L and every n-tuple

a

E

An ,

In other words,

Given B SA, we use the notation '15

= ~ lB.

Notice that the parameters

"a" range over the universe A rather than the subset B. We shall prove a series of Lgwenheim-Skolem theorems of the following

HYPERFINITE MODEL THEORY type.

For certain internal sets rl

for a large proportion of the B

of subsets of A , we shall show that

rl ,

€

51

mIB

is an elementary submode1 of

m.

To state our results precisely we introduce the notions "almost all" and "nearly all". Consider a hyperfinite probability space (rl, v). Loeb measure on rl S

~

rl

generated by v. An event in (rl, v ) is a set

which is measurable with respect to

a statement s(w) with the event {w

sure, or holds for nearly all w

€

o

v.

Sometimes we identify

An event S is

rl : s(w)} .

€

almost sure, or holds for almost all w

such that v(So)

Let °v be the

€

rl • if °V(S) = 1.

S is nearly

rl , if there is an internal set So ~ S

1. Thus "nearly sure" implies "almost sure".

"oJ

Our

Lgwenheim-Sko1em results will be strong "nearly sure" assertions. 4.2.

The intersection of countab1y many

PROPOSITION.

nearly sure events is nearly sure. Proof:

Let SO' Sl' . ..

event such that T

m

~

be nearly sure and let T be an internal m Sm and v(Tm) "oJ 1 .

By w1- saturation there is an internal event T such that for each O<m<w, v(T)

2:,1 -

I m'

Then v(T) "" 1 and Ten S -m<w m Example ,

Let T1

sets such that v(T

m)

"oJ

~

1

T2

T ~ TO so

~

_.!.. . m

n... n Tm

.

n S is nearly sure. m<w m

."

be a countable chain of internal

Then

U T is almost sure but not m<w m

nearly sure.

Because if T is internal and v(T) "" 1 , then by wI-saturation

there is a w

€

T such that w

1/

T for all m < w , hence T is not a m

subset of U T m<w m By an internal random variable on a hyperfinite probability space

(n , v} we mean an internal function X: n -- *R. If H

€

~'N

, an

52

H.KEISLER 0 :0; m < H) of (necessarily internal) random m: is independent if for all internal r E RH ,

internal sequence (X variables X m

n

rn
v { W: X (w)::=: r } m m

.

The mean of X is the hyperreal number E(X)

=

L.

X(w) yew) ,

WEn

and the variance of X is defined by

Since n is a hyperfinite set, the mean and variance always exist. X is said to be finite if for all

WEn,

X(w) is finite.

Our results will

depend on the following basic inequality (cf. [3]). 4. 3.

(X

m

: 0

:0;

Let H

BERNSTEIN'S INEQUALITY.

E

'~N

and let

m < H) be an internal sequence of independent finite random

variables on a hyperfinite probability space (rz • v) . Assume that for each m < H, E(X )

m

=0

and V(X )

m

a < 1 and every hyperreal y

v {w r]

L:

E

:0;

Then for every real number

(1"2 •

o(erft) ,

Xm(w) I::=: y(l"y'H }

_.!... a y 2

:0;

2e 2

m
p[ (I

L:

X I ::=: yeryH)] m

:0;

2e

I 2 --ay 2

m
We use Chebyshev's inequality in the form P(Y::=: a)

:0;

for Y a random variable and a > 0..

E(y)/a

Since each Xn is finite there is a

HYPERFINITE MODEL THEORY finite bound B

€

N such that for all n < H, w

€

53 rl,

IX n (w) I -s B . Let

X (w) , and define n

us write X( n, w)

Yew)

L X(£, w)

=

£< n Consider a positive hyperreal 0 . Since

e

x

122 22 = (l+x+ZX [l+TIx+4TX + ... ]:s

12x l+x+zx e

we have e OX( n, w)

Using E(X(n, w))

:S

1+

=0

sxr n, w) +

i (oX( n, w))2 e oB

,

The X I s are independent, so n E(eoY(w))

=

E( TTeoX(n,w)) n
Let

01

be a positive hyperreal number.

Replacing

01

by e 01

P(Y(w):2:

01/

,

5)

By the same argument

By Chebyshev's inequality,

we have :S

1 2 2 sn exp(ZH5 (J" e -

01)

•

54

H.KEISLER

so

Now let y

P( I Yew) [a:

a/oj

vB).

Put

€ 0(0-

-s 2exp(

21

2 20B

HI) 0- e

- a )

Since a < 1 is real and 0:0 y/o-ft ~ 0 , we have

21

H£2 2 oB

uo-e

-O!

Thus I 2 P( I Yew) I a: yo-ft ) os 2exp( -2 ay )

We shall use the following consequence of Bernstein's inequality here. LEMMA.

4.4. M -s

(-.£L) log A

1/

4

Suppose 1 -s 10gA

Let (X

m

: m :s H )

€

o(H) , ana let

be an internal sequence of

independent finite random variables on a hyperfinite probability space

(n, v) with E(Xm) :0 0 . Then P( I B-

1

\' X I a: .!..) w m M

:s

A-M .

m
Proof:

Since each Xm is finite, each V(Xm) is finite.

Let

55

HYPERFINITE MODEL THEORY 2 Then (l" is finite.

Let cry

M.jlogA (l"2,/H

-y(l",/H

P(I H-

logAl/ 4

-s

L:

m
m I 2:

X

L:

. We have

:s

(l"2 Hl/4

l

P(I H-

= M/logA

, so y

o«l"fi)

By 5.3,

I 2 --ay

Y (l"vB ) :s 2e 2

1

xmt

E

M(l"2

2:

~

2:

~)

)

:s 2e

2

-"2 aM 10gA/(l"

2

m
P( I H-

L:

X I m m
In the above proof we could have taken M es (H/IogA)b for any real b < 1/3 .

Our first L1:\wenheim-Skolem theorem concerns an internal partition 'T1" of the universe set A, that is, an internal set 'T1" of nonempty disjoint internal sets with union A. By a choice set for 'T1" we mean an internal set B C A which contains exactly one element of each partition class D

E

'T1".

Let A'T1" be the set of

all choice sets for 'T1" . We shall use the following convention: Given a hyperfinite set A, the cardinality of A is also denoted by A. 4.5. model.

Let

PARTITION THEOREM.

~

be an infinite uniform

Let 'T1" be an internal partition of A with H partition classes,

such that (i)

log A

(il)

For each partition class D

€

o(H) .

Then for nearly all choice sets B measure,

~

E

E

'T1" , HD/A'" I .

A'T1" with respect to the uniform

I B is an elementary submodel of

~.

H.KEISLER

56

Before proving the Partition Theorem we shall discuss its relation to the weak law of large numbers.

In our present context the weak law of

large numbers may be stated as follows. Let (X

m

: 0 :s m < H) be an internal sequence of independent random

variables on a hyperfinite probability space (11. v). infinite,

E(Xm)

=0

, and V(Xm)

H-

l

o(H).

€

~

Suppose H is

Then for nearly all w

€

11 ,

Xm(w) "" 0 .

m
When the X

are finite, Lemma 4.4 gives a much stronger conclusion. m The weak law of large numbers implies the following analogue of the Partition Theorem. Assume the hypothesis of 4.5 but with "log A "H is infinite".

and each a

€

€

o(H)" weakened to

Then for each term T(x) with at most one quantifier

An , the event T~ I B [a] se T~ [a] is nearly sure.

not follow that the event

(Va E An) T~ I B [a] "" T~ [a]

The weak law is used for the case U(x) no quantifiers. We let 'IT

= {D m:

be the unique element of B

n Dm

consider the random variable

and let

Then for nearly all B E A'IT

r

is nearly sure. )

= ST(X,y)dy

0 :s m < H} , and for B '

~

n

Fix a EA.

(It does

where T has €

'IT we let b

For each m we

m

HYPERFINITE MODEL THEORY

u'P [a]

H-

l[

H-

L;

Ym(B) m
L; Xm(B) m
H- l

l

57

+

L;

E(Y )]

m
m

T~[a,b] D - l

m
m

R<

A-I

L;

T~[a,b]

bEA

We now give the proof of the Partition Theorem 4. 5. Proof:

SL (1)

We show by induction on the complexity of terms T(x) of

that there is an infinite hyperinteger K(T) such that for all 'P~]

1T

P { BEA :IT [a

~ ~ I} -T [a]l2: K(T)

:0;

a E An ,

-K(T) A .

Observe that making K(T) smaller preserves the truth of (1). For atomic terms T(X), we always have T'P [a]

= T!ll [a] ,

holds for any K(T) . Let C be an £-ary connective and assume (1) holds for Tl(X), ... , T£(X) .

and for all

t .S

Choose an infinite K such that

E TI [ - IIT II, IIT II] i i i:o;£ '

K exists because C is continuous.

Then for all

a E An ,

so (1)

58

H.KEISLER :s;

L A-K(Ti)

:s; 2A- K :s; Al-K

i:s;2

Therefore (1) holds with K(T)

= K- 1

.

Assume (1) holds for T(X, y) and let U(x) ~

for all a

€

n

A

(max y) T(x, y).

Then

,

'+l ~ ~ ~ 1 p{B:IU [a] - U [a]I"" K(T)} :s; p{ B: (3b

'tI ~ ~ ~ 1 A) IT [a, b] - T [a, b] I "" K(T)}

€

Therefore (1) holds with K(U)

=

K(T) - 1 .

The same argument works for (min Y)T(X, y) . Finally, assume (1) holds for T(x, y) and let U(X) ~

Let a

€

n

A

.

Let (D

For each choice set B

m: €

0 -s m 1T

A

r

= ST(x, y)dy

< H) be the partition classes of

let b m be the unique element of B

.

1T •

n D m,

so that B

= {bm : 0

:s; m < H}

Consider the random variables Y on (A1T, v) , where v is the uniform m internal measure on A1T , defined by

Then (Y m : 0 -s m < H) is an internal sequence of independent finite . random variables.

We shall estimate the differences ~

~

U [a] - H

-1 \' LJ Ym '

m
U'tI[a] - H- l

L;

Y m m
HYPERFINITE MODEL THEORY

59

Let

=0

Then the X are still independent and E(Xm) m m and B,

IXm(B)1

211UII

:5

.

Moreover, for each

so Xm(B) is finite.

By the Bernstein Inequality Lemma 4.4, there is an infinite M with P[H-Il

~

Xml

~]

2:

A-

:5

M

r

m
l

~

Y - E(Hm

m
The expected value E(H-

l

~

l

~

Y )I m

2:

~]

:5

A-M .

m
Y ) is infinitely close to uQl[aJ

m
m

whence

~

~ TQl[a, bJ (A/DmH)

m
~

.~

m
m T

"'" 1 .

Ql

[a, b]

m

Therefore there is an infinite M' such that for all a € An, P[lH-

l

~ m
Y - Um[a] m

I

:5

~,]

:5

A-

M'

' because

60

H.KEISLER

For each ':ll, each b

B has measure H-

€

L:

H- l

T':ll[a, b

m
l,

so

m]

An ,

€

':ll -1 '\' LJ Y m - U [a]

1 I 2: K(T)]

m
l

L:

T~[a,bm]

- H-

m
:s;

P[ (3b

€

l

L:

T':ll[a,b m] m
- b] - T':ll [a, - b] A) IT ~ [a,

12: K~T)]

1 ] I 2: K(T)

Therefore if K = min(M' • K(T) - 1) ,

Hence (1) holds with K(U) zs K/2 . This completes the proof of (1). Let Co be a countable dense set of connectives. countably many symbols,

SL(C ) is countable.

conclude from (1) that

Then for each T(x) we have

O

Since SL has

For each term T(X), we

HYPERFlNlTE MODEL THEORY for nearly all B

€

A'IT .

for nearly all B

€

A'IT.

61

By Proposition 4.2,

By the Stone-Weierstrass Lemma 3.11, the above

event is the same as the event

'tI

The above theorem shows that

<~ ,

so 'tI

<~

for nearly all B

A'IT .

€

has a large collection of elementary

~

submode1s 'tI of each cardinal H where 10gA Since 10gA

o(H),

€

H:o; A/2

o(A) , we may choose H so that H

€

€

o(A).

There is a more

natural Lowenheim-Skolem theorem corresponding to the set A[H}

of all

internal subsets B S. A of internal cardinality H with the uniform measure. We shall prove later on that if log A submodel of

replacement. B

o(H) then 'tI is an elementary

~ for nearly all B € A[H] . The difficulty in proving this

result is that the subsets B

order,

€

€

A[H}

correspond to sampling without

If we order A and write the elements of B in increasing

= {bm:

because the b

0

:0;

m < H} ,

b

m I s must be distinct.

is not independent of {be: £ < m }

We first prove an auxiliary result m corresponding to sampling with replacement.

H

Let A

~

be the set of all internal sequences b

= (b m:

0

:0;

m < H)

of elements of A. By the frequency measure fl b

corresponding to a sequence b

€

AH

we mean the internal probability measure on A given by {m: b

m

€

D}

H

Thus

a

€

fl~(D)

A,

b

is the relative frequency of b

m

's which belong to D.

For

62

H. KEISLER

J.Lb(a) In particular,

J.L ~(a) b

=:

0 if a

By the subsequence

b

frequency of a in H

=:

r/ range of

~~

b

(b) .

m we mean the model

We say that m~ is an elementary subsequence of b

(m~ ,

b

4.6.

a) a

A "" (m. ala

E

E

o(H).

respect 'to the uniform measure,

a

E

b

< m,

if

A .

Let

Then for nearly all

bE

m be a

AH with

mI:) is an elementary subsequence of m.

The proof is similar to the proof of 4.5.

occurs in the induction step from T(x,y) to U(x) ~

m~

ELEMEmARY SUBSEQUENCE THEOREM.

uniform model and let logA

Proof:

E

m,

H

n

A , let Y be the random variable on A m

=:

The only difference

ST(x,y)dY .

Given

defined by

Then E(Y ) m

= ~ T~ [a, c] A-I cEA

and hence by the Bernstein Inequality Lemma 4.4, P{bE AH:IH-

I

~ Ym(b) - U~[a] /0: ~}:S A-M . m
The remaining computations are exactly as in 4.5, and we show that for

HYPERFINITE MODEL THEORY each T(X) there is an infinite K(T) such that for all

63

a E An , -t

4.7.

LEMMA. ~

Let G:5 A . Then

G ~ ,

P{ b E A : b

1S

one-one

}

2:

2 -1 1- G A

Proof: For each m < G , the probability that b E {b n : 0 :5 n < m} -m x, l. is at most GATherefore the probability that b is not one to one is at Z most G A-I . -t Recall that A[H]

is the set of all internal B c A of internal

cardinality H. 4. 8.

ELEMENTARY SUBMODEL THEOREM.

Let 21 be a

uniform model and let H be a hyperinteger such that log A E o(H) , Then for nearly all B E A[B]

H:5 A .

with respect to the uniform measure,

'13 " 211 B is an elementary submodel of 21 .

A(H) " {b E AH : For each BE A[HJ B" range

cbl .

b

is one to one}

there are exactly HI

Whenever B

sequences bE A(H) with

= range (b),

bE A(H) , we have ~b

"

211 B .

By Lemma 4.7, A(H) has internal measure at least 1- HZA- l "" 1 in AH . By Theorem 4.6, the event A(H) is nearly sure.

n {b

EA H:

u., b

< !ll}

It follows that for nearly all B E A[H]

r

21 I '13

<

21.

-t

The proof of Theorem 4. 8 in the general case is more technical and and is postponed to the Appendix. Our L~wenheim- Skolem theorems can be interpreted in the opinion poll

H.KEISLER

64

model

m is an infinite uniform model with

introduced in Section 2.

jjl

y).

countably many internal relations Sj(X, hyperfinite set of people and Sj (a,

b)

is true if the person a agrees with

the statement Sj about the n-tuple of people procedures for taking an opinion poll.

The set A represents a

b.

We consider various

Every elementary submodel

<m

'.J3

is a sample which is good in the sense that for any statement Sj and any

b€

n-tuple

An, the percentage of a

B who agree with S. about

€

J

b

is

infinitely close to the percentage of a in the total population A who agree

b. The Partition Theorem shows that we are nearly sure to

with S. about J

get a good sample by taking an internal partition of A into H equal classes where logA class.

€

o(H) , and picking one person from each partition

The Elementary Submodel Theorem shows that we are nearly sure

to get a good sample by picking an arbitrary set of H people from A if logA

€

o(H) . 4.9.

Then nearly all submodels of

* peA) ,

jjl ,

Let log A

be an infinite uniform model.

~

with respect to the uniform measure on

are elementary submodels of

Proof: all B €

Let

COROLLARY.

~

.

O(H , H € o(A). Then HO:S B for nearly O) O * P(A). For HO :s H -s A we have '.J3 < jjl for nearly all B € A[H] be infinite. For T(X) € SL and Ho:S H :s A let K(T, H) be the €

Let K O greatest hyperinteger K :s K such that O

Then K(T, H) is always infinite.

Let

and put E

Suppose

= max] fL(b) :

,::j

0

is an internal partition of A with H partition classes such

IT

that (i)

b € A}

E

AlogA

€

o(H) .

HYPERFINITE MODEL THEORY For each partition class D

(ii)

Then for nearly all choice sets B mea sure on A1T, 4.6'.

~ ~

mI B

1 and

HD /A ~ I

€ 1T ,

€

65

A1T with respect to the uniform is an elementary submodel of

ELEMENTARY SUBSEQUENCE THEOREM.

hyperfinite model and let logA

€

o(H) .

m be

Let

Then for nearly all

b€

m. a

AH with

respect to the measure v (b)

=n

m
H, on A of

fJ.(b ) m

m~ with the frequency measure is an elementary subsequence b

a,

4. 8'.

ELEMENTARY SUBMODEL THEOREM.

Let

m be

an

atomless internal model with

e

= max] ....(a):

a

E

A} ~ 0 .

Suppose f;AlogA K(T)

E

o(H) ,

= min{K(T, H):

H

H:o; A .

O

-s H sA}

Let X(T) be the set of all internal B C A such that H

a

all

E

~

o -s B

,

Then X(T) is an internal set whose measure in each A[H] , H is at least I - K(T)-1. {B

€

and for

O

s H sA,

It follows that the measure of X(T) in

'~r(A): HO :0; B} is at least 1- K(T)-l , and therefore X(T) has

measure

~

I in "r(A).

Let Co be a countable dense set of connectives.

Then

H.KEISLER

66

is nearly sure, and by the Stone-Weierstrass Lemma 3.11, 'll nearly all B

EO

for

X.

4.10.

-j

Let

COROLIARY.

internal subset of A, and log A the set of all B

EO

€

m be a

o(H) , D

€

uniform model,

D an

o(H) , H -s A.

Let

such that D ~ B. Then for nearly all B

A[H]

with respect to the uniform measure on model of

<~

Q

,

~

be

Q

€ Q

I B is an elementary sub-

m.

The above result is analogous to the downward L~enheim-Skolem result of Tarski and Vaught (see [6 [). The next three results concern models which are atomles s but not necessarily uniform. 4. 5'.

PARTITION THEOREM.

Then for nearly all B

~

"'!

1 and

mI B

EO

A[H]

Let l!I be an atomless model

with respect to the uniform measure on A[H] ,

is an elementary submodel of ~ .

We tum now to the subject of elementary chains. An elementary chain in an internal model ~ is an ordered set ('llm: m

€

S) ,

s C {o, 1, ... , H} , of submodels of l!I such that whenever m , n

EO

S

and m < n ,

That is,

B m

~

Bn

~

A and

The set of all internal linear orderings < of A may be identified with the set A(A) of one to one sequences

b= with range A.

For each

b

EO

(b

m : 0 -s m < A)

A(A) , let

HYPERFINITE MODEL THEORY

67

(':j>m:o~m
be the chain of submodels given by

Thus ':j>A-l is W itself. 4. ll. model.

ELEMENTARY CHAIN THEOREM.

Then for nearly all internal linear orderings

Let

b

~

be a uniform

of A with respect

to the uniform measure on A(A) , ( ':j>m: logA is an elementary chain in .

Proof:

nearly all

~

€

o(m), m < A)

.

At this point we shall only prove the weaker result that for

b€

A(A) , (':j> : logA m

is an elementary chain in

ojrn) , m

€

€

O(vA))

The full result is proved in the appendix.

~.

It follows from the proof of the Elementary Submodel Theorem that for each

fixed

m

€

O(vA) with logA

€

ojm) and each term

rex)

of

SL,

there

is an infinite K(T, rn) such that

Thus for all infinite H and finite k > 0 , we see that for all hyperintegers m between H log A and vA

IH ,

It follows that for all finite k > 0 and all sufficiently large finite h , (1) holds for all m between h log A and h is sufficiently large, the event

vA/h

. Thus whenever k > 2 and

68

H.KEISLER ('vIm)[h log A :s m -s vA" Ih _ ('via

has probability

2::

€

An) IT 't5m[ a] - TlU [a]

l-k 1- A and hence is nearly sure.

I<

t]

Let D be the

internal set D

{rn

€

*N: 10gA

€

oim) and m

O(.!A")}

€

.

Then the event ('vim

€

~ n 't5m ~ lU ~ D)( 'via € A ) I T [ a] - T [a]

1

I< k

is nearly sure.

Recall that a countable intersection of nearly sure events

is nearly sure.

Thus for each term T(x) (Vm

is nearly sure.

€

€

SL(C o) , the event

~ n 'llm ~ 'lI ~ D)( 'via € A ) T [a] "" T [a]

Since SL(C ) is countable, we see by the Stone-Weierstrass O

Lemma 3.11 that the event

-t

is nearly sure.

The Elementary Chain Theorem is analogous to the strong law of large numbers.

In our context the strong law may be stated as follows.

Let (X : 0 :s m < H) be an internal sequence of independent random m

variables on a hyperfinite probability space (n , v). infinite,

E(Xm)

=0 ('vi

, and V(Xm) is finite.

infinite K

:S

H)K-

l

Then for nearly all w

L:

Xm(W) "" 0 • m
This can be proved from Kolmogorov's inequality

L:

P[(3K:s H) I Xml m
Suppose H is

2:: tKlT] :S

t

2

€

n ,

HYPERFINITE MODEL THEORY where each V(Xm)

S

0"2.

69

The strong law of large numbers implies the

following analogue of the Elementary Chain Theorem. Let ~ be a uniform model. quantifier and each

a c An,

Then for each term T(X) with at most one

the event

(for all infinite m < A) T':pm [ -a] "" T~ [a] is nearly sure with respect to the uniform measure on A(A) . Again, the proof is easier for m

E

o(fi) .

We state without proof an Elementary Chain Theorem for atomless models. 4.11'.

ELEMENTARY CHAIN THEOREM.

Let

~

be an atomless

model with s = max] f.L(a): a Then for nearly all linear orderings

b

E

A} "" 0 .

of A with respect to the uniform

measure on A(A) , (':p

m

is an elementary chain in

: eA log A ~

E

o(m) , m < A)

.

V. PROBABILITY QUANTIFIERS REVISITED

In our first look at probability quantifiers we could not go very far because the interpretation of the quantifiers depends not only on the relations of

~

but also on the internal probability measure f.L.

Our

study can now be carried further with the aid of the many-valued logic 5L, which codes up the connection between the relations of ~ and the measure

f.L . Before proceeding further we need to study a slightly

more general class of models for L.

H.KEISLER

70

Bya Borel model for L we mean a two-valued structure

where (A, .... ) is a hyperfinite probability space, each a

i

€

A. and each

Sj in a Borel relation on A, Sj E B(Ali(j». We now have three types of models.

Remember that:

(a)

Hyperfinite models for L are two-valued.

(b)

Borel models for L are two-valued.

(c)

Hyperfinite models for

SL

are many-valued.

Each hyperfinite model for L is a Borel model. model

SL

m for

induces a Borel model

universe, measure, and constants as

u,

Each hyperfinite

mo for L. mO has the same and the relations Sj of

mo

are the Borel relations S.

1

When

~

= {aE Ali(j):

is two-valued,

mo

st(F.(a)) > o} 1

u .

5. 1. PROPOSITION,

All the results of Section 2 hold for

Borel models. We wish to show that every formula of

L P which is satisfiable

WI

in a Borel model is satisfiable in a hyperfinite model. Consider a pair of Borel models

m=

(A ...... a.,S.). I. l'E J' I lIE

m'

with the same universe A, measure .... ' and constants ai' that

~

and

We say

m' are equal almost everywhere if for every atomic formula

cp(x) the sets

differ by a set of Loeb measure zero, and for every atomic sentence cp

HYPERFINITE MODEL THEORY wehave

~

iff m'

~
Two Borel models

~ ~

m' are said to be L p-equivalent if they "1.

and

satisfy exactly the same sentences of L

WI

Let ill and

THEOREM.

5.2.

71

which are equal almost everywhere.

p'

m' be Borel models for L

Then ill and ill' are L p-equivalent.

WI

Moreover, for every formula ,!,(X) of L P' the sets

WI

differ by a set of measure zero. We argue by induction on the complexity of rp(X).

Proof:

result holds for atomic /\ are routine. (Px > r)
by definition, and the steps for 1,

Assuming the result for

Y).


y) ,

The 1\,

and

we shall prove it for

Let 8

{b:ill ~(Px> r)
8'

{b:~'~(px>r)
8uppose 8 A 8' has measure

> 0 in 0lJ.n. Then either 8 - 8' or 8' - 8

has measure > 0 , say 8 - 8'.

Then there exists s > r and a set

T c 8 - 8' of measure m > 0 such that for all b

€

T,

ill j: (Px ~ s) r)
{ (a , b) : has measure

~

b€

T,

m(s - r) > O.

m 1= ",[ a ,b] ,

el'

l= 1 ",[ a . b]}

This contradicts the hypothesis that the sets

{(a,b): ill ~
m'

~rp[a,b]}

This completes the induction.

H.KEISLER

72 5. 3.

THEOREM.

Let

countably many constants, L e., hyperfinite model Proof:

~'

~

be a Borel model for L with at most

I is finite or countable. Then there is a

for L which is equal to 21 almost everywhere.

8ince each atomic formula contains at most one relation

symbol, we may assume without loss of generality that

~

has only one

relation symbol, sayan n-placed symbol 8. Assume first that m has only finitely many constant symbols, say c l"'" c We form 8' by m' piecing together internal sets along various diagonals. Partition An

a

into finitely many classes B ... , B such that and b belong to the l, k same partition class if and only if they satisfy the same equations involving c l"'" c m ' xl"'" x n' For each Be' let ~ be a maximal subsequence of such that for b € Be ' the assignments of are

x

Yi

distinct from each other and from the constants.

The length of y£ is a

number f(£) :s n representing the number of degrees of freedom of B£ (In case B£

£ :s k

is a single n-tuple of constants,

and b

€

f(£)

=0

.) For each

B£ ' let b£ be the subsequence of b corresponding to

y£ ' and let T£

= {~ : b €

f(£) Then T£ is Loeb measurable in A f(£) set T£ A which differs from T£

.s

8

n

B£} .

Therefore there is an internal by a set of measure zero in the

Loeb measure on Af(£) . Let

Then 8' is an internal relation in An.

is equal to

m

almost everywhere.

The hyperfinite model

HYPERFINITE MODEL THEORY For the case of w constants c

~

c ' ... , l' 2 with only the first m constants, and let

let

73 ~

m

be a hyperfinite model equal to mm almost everywhere.

be the reduct of

By wI-saturation

the sequence (S~.: m

€

N)

may be extended to an internal sequence of relations (S'

m

: m

€

~'N)

.

For all finite m < p < wand each atomic formula
{b: -;

j=


has internal measure less than p

-1

j=


. Therefore for some infinite p, the

model

has the property that for every m < wand atomic formula
{b: ~~

j=


has internal measure less than p

-1

. Since p

j= -1


~

is almost everywhere equal to ~'p

5.4.

COROllARY.

If a set T of sentences of L p with wI

at most countably many constants has a Borel model, it has a hyperfinite model. We now use the many-valued logic

SL

to expand L wp.

The logic

74

H.KEISLER

Lwp has a canonical expansion to the two-valued logic L(5)wp described as follows.

The constant symbols of L(S) are the same as the constant

symbols of L.

The relation symbols of L(5) are the expressions of the

form [T(X) > 0] where T(x) is a term of the logic 5L. L(S)wp has the same connectives A, v,

(Px;:: r) as L

wp

The logic

and quantifiers (Px > r) ,

l

' It is understood that 5L has the same signature as

L, and we identify each relation symbol symbol [Pj (x) > 0]

of L(S).

8 j (X) of L with the relation

Thus

Lwp

S

L(S)wp

We define the infinitary logic L(S) p by adding the countable conjunction WI 1\ and disjunction V .

Each hyperfinite model

m(S) for

L(S) .

m for

5L induces a unique two-valued model

However, in general ~ (S) is not a hyperfinite model but

a Borel model.

m(S) has the same universe, measure, and constants as relation symbol [T(X) > 0]

~.

Each

of L(5) is interpreted in ~(5) by the

relation

{ a~

E

n

~

~

A : steT [a]) > o} .

Thus

if and only if

Since all the results of Section 2 hold for Borel models, they hold for the logic L(5) p and the Borel models ~(S). In particular, the satis- . WI faction relation

HYPERFINITE MODEL THEORY makes sense for formulas
WI

L(S)l<Xp'

75

P and even of the Suslin logic

From the definitions we see that the reduct of ~l(S) to L is

.

the Borel model ~O. If ~ is two-valued, then ~o" ~ Bya

S-model of a ----

model ~ for

SL

sentence

such that ~ (S)


F

L(S)WI P

of

Thus


we mean a hyperfinite

S-models of


are in

general many-valued.

is

S-valid if every hyperfinite model

for

SL

is a

Two formulas '1'(5<), tjJ(x) of L(S) p are said to be

the sentence

'P(c" -

WI

tjJ(c" is

Consider a sentence a

~

S-model of


WI

A hyperfinite model ~ for SL is

if and only if ~o is a Borel model of

is the reduct of ~(S) to L.

S-equivalent if

S-valrd,

of L p'


S-model of

Thus if rp has a

'1', because ~o

S-model it has a

Borel

model. 5.5.

COROLLARY.

A sentence rp of L

wl

has a

S-rnodel if

and only if it has a hyperfinite model. Proof:

If ~ 1s a

and by Corollary 5.4,

S-model of

rp

r

then ~O is a Borel model of v ,

'I' has a two-valued hyperfinite model

is a hyperfinite model of


then ~ is a

S-model of

~.

If

~

because


-j

~ " ~O'

5.6.

COROLLARY.

If a sentence

(in all hyperfinite models for L(S) ) then it is Proof:


of L(S)

S-valid.

WI

p is valid

If 'P is valid, then by 5.4 'P holds in all Borel models for

L(S) , and hence 'P holds in ~(S) for all hyperfinite models ~ for SL. -j The logic L(S) p is adequate for expressing most of the classical

WI

properties of stochastic processes.

To illustrate the expressive power of

this logic we describe a hyperfinite model representing Brownian motion, based on the paper of Anderson [1].

It is convenient to use a many-valued

hyperfinite model with two sorts of elements.

76

H.KEISLER Let H be a positive infinite hypertntecer, let at

*[ 0 , 1]

S be the hyperfinite sub set of

= (HI) -1 ,

and let

defined by

S = {a t , za t , ..• , H I at}

Then each standard rational number q

€

Q

n [0, 1]

the standard part function maps S onto [0. 1] .

belongs to S, and Let n be the set of

all internal functions vr : S ..... { .i .rj . Our model will have two uniform hyperfinite probability spaces (o , fJ.)

and (S. v ) •

We think of elements of S as points of time and elements w

€

n as

representing the paths

L;

B(w, t) =

w(S)v!.TI

s:st B(w, t) is an "infinitesimal random walk". function

B ; n X [0, 1] ..... R

Anderson proved that the

defined by

O

iff

(Yt"" r)B(w,t) "" s

is a Brownian motion on the Loeb space (n , 0 fJ.) in the classical sense.

m with the function

We would like to form a model

B(w, t) : n X S ..... *R, but this is not permitted because the values of B(w, t) are not always in *[ -1,1] , or even finite. truncated version of B(w, t).

For

fl €

wand r

€

*R let

n_niiff r > n -ri es r es n r

r xn { We now describe the model

To get around this we use a

~.

if r < -n

m has

the form

m= (n , fJ., S, v r an' F, Bn )0< n < w We let x, y be variables of the sort ranging over over S.

The constants an of

n , and s, t, u ranging

m all belong to S, and

HYPERFINITE MODEL THEORY { an : 0 < n < w} Thefunctionsare F: 8--*[0,1]

=

Q

77

n [0, 1]

and Bn:nXS 2 ...... *[0,1] where

F(s) = s , Bn(X, s, t)

1 = Ii" [Brx,

s) - B(x, t)] /\ 1 .

We shall express some basic properties of Brownian motion in the language L(S)

WI

T(X):S r means l T6~) - r > 0 , and other inequalities

p'

are used in a similar way.

For 0 < n <

[B(x, s) - B(x, t)] /\ n

00 ,

= n.Bn(x, s,

t)

The inequality I Bpc, s) - Btx, t)1 -s r where r is real and [r] < n is expres sed by [Bpc, s) - B(x, t)] An -s r .

The fact that for s, t

E

8 , B(x, s) - B(x, t) has expected value zero is

expres sed by 1\ (max s)(maxt)IS[B(X,s) - B(x,t)]l\ndxl:s 0 . n

The fact that B(x, s)

= B(x, s)

- B(x, 0) is almost surely finite is

expressed by (Px 2:: 1) V (max sj] B(x, s)A(n+ 1) I :s n . n

The fact that Brownian motion is almost surely continuous is expressed by (Px2::l) 1\ V (max s)(max t)![B(X, s)-B(x,t)]"lloC (F(s)-F(t»:s m' m n

n

where C (r) is the polygonal function which is zero for n for [r] :S 2~

1\

1\

n CEC O DECO

2:

~ and one

The fact that B(x, t) - B(x, s) is independent of

B(x, u) - B(x, t) when s < t 1\

Ir]

< u is expressed by

(max s)(max t)(max u)(F(t) - F(s)). (F(u) - F(t)) 0

H.KEISLER

78

The law of the iterated logarithm is expressed by (PX == 1) 1\ V m n

(Px

[B(X, q)"l] -s jZqloglog.!.. q

1\

qEQ

(1+1.. ) ,

m

1 O
== 1) 1\ 1\

qEQ 1 0< q
m n

[B(x,q)"l] ==

j

Zq log log~

1 (1 - m)

The fact that B(x,l) is normally distributed with mean zero and variance 1 is expressed by

1\

1\

(Px

qEQ

n

== 4> (q»)[B(x, l)"n] < q ,

Iql < n 1\

1 (Px > 4> (q»[ B(x, 1)1\ n] < q ,

1\

qEQ

n

Iql < n where 4> (r) is the normal distribution function. We now return to our general discussion. A formula

WI

P

is said to be quantifier-free if it contains no probability quantifiers (Px > rj , (Px

== r) . The next theorem allows us to use L(S) P in the

WI

same manner as a Skolem expansion is used in classical model theory. 5.7.
1(S)

WI

L(S)

WI

QUANTIFIER ELIMINATION THEOREM.

p is

Every formula

S-equivalent to a quantifier-free formula

.p(x) of

P .

Proof:

It suffices to prove that for every quantifier-free formula

.p(X,y) of L(S)

WI

P' the formulas (Py> r).p(x,y) • r E [0,1], are

HYPERFINITE MODEL THEORY

79

S-equivalent to

quantifier-free formulas cp(X) of Ld) p Let M be wI the set of all quantifier-free monotone formulas of L(S) p' and let Q

WI

be the set of quantifier-free formulas of L(S)

WI

P'

By the Monotone

Formula Theorem Without Quantifiers, every formula in Q is equivalent

S-equivalent

to a formula in M.

By Corollary 5.6, every formula in Q is

to a formula in M.

We show by induction on monotone complexity that

for every formula IJ!(X,y)€ M and r€[O,l],

S-equivalent to

(Py>rjlJ!(x,y) is

some cp(X) € Q .

Consider first the case where IJ!( X, y) is a single relation [T( X, y) > 0] Then (Py > r)4J(x, y) is satisfied by

a in

meS)

if and only if for some

positive m , n < w ,

Let C

n

be the polygonal function with Cn(s) = 0 for s :s zln '

Cn(S) = 1 for s

2:

~

Then for each n ,

Therefore (Py > r)4J(x,y) is S-equivalent to

v

V

O<m<w O
JC n (T(x, y»dy > r +1..m ,

or V V [ O<m<w O
JC

n

(T (

x, y»dy -

(r + 1..)] > 0 m

The last formula belongs to Q . Now suppose 4J(X, y) € M has monotone complexity zero, that is, IJ!(X, y) is a quantifier-free formula of L(S)wp' 4J is S -equivalent to a

80

H.KEISLER

formula in disjunctive normal form,

.v /\ ((Tij(X,y) > O]Al(Uij(X,y)> 0]) 1=1 J=1 p

This in tum is

q

~

S-equivalent to

p q ~ ~ 1 V I\«T'j(X,y»O]A 1\ (Ui,(X,y)<-]) i=l j=l 1 O
p

~ I ~ q 1\ 1\ ((Tij(X,y»O],,[--u .. (X,y»O]), i =1 j = 1 0 < n <w n 1J

V

p V

1\

i=lO
0< n<w

1

~

~

((Ti(x,y) > O]A[n: - U,(x.y) > 0]) , 1 ~ 1 ~ «(T(x, y) > 0],,(- - U(x, y) > 0] , n

1\

O
(V (x, y) > 0] n

Here

T(;;, y)

U(;;, v)

The terms Vn(X, y) monotonically decrease in value as n increases, Therefore (Px > r) ljJ (x, y) is V

S-equivalent to 1\

0< m<w 0< n<w

(Px

> r + 1.)( V m

n

We have already seen that the inside formulas are

(x, y)

> 0]

S-equivalent to formulas

81

KYPERFINITE MODEL THEORY of Q , so (Px > r)ljJ(x, y) is

S-equivalent to a

formula of Q .

Now assume the result for monotone complexity less than ~ 'P (x, y) in M have monotone complexity

01.

n monotone,

F (Py > r)

I\cP

n

Hence (Py > r) 'riCPn is

n

0',

and let

Since the conjunction is

1 ~ V 1\ (Py>r+-)cpn(x,y). O<m<wn m

(x, y)

S-equivalent to a

V'P (x) in M have monotone complexity

n n

formula in Q. 0'.

Finally, let

Then since the disjunction

is monotone,

F (Py > r) V cP (x, y) n n Thus (Py > r)

X'Pn

V (py

n

>

r) 'P (x, y)

n

is S-equivalent to a formula in Q.

This completes

our induction. The proof of the Quantifier Elimination Theorem gives an algorithm which, given a formula of L

wp

formula of L(S)

WI

p'

S-equivalent quantifier-free

The exact sense in which the algorithm is effective

will be discussed later. be quite complicated.

, produces a

The translation of a simple formula of Lwp may

For example, given the sentence (Px> r)(Py

~

of Lwp ' the algorithm produces the

s) F(x, y) > 0

S-equivalent

sentence

VI\VVI\I\VI\VV abcdefghij

1 1 1 1 - (r+-+-+-+-)] > 0 d g i a

5.8.

SL,

COROLlARY.

For all hyperfinite models

the following are equivalent.

~

and

'fI for

H.KEISLER

82 (i)

'13(S) are L(S) p-equivalent, 1. e., they satisfy the WI

(11)

same sentences of L(S)

WI

Proof:

If m,.,

P .

13 then m(S) and 13 (S) satisfy the same atomic

sentences, and hence the same quantifier-free sentences of L(S) p

WI

By the Quantifier Elimination Theorem they are L(S) P -equivalent.

WI

Thus

(i) implies (11). If (11), then for every constant term T and real r we have st(Tm) > riff iff '13(S) and hence m,., 5.9.

r] > 0 iff st(T'13) > r ,

13 .

WI

m and

13 are the reducts of ~(Sl and 13 (S) to L p' WI

eq symbol does not occur. ~.

13 -

and

Let If

tp

tp

-f

be a sentence L(S) P in which the

WI

has as-model m then it has a uniform

If ~ is two-valued we may take It to be two-valued.

Let ~ (J)

Proof: f:

13 are two- valued hyperfinite

13 , then m and 13 are L p -equivalent.

5.10. THEOREM.

S-model

If m,

COROU.ARY.

models and m,., Proof:

F [T -

m(S) l= [T - r] > 0

F

tp •

By Proposition 3.30 there is a refinement

=C m,., By 3.24, T T'13 for all constant terms T without the eq

~ and a uniform hyperfinite model It for SL such that B

13'" ~.

symbol. Therefore m(S) and '13(S) satisfy the same quantifier-free sentences of L(S) that

tp

WI

P without eq.

From the proof of Theorem 5.7 we see

is S-equivalent to a quantifier-free sentence IjJ of LcS) P

without eq , Therefore UI is a

S-model of

S-model of

WI

tp

and hence of 1jJ,

IjJ and hence of tp , and finally It is a uniform

13 is a

S-model of tp .-f

HYPERF1N1TE MODEL THEORY

83

We now consider the effect of changing the hyperreal universe '~[V(R)]

.

If the signature (I, cr) of L belongs to V(R) , then the set of

P belong to V(R). Let us write ~ E ':'[V(R)] if every reduct wI of m to a finite sublanguage of L belongs to *[ VCR)] . formulas of L(S)

s.u, Let

~<[V(R)]

THEOREM.

Suppose SL has a countable signature in V(R).

and #[V(R)] be two (wI-saturated) hyperreal universes.

every hyperfinite model 'J3 € #[V(R)] for Proof:

SL

Let

m€

"~[VCR)] for

such that

For

SL there is a hyperfinite model

m~ 'J3 .

Co the set of polynomials with rational coefficients,

plus the minimum function mirur , s).

Let g be the set of constant terms

T

€

SL(C such that Tm 2: O. Then g is a countable set. For each o)

T

€

g, the statement (3 '.jl)['.jl is finite and T'.jl 2: 0]

is expressible by a bounded sentence in (V(R), 8).

This sentence holds

in (,~[ VCR)] ,8) because T~ 2: 0 , and therefore it holds in (V(R), 8) . E s then min(Tl, m if and only if Tl2: 0 for /, = 1,

If Tl, ... ,T

,T

m)

E

, m.

g.and [min(T

l,

... ,T

m)]132: Thus for all T .. , T E g m l,·

0

there is a finite model 13 such that

It follows from the Compactness Theorem 3.23 that there is a hyperfinite

model '.jl € #[V(R)] such 'that T'tl2: 0 for all T

E

g.

Using the Stone-

Weierstrass Theorem we see that for any constant term U so

~ ~

E

SL,

U~ ~ U'J3 ,

'.jl . 5.12.

ABSOLUTENESS THEOREM.

Let '<[VCR)]

and #[V(R)]

be two hyperreal universes, and let the signature of L belong to VCR) • A sentence rf[V(R)] .

cp

(!)f L P is valid in

WI

~'[V(R)]

if and only if it is valid in

H.KEISLER

84 Proof:

S-model in

We show that every sentence *[V(R)] has a

many symbols occur in signature.

WI

#[V(R)].

Since only countably

S-model of

'I' in *[V(R)] . By 5.11 there is a

'13 E # [V(R)] for SL such that '13 "" Ul.

Therefore

'13(S) satisfy the same quantifier-free sentences of

P WI The proof of the Quantifier Elimination Theorem uniformly gives us a Ul(S)

and

of L(S) P which has a

we may assume that L has a countable

q>,

Let Ul be a

hyperfinite model

S-model in

q>

sentence ljJ of q>

L(S, max)

WI

P which is quantifier-free and

in every hyperreal universe.

~ is a

S-model of

L(S)

S-equivalent to

Since Ul is as-model of 'I' in *[V(R)],

ljJ in *[V(R)], hence '13 is as-model of ljJ in

#[V(R)], and '13 is a S-model of

q>

in #[V(R)] .

-j

Our next result is related to the completeness problem for Lwp' We assume the reader is familiar with the notion of an admissible Barwise [21 or Keisler [13]), Theorem: If

a

~

a

(see

We shall use the Barwise Completeness

is an admissible set with all a

a -C H C), then the set of valid a-recursively enumerable.

E

a

countable (that is,

sentences in the infinitary logic L

a

is

By Lap we mean the set of all formulas of L P which belong to the

WI

admissible set (Px

> r) , (Px

~

a.

Notice that formulas in Lap contain only quantifiers

r) such that r e

n.

5. 13. ABSTRACT COMPLETENESS THEOREM. admissible set with

W

E

aC

Let

a

be an

HC , and let L have an a-finite signature.

Then the set of all valid sentences of Lap is a-recursively enumerable. Proof:

We give the proof in the case that L has a finite signature.

The a-finite case is similar.

Let

Co be the set of connectives of SL

consisting of the linear functions with rational coefficients, the rational polygonal functions in one variable, and the max and min functions. Then the set of terms of SL(C ) is a-finite. O

In order to obtain the result for arbitrary

a

with

W

E

a~

HC we

HYPERFINITE MODEL THEORY

85

need a weakening of the notion of a monotone formula.

A conjunction

/\ q, is said to be directed if for all 'PI' 'P2 "q, there exists 'P" q, such

that 1= 'P -

'PI" 'P2 • Directed disjunctions are defined dually.

formula 'P of

L(C O'

S)wrP

A

is directed if it is built up from finite quantifier-

free formulas, using only directed conjunctions and disjunctions and probability quantifiers.

Let J

a. Ja

which belong to

be the set of sentences of

a

is an a-recursive set.

L(C

o ' S)w 1p

By effectivizing the

proof of the Monotone Formula Theorem it can be shown that there is an

a- recursive function fl: Ja -

Ja such that for each sentence

fl'P is a directed sentence which is equivalent to 'P.

'P"

Ja '

The details of this

effectivization, which require some care in the inductive definition of f

are given in Keisler [21]. Let K be the set of quantifier-free a l, sentences in J a' The proof of the Quantifier Elimination Theorem gives

an a-recursive function

f 2: J

a

-

K

a

such that whenever ljJ is a

directed sentence of Ja' f 2ljJ is a sentence lent to ljJ •

e" K a which is

S-equiva-

Let f be the composition

Then f: Ja lent to 'Po

K

a is a-recursive and for each

'P"

Ja' f'P is

S-equiva-

(This is the precise sense in which the algorithm of Theorem

5. 7 is effective. ) Call a model ~ for SL hyperrational if all the values of are hyperrational numbers.

If

f1 and

Fj

~ is a finite model for SL and the values

of f1 and F. are all rational, we call J

m rational.

The rational models

are elements of HF, the set of hereditarily finite sets. For each constant term T " SL(C in the language of (HF, 8)

o) ,

such that

there is a finite formula eT(u, v)

H.KEISLER

86 if and only if

~

function T -

aT is a-recursive.

is a rational model for

(HF, e)!= if and only if ~ (S) 1= [T > 0]. K

a

SL

and

T~ ~

-k- '

and the

Moreover, V a ( ~ , m) 0< m < w T

Let g be the

a-recursive function from

into La such that g[T> 0]

V

0< m < w

aT(u, m)

Then g(l1J) has one variable u and is built from finite formulas using only finite and infinite connectives (no quantifiers).

Let

r be the set of all

L which hold in (HF. e). Since w € a, r ww is a-recursively enumerable. We claim that for each sentence tjJ EO K finite sentences y

€

a'

the following are equivalent. (1)

(gtjJ)(u) is consistent with

(Z)

tjJ has a

r.

S-model.

(1) holds if and only if (g tjJ)(u) is satisfiable in some model of

r.

Since (gtjJ) is built from finite formulas without quantifiers, (1) holds if and only if (g
r. For each constant term T

EO

fL(eo) and 0 < m < w , we have

if and only if ~ is a hyperrational model in ~'HF and T~[ follows that for hyperrational

~ EO

a]

*HF ,

Therefore (1) holds iff tjJ has a hyperrational

S-model.

By the

~

-k.

It

87

HYPERFINITE MODEL THEORY Hyperrattonal Approximation Theorem 3.13 and by 5.8, (1) is equivalent to (2).

Thus for sentences 'P of Lap' 'P has a hyperfinite model if and

r.

only if g(f('P)) is consistent with

r

~

g(f(rp)).

By the Barwise Completeness Theorem, the set of conse-

r

quences of

Hence 'P is valid if and only if

in La is a-recursively enumerable.

Since f and g are

a-recursive, the set of valid formulas of Lap is a-recursively -j

enumerable.

5.14. signature and let (Px

> q),

COROLIARY.

where q is rational.

of valid sentences of Proof:

be the set of formulas of Lwp with only quantifiers

L~p

(Px 2: q)

a

Let

Suppose L has a hyperarithmetical

L~p

is a

1Ti

Then the set of Qjdel numbers

set of natural numbers.

be the smallest admissible set such that w

set X C w is hyperarithmetical iff it is a-finite, and

1Ti

€

G.

A

iff it is

a-recursively enumerable (see Barwise (2 J).

-j

We conclude with a result about the expanded Sus lin logic 1(S)\O( p .

5.15. has <

IC

THEOREM.

Suppose

'~[V(R))

constant and function symbols, where

cardinality of '
Let

m, '+'

is IC-saturated and L IC

is the standard

be hyperfinite models for SL with

Then W(S) and ,+,(S) are L(S)XXp-eqUivalent.

If Wand

m""

'j) •

tI are two-

valued, they are !xxp-equivalent. Proof: f:

By the Isomorphism Theorem there is a near isomorphism

m.::: '+'. Form K by adding a constant 0a to L for each a

E: A

.

We have

and hence (m(S), a)

a

E:

A and (tid). fa)

a

E:

A are KcS) p -equivalent. WI

For each n , let D(An) be the set of all subsets of An defined by n) n) formulas 'P(x) of K(S) r : and define D(B analogously. Then D(A WI n) n) and D(Bn) are rr -algebras, and X E: D(A if and only if fX E: D(B .

88

H.KEISLER

Let fl be the completion of the restriction of °fl to D(An ) , and n n similarly for Y Then X is fl measurable iff fX is "n measurable. n n' Also flnex) = Yn(fX) , because the probability quantifiers insure that each section of X has the same measure in 0fJ. as its image in °Y.

It

follows from Lemma 2.20 that for each formula
is fJon-measurab1e.

Then by induction on complexity we see that for each formula
S

~

K( )xxp and each a

E

n

A ,

-t

VI.

STANDARD PROBABILITY MODELS It is natural to consider models for the logic L

w P 1

u

= (A,

1',

of the form

ai'S.J ).IE I ,JE . J

called standard probability models, where

l'

is a standard IT-additive

probability measure on A and each S. is a relation on AlT(j) which is J

measurable in the product measure with the hyperfinite models.

1'lT

O) . We shall compare such models

It turns out that for standard probability

models, the logic L P and the logic L(S) P without the (max x] and w1 w1 (min x] quantifiers can be interpreted in the natural way, but we do not have an interpretation for the full expansion L(S) p' w1

The difficulty is

2 that given a measurable bounded real-valued function F( x, v) : A -- R,

the new function (sup x] F( x, y): A -- R is not necessarily measurable, so that S(sup ~F{x, y)dy may be undefined.

The logic L(S)

P without

wI

the (max x) and (min x) quantifiers is denoted by L(r)

P .

wI

The main result of this section is that every standard probability

HYPERFlNlTE MODEL THEORY model is L(S-)

wI

89

p equivalent to some hyperfinite model.

This gives a

precise mathematical form to the intuitive idea that standard probability structures can be replaced by hyperfinite structures. Bernstein's Inequality.

The proof uses

The advantage of hyperfinite models over standard

probability models is that in the former we can "count" elements and submodels, and we also have the full language L<5) p available. wI

We have

seen that the almost sure continuity of Brownian motion can be expres sed in L(S)

wI

p using the (max x] quantifier.

This is related to the fact that

the almost sure continuity of Brownian motion is obtained more easily in Anderson's hyperfinite model than in the standard (Wiener) model. We now introduce a general class of probability models which includes both the hyperfinite models and the standard probability models. If

JJ

is a IT-additive (hence real-valued) probability measure on A,

we let In(A, v) be the set of v -measurable sets. sequence of IT-additive probability measures v

n

Given a set A, a on An is said to have

the rubini property if : For all m, n , v +n is an extension of the product m

(6.1) .

measure v m

vn

X

(6.2).

JJ

n

that is, for all

S

€

is, preserved under permutations

1T

of {L, ... ,n},

n

In(A , fL n ) , n - lIl(A ,vn)

(6.3). S

€

(i)

lIl(A

m+n

,v

For all

The analogue of Theorem l.12 holds.

+ ) we have: m n

b€

An , { a : S( a, b)}

~

(ii)

For all r

€

R,

~

~

€

m

In(A . v

m) .

That is, for all

90

H.KEISLER

o

(iii)

then

v {b: v {a: S(a,b)} > o} n m

=0

.

Bya two-valued probability model we mean a structure 21=(A,v n ,ai'S.) i EI , E j J J n<W ,

where v

n

a

i

£

is a sequence of probability measures with the Fubini property,

A, and each Sj is a

vIT(j) -measurable relation on AIT(i)

The two-valued hyperfinite models, the Borel models, and the standard probability models are all probability models in the above sense. For hyperfinite models we take v n to be the restriction of the Loeb measure o(fln) to the Borel sets

n

a(A ) .

The results of Section 2, suitably formulated, hold for two-valued probability models 21.

21 po ",[ a]

The satisfaction relation

for '" (x) in

L p is defined exactly as before. We state the main facts from Section

WI

2 in the general setting.

A real-valued probability model is a structure 21=(A,v n,ai,F.) · J n < W , l.£ I , j

where v

n

has the Fubini property,

£

J

IT(i) a E A, and each Fj: A _ [-1,1] i

is a fl -measurable function. n Each many-valued hyperfinite model 21 induces a real-valued probability model

st 21

= (A, 0 (fln) • ai' stF.) < i J n w, £

called the standard part of 21, where

I , J. £ J

HYPERFINITE MODEL THEORY

91

The functions (st F ) are Loeb measurable by Theorem 1. 8. j

Let SL- be the set of terms of the many-valued logic SL in which the quantifiers (max X), (min x] do not occur. real-valued probability model defined in the natural way.

~,

Let T(~) E SL-. ~

In a

~

the interpretation T [a] can be

The quantifier clause is

( T( x~ , y)dy) ~~ [a]

J

=

f~~

T [a, b ] d

lJ

(b) .

For each formula q'(X) of L and wlP two-valued probability model ~, the relation 6. 4.

PROPOSITION.

is IJ. n - measurable. 6.5.


q'(X) -

q"

6.6.

For every formula

PRENEX NORMAL FORM THEOREM.
wp such that

(x) is valid in all two-valued probability models.

MONOTONE FORMULA THEOREM.

of L P there is a formula

wI

q"

For every formula
(X) of L p such that
wI

and equivalent to
q"

If q'(X) has

(X) .

We do not have a natural interpretation of terms of SL containing the quantifiers (max xj , (min x) within the scope of an integral quantifier. Let L(r) be the sublanguage of L(S) with the function symbols [T(X) > 0] for T(X)

E

SL- , and form L(r)

real-valued probability model

WI

~

p in the usual way.

Each

determines a two-valued probability

model~(S-) for L(S -) , and formulas of L(S -) P can be interpreted in

WI

the natural way in ~(S-). We do not have a natural expansion to a model for the full language L,(S).

92

H.KEISLER The standard part st ~ of a hyperfinite model is an especially well

behaved real-valued model, in which the full logic 6.7.

SL,

tenus T(x) of

Tst~[a]

the inductive definition of T

st~

SL.

For all

~

[a] makes sense and

= st(TU[a]) .

We argue by induction on the complexity of T(x).

Proof:

result holds for atomic tenus. U(x)

can be interpreted.

Let ~ be a hyperfinite model for

THEOREM.

~

SL

= ST(x,y)dy.

Assume the result for T(X, y).

Let a € An. fey) = T

st a

The

Consider

The function

~

~

~

[a,y] = steT [a,y])

is the standard part of an internal function and hence is measurable.

For

internal g(y) ,

J( st g (y» dOf.1(y)

st( ~b € A g(b) f.1(b» .

Therefore

U

st U

~ [a] =

f T st U[a,y]d ~

0

f.1(y) =

\' ~~ = st(Lib€AT [a,b]f.1(b» =

Now consider W(X)

= (max

v) T(

has a maximum value r , the set value st r.

x, y).

JsteT U[a,y])d f.1(y) ~

~~

st(U [a]) . Since the set {T U [a, b ] : b € A} ~

~

{st T [a, b] : b € A} has maximum

Therefore st r

~

~

st(IN [a]) .

The term (min Y)T(X, v) is similar. Assume the result for Tl(X), ... , Tm(X) and let

Then since C is continuous,

0

HYPERFINITE MODEL THEORY

C(stT

l

~

, ... , stT

~

m

93

)

-t 6.8.

COROLLARY.

Let ~ and 'j.\ be hyperfinite models for

Then ~ "" 'j.\ if and only if for every constant term T of

SL.

SL,

Two hyperfinite models for SL have the same standard part, st m " st'.jl, if they have the same universe A" B, the internal measures f.L

and

II

generate the same Loeb measures

they have the same

0f.L" 01',

constants a " b ' and the functions are infinitely close, i i

By 6. 8, if st m " st '.jl then

~

R!

'.jl.

Here is an easy characterization of models which are nearly twovalued. 6.9.

PROPOSITION.

Let m be a hyperfinite model for

following are equivalent. (i)

st ~ is a two-valued model.

(ii)

Fj(a)

(iii)

There is a two-valued hyperfinite model 'j.\ R!

(iv)

For all j

and

R!

0 or Fj(a) "" I for all j

€

J,

€

J, iii € AlT(j) • ~

•

SL.

The

H.KEISLER

94

We digress briefly to discuss the effect of removing the integral quantifier from 5L while retaining the (max xj and (min x) quanttffers. Let M be the set of all terms T(X) of 5L in which the integral quantifier 5 .. dx does not occur.

Each T(X)

€

M has a natural interpretation

T m[a] in a real-valued probability model m which is independent of the measures

jJ

measures.

•

n

We are thus led to consider real-valued models without

The model theory "for such models with the many-valued logic

M has been studied in [7] (with a compact Hausdorff space in place of the space of values [-1, 1]).

If m is a hyperfinite model and

x-saturated, the real-valued model st(m) turns out to be the sense of [7].

*[V(R)]

IC_ saturated

is in

The Isomorphism Theorem Without Integral Quantifiers

3.12', which we stated in this papeljfollows from the uniqueness theorem for saturated models in [7] . We now return to our study of the logic L(r)

WI

the proof that every probability model is L(r)

WI

p'

We work towards

p-equivalent to a hyper-

finite model. It is easy to check that each term T(x) liTII

a

€

which is the supremum of An.

Tm[a]

€

5L-

has a finite norm

~ and

for all probability models

Moreover, the Stone-Weierstrass Theorem holds as before.

The

Quantifier Elimination Theorem and Bernstein's Inequality can be proved in the following formulations using the arguments from Section 5. 6.10.

QUANTIFIER ELIMINATION THEOREM.


WI

For every formula

p there is a quantifier-free formula tJ!(X) of L(S-)

such that
WI

p

tJ!(X) is valid in all real-valued probability models.

Let X , n < W be a sequence n of independent random variables on a probability space (0, jJ). Assume 6.11.

BERNSTEIN'S INEQUALITY.

that the X are uniformly bounded by K, that is, n sup{ I X (w) I : n n

< cc .

W €

n} -s K

<

00

HYPERFINITE MODEL THEORY Suppose that for all n < w , E(Xn) a < 1, every

HEW,

95 2

=0

and V(Xn) :S iJ . Then for every 1(J".jHI09(2 and every y:s K- a) ,

p(1 L: Xnl2:: YiJVR)

:S

1 2 --ay 2e 2

n
COROLIARY.

dent random variables on

Let X n < W be a sequence of indepenn, (0, v) . Assume the X are uniformly n

bounded by K 2:: 1, and suppose each E(Xn) let

2

3K

:s (H flog HF 4 .

If

P(IH-

1L:X n

! 2:: ~):S

Suppose L has a finite signature.

Let

m for

b E AW

SL- ,

Let 2:s H <

H

-M

.

Consider a real-valued probability

For each m <

W

,

let ':fI(m) be

the finite model ( B(m), fJ.(m) , a. ,Fj~B(m»)i 1

E

1 .

,J E

J

where B(m)

= {be:

£

:s;

m} U A o

and fJ.(m) is the frequency measure fJ.(m)(a) = We write ':fI(m) -

and

and let A be the finite set of constants o

be a countable sequence.

':fI(m)

W ,

M:s (H flog H)1/4 ,

n
model

= O.

U

:S

m . b£ = a}/m.

m if for each constant term T in m_a>

SL- ,

96

H.KEISLER Given a probability model ~, the probability measure "won AW is

generated in the usual way be the measures "n on An . 6.13. ~

=:

(A,,,

'j3(m) --

LEMMA.

Suppose L has a finite signature.

Let

, •.. ) be a real-valued probability model for SL-. w for -almost all sequences b € A •

m

~

"w

Proof:

For T(x) in

SL-,

a€A

n,

b€A w ,and H<w, we may

define T'j3(H)[a), even when a iB(H)n , as follows.

T~[a]. If T(X)

T']>(H)[a]

=:

If T(x)

JU(X, y)dy,

=:

m
iii" €

If T(X) is atomic,

C(Tl(X), ... ,Tk(X)),

H- l ~ u']>(H)[a, b

T']>(H)[a]

Thus when

Then

~

m]

B(H)n, T']>(H)[a] coincides with our previous definition.

We prove several facts by induction on the complexity of terms T(X) in

SL-.

In each case the steps for atomic terms and for T

=:

C(T

l,

" ., T

k)

are routine. Claim 1.

For every T(x) and G < w there exists K (T, G) < w l

such that whenever Kl(T, G)

S

H < w, a € An , and b,

C€

AW differ

on at most one place,

Assume Claim lfor U( X, y) and let T(x)

=:

SU(X, y)dy.

If b.

differ on at most one place and H == Kl(U, G) , and H == 1, then

c

HYPERFINITE MODEL THEORY

97

Therefore Claim 1 holds for T with KI(T, G) = max(K1(U, ZG), 4GIIUII) . Claim Z.

For every

rex)

and G < w there exists Kz(T, G) < w

such that whenever Kz(T, G) -s H < wand -a

€

An ,

The proof is similar to the argument for the Elementary Subsequence Theorem 4.6 and uses

6.IZ. Assume the claim for U(X, v) , let

T(X) = SU(x,y)dY and let KZ(U, G) -s H.

a

€

An, G < eo , Kl(U,G) -s H < co , and

Let Y be the random variable m

on the space (AW

,

vw ) . and let

x m (b) =

Y

m

(b) -

E(Y

)

m

Then

and the X are independent. By Claim I, for each b m conditional probability inequality

€

A we have the

H.KEISLER

98 Therefore, putting r" ZiG,

We now obtain a bound for the probability that using Corollary 6.12. G :s (H Ilog H)1/

4

Assume H is large enough so that

- Tm[a]

I ~ 3/G

3(211T 11)2 ,

. We have

P(I

:s P(

I T~(H)[a]

L:

H-luNH)[a, b - Tm[a] I <: ~) m] m
L: I H- 1 U'+l(H)[8,b m]

- H-IYml

m
+ I L H-IYm -

Tm[a]

I~ ~)

m
:s

peL:

IH- I u '+l (H)[ 8 , b m ] -Yml~-&)

m
+

P(I

L H-IXm I ~ ~)

:s 2H- G :s H- G/ 3

.

m
Therefore Claim Z holds for T(x). Then for all constant terms T in G

~

2,

SL-

and all H ~ K (T, G) where 2

99

HYPERFINITE MODEL THEORY

Since

2 ~ LJ H- is finite, it follows (by the Borel-Cantelli lemma) that for

H=l

each 2:$ G < co , the probability that there are infinitely many H < w such that

is zero.

Therefore for each T, the event lim T~(m) m_oo

has probability one.

call this event

set of continuous real functions.

= T'!I ~.

Then

Let

Se(Co)

Co be a countable dense

is countable.

Thus

has probability one. By the Stone-Weierstrass Theorem, if T

€

b€

SL- , W

A

whence ~ (rn) - '!I.

b€ e

Therefore '.Jl (m) -

then b

€

eT

for all

'!l for almost all

-f

•

The above argument still holds under the weaker assumption that L has a countable signature and finitely many constants. 6.14. THEOREM.

Suppose L has a countable signature.

every real-valued probability model '!I there is a hyperfinite model

SL

such that '!I and ~ are L(f)

WI

Proof:

P -equivalent.

Write L as the union of a chain of sublanguages with

finite signatures,

L

U Ln' Let '!In be the reduct of ill to Ln . n<w

For ~

for

100

H. KEISLER

b E AW ,

For

let '+> n (m) be the reduct of '.fl(m) to 1 n .

'J3 n (rn) --

we have, by 1emma 6.13 W

~

for almost all bEl\.

mn

for almost all

For each n < W

b E AW.

Therefore

we have

(1)

Choose

b E AW

such that (1) holds, and consider the sequence of finite

models '.flm(m) for 1

By WI -saturation this sequence can be extended

m,

to a sequence of hyperfinite models '+>H(H) , H E >'N.

For infinite H let

be the reduct of 'J3 (H ) to 1. As usual let Co be a countable H dense set of connectives. Using WI-saturation again, there is an infinite ~(H)

H such that for all constant terms T E S1- (CO) and all 1

I TIE (H)

m _ T I <

fz.

<

k

<W

,

By the Stone-Weierstrass Theorem, this holds for

all constant terms T E Sc , and therefore TIE (H) "" Tm for all T E S1~(H)(S)

It follows that rp of 1(S)

WI

p'

WI

r

'P iff m(S)

l=

'P for all quantifier-free sentences

Finally, by the Quantifier Elimination Theorem,

Ii(H) are 1(S)

6.15.

l=

m and

p-equivalent. Let 1 have a countable signature, and let

COR01lARY.

be a set of sentences of

1(S)

WI

model it has a hyperfinite model.

p' If If

r

has a real-valued probability

r has a two-valued probability

model it has a two-valued hyperfinite model. Proof:

By 6. 14 and 6. 9.

VII. APPENDDC

This appendix contains the proofs of the Elementary Submodel Theorem 4.8 and the Elementary Chain Theorem 4.11 in the general case. throughout that 1 has a countable signature. hyperfinite model.

We assume

"Model" means many-valued

HYPERFINITE MODEL THEORY

101

Let Co be the set of polynomials with rational coefficients. hyperintegers A, H and all terms T(x)

E

SL(C

o)'

For all

define K(T, A, H)

inductively as follows. (7. 1)

For T(X) atomic,

(7.2)

If T(x)

=

K(T, A, H)

=:

(H Ilog A)1/4

(maxy)U(X,y) or (miny)U(X,y),

K(T,A, H)

=:

(7.3)

If T(x)

=:

(7.4)

If C

Co and

K(U, A, H) - 1

jT(x, y)dy , 1 K(T, A, H) = "2 (K(U,A, H) - 1) E

C(TI(X), ... , T,e (x»)

T(X) then K(T,A,H)

The point of the above definition is that K(T, A, H) does not depend on a model

but only on the hyperintegers A, H.

~

are fixed, we shall write K(T)

7.5.

=:

When A and H

K(T,A, H) .

If logA E o(H) then for every term T(x) of

LEMMA.

SL( CO), K(T, A, H) is infinite. By induction on the complexity of T(x) .

Proof:

7.6.

Let !ll be an infinite uniform model and let

LEMMA.

H be a hyperinteger with logA E o(H).

Then for every term T(x)

and -a E An , ....

H

~

-

!ll -

1

P{b EA :IT [a] - T [a]l2: K(T)} ~ A

(i)

Proof:

E

SL(C

o)

-K(T)

This is a sharp form of the Elementary Subsequence Theorem

4.6, and has a similar proof.

We only mention the case

H.KEISLER

102

in the induction.

Let t be the maximum of

I1T II , i :=; £. Since C i is a polynomial, its partial derivatives are bounded in [-t, t]£ . Therefore there is a finite k such that for all r,

K(T) :=; ~K(Ti) < K(Ti)

By definition, whenever

I ri - si I

S

Ik

I

Co

[-t, t]£ ,

€

for all i :=;

I/K(T ) for all i -s: i

I C(r) - CIs)

S

€

s.

It follows that

s,

-s k/min(K(T.): i :=; 1

n

:=; I/K(T)

Therefore if (i) holds for each T then (i) holds for T. i 7. 7.

and let

~

LEM:MA.

Let log A

ot H) , let !II be a uniform model,

€

be a submodel of !II such that D

=A -

B

-I ¢

and for every

(i)

(ii)

Proof:

We argue by induction on the complexity of T(i).

The

result holds trivially for atomic T(i) . The steps for the quantifiers (max y) and (min y) are easy. T(i)

= C(TI(i), ... , Tt (i)),

Tl(i) , ... , T (i) . t

€

Co' and assume (i) holds for

For any positive

K(T)

•

and any r,

Iri - si l :=; K(T.) 1

,we have

c; c;

Iril , Isil :=; I1T i li and because by 7.4,

C

€

Let

o(K(T for i:=; i))

S

with ~

c;

~

I C(r) - C(s)1

S

K(T)

t. Therefore, taking a = 3B ID

we see that (ii) holds for T(i) . Assume (i) holds for uri, y) and let T(i) =

SUfi, y)dy.

To simplify

,

HYPERFINITE MODEL THEORY

103

notation we consider the case where x is the empty sequence.

IT~- T~I

s

U

t

L:

U~[b] I

bED

l \' UU[b] / _ -D

I TU

IT

IT~-

=

6

+

3B 1 D"K(U)

+

~~

bED

~ ~

-

L:

Then

U~ [b]

bEA

L

UU[b] [ + bEB

~" K(lU)

U B 'j) B 4B 1 U 'j) B 4B 1 IT (-i)+T "j)[+D"K(U)s!T -T 1"j)+D-K(U) 3B

1

D" K(T) Therefore (ii) holds for T , This completes the induction. 7.8.

LEMMA.

Let U be a uniform model and let D be an

internal subset of U such that D /A

RJ

1.

Then :tl = U I D is an

elementary submodel of U. Proof:

for all a EA.

By induction on the complexity of terms T(X) we show that

The only nontrivial step in the induction is the step from

U(X,y) to T(X)

= SU(x,y)dY,

Assuming the result for U(X,y), we

have [TU[a] -

T~[a]1 =

Ik L:

UU[a,b] bEA

t

L:

U:tl[a,b]/ bED

H.KEISLER

104

os

Ii ~

U~[a,b] -i~ ~ u~[a,b]l+

b€A

It ~ U~[a,b]1

b€A

b€A-D

Therefore the result holds for T(X) Let

ELEMENTARY SUBMODEL THEOREM 4.8.

~

be a uniform

model and let H be a hyperinteger such that log A € o(H), Then for nearly all B € A[H] , Proof:

'13

H os A

= ~ IB

is an elementary submodel of

Since log A € o(A), A is infinite.

~.

Choose a hyperinteger

G such that logA € o(G),

G € O(.jA),

G € o(H) .

To simplify notation we shall assume that H /G is a hyperinteger, although this assumption is not necessary. We shall work with the set A(H) of one to one sequences in A of length H.

o os

For

b € A(H) and

m < H / G, let

Thus the sets Am depend only on Dm €

~( G ]

and

-\n

b restricted to mG. For each m ,

has internal cardinality A - mG.

Let Co be the set of rational polynomials, and consider a term T € SL(C ) ' To avoid difficulties arising when Am is too small, we shall o choose a hyperinteger M < H /G such that

HYPERFlNlTE MODEL THEORY

105

Such an M may be found by choosing e "" 0 large enough so that

e~€ ~'N,

G€O(-!A-H(l-e)),VAsA-H(l-e),

H and putting M = Ci( I - e) •

AM

= A-

MG

K~T)€O(E)

For then

= A-HO-e),

G € O(~)

and

Ei - ell

MG 1 AM· K(T)

H(lK(THA - H(

-c

-

K(T)e

From the definition of K(T, A, G) we see that K(T,Am,G)

2:

K(T,A,G).

7.6 that for all m

S

M,

S K(T)e 1

1!..::..£)

J\n S

b

€ A(mG), and a € An,

for m .s M, so

vA

s AM' in the space A (MG) the event

~ n ~ ~m ~ (3m< M)(3a €A) IT mea] - T [a]1 i': has probability at most

.

A implies

Thus putting K(T) = K(T,A,G), we see from

By Lemma 4. 7,

Since

"" 0

1

if(Tj

H.KEISLER

106

L:

2(AmfK(T) An :s 2MAn

(AM-K(T)

:s 2M(A )2n - K(T) "" 0 . M

m<M

Thus for nearly all bE: A(MG) with respect to the uniform Loeb measure, we have (VT

(1)

10

~ n ~m ~ ~m ~ 1 jL(CO))(Vm< M)(Va lOA) IT [aJ - T [aJI < K(T)

We shall show that (4) implies

where D

=A -

AM

=

range(b).

Assume that bE: A(MG) T(X)

10

SL(C

o) ,

satisifes (1).

and m < M

By Lemma 7.7, for all

we have

1 e_

K(T)

3G

1

--e-

Amt 1

K(T)

:s

3G

Arvr e

1

K(T)

.

Therefore for m :s M we have

3MG

AM

By assumption,

MG

1

AM • K(T) ~

Hence for all a and m es M ,

Using (1) again,

""

o .

1 e

K(T)

a

10

An ,

107

HYPERFINITE MODEL THEORY T

!i)m ill [a] "'T[a]

forallm<M.

It follows by induction on the complexity of T(x) € !i),.~

St(e o) that

~ ~

T La] '" T [a]

(3)

The only non-trivial step of the induction is from Vex, y) to T(X)

=

Assuming (3) holds for V(X,y) , we have

SV(x,y)dy.

!i)-

=

T [a]

1 LJ V [ a , b ] • MG bED

1"

'" M

,,!i)-

LJ

~

V

!i).e -

1

[a,b]·G

.e<M b €D.e

b Thus (2) holds for Now suppose D

z

T!i)£[a] '" £<M

S. S

b L:

T~[a]

£<M

Since (1) is nearly sure, so is (2).

= range( S) , D IB = MG IH '"

€ A(H) and let B

=A -

AM = {b k <: MG}. Then k: !i) is an elementary submodel of 'jl.

7 . 8,

I, so by Lemma

We conclude that the event (4)

holds for nearly all

b

€ A(H) and hence for nearly all B € A[H] .

By

the Stone-Weierstrass Lemma 3.11, every 'jl satisfying (4) is an elementary sub model of

~.

ELEMENTARY CHAIN THEOREM 4.11. model.

Then for nearly all linear orderings

b

Let € A(A) ,

~

be a uniform

108

H.KEISLER

( 'l\: log A € is an elementary chain in Proof: H

=A

otk) , k

< A)

~.

We use the proof of the Elementary Submodel Theorem.

and let G and M be as in that proof.

Choose G

We have already shown in Section 4 that for nearly all

b€

l

Let

such that

A(A) ,

(1)

is an elementary chain in aI.

The proof of the Elementary Submodel

b e A(MG)

Theorem shows that for nearly all

( 'tlmG: 0 ~ m

(2)

is an elementary chain in ~.

Let

b€

,

< M)

A(A) be any ordering of A such

that both (1) and (2) are elementary chains in

~.

For any k where

G -s k < MG, we have l mG

~

k

< (m+l)G

for some m < M.

Since G c O(G , we have G € o(k) and hence l) mG/k "" 1. Therefore by Lemma 7. 8, 'tl is an elementary submodel of mG Similarly, if MG ~ k

< A 'fle have MG/k "" 1 and hence 'tlMG is an

elementary submodel of \ . (3)

( 'l\: logA €

is an elementary chain in aI. elementary chain in aI.

It follows that otk) , k < A)

Therefore for nearly all

b€

A(A) , (3) is an

--I

HYPERFINITE MODEL THEORY

109

REFERENCES

(1]

Anderson, R. A non- standard approach to Brownian motion and its integration. To appear, Israel J. Math.

[2 J

Barwfse, J. Admissible sets and structures. 394 pages. SpringerVerlag, 1975.

(3 J

Bernstein, S. N.

[4]

Bretrnan, L. Probability. 421 pages. Addison-Wesley, 1968.

(5]

Brown, D. and Robinson, A.

Probability Theory (Russian). Moscow, 1946.

Nonstandard exchange economies.

Econometrica 43 (1974), pp. 41-55.

[6 J

Chang, C. C. and Keisler, H. J, Model Theory. 550 pages. NorthHolland Pub!.

cc.,

1973.

Chang, C. C. and Keisler, H. J. Continuous Model Theory.

[7]

165 pages. Princeton Univ. Press, 1966.

Ellentuck, E. The foundations of Suslin logic. J. Symb. Logic 40

[8J

(1975), PP. 567-575. [9 J

Fagin, R. Probabilities on finite models. J. Symb. Logic 41 (1976), pp. 50-58.

[10

J

Fenstad, J. E.

Representations of probabilities defined on first

order languages. Pages 156-172 in Sets, Models, and Recursion Theory, edited by J. Crossley, North-Holland Publ. Co., 1967.

[11]

Ga1fman, H.

Concerning measures on first order calculi.

Israel J.

Math 2 (1964j, pp. 1-18. [12 J

Halmos, P.

Measure Theory. 304 pages. D. Van Nostrand Inc.,

1950.

[13J

Keisler, H. J. Model Theory for Infinitary Logic. 208 pages. North-Holland Publ,

ce.,

1971.

110 [ 14]

H.KEISLER Keisler, H. J. Foundations of Infinitesimal Calculus. 214 pages. Prindle, Weber, and Schmidt Inc., 1976.

[15]

Kuratowski, K. and Mostowski, A.

Set Theory. 514 pages. North-

Holland Publ , Co., 1976. [16]

Loeb, P.

Conversion from nonstandard to standard measure space

and applications to probability theory. Trans. Amer. Math. Soc. 211 (1975), pp. 113-122. [17]

Los, J. Remarks on the foundations of probability.

Proc. Int.

Conq, for Math. of Stockholm, 1962, pp, 225-229. Almquist and Wiksells, 1973. [ 18]

Robinson, A. Pub!.

[19]

co.,

Non- standard analysis. 293 pages. North- Holland

1966.

Scott, D. and Krauss, P. Assigning probabilities to logical formulas. Pages 219-264 in Aspects of Inductive Logic, NorthHolland Publ. Co., 1966.

[20]

Stroyan, K. and Luxemburg, W. A. J. Introduction to the Theory of Infinitesimals. Academic Press, 1976.

[21]

Keisler, H. J. The Monotone Class Theorem in Infinitary Logic. To appear,

R. Gandy, M. Hyland (Eds.), LOGIC COLLOQUIUM 76 © North-Holland Publishing Company (1977) ON THE KIND OF DATA NEEDED FOR A THEORY OF PROOFS

G. Kreisel

Part I I of this article responds to the question impl icit in the title: there are reasonably adequate data for a natural history of proofs, but not for a systematic science. The distinction between natural history and systematic or fundamental science is elaborated in the Introduction to Part II. Part I (§§1-3) prepares for Part I I by 1isting some striking successes of early proof theory and the diminishing returns of later elaborations. The present article complements recent publ ications (Kreisel (1976) and (in press» which stress negative aspects of current proof theory. PART I:

THE PAST

INTRODUCTION Over the last half century proof theory has made a good deal of progress of permanent interest. One need only compare what WP. know now with the impressions current in the twenties concerning such general issues as Hilbert's programme; cf. §l below for two extreme examples. Also quite specific questions raised by mathematicians about the content of thp.ir own rroofs; have been answered with the help of early work in proof theory; cf. §2. This was considerably elaborated in the sixties, both w.r.t. the analysis of various informal notions of proof in the foundational I iterature and w.r.t. the mathematical techniques used. Some of the elaborations show unquestionable logical or mathematical wit, but none has excited broad (active) interest in the si lent majority of logicians; or, perhaps, a ) ittle more pointedly, what authors regarded as interesting about their elaborations, that is, the results stated, left the silent majority unimpressed. Of course, every science produc.es ,ome duJ I stuff; the point here is that the harder results in proof theory are so to speak systematically dull -- with reason (cf. §3and below). The common-place view of the (sociological) facts just described jy that the silent majority lacks the philosophical sensibility needed to appreciate the full inwardness of those elaborations. It can hardly be expected that the silent majority expresses its sensibility very well. But -- and this is perhaps the main point of Pa4t I of this article -- there is also another side to the matter. The elaborate results are stated in language proper to (philosophical) aims which themselves make dubious aeeumptione, For example, the aim of Hilbert's programme was to el iminate abstract methods because it assumed that these methods are unrel iable or otherwise 'unjustified' (or, at least, more so than elementary methods). Now, expectations derived from this assumption have been refuted explicitly by work on Hilbert's programme, and impl icitly by general mathematical experience. This has corrected false first impressions, and thus constitutes philosophical progress. Recognition of such progress (by the silent majority) would show philosophical acumen, not insensibility. Put differently, the concepts which existing foundational schemes (suggested by first impressions) consider to be basic are simply off the mark. To get some perspective on future work, it is natural to look at the past of some successful sciences, and to match up the early stages in their development with those of proof theory over the last 50 years. Two old branches of physics,

111

112

G.KREISEL

kinematics and the mechanics of continuous media, illustrate the kind of thing that has happened in the history of several sciences. Initially only qual itative impressions of the world are available, but this is often sufficient to speculate on the true nature of the phenomena involved. In the branches of physics mentioned, those early ideas were expressed by phrases I ike perfect shape or ideal fluid, and made more expl icit by use of the corresponding branches of mathematics (Euclidean geometry, Laplace's partial differential equations). As time went on, these ideas or conceptions of the world were elaborated by more advanced mathematics -- or else radically revised. Sometimes a conception can be rejected on purely mathematical grounds simply by developing it to a point where it confl icts with very general qualitative or otherwise familiar experience. In such cases, the 1 iterature speaks of imagined (Gedanken) experiments: they need only be mentioned, not carried out since their outcome is not in -- genuine -- doubt. In the theory of ideal fluids the standard example is the result that a steady current exerts no drag (on objects of arbitrary shape). For similar negative examples in logic see §l; for positive ones, see §2, and successful appl ications of elementary geometry or mechanics. As to the elaborations of §3 they may perhaps be compared to the notorious appl ied mathematics of the Cambridge Tripos at the turn of this century; cf. Littlewood (1953). This comparison is of course immensely optimistic: present-day geometry and mechanics have gone well beyond the level of the Cambridge Tripos. The para] lel above will be continued in the introduction to Pant II. -- In Pant I the main use of the parallel concerns the passage from (informal) proofs to adequate formal izations; for example, it can no more be assumed that mathematical texts provide exactly those data which are significant for proofs than that descriptions by sailors of waves in the sea provide the data which are hydrodynamically significant. EARLY SUCCESSES:

FORMAL SYSTEMS AND FORMALIZATION

1. About 50 years ago there was widespread interest in the need for analytic methods in number theory; cf. Ingham (1932). Opinions varied. In accordance with Hilbert's programme some thought that references to reals, sets of reals, etc., were mere shorthand for appropriate approximations, and straightforwardly el iminable. Others thought that the opposite was true. And, above all, both sides thought that it was a matter of 'opinion' that would never be settled. Work in proof theory showed that they were especially wrong where they agreed. (a) Codel's incompleteness theorem, and especially his interpretation of it in footnote 4Sa of Godel (1931), corrected the assumption about the general innocence, that is, el iminabil ity of set theoretic methods, even for proving number theoretic results. Incidentally the assumption was widespread even among those who did not know its precise formulation in the form of Hilbert's programme. -- NB. The correction was discovered in connection with Hilbert's proof theoretic programme. Today it is best to use different results, spl itting the notion of formal system into 2 parts: (definabil ity in a) formal language and formal rules (for the consequence relation). The results for the restriction to formal operations are corollaries of general results in recursion theory; (invariant) definability and the consequence relation are best studied in model theory without this restriction. Also the implications of axioms of infinity asserting the existence of sets of high type On Godel 's footnote 4Sa ) are val id without any restriction to formal systems; cf. p. 182. t.-10 to t.-6 of Kreisel and Krivine (1971) . (b) Actual mathematical practice, in contrast to the possibilities pointed out by Godel, turned out not to use analytic methods in an essential way. As eariy as 40 years ago, Gentzen pointed out that number theoretic practice did not use the full force of first order arithmetic (and so his consistency proof was not needed to 'justify' actual, only possible number-theoretic reasoning).

THEORY OF PROOFS

113

cf. p. 136, 170 and 200 of Gentzen (1969). In the fifties I reformulated his point more generally as fol lows: one uses the language of set theory or analysis, but 'weak' (existential) axioms> conservative over arithmetic. In fact, imagination was needed to find any specific result of analytic practice which cannot be proved by quite elementary methods: I had to turn to such curiosities as the theorem of Cantor-Bendixson (which will come up again at the end of §4 below). In short, the pious business about the collapse of mathematics if set theoretic or even all non-constructive methods are excluded, turned out to be hollow. Historical Remark. The logical status of set theoretic principles actually used in mathematical practice is not widely known even now; cf. Bishop (1967) and Stolzenberg's review, which quote the pious business mentioned above and other impressions current in the twenties, that is, before the work of Gentzen and GOdel. By implication, Bishop's exposition is presented as being the first refutation of those impressions! -- NB. Of course, as Gentzen certainly real ized, these facts about (the limitations of) current practice are not relevant to Hilbert's original programme, which was concerned with the nature of all (possible) mathematics, not with perpetuating current defects. At most, those facts added to the plausibil ity of the programme (provided one started off with the relevant general assumptions of formal ist or nominal ist epistemology). EARLY SUCCESSES: UNWINDING INDIRECT PROOFS 2. Though the incompleteness and conservation results in §l (a), resp. §l (b) were discovered in proof theoretic contexts, they do not refer to the proofs themselves but only to the set of provable theorems. As mentioned already, many such results are better treated by use of recursion theory (in the case of formal systems) or model theory (for general izations to arbitrary sets of axioms) . We now turn to results that involve operations on proofs. As is to be expected from experience in mathematics, especially category theory, close attention to the choice of data used to represent proofs is needed here, for example, to ensure efficiency of the operations involved. More generally, as mentioned at the end of the Introduction, the details of the passage from informal proofs to t~eir formal izations are more significant here than in §l. Historical Remark. The methods used below were discovered in connection with doubts about the 'legitimacy' of non-constructive proofs, sometimes called 'indirect' in the case of existential theorems 3 x F. The purpose was to show (for example, for Ej O formulae) that for some t, F[xlt] can be proved by 'legitimate' methods. This purpose is out of date as soon as the 'legitimacy' of the indirect methods is recognized. But the question remains whether or not a given (indirect) proof n of 3 x F provides an instance t:F[x/t]; and, if, it does, to determine t mechanically> that is, to find mechanical or, equivalently, formal rules: rr ~ trr' In short, the original question of justifying proofs (that is, principles of proof) is replaced by questions of handling proofs (mechanically). (a) Two examples of mechanical unwinding. They need only proof theory of first order predicate logic. They span a period of nearly 30 years. They answer genuine questions, (actually) asked by excel lent mathematicians about their own work.

(i) A theorem of Littlewood (proved in 1914, refuting a guess of Riemann): rr(x) - £i (x) changes sign infinitely often where n(x) is the number of primes < x and £i (x) = fX dx/log x . e

Of course, this means that there is some (recursive) method of determining a, in fact the least natural number n such that n In) > £i (n); since £i (n) is not l ntecr a l , we can compute n In) - £i (n) to sufficient accuracy to determine whether n(n) ;;; ~i (n), and then take the least N:rr(N) > 1i (N). Littlewood's proof consists of two

G.KREISEL

114

parts, one assuming the (still open) Riemann hypothesis, the other its negation. As Littlewood (1953), p. 113 puts it, 'it appeared later that this proof is a pure existence theorem and does not lead to any explicit numerical value [of N].' Unquestionably, the proof is indirect, since it appl ies the law of the excluded middle to an undecided proposition. Nevertheless, an essentially routine appl ication of proof theory (E-theorems or cut el imination) appl ied to Littlewood's original proof, extracts a bound for N; c f , Kreisel (195Z). Of course, this use cannot be expected to give optimal bounds for which further ideas are needed, and it doesn't: it gives a bound of about 10

10

10 34

compared to (1.65)10

1165

found in Lehman (1966). Actually, as Littlewood real ized (p , 113, 1. 13 - 14), the size of N is not the only issue here: by (iv) on p. lIS, a 'further idea' was alleged to be needed to extract a bound from Littlewood's proof, for example, by the switch from TI to W. This impression is refuted, beyond a shadow of doubt, by the proof theoretic analysis which makes a ~outine appl ication (to Littlewood's original proof for TI) of a gene~aZ (logical) method. Historical Remark. To be precise, the ideas involved in the general proof theoretic procedure were applied, not the procedure itself. For one thing, such a procedure operates on a formalized proof, and, of course, Littlewood's original proof was not. But also, at the time the relevant procedures had not been worked out for formal systems very close to the language of ordinary analysis. It turned out that what were obviously the only critical steps in Littlewood's proof could be transcribed into the formal ism of first order predicate logic, to which thencurrent proof theory appl ies; cf. Remark 5.Z p. 171 of Kreisel (1958). A significantly more systematic use of proof theory here would have to make the passage from informal proofs to their formal izations a principal object of study: this was premature before the advent of high-speed computers (at least, if the general impression is right that formalizations of such proofs as Littlewood's are too complex for humans); cf. §4(b). (i i) Unwinding a p~oof of Roth. in a conversation about Roth's theorem: \fn\fa3q (\fq;;'q )\fp(ia _ p/q[> q o

0

A. Baker brought up the following point - (Z+.!..)

n),

where a ranges over the irrational algebraic numbers (and the other variables over the natural numbers). Baker felt morally certain that the proof in Roth (1955) could be 'unwound' to yield some bound for the numbe~ of exceptionally close approximations to a, that is, the number of the set E (of rationals r): 1 n,a -(Z+-)

{r:[a- rl,,;;; q

n}

where r ~ p/q (in its lowest terms),

the bound depending on n and the height of a. But he also felt that this bound would be insufficient for the use he made in Baker (1964) of the bound by Davenport and Roth (1955), which requires a 'further idea.' As it happened the matter of unwinding had been considered in the 1iterature, in Kreisel (1970), pp. 135-136, as an application of Herbrand's analysis of logical theorems of the form 3x\fy R (x,y), R(t l, Yl) v R(t Z' yZ) v ... R(t k, Yk) , where the terms ti do not contain any variable Yj with j ;;. i. This appl ies in an obvious way to Roth's theorem for fixed n and a; with q in place of x and the pair (q,p) in place of y. Inspection then shows that k ?ields a bound on the

THEORY OF PROOFS

115

the number of En u' This supports the first part of Baker's impressions. Modulo the uncertainty about the passage from Roth's informal proof to its formal ization, discussed at the end of §la(i), I have convinced myself that the kn a suppl ied by Herbrand's analysis are indeed too large for the application in Baker (1964). Incidentally, this example provides one of the few useful appl ications of Herbrand's own formulation, specifically, in contrast to the no-counter example -- interpretation which introduces fuctionals; in the case of 3x Vy R(x,y) above, a new function symbol f is introduced (as a 'counter example' \fx-,R[x,f(x)]) and one gets Terms T containing f, such that i,

It will not have escaped the reader's notice that the interests of both Littlewood and Baker were so to speak contemplative, not activist. They had bounds -- and (as mentioned) there are better ones than those suppl ied by the original proofs considered. Contrary to activist propaganda, those interests are no less permanent than the search for better results, which, after all, will be superseded, too. (b) Hilbert's 17 th problem: a case of a knotty unwinding. Artin's solution of Hilbert's problem was stated for archimedean formally real, that is, orderable fields K in which every positive element is a sum of";;; k squares. If P is a polynomial with coefficients in K and p p

=

~(Pi/qi)2,

1";;;

~

0, then

i ,.;;; N, where Pi and qi are also polynomials with

coefficients in K -- How large is

N?

Because of the algebraic character of Artin's proof, it was clear that some such bound for N would depend only on k, the number n of variables and the degree of p; and not on, say, arithmetic properties of K (which might, however, permit 'better' bounds; cf. the end of §2). Historical Remark. In seminars Artin had raised the problem of finding bounds, ever since his original proof in the twenties. He mentioned the problem to me some 20 years ago. As it happened, around the same time, the matter was also considered by A. Robinson, but in model theoretic terms. His proof appeared in Robinson (1955), not long after the proof theoretic solution had been found. Model theory suppl ied a recursive method of determining a bound (by trial and error appl ied to derivations in predicate logic), proof theory a bound involving n-fold exponentiation. As in §l (a), or, for that matter, also in comparable uses of model theory in the fifties, Artin's proof was first transcribed into (first order) predicate logic, but with this difference: two not altogether trivial modifications had to be made, (i) Restatement of Artin's theorem. In contrast to the cases in §! (a), Ar t l n ' s formulation has to be modified since the condition that K be archimedean is not of first order. Inspection of Artin's argument showed that it was only used to ensure that p ~ 0 in all real closed extensions of K. So it could be replaced by this weaker condition. Incidentally, as observed in Henkin (1960), given this replacement a more elegant formulation is obtained for ordered K without the assumption that positive elements are sums of";;; k squares: U~der the new hypothesis, or, equivalently, if p ~ 0 in the (unique) real closure K of K, then c.sK I

and

c. > O. I

As.usual, the proof was uniform for given nand d. So there are polynomials Pi J, qi J in bo~h t~e variables and coefficients of the general polynomial p, and polynomials coJ ,c i J in the coefficients only (J ,.;;; j ,.;;; j) such that, for each j,

G.KREISEL

116 p

•

~

•

.

• 2

I(c.J/c J) (p.J/q.J) . and if P > 0 in I 0 I I J

K,

for some j, each (c.j/c j) > 0 I a (ii) Restatement of Artin's proof. Very much in contrast to §l (a), it can hardly be claimed that the passage from Artin's own proof to a proof in the language of predicate calculus is altogether obvious, even after a reformulation of his theorem is given,as in (i) above. Artin uses an infinite tower of field extensions; determining a bound on the relevant finite portion of it, is tantamount to finding N: and one needs that finite portion before one can even begin to formalize a version of his proof in predicate logic. -- Actually two quite different (first order) proofs of (i) are sketched in Kreisel (1960). One keeps quite close to Artin's proof, the other combines the algebraic identities used in his proof with the completeness of the axioms for real closed fields and the so-called uniformity theorem of logic, a corollary of Herbrand's theorem. The last five years have seen a considerable extension of proof theory; in particular, cut-el imination was extended to (impredicative) type theories in such a way that all cut-free proofs of first order theorems are themselves of first order. In other words, to an arbitrary (higher order) derivation of a first order theorem is associated a particular first order derivation: its cut-free (normal) form. Cut-elimination is then the procedure which mechanizes the mathematician's unwinding of proofs (or is, at least, a candidate for such a procedure). But the matter is not yet settled. Sure, if one insists on replacing Artin's ordinary (algebraic) language by the 'logical' language of type theory mentioned in the last paragraph, the passage from his own exposition to a formal ization of his proof in type theory is pretty unambiguous; certainly as much as for the parallel passages in §l(a). However, this insistence may involve a strategic mistake, in accordance with the ordinary mathematician's instinctive resistance to the language of type theory and to other 'foundational' languages. This issue is a principal topic of §4(b) below.

Remark. At least at present the issue is also contemplative, and not activist (in the sense of the remark at the end of §2(a) above). In particular, numerically better bounds for N itself have been obtained in Pfister (1967) independently of d when K itself is real closed. (Both the proof theoretic and model theoretic methods yield bounds simultaneously for N and the degrees of everything in sight.) More interestingly, for suitable topologies on (suitable) K, one would expect topological versions of Artin's solution, where the p.,q.,c.,c depend continuously on the coefficients of p or where c and q~ Aev~r ~anish 0 I at al I when p ~o in K. Such results are certainly not to be obtained by unwinding Artin's proof. ELABORATIONS AND DIMINISHING RETURNS 3. The main proof theoretic results on first order' predicate calculus used in §2 are. cut el imination and functional interpretations. They have been extended to stronger systemS (including infinitary ones), usually by use of some kind of transfinite iteration or hierarchy, for philosophical reasons discussed in (a) below. When the process which is iterated is 'elementary', the dOminant factor is ti) the 'length' or ordinal of the iteration; tacitly, together with a 'control' at limit ordinals by means of so-cal led fundamental sequences. When already each step of the process involves some kind of 'collection', one has (i i) functionals of higher type; tacitly, together with some defining schema for the latter. -- NB. The distinction between (L) and (t i ) is quite well illustrated by the familiar (i) constructible and (i i) natural hierarchies of sets generated by iterating the socalled predicative, resp. the full power set operation; 'quite well', not perfectly since in the case of proof theory there are additional relations. Specifically, the computability of terms (for functionals) in (ii), is proved by transfinite induction on (orderings with) the ordinals in (i); conversely, 'definitions' by transfinite recursion in (i) are, demonstrably, satisfied by suit-

THEORY OF PROOFS

117

able terms in (ii). -- NB. Many of the elaborations contain material of considerable mathematical charm, as illustrated very well by the lectures of Girard and Ershov at this conference, which state results in a form independent of the original proof theoretic aims. Make no mistake about it: though perhaps exceptionally polished, Girard's material is quite typical of the kind of ordinal structures developed in proof theory (categories of not necessarily well-founded orderings with suitable maps and functors between them), and Ershov's is typical of -- what are nowadays cal led -- cartesian 'closed categories used for functional interpretations when certain informal continuity properties are needed (made precise by Ershov's topology or by appropriate limit space structures); cf. footnote 1 of Kleene (1959) or the 'principal result' (2.4 and 5.1) of Kreisel (1959). Two obvious questions are prompted by what has just been said: (i)

Have we simply not elaborated enough? not put enough resources into the traditional aims? or,

(ii) Have the elaborations only heuristic value? by having led (some of us) to ideas which are effective only when divorced from the original, misconceived aims? Before giving -- what seems to me convincing evidence for (i i) in the case of current proof theory, readers may wish to have some

Examples concerning (i) and (ii) from successful parts of logic.

Ad (i): Mcintyre's lecture at this conference illustrates very well (my) misjudgments on the use of elaborations in model theory, specifically, in connection with Morley (1965), usually considered to be the first piece of 'hard' model theory. Some ten years ago, I had high hopes of glamorous appl ications; cf. p. 152 of the original (French) version of Kreisel and Krivine (1971). These were not fulfil led in the next five years, and so the reference to Morley (1965) was dropped from (p. 186of) the latest (German) version. Mcintyre's results more than fulfill the original hopes, when the new 'hard' model theory is combined with results for suitable mathematical structures (groups, not, e.g., rings, as Mcintyre mentioned), and emphasis is shifted away from simple-minded 'logical' cardinal ity properties to stability and the like. I -- Ad (ii): Gadel 's work on the constructible hierarchy, L, mentioned earl ier, provides an 'excellent object lesson here. L is an (elegant) extension of the ramified hierarchy introduced by Poincare and Russell for their programme of predicative foundations. But as stressed in the clearest possible terms on pp. 146-147 of Gadel (1944), his work drops a requirement which is absolutely essential to that progr9mme, since the relevant properties of the iteration process are not proved predicatively (but in axiomatic set theory itself, which is enough for Gadel's relative consistency results). A somewhat similar twist w.r.t. fundamental sequences for 'proof theoretic' ordinals was used successfully in Jensen (1972). Gadel 's somewhat nostalgic description of his twist, loco cit. as having primarily mathematical, not its original, .philosophical interest seems (to me) to overlook the principa) philosophical issue: Is the predicative programme, admittedly a philosophical affair, also of philosophical interest? And one test is to compare the results obtained by modifying it (as Gadel did) with those obtained by respecting it, as in certain autonomous transfinite progressions; cf. the survey in Feferman (1968). Of course, outside logic there are much more impressive examples of (i i). for example. from the famil iar theory of ideal fluids, The need for such combinations is also fami1 iar from applications of '50ft' model theory (p-adic fields) and of recursion theory (finitely generated groups), and of course from §2 above. -- Warning. The 'contemplative' results of §2 are to be compared to appl ications of model theory in the fifties which consisted in easy (general) proofs of easy (general) theorems, not to the more substantial applications in the sixties. In the fifties, model theorists familiar with the material of §2 found it to be as promising as then-current model theory (an impression which has not been justified so far); cf. A. Robinson's review of Kreisel (1958).

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mentioned in the Introduction. This theory -- or, as one says, this' ideal ization' -- is simply not adequate for its original aim, for hydrodynamics. But the ideas that came from the development of the theory, especially in the two~dimen­ sional case, have permanent value provided they are suitably separated from the original aim (functions of a complex variable or harmonic functions, used to describe the potential and the flow of -- hydrodynamically pretty useless -- ideal fluids). After these illustrations concerning the alternative between (I) and (iii auuve we return to our principal concern, proofs. Here, as promised, is evidence for (i i), both w.r.t. (a) the particular hierarchies mentioned at the beginning of §4, and (b) traditional proof theoretic aims in general. (a) Two strategic assumptions in the construction of hierarchies. The first concerns justifications (of principles P of proof), and is restrictive: P should be justified 'from below', via a hierarchy, by some kind of reduction to more elementary principles than P. The second is permissive: with any (reduction) step, an arbitrary finite iteration is taken to be 'given' too; in short, w-iterations are not counted, nor their w-iteration, and so forth; for a fairly, but by no means absurdly broad sense of 'and so forth', cf. Girard!s lecture. -- NB. Trivially, idealizations are involved here. This Is not the issue at all. What is questionable is the impl icit assumption that they are even approximately adequate, for example, for studying rel iabil ity of proofs. In fact, there is a radical alternative: Don't we do better by reversing the strategy altogether? specifically, by not building up P from below at al I, but by reducing length, the (finite) number of iterations of anyone step. For example, suppose that, for some given P, the passage, in §2, from derivations d to explicit real izations td becomes compl icated. What use is then the possibility of such a passage? One would actually look for P+ which permit -- real istically -- simpler proofs than P, at the -- real istically negl igib!e -- cost of losing that possibi lity altogether. A less extreme example is famil iar enough from elementary arithmetic, where it is certainly futile to reduce numerical terms to numerals (0, sO, ssO, •.. ); instead one looks for new, more efficient notations; for an instructive appl ication in 'advanced' arithmetic, cf. Feferman (1971) and its review where the aspect relevant here is pointed out. -- For reference in §4: Statman (1974) reverses the -- usual -- aim of eliminating cuts in order to reduce not length, but genus. More generally, one might try to mechanize a good deal of the related, fami! iar routine of introducing suitable lemmas for 'cleaning up proofs': after al I, we learn to do this sort of thing almost automatically. Viewed in the I ight of the considerations above, the successes of §2 may well constitute a kind of I imit to useful apptications of the guiding ideas of current proof theory; a kind of optimal value for the ratio: additional information/additional effort in a traditional proof theor.etic analysis. To be a I ittle more specific, we conclude PaAt I of this article with some general ities about proof theoretic aims. (b) Mathematical reasoning and mathematical objects: a discovery. Trivially, the moment we make it our business to be self-conscious about our knowledge (of anything!), the so-called subjective elements of this knowledge become most prominent: they are thought of as particularly close to the thinking subject. In the case of mathematical knowledge, definitions and proofs -- as opposed to the objects defined or to the theorem proved -- are among those elements. As a matter of historical fact, whenever some branch of mathematics began to be analyzed, the first distinctions that came to mind concerned methods: projective and metric methods in geometry, algebraic and differential ones in analysis, and the like. It was a discovery that the particular differences mentioned were more profitably interpreted

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by reference to the different, possibly novel notions or structures for which results proved by different methods are val ido 2 In other words, we have discovered an unexpected adequacy of 'objectivist' analysis. Historical Remark. Traditionally, and more dramatically, one speaks here of confl icting views of the nature of (mathematical) real ity, of a grand conflict between: objective and subjective. This becomes much less dramatic when specialized to famil iar examples: after all, there are projective and metric planes on the one hand, and there are projective and metric methods on the other. Far from being presented with a confl ict, with a choice between different views on what there is, we have a very close relation between methods and objects, so close that the objects concerned can be characterized in terms of the methods; cf. the distinction between those physical objects which are, and those which are not visible to the (ideal i zed ) naked eye. The distinction is objective enough (and not particularly hard to make precise). But -- on present evidence -- it is weak simply because visibi 1ity is not a significant factor in most physical phenomena at all (for which we have viable theories). In short, there is a very real issue here, but much more subtle than the hackneyed business of real ity. Do the reservations (a) and (b) above finish the subject? (of proof theory) Surely not, provided we look for phenomena of mathematical reasoning in which proofs are -- 1 ikely to be -- principal factors; in short, if proofs are to be principal objects of study; if we do not insist on standing on our heads, and think of proofs principally as a means, for example to analyze the 'meaning' of theorems. 0

PART II.

A FRESH START

INTRODUCTION To continue the view of scientific progress presented in the Introduction to P~ T, readers should recall that, at least in existing sciences, the mathematical elaboration of early conceptions and the restriction to imagined experiments soon reached a point of diminoishing returns. Generally -- this seems to be a fact of scientific 1 ife -- progress in the successful sciences really picked up only when striking laws were discovered which had been in doubt or had not even been suspected. In other words, they were found by genuine, not merely imagined experiments. For the present purpose it is not necessary to distinguish between such experiments and what are called observations: the latter concern phenomena that turn up in the course of nature, while experiments involve observations of phenomena in specifically designed situations. Of course, as in all matters of knowledge, observers are not passive: in experiments the external circumstances are manipulated (so to speak' interfered with' before the observation), in observations a selection is made among the raw data. This selection involves, in effect if not by intention, .the notions to which early speculations, discussed in the Introduction to p~ T, had drawn attention; in particular, in the branches of physics considered there, one looks for phenomena exhibiting 'perfect' shapes, 2 For readers interested in constructive mathematics, the corresponding reinterpretation concerns the particular (new) species of operations for which the theorems hold -- as opposed to the methods of proof used to establ ish that the identities hold which the operations are claimed to satisfy. -- Occasionally, there are candidates for the reverse procedure, for example, when Brouwer came up with a characterization of functions F on choice sequences f in terms of his 'fully analyzed' proofs of Vf3n R(f,n), and F satisfies the identity VfR[f,F(f)]. As at the end of (a) above, one expects only a narrow class of cases where the possibility of 'extracting' F is useful: in general, F must be so simple that it is worth writing out a definition in full.

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resp. properties of 'ideal' fluids. Incidentally, even if this normative terminology, of perfection and ideals, is inappropriate if taken to mean what 'ought' to be the case, it occasionaly makes good sense as a hint on what ought to be studied. In short, early speculations come up with such hints.

Natural history wil I be used below for those parts of science which formulate laws close to our ordinary conception of the phenomena involved. This is somewhat broader, but more appropriate than the usual meaning of 'natural history' which excludes all but the most rudimentary mathematical developments of the (ordinary) conception.

As is well known, some early speculations introduced also extraordinary views. The best known such view goes back to the Greeks: the atomic structure of matter. (It is so far removed from our ordinary views that we do not see the world as atomic even after we have absorbed atomic theory). However -- and this is perhaps the principal point of p~ II -- the mere introduction of this extraordinary view was sterile. What was needed was massive progress in natural history,specifical ly the discovery of laws relating data belonging to sciences which are quite different for our ordinary conceptions, for example, laws in spectroscopy relating data from optics and chemistry. (Massive progress was needed, and not merely isolated relations of this kind; for example -- the substance with the composition of -- glass has long been known to be transparent). Naturally, a systematic science, covering phenomena from different branches of natural history has to use extraordinary conceptions(or combinations of ordinary conceptions, which are so compl icated that the difference in degree beCOmes enough of a difference in kind; enough to be consistent with the assumption in force, that we have to do with different branches of natural history). In general, laws relating different branches do not determine a particular extraordinary conception. But they pinpoint an area where such a conception may be tested, The simple observations above will guide the exposition and the interpretations of p~ II. PROOFS AS PRINCIPAL OBJECTS OF STUDY:

NATURAL HISTORY

4.

Explicit definitions have turned out to be fruitful here. They provide the sort of striking phenomena (in the area of mathematical proof) which are needed for natural history in the sense of the Introduction above, And, as required at the end of §3, these phenomena cannot be analyzed in 'objectivist' terms, that is, by anything 1 ike current reinterpretations of the theorems proved. -- NB. This article confines itself to val id proofs though, presumably, fallacious arguments are also important fora study of reasoning (as illusions are for perception). Incidentally, fallacies are often a~equately analyzed in objectivist terms, for example, in those paradoxes which come about when some of the axioms and inferences are val id for one notion, say, of 'set' or 'predicate', others for another notion; cf. the discovery discussed in §3(b). Uses of explicit definitions are conveniently grouped into those where (a) notions are introduced by means of such definitions, and (b) already understood notions with many famil iar properties are analyzed by being shown to be equivalent to others that are expl icitly defined (in some given language). Roughly speaking, in (a) it will often be obvious, in (b) some 'foundational' work is needed to show that the expl icit definitions can be el iminated (at all): but the quantitative differences between (a) and (b) will turn out to be quite del icate.

Examples. (a) An important strategy in applying modern abstract mathematics, say group theory to number theory, is this: (i) An operation 0 is defined number theoretically, say on lZ.xlZ. or modulo some prime. (ii) 0 is shown to satisfy the group laws. (iii) Special izing general facts about groups to 0 leads to number theoretic results. As a matter of empirical fact this strategy has often

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been strikingly successful; probably the most spectacular use is in Weil (1967) which brought to I ight a computational oversight in Siegel (1952). At the same time, here and elsewhere in current practice, the strategy is logically tpivial, that is, el iminable in the following precise sense: Proofs of (ii) have turned out to be formal izable in elementary number theory (though, by incompleteness, this has to be checked), and the general facts used in (i ii) are naturally proved in conservative extensions of that theory (by completeness of predicate logic this is automatic if the facts involved are of first order). Thus the uses of this strategy in current practice are candidates for the~rogramme at the end of §3. -- (b) Much of traditional mathematics, for example, Eucl id's geometry was dominated by the search for correct definitions of famil iar notions (circle, tangent) in Eucl idean terms, for which fami 1iar (geometric) properties of those notions can be formally derived from Euclid's axioms. Clearly, these properties correspond to the facts about groups in (i i i) above, though -- in contrast to the case of groups -- geometric properties were used long before there was an 'official' list of axioms for those familiar notions. Bibl iographical remarks. The examples (a) and (b) above playa basic role in the particular brand of natural history (of mathematics) described, somewhat grandly, on p. 42 of Bourbaki (1948) in terms of 'intuitive resonances' (to mathematical structures). The discovery of (expl icit) definitions which are easy to grasp and to handle provides of course critical evidence for such 'resonances' (and for the significance of the notions defined): evidently, this appl ies to groups in (a) above. Presumably in order to stress the (obvious) similarities between the two kinds of uses of explicit definitions in (a) and (b), Bourbaki play down the differences. Finally, with the relentless logic of epigones, some followers want to suppress altogether the uses quoted in (b) above, for example, in the school curriculum. -- Less grand, but no less emphatic are the references to the importance of expl icit definitions in Wittgenstein (1976), for example. p. 33 or p. 111. The idea, in (b) of pecognizing the equivalence between an understood notion and an explicit definition,that is, of the correctness of the definition is as much anathema to Wittgenstein as to Bourbaki. It is perhaps worth noting that -- apart of course from the obvious literary differences -- there are several other significant similarities in the general views of Bourbaki (1948) and Wittgenstein (1976), above al I, concerning the superficial character of the properties of proofs stressed in traditional foundations. Wittgenstein illustrates his points by means of examples from the most elementary kind of numerical arithmetic, Bourbaki from 'advanced' mathematics. Wittgenstein's skepticism extends to set theoretic foundations, while Bourbaki (seem to) pay I ip service to the language of set theory on p.37. However, on p. 40, footnote (2), they stress the importance of their basic structures without eve~ mentioning the definability of those structures in set theory. These matters wi 11 be elaborated in a review of Wittgenstein (1976) for the Bull. A.M.S., including the errors of both Bourbaki and Wittgenstein due to their ignorance of logic (of the sixties). Granted then the -- obviously -- striking ro~e of expl icit definitions, one looks for specific factors involxed in the effective use of such definitions; tacitly, always together with the pertinent axioms; cf. (ii) in Example (a) above. The reader should recall here from the Introduction (to p~ II) that superficial or 'phenomenological', not 'hidden' factors are treated in (any) natural history; further, that their relevance is not expected to be establ ished purely mathematically from fami 1iar experience: the latter only helps one choose a sensible candidate for a (relevant) factor. -- NB. As in other scientific practice, the facts will be stated in terms of any notions we have learnt to use, without insisting on a premature analysis (of 'how' we know, for example, that a particular spectral I ine is blue); an extreme case is the use of 'metal I ic look' in early mineralogy. (a) Genus of proof figures and the particular use of expl icit definitions illustrated in Example (a) above. Fi rst of all, it is to be remarked that none of the famil iar measures of length is a particularly decisive factor here. The number

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of steps (nodes) is reduced only (roughly) linearly by introducing new notions by explicit definitions. The number of symbols (in formulae) can be reduced exponentially, but striking uses of the strategy in Example (al occur also without such shortening. In any case many arguments are strikingly affected by mere rearrangement, if thereby, say intricate cross references are avoided. This observation is, qual itatively, consistent with using genus as a factor, introduced in Chapter 1 of Statman (1974) where details are to be found. (Roughly speaking, the genus of a proof figure is the least genus of any surface in which the figure can be embedded without crossing). Here are a few rather general features of genus which are not discussed in Statman (1974), but illustrate the kind of considerations needed for the natural history of proofs (and which tend to be disturbing to someone brought up on traditional mathematical logic). (i) Genus is sensitive to the style of formal ization. Specifically, all derivations in a calculus of sequents have genus 0, in contrast to natural deductions (provided the nodes are joined where an assumption is introduced and used).3 Far from being a defect, this sensitivity allows one to choose between the styles; cf. the shift to hel iocentric coordinates in astronomy at the time of Kepler (which was essential for natural history, despite all the business about the relativity of space). (ii) Genus is hard to calculate from the data: one needs a computer even for quite elementary derivations (and a good deal of effort to convert ordinary proofs into data suitable for high-speed calculation). An anecdote concerning such calculations: At Stanford a body of (natural) deductions had been stored in a computer, left over from the homework by students in a computer-assisted course on axiomatic set theory; information on the course can be found in Suppes (1975). Tarjan's algorithm was efficient enough for dividing this ready-made material lnto planar and non-planar deductions. The job would be formidable if instead written homework for an ordinary course had to be prepared for the computer. Far from being a drawback, (i i) seems (to me) promising, granted that genus is a significant factor at all. For one thing, since computers are new, only a short while ago it would have been simply premature to study our subject empirically at all. Less trivially, (ii) reduces a, if not the principal complication in studying human behaviour, whether it be psychological or physiological (which is in practice rarely an important difference). Specifically, knowledge of a theory is 1iable to influence the subject of the observation; in contrast to other sciences where of course the influence of such knowledge on the observer, not on the object of the observation, can be critical, and where one needs safeguards against conscious or unconscious faking (by the observer). If a theory involves notions 1ike genus, which are hard to calculate, there is a better chance that the subjects will not even try -- and so simply will not know the theoretical prediction (even if they know the theoretical Jaws). At the very least, the need for statistical tests is reduced in the case of such intractable theories. Finally -- and quite superficially -- the phenomena (of reasoning) considered here strike us as obscure: so if a theory uses superficial (phenomenological) notions 1ike genus such obscurity would go well with a difficulty in actually applying the theory (to the data); cf. also the last paragraph of the Introduction to Panx II. -- Incidentally, the actual computations of genus for the corpus of proofs by Stanford students mentioned earlier, were not particularly conclusive; at least partly because the whole material was just a bit too elementary to present any really striking differences to start with. What has been said so far is quite sufficient to show that there are plenty of data for a natural history of proofs; in fact, whenever we find striking phenomena 3 This distinction, in Statman (1974), pinpoints for the first time a measure for the difference between the two styles, though it had long been clear that there was some significance to that difference (as suggested in a general way by Gentzen's terminology 'natural deduction').

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which do not invite analysis in objectivist terms (in the sense of §3). The role of explicit definitions discussed above is of course only one of many such phenomena. As observed in the last paragraph, the presentation of those phenomena for theoretical treatment, that is, the choice of data, is quite critical, especially if notions 1ike genus are involved which are so hard to compute that only automatic data-processing is real istic. 4 For a conventiona1 exposition of elementary logic, one of the best known 'reductions' is the expl icit definition of an arbitrary propositional operation (tacitly, together with proofs of its axioms) in terms of (the rules for) say ~ and v. What is lost by such reductions? Statman (1974) shows (actually for the Impl icational calculus, not for --, and v) that the genus can be increased unboundedly. In a lecture at Clermont-Ferrand (in 1975) he considered propositional quantifiers, and showed how length is increased too; those quantifiers allow expl icit definitions since \fpP is 'short for' p[p/T] " p[pll].

Example.

Correction. Though Statman (1974) refers to Kreisel (1973), tacitly, p, 267, his (successful) treatment followed totally different 1ines from what I intended at the time, that is, at the congress at Bucharest in 1971. My idea was that the choice of axioms for explicit definitions would be made with essential help of some kind of traditional philosophical analysis of the meaning of the defined notions, and the combinatorial side would look after itself as it were. Indeed, as J stressed expl ici tly on p. 258 and in the PS, I was looking for sustained philosophical analysis, in contrast to its use in current foundations where it is completely overshadowed by the mathematics (needed to develop the one-l iners expressing those analyses). Statman concentrates on combinatorial, not philosophical analysis. -- Incidentally, five years ago I did not think (consciously) of Bourbaki nor of Wittgenstein in connection with expl icit definitions, though J had read Bourbaki (1948), and I knew Wittgenstein in the forties. Actually, even today I do not remember his talking about the topic in my presence. Evidently he must have done this in view of the quotations from Wittgenstein (1976) given earl ier, and equally evidently, I was not ripe to profit from those remarks. (b) Foundational languages: artifacts. Trivially, the effectiveness of an expJ icit definition of a given notion will depend on the choice of language (to be used in the definiens). What seems to be less weI I-known is that the famil iar foundational languages tend to introduce artifacts, in particular, when notions from geometry including descriptive set theory are defined in the usual way in the language of higher order arithmetic, The examples below come from my own misjudgments. (i) There are two standard proofs of the theorem of Cantor-Bendixson; one identifies the perfect kernel of a (closed) set F (of reals) with its set of condensation points, the other with the I imit of its derived sets. If one insists on transcribing those proofs into second order arithmetic, when F is coded by its set of complementary intervals (with rational end points), then the difference between the proofs is most obviously expressed in model theoretic terms: the former uses IT 1 - comprehension (appl ied to the predicate of being a condensation 2 point), the latter uses only IT] 1 - comprehension; cf. Kreisel (1959A).5 This difference is illusory l nasrnucb as a so-to-speakmoregeometric formalization of the first proof (with additional primitives for real numbers, etc.) can be modelled

4 For example, in contrast to an exposition based on normal ization where the

data include normal ization rules, and so a 'reduction' must preserve normal ization steps, too. 5 The main result of Kreisel (1959A) does not concern the difference between the two proofs mentioned, but the relative complexity of F and of (any second order definition of) the perfect kernel of F.

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by use of TIll - comprehension too. This was discovered by Friedman in recent developments (to third order theories) of his careful formal izations in Friedman (976). Corollary. The (striking) difference between -- careful formal izations of -- the two proofs above does not invite analysis in objectivist terms after all, and is thus a candidate for the programme at the end of §3.

Remark concerning the logician's (so far practically unsubstantiated) impression of a 'need' for higher types in ordinary mathematical practice. Of course, by a footnote 48 in Godel (1931) mentioned already in §l (a), the possibility of a

logical need exists: some arithmetic theorems of the (usual impredicative) theory of type n + 1 just aren't provable in the theory of type n , But the actual: basis for that impression may not be logical at all, but structural; cf. the structural improvements introduced by Friedman's third order language with axioms that are weaker (than those in the original second order formalization). It is certainly of interest for the natural history of proofs to present alternative hypotheses for such impressions (as was done above by distinguishing between logical and structural needs). (i i) Many theorems which are formulated in the language of first order logic (and therefore, by completeness, aan be proved by the usual rules of predicate calculus), are in fact proved by use of geometric and/or analytic methods. An obvious problem for the natural history of proofs is this: Are these mathematical methods needed to make the proofs manageable (by reduction of length, genus or whatever)? In Kreisel (1976) a candidate from the theory of real closed fields was suggested: the theorem establ ished by schut te and van derWaerden (1953) in their solution of Newton's problem of the 13 points (on the surface of the unit sphere). My specific suggestion was to formal ize their solution in usual type tyeory where, in particular, trigonometric functions are expl icitly defined. But closer inspection shows that elementary facts about geometric and topological notions (area of spherical triangles, or Euler's theorem on polyhedra) have quite unmanageable proofs in the language I had in mind. In other words, a representation in that language introduces complexities that are not at all intrinsic to the solution of Newton's problem.

Remark.

It cannot be assumed that to every solution of Newton's problem in a suitable geometric language there corresponds a particular proof in first order predicate calculus (in contrast to solutions in type theory where such a correspondence is provided by normal ization); cf. the transcription of Artin's own work on sums of squares into first order logic discussed in §2(b), as an issue where the 'instinctive resistance' of ordinary mathematicians to foundational languages is relevant. PROOFS AS PRINCIPAL OBJECTS OF STUDY:

SYSTEMATIC SCIENCE

5. To put first things first: How much natural history of the kind suggested in §4 do we need? Can we not learn from the history of successful sciences how to find a systematic scheme without wasteful detours via natural history? Historical Remarks. Much work in natural history was indeed scientifically wasteful; it was superseded by later, fundamental science without being used. One example was already menioned in (the bibl iographical remark of) §4: before X-ray crystallography, the natural history of minerals was preoccupied with shapes and colours or with the distribution of particular minerals in odd corners of the globe. The information obtained was sound enough, but simply does not lend itself to theoretical study. (The same appl ies to much of botany or zoology). A more topical example is provided by the subject of 'natural languages'. It has not inspired much confidence, perhaps because -- as somebody said -- these particular languages do not tell us much about the genuine possibilities (for human language), and even less about the actual world which is supposed to be described by language. -- Cer-

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tainly, there have been bright ideas in the parts of natural history mentioned (perhaps as imaginative as much more successful ones); for example, d'ArcyThompson's on growth and form, or Chomsky's on transformational grammars. Not despite, but rather because of their familiarity and general ity, these ideas do not even smell 1ike germs for a systematic theory. -- As a memorable contrast: The superficially very special idea in genetics, to concentrate on bacteria and viruses, was not only successful for a systematic science of genetics, but immediately convincing: here one could observe -- in a humanly short time -- 50 many more generations that the difference in degree could be expected to produce a difference in kind (of the data involved). Despite all the excitement about the discovery of formalization, at present there do not seem to be any even mildly promising ideas for a systematic or fundamental science of proofs. Worse still, the two pillars of all current work, (a) and (b) below (which -- at the present stage -- are surely sound for the natural history of proofs), seem -- to me -- quite weak. (a) We look for conscious elements in (mathematical) reasoning, or, at most, the very mild extension to those elements which we are able to make conscious to ourselves. As usual, traditional philosophy discusses pedantic doubts (here: doubts about the reliabil ity of introspection, existence of mental objects and the 1ike), and thereby -- consciously or unconsciously! -- draws attention away from the critical issue,namely this: Do those (sub)conscious elements constitute adequate data for anything 1ike a systematic theory? cf. the situation in mineralogy before X-ray crystallography (in the Historical Remarks above), and more generally the inadequacy, for systematic physical theory, of the domain of phenomena that can be made visible. (b) Impressed by its unquestionably distinctive features, we separate mathematical reasoning from other intellectual activities; in humans, and of course in other species. Again, the issue is not the (official) alleged difficulty of making some such distinction precise, and even less the -- equally often alleged-prejudice against assuming intellectual abilities in subhuman species (after all, before very recent advances in electronic technology, we could not even begin to record the more intimate features of their behaviour). The issue is, once again, whether this particular part of intellectual activity, which as it were strikes the mind's eye particularly vividly, is right for (building a fundamental) theory. To repeat: (a) and (b) do not cast doubt on the natural history of proofs, where we make do with with a comparison between stages in the study of nature of) proofs and of matter. The comparison tion; of course, it is not suggested that proofs

poss i b iii ty of progress with the what we have. -- Let me conclude (what is usually called: the seems quite effective for orientaare material substances. 6

A parallel: What is proof? and What is matter? (i) An important step in the development of the atomic theory was the discovery of chemically pure substances, among the mostly impure substances which dominate natural history. This allowed the distinction between atoms and molecules. Bourbaki (1948) mentioned already in §4(a), sounds very much as if their structures - meres (basic structures) are to be compared to chemically pure substances, in particular, to atomic ones. But as stressed in §4(b), foundational analysis has not yet taken the discovery of those analogues to chemical purity into account: By insisting on one foundational language, such as set theory or type theory, one enforces a 1iteral, but purely

6 The comparison is also useful for expanding the discussion, at the Introduction, of the heuristic role of traditional philosophy. After idea of an atomic structure is one of the most striking contributions lative philosophy -- perhaps matched by the idea of the relativity of time, made prominent by critical philosophy.

end of the all, the of specuspace and

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formal unity on mathematics. (On pp. 36-37 Bourbaki too begin with a fantare on the 'unity' achieved by means of their basic structures -- but eight pages later they stress that there are several such structures, and also that quite a lot of mathematics, for example, what is left out of their treatise, is not 'composed' of those structures anyway). Bourbaki, surely quite properly, do not mention in this context the hackneyed business about an objective or subjective view of mathematical real ity. After all, even granted the objectivity of those basic structures, the fact that they contribute to the 'profound intelligibil ity' of mathematics (p. 37) concerns specifically our intellectual equipment. (ii) Building on the discovery in (i), chemical atomism could be developed, and the molecular diagrammes of elementary texts in chemistry. This required only a quite crude idea of atomic structure, the valency of particular atoms; in fancy language, needed for a pun (on the representation of proofs by graphs, as in §4(a) above}, molecules were represented by a purely graph theoretic structure. This can hardly be said to tell us what matter is (possible, as it were), since it leaves open an enormous number of combinations of atoms which never turn up at all. (i ii) A really significant advance was made by relating valency to the intepnal structure of atoms, in Rutherford's theory refined by use of quantum theory. This not only made 'sense' of valency (as is well known). It did much more when Paul ing derived from it a metric structure, specifically, the lengths of and angles between chemical bonds. This additional structure cut down the number of possible combinations sufficiently to make it an effective scientific tool; probably, the most spectacular uses were made in molecular biology. -- It would be a I ittle too facile to compare the use of an internal structure of atoms to going beyond conscious elements in (a) above; or the quite essential information provided by such unfamil iar elements as radium to the need for broader experience in (b). However, it seems quite clear that in one direction the current graph theoretic representation of proofs has to be enriched (even though, presumably not by a conventional metric) since a proof is rarely convincing if the conclusion depends on something which is too 'distant' in our memory. In short, one expects that some hypothesis about memory structure is needed here (and, as mentioned already, the difficulties are much the same whether one thinks of memory in physiological or psychological terms). But also, after we have grasped a proof of a proposition, we use the proposition more easily: as somebody said, we know more when we have proved a proposition than when we merely know it is true. The functional interpretations mentioned in §§2-3 make expl icit some such additional information (which, by §2, is occasionally useful). What is missing is a convincing test for deciding whether this particular addition even approximates the factors which are actually operative. As so often, the trouble is not at all that we cannot think of any theory; if anything, we can think of too many which are roughly consistent with famil iar experience (of mathematical proofs). HEURISTIC VALUE OF TRADITIONAL AIMS:

A OISCLAIMER

6. The view of scientific progress adopted in this article and described in the introductions to p~ I and II disregards the principal specific claims of traditional philosophy -- and the conscious or unconscious hope that something like this tradition would provide a short-cut to a fundamental systematic science, without detours via natural history. In particular, the view disregards the claim that traditional analysis is needed to coprect eppors in our ordinary conceptions, and the associated hope that corrections would so-to-speak automatically lead to the extraordinary conceptions needed for a basic systematic science. In the case of proofs, our idea of a valid argument is claimed to be in need of correction (or at least of some kind of analysis). And the hope mentioned is impl icit in the socalled logical priority of validity which, being the 'essence' of proof, is assumed to be a basic element of any systematic theory of proofs. (A pun may be involved

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here too, since unquestionably, discussions of val idity were temporally, that is. 1iterally prior to theories of proof.) Granted all this. the hope rests on the further assumption that an analysis of val idity (tacitly. in terms real istically available at the present time) would be rewarding. Of course. the parallel to validity in the case of physics is the business of real ity. which ,- in its originally intended generality -- has not been of much consequence for the progress of physics. -- The claims just mentioned are disregarded in this article, but not dismissed nor rejected. After all, we have evolved in the world in which we 1ive. So why shouldn't our built-in so-called a priori conceptions be a very good guide in science, both efficient and re l iable, and not primarily a I imitation, an obstacle between us and the Ding an sich? Certainly, a mild form of paranoia is needed to concentrate only on the limitation -- either with pride in our impotence or in horror of it. ACKNOWLEDGMENT My colleague, P. Suppes, has helped me with careful criticisms of an earl ier draft of this article. REFERENCES: Baker, A. (1964).

Acta Math. 111, 97.

Bishop, E. (196]). Foundations of constructive analysis. (McGraw-Hi 11, New York); reviewed by A. Stolzenberg (1970). Bull. Amer. Math. Soc. 76, 301. Bourbaki, N. (1948). Pp. 35-47 in: Les grands courants de la pensee mathematique ed. F. LeLionnais. (Cahiers du Sud, Paris). Davenport. H., and Roth, K. F. (1955). Mathematika 2. 160. Feferman. S. (1968). Pp. 121-135 in: Logic, Methodology and Philosophy of Science III. (North-Holland, Amsterdam). (1971). Actes Congr~s Int. Math. 1970, Vol. I. 229; rev. J. Symb. Logic 40, 625. Friedman, H. (1976). J. Symb. Logic 41, 558. Gentzen, G. (1969). The collected papers of Gerhard Gentzen. (North-Holland, Amsterdam). Godel, K. (1931). Monatshefte Math. Physik 38, 173. (1944). Pp. 123-153 in: The philosophy of Bertrand Russell (Northwestern Univ. Press, Evanston and Chicago). Henkin, L. (1960). Pp. 284-291 in: Surnrna r i es of talks presented Summer Inst. 5ymb. Logic, Cornell ·Univ. 1957. (Inst. Defense Analyses, Princeton). Ingham, A. E. (1932). The distribution of prime numbers. (Cambridge University Press, Cambridge). Jensen, R. B. (1972). Ann. Math. Logic 4,229. Kleene, S. C. (1959). Pp. 81-100 in: Constructivity in Mathematics. (North-Holland, Amsterdam) • Kreisel, G. (1952). J. Symb. Logic 17, 43. (1958). ibid. 23, 155; reviewed by A. Robinson (1966). ibid. 31, 128. (1959). Pp. 101-128 in: Constructivity in Mathematics. (North-Holland, Amsterdam) •

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G.KREISEL (l959A) . Bu11. Acad. Sc , Po1. ?, 621. (1960). Pp. 313-320 in: Summaries of talks presented Summer Inst. Symb. Logic 1957 (Inst. Defense Analyses, Princeton). (1970). Springer Lecture Notes 125, 128. (1973). Pp. 225-277 in: Logic, Methodology and Philosophy of Science IV. (North-Holland, Amsterdam). (1976). Acta Phil. Fennica 28, 166. (in press). Proc. Fourth Scand. Logic Symp. (North-Hal land, Amsterdam).

Kreisel, G., and Krivine, J.-L. (1971). Elements of Mathematical Logic. (NorthHolland, Amsterdam). Lehman, R. S. (1966). Acta arithmetica

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Littlewood, J. E. (1953). A Mathematician's Miscellany. (Methuen, London). Morley, M. (1965). Trans. A.M.S. 114,514. Pfister, A. (1967). Inventiones Math. 4, 229. Robinson, A. (1955-6). Math. Ann. 130, 257. Siegel, C. L.

(1952). Math. Ann. 124, 17 and 364.

Statman, R. (1974). Structural complexity of proofs. (Dissertation, Stanford University). Suppes, P. (1975). pp , 173-179 in: Computers in Education, IFIP, Part 1 (NorthHolland, Amsterdam). Wei1, A. (1967). Acta Math. 113,1. Wittgenstein, L. (1976). Lectures on the Foundations of Mathematics, Cambridge, 1939 (Cornell Univ. Press, Ithaca, N.Y.).

R. Gandy, M. Hyland (Eds.l, LOGIC COLLOQUIUM 76

© North-Holland Publishing Company (1977)

THE FOUNDATIONS OF MATHEMATICS IN POLAND AFTER WORLD WAR II by

, "

W. MAREK (WARSZAWA) ,

In the period after world war two, Foundation of Mathematics developed in Poland in three main directions:

Set Theory, Hodel Theory, and Recursion Theory.

Obviously the Foundations of Mathematics in Poland were not created in 1945; the roots of interest in these particular domains of sciences one must seek in the investigations of great mathematicians of the between the war period, 1918-1939 and in particular of Banach, Jaskowski, Kuratowski, Lesniewski, Lindenbaum, . ~ukasiewicz, Sierpinski, Tarski and Ulam.

Among those mentioned only Jaskowski,

Kuratowski and Sierpinski lived and worked in Poland after the war.

Banach,

Lindenbaum and Lesniewski vanished or died in the war, and~ukasiewicz, Tarski and Ulam carried over their activities outside of Poland.

It is worth mentioning

that Tarski has maintained contact with Polish scholars up to the present. From the historical point of view, the oldest among the fields listed above is set theory and its foundations.

Immediately after the war, important investi-

gations in this field were carried out by Sierpinski [35], who had already proved during the war the result (previously announced by Lindenbaum and Tarski) that the generalized continuum hypothesis implies the axiom of choice; he subsequently investigated equivalents of the continuum hypothesis. Immediately before the war, Mostowski discovered a general method of independence proofs in set theory with atoms (urelements).

This method, which is nowadays

called the Fraenkel-Mostowski method, is based on the following construction: Let

A

be a set of individuals (i.e. objects which are not sets).

a hierarchy of sets over these individuals as follows: A A U (R~ n P(R~»), for t; > 0 Finally set R = A ; Rt; O \!
We construct U

t;EORD

*The opinions expressed in this paper as well as judgments of the importance of particular papers reflect only the author's opinions and not those of persons and institutions with which he is connected.

These opinions are probably affected by

the author's own mathematical interests and his own ignorances.

"While preparing this paper the author was partially supported by NRC Grant No. A3040.

129

w.

130 Let

G be a group of permutations of

MAREK A ; then for


E G we define the exten-

(~ permutes VA) as follows: ~(a) =
of



which are hereditarily symmetric where a symmetric element is one for which G x set

{