model theory keisler

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Actes, Congrès intern, math,, 1970. Tome 1, p. 141 à 150.

MODEL THEORY

by H. JEROME K E I S L E R

1 . Introduction

Twenty years ago, A. Robinson and A. Tarski lectured on the subject of model

theory to the International Congress at Cambridge, Massachusetts. At that time the

subject was just beginning, and only two real theorems were known. Since then

progress has been so spectacular that today it takes years of graduate study to reach

the frontier. In this lecture I will try to give an idea of what the subject is like and

where it is going.

Model theory is a combination of universal algebra and logic. We start with a

set L of symbols for operations, constants, and relations, called a language; for exam

ple, L = { + , . , 0, 1, < }. The language L is assumed to be finite or countable

except when we specify otherwise. A model 1 for the language L is an object of the

form

vl

=

<

A,

+Q1,

-5J, 1 % %, < r>.

A is a non-empty set, called the set of elements of 9Ï,

+gj

and .& are binary operations

on A x A into A, 0% and lgj are elements of A, and << is a binary relation on A.

E X A M P L E S . — The field of rationals, < Q, + , ., 0, 1 >, is a model for the language

{

+,

., 0, 1 }. So is every other ring, lattice with endpoints, etc. The ordered field

< ß ,

+> •> 0,

1, < >

is

a model for the language { + , ., 0, 1, < }. Each group,

partially ordered set, graph, etc., is a model for the appropriate language.

Most results in model theory apply to an arbitrary language. We frequently

shift from one language to another, for instance a new theorem about a given language

is often proved by applying an old theorem to a different language.

Many facts about models can be expressed in first order logic. In addition to the

operation, relation, and constant symbols of L, first order logic has an infinite list of

variables

x, y , z,

v

09

v

l9

v

29

..

.,

th e equality symbol =, the connectives

A (and), V (or),

-i

(not),

and the

quantifiers

V (for all), 3 (there exists).

Certain finite sequences of symbols are counted as terms, formulas, and sentences.

The class of terms is defined as follows:

Every variable or constant is a term;

If t, u are terms, so are t + u, t- u.



142 H. J. KEISLER G

T he

formulas

are defined by the rules:

If t, u are terms, then t = u, t < u are formulas.

If

<p,

i//

are formulas and

v

is a variable, then - |

cp,

-<p

A

ij/,

<p

V

\j/,

Vvcp, 3vq>

are

formulas.

A

sentenc e

is a formula all of whose variables are bo un d by quantifiers. F or exam

ple, the sentence

(1)

Vx (x = 0 V3y (x-y=

1))

states that every non-zero element has a right inverse.

Hereafter 31 = <

A,.

.. >, 93 = <

B,

. .. > ,. . . den ote mod els for

L,

and p ,

\j/,

6,...

denote sentences.

The central notion in model theory is that of a sentence

cp

being

true

in a model

31,

in symbols

31t=

<p.

This relation between mo dels and sentences is defined ma the

matically by an induction on the subformulas of

<p.

It coincides exactly with the

intuitive conce pt. Fo r exam ple, the sentence (1) is true in the field of ration als b ut

no t in the ring of integers. A set of sentences is called a

theory.

3ïis a

model of

a

theory

T,

in symbols 31N

T,

if every sentence cp

e T

is true in 31.

E X A M P L E S . — The theory of rings is the familiar finite list of ring axioms found in

any mo de rn algeb ra text, an d each ring is a mo del of this theory . The theor y of real

closed ordered fields is an infinite set of sentences, consisting of the axioms for ordered

fields, the axiom stating that every positive element has a square root, and for each

odd n an axiom stating that every polynomial of degree n has a root.

For each model

3t,

th e theory of

31,

T h

(31),

is the set of all sentences true in

31.

M ode l theory is a rich subject which studies the interplay betw een variou s kinds

of sentences and various kinds of models.

2. Two classical theorems.

M odel theory traces its beginnings to two basic theorem s which come ou t of the

1930's. The math em atician s wh o prove d them are the founders of the subject.

C O M P A C T N E S S THEOREM. — If every finite subset of a set T of sentences has a model,

then T has a model.

This theore m was first prov ed by God ei, 1930 for cou ntab le language s. M alcev,

1936 extended the theorem to the case where

T

is a set of sentences in an uncountable

language . The compactness theorem has many applications to algebra (see R obin

son, 1963).

Example.

— Suppose the sentence cp is true in every field of characteristic zero.

Then there is an

n

such that

<p

is true in all fields of characteristic

p > n.

Proof. — C onsider the set

T

of sentences consisting of the field axioms, the sentence

~i

cp,

and the infinite set

- , (1

+ 1 = 0),

i

(1 + 1 + 1 = 0),

i

(1 + 1 + 1 + 1 = 0 ) , . . .



M ODEL THEO RY 143

By the hypothesis, T has no models, so some finite subset of T has no models, and the

conclusion follows.

By the

cardinal

of a model

31

we mean the ca rdinal of the set

A

of elements of 31.

L ö W E N H E IM -S K O L E M -T A R S K I T H E O R E M . — If T has at least one infinite model,

then T has a model of every infinite cardinality.

Example.

— L et

T

be the theory of real closed fields. Th en

T

has a model of car

dinal 2

K

°, nam ely the field of real num bers. Th ere are cou ntab le real closed fields

and also real closed fields of every other infinite ca rdinality. Th e L S T theo rem

shows that this happens in general.

Both of the theorems above assert that a certain kind of model exists, and their

proofs depend on techniq ues for constru cting mo dels. Indee d, alm ost all the deeper

results in mode l theo ry depend on the con struction of a mo del. W e shall indicate

some of the most useful methods of constructing models and state some of the theorems

which they yield.

3. The method of diagrams.

This method, d ue to Henkin, 1949 and R obinson, 1951, is the basis of Henkin's

proof of the Godei com pleneness theorem . It also has many other uses.

The

diagram language

for 31 is obtained by add ing t o L a new constant symbol

a

for each element a of A. The elementary diagram of 31, denoted by Diag

(31),

is the

set of all sentences in the diagram language of 31 which are tru e in 3Ï. Th e difference

between Th (31) and Diag (31) is that Diag (31) has new constant symbols for the ele

ments of 31 while Th (31) does not.

In many situations it is possible to construct a model of a set

T

of sentences by

extending

T

to a set of sentences

T'

which happens to be an elementary diagram of

some model 31. In this construc tion one is always wo rking

with

sentences, and cons

tant symbols are used for the elements of 31. The compactness and L S T theorems

can be proved by this method . The construction has many other app lications; we

shall state three of them without proofs.

The notation cp

N

\j/ means that every model of cp is a model of

i//.

THEOREM

1 (C raig interpolation theorem, C raig, 1957, A . R obinson, 1956). — S up

pose

cp 1=

\j/. Th en there is a sentence

6

such that cp

\= 6, 6

}= \j/, and every operation,

constant, or relation symbol which occurs in

6

occurs in both cp and i/r.

The next theorem concerns homom orphisms. A mapping

h

of

A

onto

B

is called

a homomorphism, and

23

is called the homomorphic image of

31

by h , if for all a, b, e A,

h a

w

b) = h a)

93

h b),

h(U

n

) = 1»,

a << b

implies

h a)

<©

h b),

etc.

If h is one-one and h~

1

is also a homomorphism, then

h

is called an

isomorphism.

It is obvious that every sentence cp is preserved u nde r iso m orp hic im ages, tha t is,

every isomorphic image of a model of cp is a model of cp. But which sentences are

preserved under homomorphic images?



144 H . J.

KEISLER

G

A sentence cp is said to be positive if it contains no negation symbol —i, i. e. it is built

using only A , V, V, 3.

T H E O R E M 2 (Lyndon homomorphism theorem, 1959). — A sentence cp is preserved

under homomorphic images if and only if there is a positive sentence ij/ which has

exactly the same models as

cp.

The hard direption is only if .

Examples. — The theories of groups, abelian groups, rings, and fields (if we allow

the one element field) are preserved under homomorphic images because their axioms

are positive. But the theo ry of integral dom ains is no t preserved unde r ho m om orph ic

images. It has the axiom

Vx Vy

(x

= 0 V

y =

0 V -i

x-y

= 0),

and this axiom cannot be replaced by a positive sentence.

A theory is complete if it is equal to Th (31) for som e 3Ï. L et us consider the num ber

of (non-isomorphic) countable models of a complete theory T.

Examples.

— W e have examples of complete theories with exactly one countable

model (atomless Boolean algebras); N

0

countable models (algebraically closed fields);

2

X

° countable models (real closed fields); and

n

countable models for each

n >

3

(due to Ehrenfeucht).

But the following surprising theorem is due to Vaught, 1959.

THEOREM 3 . — There is no complete theory which has exactly two countable models.

4.

Elementary chains.

This construction was introduced by Tarski and Vaught, 1957.

31

and 93 are said to be

elementarily equivalent

if Th

(31)

= T h (93), that is, they are

models of exactly the same sentences.

3Ï is said to be a submodel of 93, 3t c 93, if A c B and the operations, constants, and

relations of 31 are those of

93

restricted to

A.

31 is an

elementary submodel

of

93,

3f-<23,

if 31 c 93 an d every senten ce of Dia g (31) is tr ue in 93. A simple exercice: if 31 •< 93

then 31 and 93 are elementarily equivalent.

Example.

— Tarski, 1948 has shown that if 93 is any real closed field and

31

is a real

closed subfield of 93, then

31 -<

93. S imilarly for algebraically closed fields. S uch theo

ries are called

model complete

(R obinson, 1963).

A n

elementary chain

is a sequence of models

3 I

0

, 3 l

1

, . . . , 3 I

a

, . . . , a < y ,

where

y

is an ordinal, such that

if a < ß < y then

3I

a

-< 31^.

The

union

of an elementary chain is the model

31

= U

a y

3I

a

such that

A =

U

a<7

A

a

and each

3L

is a submodel of 31.



M O D E L T H E O R Y 1 4 5

THEOREM 4 (Tarski-Vaught, 1957). — Let

3ï

a

,

a <

y,

be an elementary chain. Then

each 31^ is an elementary submodel of U

a<y

<

U

a

.

A

typical application of this construction is the following

Löwenheim-Skolem-

Tarsk i type result for pairs of cardina ls. F or this theo rem w e assum e that the language

contains a one-placed relation symbol

U.

By a

model of type

(X

a

,

tt

p

)

we mean a

model 3Ï such that

A

has cardinal X

a

and 1% has cardinal

tf

ß

.

THEOREM

5 (Vaught, 1962). — Suppose a theory

T

has a model of type (K

a

,

K

p

)

where

N

a

>

tt

p

. Then

T

has a model of type (H

l9

X

0

).

The model

31

of type ( ^ , X

0

) is constructed as the union of an elementary chain 3I

a

,

a < (o

l9

of H

x

countable models such that all the sets l/gr

a

are the same.

M any results in model theory depend on the Generalized C ontinuu m Hypo thesis

(GC H), which states that for all infinite cardin als

N

a

,

2

K

* = X

a

+ i- O ne such result

is the following.

THEOREM 6 (C hang, 1965) (GC H). — S uppose a theory T has a m odel of type

(Ki>

N

0

).

Then

for every H

a9

T

has a model of type (N

a

+

2 s

X

a

+i).

The proof uses an elementary chain of length H

a+2

of models of cardinality

X

a + 1

.

Example

( G C H ) . — L e t

31

be the model

3I = < Ä , + ,

. , 0 , 1 ,

< , Z > .

where R is the set of real numbers and Z is the set of integers. 31 is a model of type

( N i ,

K

0

).

By C hang's theorem, Th (31) also has a model of type (N

2

, X J . But Th (31)

cannot have a model of type

X

2

,

X

0

).

5. Ultraproducts.

This construction was introduced by Skolem, 1934 to get a non-standard model

of arithmetic and in its present general form it is due to L o s , 1955.

L et

/

be a non-empty set and let 3I

{

,

iel,

be models for L . A n

ultrafilter over I

is

a set

D

of subsets of

J

such that

D

is closed under finite intersections, any superset

of a member of

D

is in D, and for all

X

a

I,

exactly one of the sets X

9

1

— X

belongs

to

D. A

statement

P(i)

is said to hold

almost everywhere

(D) if the set of

i e I

for which

P(i) holds is in D.

N ow consider the C artes ian product

H

ieI

Ai

. F o r

f, ge H

ieI

At

we write

f=

D

g iff /( 0 =

£ ( 0 a . e .

(D)

Then =

D

is an equivalence relation on n

t e I

i 4 , . L et

f

D

be the equivalence class of /,

and

Yl

D

Aj

the set of all equivalence classes.

The ultraproducl

H

D

%

is a model with the set of elements

n

fl

>4

f

.

The relation <

on this model is defined by

f

D

<g

D

iff

f(0< g<J) a.e. D) .



146 H. J . KEISLER G

The operation + is defined so that

A

go

=

h

D

iff

f(i)

+^g )

=

h(i)

a. e. D ) .

The fundamental result about ultraproducts is the following.

THEOREM

7 (Los , 1955). — For each sentence c p

9

co holds in the ultraproduct n

D

3 I

£

if and only if cp holds in almost everywhere

(D).

Ultra prod ucts can be used to give us anothe r proof of the C ompactness Theorem .

M any applications of the C om pactness The orem can be done more neatly using ultra-

products directly.

Example.

— Suppose al l the models 31̂ are fields, and form the complete direct pro

duct ring n

f 6

j3 I

£

. It turn s out tha t the set of ultra pro du cts n

l )

3 I

£

is exactly the same

as the set of quotient fields

U

ie

j LJJ

of the ring n

i e I

3 I

f

modulo a maximal ideal

J

(The

fields U

ieI

%/J were studied by He witt, 1948; see

Gillman-Jerison,

1960).

Suppose al l the models 3l

f

are the same model 31. The n the ultraprodu ct n

D

3t is

called an

ultrapower

of 31. By the theorem of L o s , 31 is elem entarily eq uivalen t to

each ultrapower n

D

3l .

Example

(non-stand ard analysis, A . R obinson, 1966). — L et

31

be the model

3I

= <R, + , . , 0, 1, < , . . . >

where

R

is the set of real numbers, and the three dots stand for a list of all the 2

2

°

operations, constants, and relations on R. L et

D

be an ultrafilter over the set

co

=

{ 0, 1, 2 , . . . } which con tains no finite set. Th en the ultrapow er n

D

3l is a non-

A rchim ede an rea l closed field; for instan ce, < 1, 1/2, 1/3, 1/4,

1/5,...

}

is a posi

t ive infinitesimal an d < 1, 2, 3 , . . . >

D

is positive infinite. Using the ultra pow er n

D

3t,

the whole subject of analysis can be based on infinitesimals in the style of Leibniz.

Fo r example, consider any real function / an d real num ber s

c

and L . Then

lim f(x)

=

L

if and only if for every b in HjyA which is infinitely close but not equal to c, f(b) is infi

nitely close to L.

Ultrapowers can also be used to give purely algebraic characterizations of model-

theoretic notions such as elementary equivalence.

THEOREM

8 (Isomorphism theorem). — Two models 31, 93 are elementarily equiva

lent if and only if there is an ultrafilter

D

such that n

D

3l and n

D

93 are isomorphic.

This theorem was proved by K eisler, 1963, using the GC H , and was proved without

the GC H by Shelah, 1971.

A mong the impo rtant to ols in model theory are the saturated models; they are

used in theorem s 6 and 8 above. The ultrap roduc t is one way of constructing such

models. L et X

a

be an uncountable cardinal .

31

is

^-saturated

iff for every set

<D

of fewer than X

a

formulas cp(x) in the diagram language of 31, if for each c p

l9

..., c p

n

e <b

the sentence

3x cp

t

x) A . . . A cp

n

x))

is true in 31, then the infinitely long sentence

3x

A

<pe0

cp x)

is true in

31.



MO DEL THEOR Y 147

THEOREM 9 . — Let J be a set of power X

a

. Th ere is an ultrafilter

D

over J such

that every ultraproduct

n

D

3ï

f

is X

a+1

-saturated.

The above result was proved under the G C H by K eisler, 1963, and w ithout the

GC H by K unen , 1970 .

X

a+ 1

-saturated

models were first constructed in another

way by Morley-Vaught, 1962.

Example. — It turns out that a real closed field is X

a

-saturated if and only if its order

ing is an

n

a

-set

9

th at is, for any two s ubsets X, Y of power < K

a

(perhaps empty), if

X < Y then there is an element z such that X < z < Y.

There are a num ber of applications of saturated models to algebra. Fo r ex am ple,

they are the main tool in the proof by A x and K ochen, 1965 of A rtin 's conjec ture:

for each positive integer d

9

the following holds for all but finitely many primes p. Every

polynomial in the field

Q

p

of p-adic numbers, with degree

d

9

more than

d

2

variables,

and zero constant term, has a non-trivial zero in Q

p

.

6. Indiscernibles.

Suppose we expand the language L by adding

n

new constant symbols c

x

,..

.

,

c„,

forming L „. Fo r each model

31

for L an d each rc-tuple a

t

,...,

a„

of elements of 31,

we obtain a model (31,

a

1

,...

,

a

n

)

for L

n

. C onsider a subset X of A and a l inear

ordering < of X

9

which is not necessarily one of the relations of 31. W e say tha t

<

X

9

<

> is a set of

indiscernibles

in 31 if for any

n

and any two increasing

n-tuples

a

x

< . . . <a

n9

b

1

< . . . < b„

from <

X

9

<

>, the models

(31, a

x

,...,

a„) and

(31, b

l9

..

.

9

b

n

) are elementarily equiva

lent. The basic result below shows that there always are models with indiscernibles.

THEOREM

10 (E hrenfeucht-M ostowski, 1956). — L et T have infinite models and

let < X

9

<

> be any linearly ordered set. The n there is a model 31 of

T

such that

< X

9

< > is a set of indiscernibles in 31.

The construction of the model

3Ï

uses the parti t ion theorem of R amsey.

Examples.

— L e t 31 be a field and

93

be the ring of polynomials over 31 with the set

X

of variables. The n for any linear ordering < of X, < X, < > is a set of indiscernibles

in 93.

L et 31 be a non-A rchim edea n real closed ordered field and let

X

be a set of positive

infinite elements such that if x < y in X then x" < y, n = 1, 2 , . . . Then X with

the natural order is a set of indiscernibles in 31.

Indiscernibles are used to prove results such as the following (Two elements a

9

be A

have the same

automorphism type

if there is an automorphism of

A

mapping

a

to

b).

THEOREM

11 (E hrenfeuch t-M ostowski, 1956). — If T has an infinite m ode l, then

for every infinite cardinal X

a

,

T

has a model of power X„ with only countably many

automorphism types.

The following very deep results use both the method of indiscernibles and saturated

models.

A theory T is said to b e ^-categorical if all m od els of T of cardinal tt

a

are isomorphic.



148 H. J.

KEISLER

G

THEOREM

12 (M orley, 1965). — If T is X

a

-categorical for some uncountable X

Œ

,

then

T

is X^-categorical for every uncountab le X ^ .

S helah, 1970, extended Theo rem 12 to unc oun table languages.

THEOREM 13 (Baldwin-Lachlan, 1970). — If T is X

x

-categorical, then either T is

X

0

-categorical or

T

has exactly X

0

models of cardinal X

0

.

W e men tion one theorem at the opposite extreme from the above.

THEOREM

14 (S helah, 1970). — S uppose

T

has a model

A

such that for some formula

cp(x

9

y)

and some infinite set

X

c

A

9

the relation

{ ( f l , f c > £ l

2

: 31 h=

cp(a,

b)}

is a linear order. Th en for every unc oun table X

a

,

T

has

2

X a

non-isomorphic models

of cardinal X

a

.

Example.

— The theory of algebraically closed fields is X

a

-categorical for every

uncountable X

a

and has X

0

coun table mo dels. Th e theory of abelian groups with

all elements of order two is X

a

-categorical for every X

a

. The theory of real closed

fields has 2

X

" models of each infinite cardinal X

a

. The theory of atom less Boolean

algebras is X

0

-categorical but has 2

X a

models of each uncountable cardinal

X

a

.

7.

R ecent trends.

The model theory of first order logic contains a number of substantial results, but

unti l recently only the compactness theorem h as had ma ny applications. This si tua

tion is changing and will change more as the subject becomes more widely known.

O ne of the bott le-necks has been that m ost properties arising in mathem atics cannot

be expressed in first orde r logic. Fo r this reaso n there is a strong mo ve towar d m odel

theory for mo re powerful logics. In the last few years there have been exciting deve

lopments in the model theory of the infinitary logic L

mim

. Th is logic is like first order

logic except tha t it allows the connectives A an d V to be applied to cou ntab le sets

of formulas, that is, if c p

0

, <p

l9

cp

2

,... are formu las of L

aia9

then so are

cp

0

A cp

x

A cp

2

A . . . , cp

Q

M cpyV cp

2

V . . .

The formulas may thus be countable in length.

Examples.

— The sentence

Vx (x = 0 V x + x = 0 V x + x + x = 0 V . . . )

is true is an abelian group G if and only if G is a tor sion gro up . Th e sen tence

Vx ( x < l V x < l + l V x < l + l + l V . . . )

is true in an orde red field if and only if it is A rchim ede an.

Both the C om pactness Theorem and the L S T Theor em in their original form are

false for L

aitD

. Fo r the lat ter, note that every A rchimede an ordered field has power

<

2**°.

N evertheless, it turn s out that all of the me thod s from first order m odel

theory can be used in L

mito

. M any of the m ain results have been generalized to



MODEL THEORY 149

L

a

a

, often in a more subtle form. For example, the L S T Theo rem take s the follow

ing form. The cardinal 3

a

is defined by the rule

J

0 — 0» -

i

a+l ~~

z

>

3

a

=

l

ip<a

'2p

for limit ordinals a.

THEOREM 15 (M orley, 1965). — L et cp be a sentence of L

ai(0

. If cp has a model of

cardinal at least 3

f f l l

, then c p ha s m odels of every infinite ca rdin al.

The proof is much deeper than the L S T Theorem . It uses the parti t ion calculus

of Erdós and R a d o , 1956, and also yields an analog of Theorem 10 on indiscernibles

for L

aia

.

Theorem s 1 and 2 above were extended to L

mita

by L opez-E scobar, 1965, The orem 5

by K eisler, 1966, various forms of Theorem 12 by C hoodnovsky , K eisler, and S helah,

1969, and Theorem 14 by Shelah, 1970.

A nothe r basic result is

THEOREM 16 (Scott, 1965). — For every countable model 31 there is a sentence cp of

L

aito

such that 31 is a model of cp and every countable model of cp is isomorphic to

31.

This result is analogous to Ulm's theorem for countable abelian torsion groups.

In fact, L

mxa

has been applied by Barwise and Eklof, 1970 to extend Ulm's theorem

to arbitrary abelian torsion groups.

The model theory for

L

aia

is greatly enriched by the use of recur sion theo ry as a

way to get a hold on infinitely long sentences (a suggestion of K reisel). Th is has led

to the Barwise C om pactn ess T heo rem (Barwise, 1969) which is the ana log for

L

aim

of the C ompactness T heorem.

A nother type of logic whe re mod el theory h as had recent successes is logic with

extra quantifiers, such as " there exist infinitely many

and

there exist uncountably

man y ". Fo r more inform ation see the pap er [12].

A m ajor recent trend is the im pac t of set theory on model theory an d

vice versa.

A

num ber of problems have been shown to be consistent or independent using C ohen 's

forcing, notably by Silver. M oreo ver, forcing itself is being used as a techn iqu e for

constructing models (see A . R obinson's lecture in this C ongress). O ther results

have been proved on the basis of strong hypotheses such as the existence of a measu

rable cardinal (R ow bottom and Gaifman, 1964, S ilver, 1966, K unen , 1970,) or the

axiom of constructibility. Fo r exam ple, Jensen, 1970 has shown that if the ax iom

of constructibil i ty holds then C hang's Theorem 6 above can be improved to :

If

T

has a model of type ( X j, X

0

) then

T

has a model of type (X

a + 1

,

X

a

).

B I B L I O G R A P H Y

The three new books [4], [7] and

[11]

taken together cover most facets of model theory

and contain up to date references as of 1970. A n extensive list of references to jou rna l articles

prior to 1963 can be found in [1]. W e have decided that a list of books on model theory and

related topics will be more useful at this time than another long list of references to the ori

ginal articles in journals.



150 H. J. K EISL E R

[1] A D D IS O N , H E N K I N and T A R S K I , edi tors . —

The Theory of Models.

A msterd am , 1965.

N o r t h H o l l an d P u b i . C o .

[2] J. L .

B E L L

a n d A . B .

S L O M S O N .

—

Models and U ltra produc ts.

A m s t e r d a m , 1 96 9. N o r t h

H o l l a n d P u b i . C o .

[3] C . C . C H A N G and H. J . K E I S L E R . —

Continuous Model Theory.

Pr inceton, 1966. Pr ince

ton Univ. Press.

[4] _ . _

Model Theory.

N e w Y o r k , 1 9 7 1 . A p p l e t o n C e n t u ry C r o ft s.

[5] P . J. C O H E N . —

Set Theory and the Continuum Hypothesis.

N ew Yo rk , 1966. W . A . Ben

jamin , Inc .

[6] P . M . C O H N . —

Universal Algebra.

N ew York , 1965. Harp er and R ow.

[7] M . A . D I C K M A N N . —

Uncountable Languages.

A m s t e r d a m , 1 9 71 . N o r t h H o l l a n d

P u b i . C o .

[8] L . G I L L M A N a nd M . JE R I S O N . —

Rings of Continuous Functions.

Pr inceton, 1960. D. Van

N o s t r a n d C o . , I n c .

[9] G. G R ä T Z E R . —

Universal Algebra.

P r ince ton , 1968. D. Van N os t ran d C o . , Inc .

[10] C. K A R P . —

Languages with Expressions of Infinite Length.

Amsterdam, 1964. North

H o l l a n d

P u b i .

C o .

[11] H. J . K E I S L E R . —

Model Theory for Infinitary Logic .

A m s t e r d a m , 1 9 7 1. N o r t h H o l la n d

P u b i . C o .

[12] — . — L ogic with the quantifier « Th ere exist unco untab ly ma ny ».

Annals of Math.

Logic,

vol. 1 (1970), pp . 1-93.

[13] G. K R E I S E L and J . L . K R I V I N E . —

Elements of Mathematic al Logic

A msterd am , 1967.

N o r t h H o l l an d P u b i . C o .

[14] R . C . L Y N D O N . —

Notes on Logic.

P r ince ton , 1966. D. Van N os t rand C o . , Inc .

[15] A . M O S T O W S K I . —

Thirty Years of Foundational Studies.

Helsinki , 1965, N ew York ,

1966.

Barnes and N oble , Inc .

[16] A.

ROBINSON.

—

Introduction to M odel Theory a nd the

Metamathematics

of Algebra.

A m s t e r d a m , 1 9 6 3. N o r t h H o l l a n d P u b i . C o .

[17] — . —

Non-standard Analysis.

A m s t e r d a m , 1 96 6. N o r t h H o l la n d P u b i . C o .

[18] J. B. R O S S E R . —

Simplified Independence Proofs.

N ew York , 1969. A cademic P ress .

[19] J. R . SHOENFIELD. —

Mathematical Logic.

R eading , M ass . , 1967. A ddison W esley

P u b l is h i ng C o .

[20] R . M .

S M U L L Y A N .

—

First-order Logic.

N ew York , 1968.

Springer-Ver lag.

[21] M . A . T A I C L I N . —

Theory of Models

(R ussian ) . N ovosibirsk, 1970.

[22] J.

VA N

H E I J E N O O R T . —From

Frege to Godei.

C am bridge, 1967. Ha rvard U niv. P ress.

U n i v e r s i t y o f W i s c o n s i n

D e p a r t m e n t o f M a t h e m a t i c s ,

M a d i s o n , W i s c o n s i n 5 3 7 06

( U . S . A . )

model theory keisler

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