Building Models by Games 0521268974, 9780521268974

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London Mathematical Society Student Texts: 2

Building Models by Games WILFRID HODGES School of Mathematical Sciences Queen Mary College, University of London

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CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney

Published by the Press Syndicate ff the University of Cambridge The Pitt Building, Trumpington Sbreet, Cambridge CB2 1RP 32 East 57th Street, New York,

i

N~

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10 Stamford Road, Oakleigh, Melb~urne 3166, Australia © Cambridge University Press 198~

First published 1985 Printed in Great Britain at the University Press, Cambridge British Library Cataloguing in Publication Data Hodges, IHlfred Building models by games. -

(London Mathematical

society student texts; 2) 1. Mathematical models I. Title

II. series

511'. 8

QA401

ISBN 0 521 26897 4 hard cover ISBN 0 521 31716

9 paperback

j:

jl

'::ONTENTS

Introduction Chapter I

iii

PRELIMINARIES

I,I

Pictures

1.2

Model theory

Chapter 2

5

References

I6

GAMES AND FORCING

I7

A way of building models

I7

2.2

Games

23

2.3

Forcing

27

References

34

2.I

EXISTENTIAL CLOSURE

35

3.I

Adjunction of elements

36

3,2

Existentially closed models

47

3,3

E.c. groups

59

3.4

Robinson forcing

72

References

80

Chapter 3

CHAOS OR REGIMENTATION

83

4.I

Mass production

84

4.2 4,3

Atomic models Finite-generic models

95 IOS

4.4

E,c, nilpotent groups of class 2

Ill

References

130

Chapter 4

Chapter 5

CLASSICAL LANGUAGES Classical omitting types

5.2

Set.-theoretical interruption:

5,3

Saturation

152

References

167

Chapter 6

I

unbound~d subsets

133 144

PROPER EXTENSIONS

170

6.1

Largeness properties

171

6.2

Definable ultrapowers

185

6.3

Uncountable boolean algebras

195

References

209

Chapter 7

GENERALISED QUANTIFIERS

211

7.1

L(Q)

212

7.2

Omitting types in L(Q)

223

7.3

Magidor-Malitz quantifiers

235

References

248

L(Q) IN HIGHER CARDINALITIES

250

Chapter 8 8.1

There is a problem

251

8.2

Completeness and omitting types

260

References

273

List of types of forcing List of open questions Bibliography Index

132 I

5,1

275 276 282 303

INTRODUCTION

Most mathematical books are about some particular structure (such as the complex numbers) or class of structures (such as Banach algebras),

This book is not,

for building structures,

Instead it is about a very general method

For a precedent one should look for a book on

Cartesian Products, or perhaps on Left Adjoints of Forgetful Functors. The method of construction is easily described. and I want to build a group.

Suppose you

I write down a partial specification:

You add some further information: b

2 c , bd " db.

I continue: a= I, df

2

g •

Your turn:

And so on.

The only constraints are that we should each write down a

finite amount at a time (try violating that!), and that we should never contradict anything which has been written down so far,

After an infinite

number of steps we, shall have assembled a set S of equations and inequations.

There will be a group G, unique up to isomorphism, which is

generated by a, b, c, ....subject to the equations in S,

Together we have

built the group G,

(The inequatioJs that went into S played an indirect

role in the definition of G: into S later,)

they.prevented us putting certain equations

Now let me ask a question,

Given a property P which the group

G might have, is it possible for you to make your choices so that G will have property P regardless of what I write down? If it is, we say that p is an enforceable property.

Clearly the property of being non-abelian is

enforceable - you can always find two new letters x, y and add xy f yx at your next turn.

Less obviously, it is enforceable that all elements of

order 7 lie in the same conjugacy class of G. (i


-

1

("not"),

1\

("and"),

("iff"), Vx ("for all x"), 3x ("there is

For example the sentence "Every non-zero element has a

multiplicative inverse" can be written as a first-order formula: Vx(lx=O

(J 0)

+

3y x.y=l).

Two important examples of non-first-order ways of building formulas are infinitary conjunction

1\

and infinitary disjunction \f:

if {~i: i E I}

is a set of formulas, then l\iEI$i is the formula which says that all the formulas ~i (i E I) are true, and \fiEI~i says that at least one of the formulas

~i

is true. The class of all formulas of L built up in the ways described

above is called L

OOUl

The first subscript

~

means that there is no bound

on the size of conjunctions and disjunctions;

the second subscript w

means that we can only have finitely many variables x , ••. , xn in a 1 quantifier Vx ••• xn or 3x ••• xn. Actually the definitions above only 1 1 allowed quantifiers Vx or 3x which contain a single variable x. But we regard Vx •.• xn as shorthand for Vx Vx •.. Vxn' and' likewise with 3. 1 2 1 On the same principle, Lww is the set of formulas of L~w in which we only form conjunctions and disjunctions of finitely many formulas at a time.

Since f.inite conjunctions and disjunctions can be built up

with /\'and v, Lww is in practice the set of first-order formulas of L.

1.2

Model theory

8

More generally, LKA is 'the set of formulas built up from the symbols of L in the ways described above, allowing conjunctions and disjunctions of fewer than K formulas at a time, and quantifiers where x, y are sequences of fewer than A variables.

Thus L

WIW

vi

and 3y

, which we

shall meet several times, has finite and countable conjunctions and disjunctions but only finite quantifiers. If we start from the atomic formulas of a signature L and allow formulas to be built according to some fixed set of rules, the resulting collection of formulas is called a language of signature L. For example L

WW

L.

, L

WIW

and L00

W

are three different languages of signature

A first-order language is a language of form Lww There will be no harm if we use the symbol L for languages as

well as signatures.

Each language L determines its signature, and hence

it makes sense to speak of L-structures when L is a language.

Similarly

if L is a first-order language, it is clear what language is meant by L

WlW

We shall freely use standard abbreviations, such as x#y for lx=y.

We write

i

for a tuple (x , ••• ,xn_ ) of variables. 1

0

xwill mean an infinite sequence of variables.) formula

~

A formula ~(x) is a

whose free variables (i.e. variables y not bound by quantifiers

VY or 3y) all occur in

x,

and when nothing is said to the contrary we

assume that x is a sequence of distinct variables; ~Cx , ••• ~(t

(But sometimes

1 0 0 , ••• ,tn_ 1)

,xn_ ) is introduced, and t , ••• , tn_ 0

1

when a formula

are terms, then

is the resulting formula when each free xi

in~

is replaced

by ti. A negated atomic formula is a formula

l~

where

~

is atomic.

A

formula is basic iff it is either atomic or negated atomic. Let L be a language and A an L-structure.

A closed term of L

is a term of L (i.e. of the signature of L) which has no variables. closed term t of L names an element tA of A. of L with no free variables.

f=

Each sentence

~

of L makes a statement about

~ ("~ is true in A~!, "A is a model of ~" or "A satisfies

A;

we write A

~ )

iff this statement about A is true.

11

Each

A sentence of L is a formula

A theory is a set of sentences.

A first-order theory is a

theory in a first-order language. If L is a signature and A is an L-structure, then the theory

1.2

Model theory

9

of A, Th(A), is the set of all sentences of Lww which are true in A.

We

say that two L-structures A and Bare elementarily equivalent, in symbols

= Th(B).

A: B, iff Th(A)

For any theory T, A~ T ("A is a model of T") means A~ all

E T;

for

when nothing is said to the contrary, we assume that T is in

some fixed language L and A is an L-structure, i.e. that A has no irrelevant symbols. T

=. Th(A).

T

1-

For a first-order theory T, A~ T means the same as

("T entails ", " is a consequence of T") means that

every model of T is a model of . We say that (i) and ~(x) are equivalent modulo T iff T 1- Vx( ++ ~).

More generally we write ~(x) for a set ~ of formulas of

form (x), and we say that ~(x) and W(x) are equivalent modulo T iff T

1-

Vx(l\~ ++

1\w).

We say that two theories are equivalent iff they have

exactly the same models. When K ·is the class of all models of the theory T, we say that T axiomatises K, or is a set of axioms for K.

If T axiomatises the class

of fields, then we say T is the theory of fields.

In a similar spirit we

talk of the theory of groups, the theory of lattices etc. dangerous terminology:

This looks

there are different possible choices of signature

for groups, and even in one fixed signature there are infinitely many ways of axiomatising the class of groups. be no problem.

But in practice I think there should

For example we shall prove a theorem about "strict

universal Horn theories" and then apply it to "the theory of groups".

If

the reader has in his hand a set of axioms for groups which isn't strict universal Horn, he should look for another set which is. always be a natural choice which works.

There will

Thus a natural signature for

fields, or more generally for any rings with I, consists of the symbols +, -, ·, 0, I;

we can call this the signature of rings with I.

A natural

set of axioms for fields in this signature is given at (11) below. The following lemma is proved for first-order formulas in elementary courses, but it is true quite,generally: LEMMA 1.2.2 (Lemma on constants). Let (x , ••• ,xn) be a 1 for-muZa of signature L, T a theory of signature Land c , •.• , en distinct 1 constants not in L. Then T ~ (c 1, ••• ,c ) iff T 1- Vx ..• x .

Proof. 33£.

n · 1 n ·For the.first-order case, cf. Shoenfield (1967) p.

[J

1.2

Model theory

10

Expansions and parameters Let Land L+ be signatures with L ~ L+. L+-structure.

Let B be an

We can convert B into an L-structure by deleting the

symbols not in L (without removing any elements of B).

Then A is called

the L-reduct of B, in symbols BIL, and B is called an expansion of A to L+.

(Note the difference between expansions and extensions:

an expansion

adds new symbols but no new elements, while an extension adds new elements but no new symbols.)

If W is the set of symbols which are in L+ but not

in L, then we can refer to L+ as L(W).

The same notation applies if Land

L+ are languages rather than signatures.

For example if L is a first-

order language and W is a set of constants which are not in L, then L(W) is the first-order language got from L by adding all the constants in W. Suppose L is a language and A is an L-structure. Maybe not all the elements of A are named by constants of L. However, we can add to L a new constant a for each element a of A.

(For example we can choose

each constant~ to be the element a itself.

These new constants will be

"symbols" only in a formal sense, but this will be no problem.)

The

expanded language is called L(A), and A can be expanded to an L(A)structure D by putting

;n

a for each a E dom(A).

We shall very often

write A ~ q such that r l~w cj>. iff there is no q ;:: p such that q l~w cj>. p ~~w 1/J.

(d) p l~w 1

(e) p l~w cj>AI/J iff p l~w cj> and

(f) p l~w 'v'xi/J(x) iff for every closed term t, p ~~w 1/J(t).

(g) Every condition weakly forces every logically true first-order sentence of L(W).

(h) If p ~~w and ~ 1/J then p ~~w 1/J.

(j) It can happen

that for some atomic sentence cj> and some condition p, p doesn't strongly force V1cj>,

[For (g) and (h), choo3e a particular proof-calculus for

first-order logic and go by induction on the lengths of proofs.]

REFERENCES FOR CHAPTER 2 2.1:

The compactness theorem for countable first-order

languages is due to Glldel (1930).

Without the cardinality restriction it

is due to Mal'tsev (1936) and Henkin (1949).

The proof given above,

emphasising notions of consistency, is based on Smullyan (1963). Exercise II: for more on this theme see van den Dries et al. (J

982);

they construct elementarily equivalent groups whose commutator

subgroups are not elementarily equivale~t.

2.2: by

s.

Exercise 4(a,b) was conjectured by S. Mazur and proved

Banach, in about 1930. 2.3:

Cf. Oxtoby (1971) Chapter 6.

Types of forcing tend to be grouped into two species,

strong and weak, according to the rules for forcing cj>VI/J and 3xi/J; Exercise 7. forced;

cf.

In weak forcing, all logically true sentences are everywhere

in strong forcing only intuitionistic laws are forced by all

conditions.

(Our forcing is weak.)

of forcing are quite different,

The motivations behind· the two kinds

Strong forcing is simply the classical-

style truth definition for intuitionist logic, and the defining clauses correspond to the meanings of the intuitionist connectives, cf. Kripke

(1965),

Weak forcing is a way of looking at dense subsets of a partial

order (cf. Exercise 6 of section 2.2).

It's a historical accident that

Cohen (1963), in his brilliant work on independence in set theory, discovered weak forcing via strong forcing and the 11-interpretation of classical in intuitionist logic.

Abraham Robinson (Barwise & Robinson

1970) adjusted Cohen's definitions to give a general way of constructing models.

Keisler's survey (1973) shows how Robinson's weak forcing

generalises omitting-types constructions, cf. Chapter 5 below.

Ziegler

(1980) finally removed the red herring of strong forcing by defining Robinson's weak forcing in terms of Banach-Mazur games.

35 3

EXISTENTIAL CLOSURE

Tant qu'on a de nouveaux eZements a introduire, on doit araindre d'avoir a reaommenaer tout son travaiZ; or iZ n 1arrivera jamais qu'on n'ait pZus de nouveaux el-ements introduire ••• Et aZors des points qui n'etaient pas definissabZes deviendront susaeptibZes d'~tre definis; d'autres qui Z 1etaient aesseront de Z 1 ~tre.

a

Poincare, La logique de 1 1 infini (1913) Poincar~ 1 s

essay, published in 1913, was part of the

metaphysical hubbub that followed the discovery of the foundational paradoxes.

It is a pity that nobody read his words mathematically.

fact he raised two excellent questions.

In

First, when and how can elements

be added to a structure so that the result is another structure of the same kind?

And second, if new elements are added, what does this do to

definability of elements or sets of elements within the structure?

To the

best of my knowledge 1 the earliest frontal attack on the first question was by B•..H. Neumann in his paper (1943) on groups';

1

Adjunction of elements to

in principle he gave a general solution for all varieties.

Progress on the second question was more gradual. Why is this relevant to forcing?

The answer is that in a

construction by forcing we are continually adding new elements - if not to a structure, then at least to a partial description of a structure.

When

a structure is compiled by games, we can enforce that it contains "all possible kinds of element", This chapter and the next are about a particularly simple kind of forc.ing invented by Abraham Robinson.

He called it finite forcing to

distinguish it from the cruder infinite forcing that we shall meet in Chapter 5.

In Robinson 1 s (f_inite) forcing it is always enforceable that

3.1

Adjunction of elements

36

the compiled structure is existentially closed.

In section 3,2 I say what

this means, after laying the groundwork on adjunction of elements in section 3.1. example;

Section 3.3 takes existentially closed groups as a concrete

the high point is Martin Ziegler's beautiful theorem on the form

of resultants in groups,

3.1

Section 3.4 develops Robinson's machinery,

ADJUNCTION OF ELEMENTS Let L be a first-order language and T a theory in L.

extension problem for T runs as follows.

The

Given a model A of T, an

existential formula ~(y) of Land a tuple a of elements of A, when does there exist a model B of T such that A c B and B I= Ha)? For example if A is a group with elements a and b, when is there a group B ~ A in which a and b are conjugate?

This is a case of the

extension problem for groups:

take T to be the theory of groups (cf. 1 section 1.2), and take ~(y,z) to be the existential formula 3x(x- yx = z). There is such an extension B iff a and b have the

The answer is known,

same order, in other words, iff A I= 1\n(x) be the

It will be enough to show

then there is a model B of T,

(For then if w(x) E Res~ we have B

Suppose that A

I=

T and A

I=

f

w(a) and

A4\(a) but there is no such

We shall derive a contradiction. The first step is to add a "free" solution of

w(a,y)

to A.

For this we introduce new constants c to name the solution, and we define c+ to be the canonical model of the set (9) { e

e

is an atomic sentence such that T U diag+(A) U {w(a,c)}

Let C betheL-reduct of C+.

Since C+

f

r e}.

diag+(A), there is a homomorphism

g: A+ C (as in the diagram lemma, Lemma 1.2.3).

Also Cis a model ofT,

I omit the proof of this (which uses the assumption that T is strict universal Horn), and instead I point to some examples.

3.I

4I

Adjunction of elements

If T is the theory of groups and w(a,y) is a conjunction of equations s.(a,y)=I (i E I), let F- be the free group generated by the L

C

elements c, and let N be the normal subgroup of the free product A*Fc which is generated by the elements si(a,c). Then C will be (A*Fc)/N. is clear what the homomorphism g is; in fact g is an embedding iff

It

AnN= {I}. If T is the theory of abelian groups and w(a,y) a conjunction of equations si(a,y)=O, then let Fe be the free abelian group generated by c, and let H be the subgroup of A$Fc generated by the elements si(a,c). Then C will be (A$Fc)/H, and again we have an embedding iff An H = {0}. The same description works for modules. In the case of commutative rings we form the polynomial ring A[c] and then factor out a suitable ideal I to get C example above. We return to the proof.

A[c]/I, as in the

F w(ga,c).

By construction, c

If g

is an embedding, then we can identify A with its image under g, and C will be an extension of A containing a solution oi w(a,y). has no such extension.

But we assumed A

Therefore g is not an embedding.

g fails co preserve some negated atomic formula. sentence 8(a,d) such that A f o8(a,d) but c

This means that

So there is an atomic

F 8(ga,d).

Since c+ was the

canonical model of (9), it follows by the compactness theorem that there is a finite conjunction x(a,d,e) of sentences in diag+(A) such that T ~ X(a,d,e) A w(a,c) + 9(a,d). Using the lemma on constants (Lemma 1.2.2) three times, this yields that T ~ vi(3yw(x,y) + v;;(x(x,;,;) + 9(x,;))). Now v;;(x(x,~,;) + 9(x,;)) is strict universal Horn, so we have proved that it lies in~. But A oV~w(x(a,;,w)+B(x,;)),

F

since x(a,d,e) and o9(a,d) are both true in A. assumption that A

F 1\~(a).

This contradicts our

Thus part (a) is proved.

For part (b) let be 3y(w(x,y)A"lOI (x,y)A,, .AIOm(x,y)) where

w is

as before and cri, ••• , am are atomic.

universal Horn formulas in Res.

Let ~(x) now be the set of all

Again it suffices to assume that A I= T

and A I= /\~(a) but there is no model B ofT such that A~ Band B I= (a), and deduce a contradiction.

We begin by constructing C just as before, by

adding a free solution of w(a,y) to A,

By tbe argument of part (a), the

homomorphism g: A+ ,C is an embedding;

without. loss

~ve

can suppose A c: C.

Now since C is not an .extension of A satisfying cp(a), we must have C

F cri(a,c)

for some i (I ~.i ~ m).

But then by the same argument as for

3.1

Adjunction of elements

42

part (a), there is a conjunction x(~,e) of finitely many sentences from diag+(A), such that T ~ x(a,e) A $(a,c)

+

constants, T ~ Vi(3y($(i,y)

+ V~,x(i,~)).

A

,cr.(i,y)) ~

cr.(a,c). ~

By the lemma on So V~,x(i,~) is in

~. and again we have a contradiction to A~/\~(~).

[J

The general theory can usefully be pushed one step further, Let us say that the theory T in language L has the amalgamation property iff the following holds: (10)

If A, B, C are L-structures such that A c B and

A~

C, and all

three are models of T, then there are a model D of T such that B ~ D, and an embedding g: C

+

D which is the identity on A.

For example T is known to have the amalgamation property if T is any of the following "theories:

groups, abelian groups, left modules over a fixed

ring, lattices, distributive lattices, skew fields. A (strict) quantifier-free Horn formula is the same as a (strict) universal Horn formula, except that it is quantifier-free. THEOREM 3.1.4. LetT be a strict universaL Horn theory in a first-order Language L whose signature has at Least one constant. Suppose T has the amaLgamation property. Then: (a) If ~(i) is a positive primitive formuLa of L, then Res~ is equivaLent.moduLo T to a set of strict quantifier-free

(b)

Horn formuLas of L. If ~(i) is a primitive formuLa of L, then Res~ is equivaLent moduLo T to a set of quantifier-free Horn formuLas of L.

Proof.

Copy the proof of Theorem 3.1.3, but instead of

extending A to C, extend A to C.

(The assumption that L has a constant

is needed to make sure that A is well-defined when a is empty.) lemma on preservation (Lemma 1.2.4(a)), A is a model ofT.

By the

By (10),

since A~ A and A~ C, there is a model D ofT such that D ~A and C can be embedded in Dover A; on preservation (Lemma 1.2.4(b)),

so if C ~ ~(a) then D I=~(~) by the lemma The effect of working with A in + -

-

place of A is that any sentences from diag A can be written as x(a) with no further constants added. in the previous proof.

This disposes of the variables ~ and w [J

3.1

Adjunction of elements

43

In the proof of Theorem 3.1.3 we formed C by adding a free solution of ~(a,y) to A.

Speaking practically, this is often the best way

to start looking for resultants, algebra

F~

with A,

A*F~,

The first step is to construct the free

in the appropriate class, and then to form its free product The feasibility of the next step depends on whether we have

a good algebraic description of

A*F~.

In commutative rings, as we saw,

A*F~ is just the polynomial ring A[~], and this makes calculations rather

easy.

Abelian groups and modules are even easier.

Nilpotent groups of

class 3 are about at the limit of what one can handle by brute force;

in

higher nilpotency classes we don't know what free products look like.

The

exercises below and section 4.4 will illustrate all this. Not all interesting theories are strict universal Horn, of course.

(Cf, Exercise S(a).)

But finding resultants for other theories

seems to be more a matter of insight and good luck. One such theory which has been studied very thoroughly is the theory T of fields.

Here every primitive formula is equivalent to a

positive primitive one, since an inequation sit can be rewritten as 3w (s-t)w =I.

(This is known in some contexts as Rabinowitsch's trick.)

One of the achievements of nineteenth century mathematics was to find a way of computing, for every positive primitive formula

~

of rings, a single quantifier-free formula equivalent to

of the language Res~

modulo T.

Vander Waerden (1950) Chapter XI describes the method in detail, remarking that "In practice the required calculations are very often too complicated to be carried out effectively".

General model-theoretic

arguments show quite easily that the resultants for T can be be brought to this form (cf. Exercises 13, 20 in section 3.2 below), but they give no useful guidance on how to compute the formulas in question.

Exercises for 3.1. I.

Let L be the first-order language of rings with I.

(a) Show that if

p(x) is a polynomial in x with integer coefficients, then the polynomial equation p(x)=O can be written as an atomic formula ~(x) of L, assuming the L-structures under discussion are rings,

(b) Show the same when p(x)

has rational coefficients, assuming the L-structures under discussion are rings of characteristic 0. 2. Show that each of ·the following classes is axiomatised by a v2 firstorder theory T, and describe the class of models of TV in each case:

3.1

44

Adjunction of elements

(a) The class of algebraically closed fields, in the signature with +, •, 0, I.

(b) The class of commutative von Neumann regular rings (i.e.

commutative rings such that for each element x there is y with xyx

= x)

the same signature as (a).

(c) The class of boolean algebras, in the

signature with A, v, 0, I.

f(a) integral domains, (b) commutative rings

in

with no non-zero nilpotent elements, (c) distributive lattices with 0 and !.]

A 'language is said to be reaursive iff its terms and formul-as are encoded as natural- numbers so that aZZ the basia syntaatia operations, suah as substitution for a free variabl-e or aonjunction of two formul-as, go over into reaursive functions. Virtual-l-y every aountabZe first-order 'language that one meets in praatiae aan be encoded as a.recursive 'language. 3.

Let L be a recursive first-order language and Tan r.e. theory in L.

(a) Show that the set U

=

{e : e is a sentence of Land T

~

e} is r.e.

(b) (Craig's trick) Show that the theory T is equivalent to a recursive theory in L.

[Replace 8 by 8A ••• A8 (k times) where k is the Gtldel number

of a computation putting

e into T.]

The definition of (cartesian) product of groups generalises at once to products rriEIAi of arbitrary L-structures; cf. Chang & Keisl-er (19?3) p ..1?? or Cohn (1981) p. 49. We say that a theory T is closed under nontrivial products iff for every non-empty famil-y Ai (i E I) of model-s of T, rriEIAi is aZso a model- of T. We say that T is aZosed under produats iff it is cl-osed under non-triviaZ produats and the one-eZement structure (whose eZement satisfies aZZ atomic formul-as) is a model- of T. 4.

Let T be a theory which is closed under non-trivial products.

Show

that if wi (i E I) are a non-empty family of atomic sentences and T ~ \I.EIW·• then T ~ ~ ~ 5.

w.

~

for some i E I.

(a) Show that every strict universal Horn theory is closed under

products, ana. every universal Horn theory is closed under non-trivial products.

(b) Show that Theorem 3.1 .3(a) remains true if we replace

"strict universal Horn theory" by "theory closed under products". 6.

Let T be a theory in a first-order language L.

We say the universal

sentence problem for T is solvable iff there is an algorithm which

v1 formula e of L, whether or not T ~ e. (a) Show that if the universal sentence problem for T is solvable, then there is an

determines, given any

algorithm for determining, given any 3

1

formula ~(i) and V formula w(i) 1

3.1

Adjunction of elements

of L, whether or not wE

Res~.

45

(b) Assuming that Tis closed under non-

trivial products, show that if there is an algorithm which determines, for any 3

1

formula ~(x) and any atomic formula w(x) of L, whether or not w E

Res$, then the universal sentence problem for T is solvable. 7.

LetT be a theory in a first-order language L with at least one

constant.

Show that if TV has the amalgamation property, then for every

formula ~(x) of L, Res~ is equivalent modulo T to a set of quantifier1 free formulas of L.

3

8.

Let T be a strict universal Horn theory in a first-order language L

with at least one constant, and let U be the set of atomic sentences 8 of L such that T ~e.

Show that U is =-closed in L, and that the canonical

model of U is a model of T.

Use this to complete the proof of Theorem

3 .I .3.

9.

Suppose T is a strict universal Horn theory satisfying the

amalgamation property.

Suppose that in Theorem 3.1.1, a group G of

automorphisms of A is given.

Show that in (a) of the theorem we can add

"and the automorphisms in G extend so that G acts as a group of automorphisms on B".

The remaining exercises caZcuZate resuZtants in various theories. need speciaZist knowZedge. 10.

Some

Let R be a ring with I and T the theory of left R-modules (cf.

Exercise 2(c) of section 1.2).

(a) Show that if $(x) is

3yA.(l:.)J .. y. = l:.v .. x.) where llij' vij are ring elements, then Res~ is l. J l.J J J l.J J equivalent modulo T to the set of all equations l: . . A.v .. x. such that the l.,J l. l.J J family (A.) of ring elements satisfies l:.A.)J .. = 0 for each j. (b) Show l. l. l. l.J that if Hx) is 3y(A.(l:.)J .. y. = l:.v .. x.) A Ah(l:.llh'.y. i' l:.vh'.x.)), then l. J l.J J J l.J J J J J J J J Res~ is equivalent modulo T to the set of equations given in (a) together with, for all h, the set of all inequations l:.(vh'.- l:.cr.v .. )x. i' 0 such J J l. l. l.J J that the family (crl..) of ring elements satisfies )Jh1 • = l:.a.)l .. for all j. J l. l. l.J (c) The ring R is said to be left coherent iff for every finite n and R-linear a: Rn + R, ker a is finitely generated.

Show that if R is left

coherent then the resultants in both (a) and (b) are equivalent modulo T to finite sets of formulas of the forms shown.

[If R is left coherent,

then for all finite n and m, every R-linear a: Rn + Rm has finite kernel, cf. Chase (1960) Theorem 2.1.]

3.1 II.

Adjunction of elements

46

Let L be the first-order language of rings, with symbols+,-, ., 0,

I, and in L letT be the theory of commutative rings.

Show the following:

(a) If ~(x,y) is xly (i.e. 3z xz = y), then Res~ is equivalent modulo T to the single formula Vt(ta = 0 + tb = 0), (b) If n ~ I and ~(x) is 2 3uw(x=u+w A u 1u A uw=O A wn=O), then Res~ is equivalent modulo T to Vt(tan+l=O + tan=O). (c) If n ~ 1 and ~(x) is 3y(y 2=y#O A xy=y A (x-y)n#O), then Res~ is equivalent modulo T to xn#O

A

Vt t(I-x)#O.

(d) If ~(x) is 3y(y 2 =y#O A xly) then Res~ is equivalent modulo T to the set of formulas xn # 0 (n ~I).

(e) If A is a commutative ring with I, a

an element of A, a is not nilpotent and 1-a is not invertible, then there is a commutative ring B ~A containing an element b # 0 such that ab = b 2 = b and a-b is not nilpotent, 12.

[Use (c) and compactness.]

Let L be as in Exercise II.

A local ring will mean a commutative

ring with I which has exactly one maximal ideal.

(a) Show that there is a

V2 theory in L which axiomatises the class of local rings.

(b) Let T be

. a theory as in (a), and for some k ~ 0 let ~(y) be the formula i 3x Ei~kyix = 0. Show that Res~ is equivalent modulo T to the set of all formulas of the following form: Vwo. • .wmuo. • .uk+m~A1~i~k+m Ep+q=i (ypwq) = uiyowo + Yowo = O) • 13.

Let L be as in Exercise 11 but without the symbol I.

the theory of rings, not necessarily with a unit element. 3uvw(xu = uv A wuv = xwu A wyu f ywu),

Show that

Res~

In L, letT be Let

~(x,y)

be

is equivalent

modulo T to the set of formulas Vzt(y f nx + zx + xt) (nan integer), (Note that

Res~

expresses that y is not in the subring g'enerated by x;

in

rings with I this is expressible by a single formula.) 14. i
(i)

of

L,

Then certainly A is a model

Let ~(x) be an 3 formula of L and a a tuple from A such that 1 1\Res~(a). We must show that A f ~(a). By Theorem 3.1 .1 there is a

of TV. A

F

model B ofT such that A c Band B f ~(a). so it follows that A f ~(a) as required.

But A is an e.c. model ofT,

:v·

(b)~ (c): Assume (b). Then A is a model of Suppose B =A, B is a model of TV' ~(x) is an 3 formula of L ·and a is a tuple 1 from A such that B f ~(a). We must show that A f ~(a). By Corollary

3.1.2 there is a model C ofT such that B 5 C, and C on preservation (Lemma 1.2.4(b)). and so A

F ~(a) (c)

Hence A

f

f

~(a) by the lemma

1\Res~(a) by Theorem 3.1.1,

by (b). ~(a):

Assume (c).

A is a model ofT (cf. Exercise 3).

To prove (a) it suffices to show that Now (c) asserts that A is a model of

TV' and so by Corollary 3.1.2 there is a model B ofT such that A c B. Since T is a v theory, a typical sentence in T is of form vi3yx(x,y), 2 where X is quantifier-free. For any tuple a from A, B F 3yx(a,y) since B is a model of T.

But A is an e.c. model of TV and B is a model of TV;

it follows that A

F3yx(a,y).

Since

a was

arbitrary, A

so

Fvi3yx(x,y).

Therefore A is a model of T as required.

c

COROLLARY 3.2.4. Let T be a v theoroy in a firost-orodero 2 Zanguage Land ZetA be an L-str>uaturoe. Then A is an e.a. modeZ·of Tiff: A is a modeZ of TV. and foi> everoy 3 forwruZa cj>(x) of L and aZZ 1 tupZes a froom A, i.f A F·l~(a) then theroe is some 31 forwruZa x(x) of L suah that A x(a) and T ~ vim Vy3zvWw(x,y,z,w) with~ 3 L L quantifiero-froee), if is a tupZe froom A and B r ~(a) thenAr ~(a). COROLLARY 3.2.5.

a

Proof. B

F ~(a).

Let ~(x) be as shown, and a a tuple from A such that

We must prove that for every tuple b from A,

c

3.2 A

Existentially closed models

F3z~(a,b,z,;),

Now by assumption B

51

F3z~(a,b,z,w),

so for some

tuple~ from B, B f ~(a,b,c,w). Since vW can be paraphrased as l3wr, and B is an e.c. model of T, by Corollary 3.2.4 there is an 3 formula

x (b) is immediate.

(b) implies that if

~(x) is an 3

formula of L and c are parameters, then for every 1 substructure A of any model of T U {~(~)}, if AfT then A f ~(~).

A

variant of the proof of Theorem 3.1.1 now shows that ~(c) is equivalent

v1 sentences. Compactness reduces the set to a single sentence e(~), and then ~(x) is equivalent modulo T to e(x) by the

modulo T to a set of lemma on constants. modulo T, every v

Thus (b) implies (c).

Using l, (c) implies also that

formula of 1 is equivalent to an 3

formula of 1, 1 1 Using this and (c), we can change blocks of existential quantifiers to

3.2

Existentially closed models

55

blocks of universal, and vice versa, until any formula is brought to v form;

this gives (d).

Finally (d) implies {a) by the lemma on

1

preservation (Lemma 1.2.4{a)).

[J

We say that a theory T is A-categorical iff up to isomorphism, T has exactly one model of cardinality A.

For example a well-known back-

and-forth proof shows that the theory of dense linear orderings without endpoints is w-categorical.

A very similar argument shows that the theory

of atomless boolean algebras is w-categorical (cf. Exercise I of section 6.3 below). Both these theories can be written as v theories, By a 2 theorem of Steinitz, the theory of algebraically closed fields of a fixed characteristic is A-categorical for every uncountable cardinal A;

this

theory is also v2.

LetT be a V2 theory in a first-order language L. Suppose that T has no finite models, and for some cardinal A~ JLJ, T is A-categorical. Then T is model-complete. THEOREM.3.2.10.

Proof.

Suppose the conclusion fails..

lemma, there are models A c B of T, an 3 from A such that B

F ~(a)

but A

F ~~(a).

1

Then by {b) in the

formula ~(~) of L and a tuple a Let L' beL with an added 1-ary

B'

relation symbol P, and let B1 be the expansion of B got by putting P dom(A).

Since T has no finite models, A is infinite,

So by a combination

of the downward LBwenheim-Skolem theorem (Lemma I .2.6) and Exercise 8(b) of section 2.1, there is a structureD' which is elementarily equivalent to B' and has JpD'

I = A,

By considering what can be said about A in

Th(B'), we find a substructure C of D' JL such that dom(C) for some tuple~ from C, D' JL not an e.c, model ofT.

f

~{~) but C

f

1~(~).

PD', C f T and

It follows that C is

But by Theorem 3.2.1 and Exercise I below, T has

an e.c. model of cardinality A.

Thus T has two models of cardinality

which are not isomorphic, contradicting that T is A-categorical.

[J

Exercises for 3.2. 1.

Suppose that Kin Theorem 3.2.1 is the class of all models of some V2 theory in L. Show that B in the theorem can be chosen to be of cardinality at most lAJ + JLJ, theorem.]

[Use the downward LBwenheim-Skolem

3.2 2.

56

Existentially closed models

Show that the statement "Every group can be extended to an e,c. group"

can be proved in Zermelo-Fraenkel set theory without the axiom of choice, [Amalgamate free solutions, cf. the proof of Theorem 3.1.3.] 3.

Show that if T c T 1 are first-order theories and A is a model of T'

which is an e.c. model. of T, then A is an e.c. model of T'. 4.

LetT be a V theory in a first-order language L. Show the following. 2 is a chain of e,c. models ofT, then U.< A. is an e.c.

(a) If (A.).

J. J. is recursive and presents a finitely generated group H.

By Higman's embedding theorem (Fact 3.3.1) His

embeddable in a finitely presented group K. presentation.

So K has a finite

Adding a finite number of equations if necessary, we can

assume that this presentation is of form where e(x,y,z,w) is a finite set of equations.

Let $(x) be the formula

3yzx'y'z'w'O(zi =I + w =I))

(14)

and a final application of Theorem 3.3.7 turns this into an 3 I(z) of L.

Thus we can take xH to be the sentence 3zx(I(z)

11

1

formula x'(x,z)),

o

Close inspection shows that if H in the corollary has r.e. word problem, then Xa can be chosen to be an 3

3

sentence.

(Cf. section

I. 2,)

Some of the results of this section go through for other classes besides the class of groups,

In the early 1970s, after Paul Cohn

had discovered a good analogue of HNN extensions for skew fields (cf. section 5.5 of Cohn (1977)), there were high hopes that e.c. groups and e.c. skew fields could be studied in close parallel,

But for skew fields

we still lack an analogue of Fact 3.3.1, in spite of the efforts of Macintyre (1979). is·known for

s~ew

The upshot is that no general result like Theorem 3.3.7 fields, and so advances have to be piecemeal, chasing

the tail of cheir group-theoretic analogues.

Exeraises for 3. 3. Show that there is an e,c, group of cardinality w which is not the 1 union of any countable chain of proper subgroups. [Using Exercise 4 of I.

section 3.2 and Fact 3.3.3, build ·a chain (Ai : i < w ) of countable e.c. 1

groups so that each Ai lies in a 2-generator subgroup of Ai+l'] 2.

(a) Show that in groups, the resultant of ;:Jz([z,x]=l

equivalent to {y#xn : n E Z}.

(b) Show that in any e,c, group G, i: G = CGCG(a).

11

[z,y]#l) u

[Use free products with amalgamation.]

a is

a tuple of elements then

(CG(X) is tne set of all elements that commute with every

3.3 x € X.)

69

E,c, groups

(c) Show that the theory of groups is not companionable.

.[Use

Exercise IS of section 3.2 with (b) above.] (a) Show that if A , .. , , An are finitely 1 generated subgroups of G then CG(A 1n.,.nAn) = G,

3.

Le·t G be an e.c. group.

(b) Suppose A is the intersection of finitely many finitely generated subgroups of G, and A satisfies the minimum condition on subgroups. that for every group H with CG(A)

== H

Show

G, there is a group K c A such

=H =NG(K). (c) Show that if A is a finite and characteristically simple subgroup of G then NG(A) is a maximal proper subgroup of G. (d) Show that if A =B are finite subgroups of G, A is a maximal subgroup

that CG(K)

of B and the identity is the only automorphism of B which induces the identity on A, then CG(B) is maximal in CG(A),

(e) Suppose H is a maximal

subgroup of G and A is a finitely generated subgroup of H. CG(A) 4.

~

H iff there are h , 1

Let G be an e.c. group.

groups, H

= = G, H

Show that

hn € H such that Ahln ••• nAhn ={I}. (a) Show that if H, K are finitely generated

K and K is embeddable in G, then there is an embedding

of K into G which is the identity on H. G is not finitely generated.

[Use Fact 3,3, 2.]

(b) Show that

(c) Show that if K is the union of a chain

(Hi)ie A+ is a modeZ of Tv" is enforaeabZe. (a) Let w(x 1 , ••• ,x) be a formuZa of L(W) in whiah at most n W!W finitel-y many witnesses oaaur, and Zet q be a aondition. Then q foraes Vx 1 ••• xnw iff for some tupZe (c , •.• ,en) of distinat witnesses not 1 oaaurring in q or in*' q foraes w(cl, ••. ,cn)· (d) Let p be a aondition and ~an 3 sentenae of L(W), and 1 suppose that T U p U {~} has a modeZ. Then there is a aondition q ~ p suah that

q

f-

~·

Proof,

(a) is left to the reader.

Then (c) follows from

3.4

73

Robinson forcing

Exercise 2 of section 2.3. To prove (b), let Vx~ be a sentence in TV, with ~(x) We show that for every tuple ~ of witnesses, ~(~) is

quantifier-free.

In a play of G(~(~);odds), let player V offer p • Since 0 has models and T H~), T U p U {~(~)} has a model B. Express

enforceable. T Up

0

1-

0

~(~) in disjunctive normal form as \f.l\.e ... Bf

1\. e ... J

than Theorem 2.3.4'(but only for Robinson forcing).

Let p be a condition and ~(x) a quantifierThen p forces Vx~ if and only if T U p 1- ¥X~.

THEOREM 3.4.2.

free formula of L(W). Proof.

By Lemma 3.4.1(c) and the lemma on constants (Lemma

1.2.2), we can replace vX~ by~(~) where~ is a tuple of distinct witnesses not occurring in

~

or p.

Suppose first that T U p

if

~(c).

Then T U p U h~ (~)} has a

model, and so by Lemma 3.4.1 (d) there is a condition q 3 p such that ·q

1-

,~(~).

Then player V wins G(~(~);odds) by putting p

to make A+ I= q.

Hence p doesn't force~(~).

Secondly suppose that T Up

= q and playing 0 This proves left to right.

r ~(c). Then T r 1\p +~(c), and r )\p +~(c). If player V

by the lemma on constants it follows that TV

begins a play of G(~(c);odds) by playing p ~ p, then let player 3 play so 0 that A+ is a model ofT U p • She can do this by Lemma 2.3.1 and Lemma 0 3.4.1(b) (and of course the conjunction lemma, Lemma 2.3.3(e)). Clearly she wins the

~ame.

c

This proves right to left.

COROLLARY 3.4.3. Let T be a v theory in L. Then the 2 property "The compiled L-structure A is an e. c. model of T" is enforceable. ~·

By the equivalence of (a) and (c) in Theorem 3.2.3,

the property in question can be rewritten as "A is an e.c. model of Tv"· Now by Lemma 3.4,1 it is enforceable that A is a model of TV.

Also by

3.4

Robinson forcing

74

Lemma 2.3.1 it is enforceable that every element of A is named by infinitely many witnesses, and hence it is enforceable that every tuple of elements of A is named by a tuple of distinct witnesses.

So by Corollary

3.2.4 it remains only to show that the following is enforceable, for any tuple ~ of distinct witnesses and any v (1)

f

If A+ A+

formula ~(x) of L:

1

~(~) then there is some 3 formula ~(i) of L such that 1 and Tv r vic~+~).

F ~(~)

Let P be the property (1), and

suppo~e-that

play of G(P;odds).

as ~(c,d) where d lists the distinct

Write J\p

witnesses which occur in p ~:

0

0 but not in ~.

r

pla:er V has chosen p

0

in a

Now there are two cases.

r

T U Po ~(~). Then TV vi(3y~(x,y) + ~(i)) by the lemma on constants, and player 3 can ensure that A+ 3y~(~,y) by playing to make A+

Fp 0 •

F

Thus player 3 wins G(P;odds).

Case 2:

T Up

choose PI= Po so that PI G(P;odds),

0

V ~(~).

r l~(~);

Then by Lemma 3.4.1 (d), player 3 can playing so that A+

FPI'

she wins o

In short, every model that we get by Robinson forcing is an e.c. model.

This was the justification for sections 3.1-3.3 above.

From

now onwards, the question is whether we can twist the screw tighter and force the model to have some further properties. ask what infinitary sentences are enforceable.

For example one might That turns out to be a

good question:

Let ~i(x) (i < w) be 3 1 formutas of Land tet be a aondition. Then the fottowing are equivaZent: (a) p doesn 1 t forae vi V.~

H there is an

since His an e.c.

4. 2 Atomic models

98

group, there is such an element h already in H. dh = fc. similar.

Then (G,a,c) :

0

(H,b,d) as required.

Choose d in H so that The proof of (8) is

By Lemma 4.2.2 it follows that each countable e.c. group is determined up to· isomorphism by its skeleton. Let us connect these notions with those of section 4.1 above. As before, L is a countable first-order language and T is a

v2

theory in

L. A model A of T is said to be 3-~ iff every tuple of elements of A realises an isolated maximal 3-type. Note that by Exercise I of section 4.1, every 3-atomic model ofT is existentially closed. LEMMA 4.2.4.

modets of the theory T. Proof.

Let A and B be finite or countabte 3-atomic Then A:;; B if and onty if A = B. 1

Left to right is obvious.

For right to left, assume

A =1 B, and let I be the set of all pairs of tuples (a,b) such that (A,a) = (B,b). We claim that I is a back-and-forth system from A to B. 1

Clauses (5) and (6) are clear. tuples, (A,a)

=1

To prove (7), assume a, bare

(B,b) and cis an element of A.

By assumption 3-tpA(a,c)

is an isolated maximal 3-type. isolates it. formula.

Then A

Hence there is d in B such that B

3-tpA(a,c) in B. (A,a,c) _

f

Let w(i,y) be an 3 -formula of L which 1 3yw(a,y), and soB F 3yw(b,y) since 3YW is an 3

Fw(b,d),

1

and thus bd realises

Since this type is a maximal 3-type, it follows that

(B,b,d) as required.

The same argument gives (8).

Hence I is a back-and-forth system from A to B, as claimed. It follows by Lemma 4.2.2 that A is isomorphic to B.

[J

We shall say that isolated maximal 3-types are dense (for the theory T) iff for each n

~ 0

and each 3

1

formula

0

~(x , •••

,xn_ ) of L, if 1

T U {3i~} has a model, then there is an isolated maximal 3-type ~(~) ofT which contains

~·

LEMMA 4.2.5. Suppose that isolated maximat 3-types are dense for the v theory T. Then the property "3-atomic modet of T 11 is 2 enforceabte. ~·

By the conjunction lemma it suffices to show that for

each tuple ~ of witnesses, it is enforceable that the tuple of elements

4.2

Atomic models

99

of the compiled structure A+ named by c realises an isolated maximal 3-type.

Let player V open with a choice p • Write Ap as .p (c,d) where 0 0 ,P(x,y) is a quantifier-free formula of Land dare the witnesses which occur in p

0

but not in c.

Then T U {3x(3y,P(x,y))} has a model, so by

assumption there is an isolated maximal 3-type $(x) lvhich contains 3y.p(x,y).

Let 1/l(x) isolate$,

By Lemma 3.4.l(d), player 3 can choose

=Po so that P1 1- 1/l(c). By Lemma 2.3.1 and Corollary 3.4.3, she can 1 play so that A+~ p and A is an e.c. model ofT, Then by Lemma 4.1 .2, 1 the tuple of elements named by c in A+ will realise a maximal 3-type

p

containing 1/J, and this 3-type must be $,

[J

Now we can separate the sheep from the goats. THEOREM 4.2.6 (Dichotomy theorem). Let L be a countable first-order language and T a v theory in L. Then one of the following 2 two situations holds for Robinson forcing with T:

(a)

For every enforceable property P of L-structures, there are continuum many pairwise non-isomorphic finite or countable models of T which have property P. (b) There is a set K of at most countably many 3-atomic models of T such that the property "A is isomorphic to a model in K 11 is enforceable. The cardinality of K is the number of isolated maximal elements of Player V can choose by his first move which of the models in K wilt be isomorphic to A.

s;.

~·

The deciding question is whether isolated maximal

3-types are dense for T. Suppose first that they are not.

Then there is an 3 1 formula .p(x) of L such that T U {3x.p} has a model but there is no isolated maximal 3-type containing .p.

Now we set players V and 3· to play the splitting

game of section 4.1.

Player V will choose p

tuple c of witnesses.

so that p 1- ,P(c) for some 0 0 At all his later moves he will split each condition

into two, so that the outcome of the play is a family (A+ : a < 2w) of a

a

L(W)-structures. Write for the tuple of elements named by c in A+. a + "' By Th eorem 4 • I , 5 , f 1ayer 3 can ensure that each Aa is a model of ,P(c), and that A is an e.c. ·model of T with property P, where P is any enforceable a property of L-structures. Then by choice of .p, each a will realise in Aa

a

4.2

100

Atomic models

a maximal 3-type

~a

which is not isolated.

3 can arrange that whenever a f

Hence by Theorem 4.1.4, player

e, Ae omits

~

a

and so A

a

f

A •

e

On the other hand suppose that isolated maximal 3-types are dense for T. model of T.

Then by Lemma 4.2.5 it is enforceable that A is an 3-atomic Let K be a set consisting of representatives of the

isomorphism types of finite or countable 3-atomic models of T, one representative for each isomorphism type.

By Lemma 4.2.4 and the downward

LBwenheim-Skolem theorem, the number of elements of K is the number of

=1-types 3

of 3-atomic models of T.

For an 3-atomic model B, the set of all

sentences of L which are true in B is an isolated element of S~.

1 player V can determine it by taking an isolating sentence choosing p

~

So

and then

so that p ~ ~. 0 0

c

EXAMPLE of (b) in the theorem.

Let T be the theory of fields.

It is enforceable that the compiled model A is an e.c. field, that is to say, an algebraically closed field.

By Exercise 3(e) of section 3.4, it

is enforceable that A is the algebraic closure of a finite field.

Player

V can use his first move to choose any prime characteristic.

I should add two remarks which refine Theorem 4.2.6.

First,

the continuum many models in (a) are non-isomorphic in a very strong sense:

if a f

e then

Aa ~s ' no t even emb edd a bl e ~n . A • Th'~s ~s ' b ecause 6 any 3-type realised in Aa is also realised in every structure in which Aa is embedded, by the lemma on preservation (Lemma 1.2.4(b)). Second, s3 has a unique maximal element if and only if T has 0 the joint embedding property (cf. Exercise 4 below). Hence when T has the

joint embedding property, we can sharpen (b) as follows:

there is an e.c.

model B of T such that it is enforceable that the compiled structure A is isomorphic to B.

We call such a model B an enforceable model of T.

As a general principle, whenever we can show that a certain class of structures has few members, we should hope to be able to find a good structure theory for the structures in the class.

This is vague, but

it has to be. In particular suppose (b) holds in Theorem 4.2.6. say about the structures in the class K? answers to some

ma~imal

3-type in

What can we

First, each structure in K

~which

is isolated by a

sentence~·

4.2

Atomic models

101

If we replace T by T U {1}!}, this replaces K by a !-element class.

isolating sentences

1}i

So the

serve as invariants of a sort, to label the elements

of K. We are reduced to the case where T has an enforceable model. In this case, let us define a relation < between isolated maximal 3-types, as follows: ~(x) < ~(x,y) iff for some n