Marker Introduction to Model Theory

8/8/2019 Marker Introduction to Model Theory

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Model Theory, Algebra, and GeometryMSRI PublicationsVolume 39 , 2000

Introduction to Model Theory

DAVID MARKER

Abstract . This article introduces some of the basic concepts and resultsfrom model theory, starting from scratch. The topics covered are be tailoredto the model theory of elds and later articles. I will be using algebraicallyclosed elds to illustrate most of these ideas. The tools described are quitebasic; most of the material is due either to Alfred Tarski or AbrahamRobinson. At the end I give some general references.

1. Languages and Structures

What is a mathematical structure? Some examples of mathematical struc-tures we have in mind are the ordered additive group of integers, the complexeld, and the ordered real eld with exponentiation.

To specify a structure we must specify the underlying set, some distinguished

operations, some distinguished relations and some distinguished elements. Forexample, the ordered additive group of integers has underlying set Z and wedistinguish the binary function +, the binary relation < and the identity element0. For the ordered eld of real numbers with exponentiation we have underlyingset R and might distinguish the binary functions + and , the unary functionexp, the binary relation < and the elements 0 and 1.

Here is the formal denition.

Definition 1.1. A structure M is given by the following data.

(i) A set M called the universe or underlying set of M .(ii) A collection of functions {f i : iI 0}where f i : M n i M for some n i 1.(iii) A collection of relations

{R i : i

I 1

}where R i

M m i for some m i

1.

(iv) A collection of distinguished elements {ci : iI 2} M .Any (or all) of the sets I 0 , I 1 and I 2 may be empty. We refer to n i and m j asthe arity of f i and R j .

Here are some examples:

15


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16 DAVID MARKER

(1) The ordered eld of real numbers has domain R , binary functions + , , ,relation < , and distinguished elements 0 and 1.

(2) The valued eld of p-adic numbers has domain Q p, binary functions + , , ,distinguished elements 0 and 1, and a unary relation Z p, for the ring of inte-gers.

In mathematical logic we study structures by examining the sentences of rstorder logic true in those structures. To any structure we attach a languageL where we have an n i -ary function symbol f i for each f i , an m i -ary relationsymbol R i for each R i and constant symbols ci for each ci .

An L -structure is a structure M where we can interpret all of the symbols of L . For example, let L be the language where we have a binary function symbol and a constant symbol 1. The following are examples of L -structures:(1) M 1 has underlying set Q . We interpret

as

and 1 as 1.

(2) M 2 has underlying set Z . We interpret as + and 1 as 0.Of course we also could take the natural interpretation of L in Z .

(3) M 3 has underlying set Z . We interpret as and 1 as 1.Definition. If M and N are L -structures with underlying sets M and N ,respectively, an L -embedding : M N is an injective map that preserves theinterpretation of all function symbols, relation symbols and constant symbols of L . An L -isomorphism is a bijective L -embedding.

We say that M is a substructure of N (and write M N ) if M N and the

inclusion map is an L -embedding .

Formulas in our language are nite strings made from the symbols of L ,the equality relation =, variables x0 , x1 , . . . , the logical connectives ,,,quantiers and and parentheses. (See the appendix on page 34 for precisedenitions.) We interpret ,,as not, and and or and and asthere exists and for all. I will use x ,y, z . . . and their subscripted varietiesas variables and not use the symbol when no confusion arises.

Let L r be the language of rings, where we have binary function symbols + , and and constant symbols 0 and 1. The language of ordered rings, L or is L rwith an additional binary relation symbol < . (As usual we will write x+ y insteadof +( x, y ) and x < y for < (x, y ).) Here are some examples of L or -formulas:

x1 = 0 x1 > 0

x2 x2 x2 = x1

x1 (x1 = 0 x2 x2 x1 = 1)Intuitively, the rst formula asserts that x1 0, the second asserts that x1 is

a square and the third asserts that every nonzero element has a multiplicativeinverse. We would like to dene what it means for a formula to be true in a


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INTRODUCTION TO MODEL THEORY 17

structure, but these examples already show one difficulty. While in any L or -structure the third formula will either be true or false, the rst two formulas

express a property which may or may not be true of elements of the structure.We say that a variable occurs freely in a formula if it is not inside the scope

of a quantier; otherwise we say it is bound . For example, x1 is free in the rsttwo formulas and bound in the third, while x2 is bound in both formulas.

We call a formula a sentence if it has no free variables. For any L -structureeach sentence of L is either true or false. Let be a sentence. We say that M isa model of , and write M |= , if and only if is true in M .

We often write (x1 , . . . , x n ) to show that the variables x1 , . . . , x n are freein the formula . We think of (x1 , . . . , x n ) as describing a property of n-tuples from M . For example, the L or -formula x2 x2 x2 = x1 has the singlefree variable x1 and describes the property x1 is a square. If a1 , . . . , a n areelements of M we say M

|= (a1 , . . . , a n ) if the property expressed by is true

of the tuple ( a1 , . . . , a n ).

We say that two L -structures M and N are elementarily equivalent if for allL -sentences M |= N |= .Proposition 1.2. If M and N are isomorphic , then they are elementarily equivalent .

Proof. Show by induction on formulas that if (x1 , . . . , x n ) is a formula, :M N is an isomorphism and a1 , . . . , a n M , then

M |= (a1 , . . . , a n ) N |= ((a1), . . . , (an )) .We say that an L -embedding f : M N is elementary if for any a1 , . . . , a n M and any formula (x1 , . . . , x n )

M |= (a1 , . . . , a n ) N |= (f (a1), . . . , f (an )) .If M

N we say that M is an elementary substructure if the inclusion map iselementary.

Definition. We say that a set X M n is denable in the L -structure M if

there is a formula (x1 , . . . , x n + m ) and elements b1 , . . . , bm M such that

X = {(a1 , . . . , a n ) : M |= (a1 , . . . , a n , b1 , . . . , bm )}.We say that X is A-denable or denable over A, where AM , if we can choosethat b1 , . . . , bm A. For example, if m = 0 we say X is

-denable.

For example, {x : x > } is denable over R but not -denable, while {x :x > 2} is -denable by the formula x x > 1 + 1 x > 0. In the eld(Q p, + , , , 0, 1) if p = 2 we can dene the valuation ring Z p by the formulay y

2 = px2 + 1.We can give a very simple characterization of the denable sets.


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18 DAVID MARKER

Proposition 1.3. Suppose that Dn is a collection of subsets of M n for all n 1 such that D = ( Dn : n 1) is the smallest collection satisfying the following conditions :(i) M n Dn .(ii) For all n -ary functions f of M , the graph of f is in Dn +1 .(iii) For all n -ary relations R of M , RDn .(iv) For all i, j n, {(x1 , . . . , x n )M n : x i = x j } Dn .(v) Each Dn is closed under complement , union and intersection .(vi) If X Dm and : M

n M m is the projection map (x1 , . . . , x n ) (x i 1 , . . . , x i m ), then 1(X )Dn .(vii) If X Dn and is as above , then (X )Dm .(viii) If X Dn + m and bM

m , then {aM n : (a, b)X } Dn .Then X

M n is denable if and only if X

Dn

.

2. Theories

An L -theory is a set of L -sentences. Theories arise naturally as we attempt toaxiomatize the properties of mathematical structures. For example, if L r is thelanguage of rings we can write down the eld axioms as L r sentences. We cangive the theory of algebraically closed elds (ACF) by taking the eld axiomsplus, for each n 1, the axiom

x0 x1 . . . xn 1 y yn + xn 1yn 1 + x1y + x0 = 0 .

If T is an L -theory, we say M

|= T if M

|= for all

T . We say that anL -sentence is a logical consequence of an L -theory T (and write T |= ) if andonly if M |= for all M |= T . For example, ACF |= xyz x2 + y2 = z2 .Theorem 2.1 (G odels Completeness Theorem, first version). T |= if and only if there is a formal proof of using assumptions from T .

This has a very useful reformulation with an important corollary. We say that anL -theory T is satisable if and only if there is an L -structure M with M |= T andwe say that T is consistent if and only if we cannot formally derive a contradictionfrom T .

Theorem 2.2 (Completeness Theorem, second version). T is satisableif and only if T is consistent . Moreover if T has innite models then T has a model where the underlying set has cardinality , for all |L |+ 0 .This has an easy consequence, which is the cornerstone of model theory.

Theorem 2.3 (Compactness Theorem). If every nite subset of T is satisable , then T is satisable .


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20 DAVID MARKER

Proof. The equivalence of (i)(iii) is just the completeness of ACF 0 and (v) (iv) is obvious.

For (ii) (v) suppose ACF0 |= . By the completeness theorem, there is aproof of from ACF 0 . That proof only uses nitely many assertions q; thus,

for large enough p, ACF p |= .For (iv) (ii) suppose ACF 0 |= . By completeness ACF 0 |= . By the

above argument, ACF p |= for sufficiently large p; thus (iv) fails.This result has a striking application.

Theorem 2.6 (Ax). Let F : C n C n be an injective polynomial map . Then F is surjective .

Proof. Suppose not. Let X = ( X 1 , . . . , X n ). Let F (X ) be a counterexample,with coordinate functions F 1(X ), . . . , F n (X ), each F i

C [X ] having degree atmost d. There is an L -sentence n,d such that, for K a eld, K

|= n,d if and

only if every injective polynomial map G : K n K n whose coordinate functionshave degree at most d is surjective. We can quantify over polynomials of degreeat most d by quantifying over the coefficients. For example, 2,2 is the sentence

a0,0a0,1a0,2a1,0a1,1a2,0b0,0b0,1b0,2b1,0b1,1b2,0

x1y1x2y2

a i,j x i1yj1 = a i,j x i2y

j2 bi,j x

i1y

j1 = bi,j x i2y

j2 x1 = x2y1 = y2

uvxy a i,j x i yj = u bi,j xy yj = v.If K is a nite eld then K |= n,d . It follows that n,d holds in any increasing

union of nite elds. In particular the algebraic closure of a nite eld satises

n,d . Hence, by Corollary 2.5, C |= n,d , a contradiction.Originally logicians looked for completeness results because they lead to decid-ability results.

Corollary 2.7. The theory ACF p is decidable for p 0. That is , for each pthere is an algorithm which for each sentence will determine if ACF p |= .Proof. By the completeness of ACF p and the completeness theorem eitherthere is a proof of or a proof of from ACF p. We can systematically searchall nite sequences of symbols and test each one to see if it is a valid proof of either or . Eventually we will nd one or the other.

3. Quantier EliminationLet F be a eld. If p(X 1 , . . . , X n ) F [X 1 , . . . , X n ], the zero set {x F n :

p(x) = 0}is dened by a quantier free L r -formula. We say that a subset of F n is constructible if it is a boolean combination of zero sets of polynomials inF [X 1 , . . . , X n ]. It is easy to see that the subsets of F n dened by quantier free


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formulas are exactly the constructible subsets of F n . If F is an algebraicallyclosed eld then Chevalleys theorem from algebraic geometry asserts that the

projection of a constructible subset of F n +1

to F n

is constructible. Restatingthis model-theoretically, this says that every denable set is constructible.

This is of course not true for non-algebraically closed elds. In the reals wecan dene the ordering by

x < y z z = 0 x + z2 = y,but this is not a constructible subset of R 2 . Here this is the only problem. Wesay that a subset of an ordered eld is semialgebraic if it is a boolean combina-tion of zero sets of polynomials and polynomial inequalities (like {x : p(x) > 0}).It is easy to see that the semialgebraic sets are exacty the sets dened by quanti-er free L or -formulas. The TarskiSeidenberg theorem says that in the reals (ormore generally in a real closed eld) the projection of a semialgebraic set is semi-algebraic. Thus in the real eld the denable sets are exactly the semialgebraicsets.

In model theory we study the denable sets of a structure. Quantier elimi-nation results are very useful, as often one can show the quantier free denablesets have good geometric properties while the denable sets have strong closureproperties. For example, suppose A

R n is semialgebraic. We want to showthat the closure of A is also semialgebraic. Since A is denable there is anL or -formula (x1 , . . . , x n , a 1 , . . . , a m ) that denes A. Then the formula

> 0y1 . . . yn (y1 , . . . , y n , a 1 , . . . , a m )(x1 y1)2 + +( xn yn )2 <

denes the closure of A. Since the closure of A is denable it is semialgebraic.In the structure ( Q , + , , 0, 1) we can also dene the ordering by sayingthat the nonnegative elements are sums of four squares. Julia Robinson showedthat the integers are denable in the eld of rational numbers. By G odelsincompleteness theorem this implies that the theory of the rational numbers isundecidable and the denable subsets are quite complicated.

There is a useful model-theoretic test for quantier elimination.

Theorem 3.1. Let L be a language containing at least one constant symbol .Let T be an L -theory and let (v1 , . . . , v m ) be an L -formula with free variablesv1 , . . . , v m (we allow the possibility that m = 0). The following statements areequivalent :

(i) There is a quantier free L -formula (v1 , . . . , v m ) such that

T |= v ((v) (v)) .(ii) If A and B are models of T , C

A and C

B , then A |= (a) if and only if B |= (a) for all aC .


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22 DAVID MARKER

Proof. (i) (ii): Suppose T |= v ((v) (v)), where is quantier free.Let a

C where C is a substructure of A and B and the later two structures

are models of T . Since quantier free formulas are preserved under substructureand extension

A |= (a) A |= (a)

C |= (a) (since C A )

B |= (a) (since CB )

B |= (a).(ii) (i): First, if T |= v (v), then T |= v ((v) c = c). Second, if

T |= v (v), then T |= v ((v) c = c). In fact, if is not a sentence wecould use v1 = v1 in place of c = c.

Thus we may assume that both (v) and (v) are consistent with T .Let ( v) =

{(v) : is quantier free and T

|=

v ((v)

(v))

}. Let

d1 , . . . , d m be new constant symbols. We will show that T + ( d) |= (d). Thusby compactness there are 1 , . . . , n such that T |= v i (v) (v) .So T |= v i (v) (v) and i (v) is quantier free. We need only provethe following claim.

Claim. T + ( d) |= (d).Suppose not. Let A |= T +( d)+ (d). Let C be the substructure of A generatedby d. (Note: if m = 0 we need the constant symbol to ensure C is non-empty.)

Let Diag( C ) be the set of all atomic and negated atomic formulas with pa-rameters from C that are true in C .

Let = T + Diag( C) + (d). If is inconsistent, then there are quantier

free formulas quantier free formulas 1(d), . . . , n (

d)Diag(

C), such that T |=v ( i (v) (v)). But then T |= v ((v) i (v)) . So i (v)

and C |= i (d), a contradiction. Thus is consistent.Let B |= . Since Diag( C), we may assume that C B . But since

A |= (d), B |= (d), a contradiction.The next lemma shows that to prove quantier elimination for a theory we needonly prove quantier elimination for formulas of a very simple form.

Lemma 3.2. Suppose that , for every quantier free L -formula (v, w), thereis a quantier free (v) such that T |= v (w (v, w) (v)). Then every L -formula (v) is provably equivalent to a quantier free L -formula .

Proof. We prove this by induction on the complexity of . The result is clearif (v) is quantier free.

For i = 0 , 1 suppose that T |= v (i (v) i (v)), where i is quantier free.If (v) = 0(v), then T |= v ((v) 0(v)) .If (v) = 0(v)1(v), then T |= v ((v) (0(v)1(v))).In either case is provably equivalent to a quantier free formula.


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Suppose that T |= v (v, w) 0(v, w) , where is quantier free. Sup-pose (v) = w (v, w). Then T |= v (v) w (v, w) . By our assump-tions there is a quantier free (v) such that

T |= v (w 0(v, w) (v)) .But then T |= v ((v) (v)).Thus to show that T has quantier elimination we need only verify that condition(ii) of Theorem 3.1 holds for every formula (v) of the form w (v, w), where(v, w) is quantier free.

Theorem 3.3. The theory ACF has quantier elimination .

Proof. Let F be a eld and let K and L be algebraically closed extensions of F . Suppose (v, w) is a quantier free formula, aF , bK and K |= (b, a).We must show that L

|=

v (v, a).There are polynomials f i,j , gi,j F [X ] such that (v, a) is equivalent to

l

i =1

m

j =1

f i,j (v) = 0

n

j =1

gi,j (v) = 0 .

Then K |=mj =1 f i,j (b) = 0

nj =1 gi,j (b) for some i.

Let F be the algebraic closure of F . We can view F as a subeld of both K and L. If any f i,j is not identically zero for j = 1 , . . . , m , then bF L andwe are done.

Otherwise, sincen

i =1

gi,j (b) = 0 ,

gi,j (X ) = 0 has nitely many solutions. Let {c1 , . . . , c s }be all of the elementsof L where some gi,j vanishes for j = 1 , . . . , m . If we pick any element d of Lwith d / {c1 , . . . , c s}, then L |= (d, a).The next result summarizes some simple applications.

Corollary 3.4. Let K be an algebraically closed eld .

(i) If X K is denable , then either X or K \ X is nite . (This property iscalled strong minimality ).(ii) Suppose f : K K is denable . If K has characteristic zero , there is a rational function g such that f (x) = g(x) for all but ntely many x. If K hascharacteristic p there is a rational function g and n

0 such that f (x) = g(x)1/p

n

for all but nitely many x.Proof. (i) X is a boolean combination of sets of the form {x : f (x) = 0 }andthese sets are nite.

(ii) Assume K has characteristic 0 (the p > 0 case is similar). Let L be anelementary extension of K and let a L \ K . If is any automorphism of L


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24 DAVID MARKER

xing K (a), then (f (a)) = f (a). But then f (a)K (a). Thus there is a rationalfunction gK (X ) such that f (a) = g(a). The set {x K : f (x) = g(x)}iseither nite or conite by (i). If it has size N , then the fact that it has exactly N elements is expressed by a sentence true of L and K . Then L would not containany new elements of this set. Thus f (x) = g(x) for all but nitely many x.

Indeed, (in characteristic zero) if f is denable there is a Zariski open O and arational function g such that f |O = g|O.

The quantifer elimination test has many other applications. For example,consider RCF, the theory of real closed ordered elds in the language L or . Theaxioms for RCF consist of:

(i) the axioms for ordered elds;(ii) x > 0y y

2 = x;(iii) the axiom

x0 . . .

xn 1

y yn + xn 1yn 1 +

+ x0 = 0 for each odd

n > 0.

Clearly RCF is part of the L or -theory of the real eld. We will see shortly thatthis theory is complete and hence axiomatizes the complete theory of the realeld. First we show RCF has quantier elimination. We use the algebraic factsthat every ordered eld has a unique real closed algebraic extension and over areal closed eld any polynomial in one variable factors into a product of linearand irreducible quadratic factors (see [Lang 1984, Section XI.2], for example).

Theorem 3.5. The theory RCF has quantier elimination in L or .

Proof. We apply Theorem 3.1. Let F 0 and F 1 be models of RCF and let ( R, < )be a common substructure. Then ( R, < ) is an ordered domain. Let L be the real

closure of the fraction eld of R. By the uniqueness of real closures we can mayassume that ( L, < ) is a substructure of F 0 and F 1 . Suppose (v, w) is quantierfree, aR, bF 0 and F 0 |= (b, a). We need to show that F 1 |= v (v, a). Itsuffices to show that L |= v (v, a).

As in the proof of Theorem 3.3 (and fooling around with the order), we may as-sume that there are polynomials f 1 , . . . , f n , g1 , . . . , g m R[X ] such that (v, a) is

n

i =1

f i (v) = 0

m

i =1

gi (v) > 0.

If any of the f i is not zero, then since (b, a), b is algebraic over R and thus inL. So we may assume (v, a) is

m

i =1

gi (v) > 0.

Since L is a real closed eld, we can factor each gi as a product of factors of theform (X c) and ( X 2 + bX + c), where b2 4c < 0. The linear factors changesign at c, while the quadratic factors do not change signs. If follows that we can


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nd 1 , . . . , l L{}and 1 , . . . , l L {+ }such that, for vF 0 ,(v, a) if and only if

l

i =1

i < v < i .

Since F 0 |= (b, a), we have i < b < i for some i. Then L |= ( 12 ( + ), a).Corollary 3.6. RCF is complete and decidable .

Proof. Let be a sentence. By quantier elimination there is a quantier freesentence such that RCF |= . We can embed the rational numbers inany real closed eld F and F |= if and only if Q |= . Thus F |= if and onlyif Q |= . In particular if F 1 and F 2 are real closed elds then F 1 |= if andonly if F 2 |= .

Hence RCF is complete. Decidability follows as in 2.7.

We also have an analog of Corollary 3.4.

Corollary 3.7. If R is real closed and X R is denable , then X is a niteunion of points and intervals (this is called o-minimality ).

Proof. Denable subsets of R are boolean combinations of {x : f (x) > 0}which are nite unions of intervals.

4. Model Completeness

We say that a theory T is model complete if whenever M and N are modelsof T and M

N , then M is an elementary substructure of N .

Proposition 4.1. If T has quantier elimination , then T is model complete .

Proof. Let M N . Suppose (v) is a formula and a M

n . There is aquantier free formula (v) such that

T |= v ((v) (v)) .Since is quantier free, M |= (a) N |= (a). Thus M |= (a) N |= (a).Model completeness can arise in cases where quantier elimination fails. Forexample, let T be the L r -theory of the real numbers (without a symbol for theorder). The formula y y

2 = x is not equivalent to a quantier free formula(recall that quantier free denable sets in L or are constructible), so T does nothave quantier elimination. On the other hand the ordering of a real closed eldis denable in the eld language, thus if F and K are real closed elds and F isa subeld of K , then the ordering on F agrees with the ordering inherited fromK (that is, F is an L or -substructure of K ). Thus, by quantier elimination inL or , F is an elementary substructure of K .


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26 DAVID MARKER

This example shows that quantier elimination is sensitive to the exactchoice of language. In general we can always enrich the language so that we

have quantier elimination. If T 0 is an L -theory, we can add to our languagean n-ary relation symbol R for each formula with n free variables and wecould let T be the theory where we add to T 0 axioms x ((x) R (x)) foreach formula . The theory T will have quantier elimination, but this wouldbe useless as we would not be able to say anything sensible about the quantierfree formulas. The goal is to show we have quantifer elimination in a languagewhere the quantier free formulas are simple.

Wilkie showed that the theory of ( R, + , , , < , exp) is model complete, buty > 0 w (wy = xz = y exp( w))

is not equivalent to a quantier free formula in the language

{+ , , , < , 0, 1, exp}(or any expansion by total real analytic functions). Van den Dries, Macintyre andI showed that you can eliminate quantiers in a much more expressive language,but we do not know the simplest language for quantier elimination.

Model completeness itself has useful consequences. For example, the modelcompleteness of ACF leads to an easy proof of a version of the Nullstellensatz.

Theorem 4.2. Let F be an algebraically closed eld and let I F [X 1 , . . . , X n ]be a prime ideal . Then there is aF

n such that f (a) = 0 for all f I .

Proof. Let K be the algebraic closure of the fraction eld of F [X 1 , . . . , X n ]/I .If x i K is X i /I , then f (x1 , . . . , x n ) = 0 for all f I . Choose f 1 , . . . , f m

generating I . Then

K |= y1 , . . . , y nm

i =1

f i (y1 , . . . , y n ) = 0 .

As this is a sentence with parameters from F , by model completeness this sen-tence is also true in F . Thus there is aF

n such that f i (a) = 0 for i = 1 , . . . , mand hence f (a) = 0 for all f I .

A very similar argument can be used to reprove Artins solution to Hilbertsseventeenth problem.

Theorem 4.3. Let F be a real closed eld . Suppose that f (X 1 , . . . , X n ) F (X 1 , . . . , X n ) and that f (a) 0 for all aF n (we call f positive semi-denite ).Then f is a sum of squares of rational functions .Proof. If not, we can extend the order of F to F (X 1 , . . . , X n ) such that f < 0(see [Lang 1984, Section XI.2]). Let K be the real closure of F (X 1 , . . . , X n ) withthis ordering. Then

K |= y1 . . .yn f (y1 , . . . , y n ) < 0,


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since we can use X 1 , . . . , X n as witnesses. By model completeness

F

|=

y1 . . .

yn f (y1 , . . . , y n ) < 0.

Definition. We say that a theory T is the model companion of a universialtheory T 0 if:

(i) every model of T is a model of T 0 ,(ii) every model of T 0 can be extended to a model of T , and(iii) T is model complete.

For example, the theory of algebraically closed elds is the model companion of the theory of integral domains and the theory of real closed elds is the modelcompanion of the theory of ordered domains. More interesting examples can befound in [Chatzidakis 2000].

Model theoretic methods can sometimes be used to obtain effective bounds.Compactness arguments alone can lead to crude bounds.Proposition 4.4. There is a computable function (n, d ) such if F is a real closed eld and f = g/h F (X 1 , . . . , X n ) where f and g are polynomials of degree at most d and f is positive semidenite then f is the sum of squares of at most (n, d ) rational functions with numerator and denominator of degree at most (n, d ).

Proof. Fix n, d . We rst claim that there is an M such that any positivesemidenite rational function in n variables with numerator and denominatorof degree at most d is a sum of at most M squares of rational functions withnumerator and denominator of degree at most M . Suppose not. Let c1 , . . . , c N be new constants which will be coefficients of a rational function f in n-variableswith numerator and denominator of degree at most d. Let M be a sentenceasserting f is not a sum of at most M squares of functions of degree at mostM . Then RCF + f is positive semidenite + {M : M 1}is satisable,contradicting Hilberts 17th problem.

Given n and d, let (n, d ) be the least M as above. Since RCF is decidable,we can compute (n, d ).

5. Types

Suppose M is an L -structure and AM . LetL A be the language obtained

by adding to L constant symbols for all elements of A. Let Th A (M ) be the setof all L A -sentences true in M .

Definition. An n -type over A is a set of L A -formulas in free variables x1 , . . . , x nthat is consistent with Th A (M ). A complete n -type over A is a maximal n-type.In other words a complete type is a set p of L A -formulas consistent with Th A (M )in the free variables x1 , . . . , x n such that for any L A -formula (x) either (x)por (x)p. Let S n (A) be the set of all complete n-types over A.


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28 DAVID MARKER

We sometimes refer to incomplete types as partial types .

By compactness every n-type over A is realized in some elementary extensionof M .

There is one easy way to get complete types. Suppose N is an elementaryextension of M and bN

n . Let tp( b/A) = {(x)L A : N |= (b)}. It is easyto see that tp( b/A) is a complete type.

If pS n (A) we say that b realizes p if tp( b/A) = p.

What do types tell us?

Proposition 5.1. Suppose a, bM n and tp( a/A ) = tp( b/A). Then there is an

elementary extension N of M and an L -automorphism of N which xes A and maps a to b.

Proof. We carefully iterate the following lemma.

Lemma 5.2. Suppose M is an L -structure , AM and f : A M is a partial elementary map (i .e ., M |= (a1 , . . . , a n ) M |= (f (a1), . . . , f (an )) . If bM , we can nd

N an elementary extension of N and extend f to a partial elementary map from A{b}into N .Proof. Let c be a new constant symbol. Let

= {(c, f (a1), . . . , f (an )) : M |= (b, a1 , . . . , a n ), a 1 , . . . , a n A}Th M (M ).Suppose we nd a structure N and an element cN satisfying all of the

formulas in . Since N |= Th M (M ), N is an elementary extension of M . It isalso easy to see that we can extend f to an elementary map by b c.

So it suffices to show that is satisable. By compactness it suffices to showthat every nite subset of is satisable. Taking conjunctions it is enough toshow that if M |= (b, a1 , . . . , a n ) then M |= v (v, f (a1), . . . , f (an )). But thisis clear since M |= v (v, a 1 , . . . , a n ) and f is elementary.The type space S n (A) can be topologized as follows. For each L A -formula(x1 , . . . , x n ) let B = { pS n (A) : p}. The Stone topology on S n (A)is the topology generated by using the sets B as basic open sets. 1

Proposition 5.3. S n (A) is compact and totally disconnected .

Proof. Suppose {B i : iI }is a cover of S n (A) by basic open sets. Supposethere is no nite subcover. Let = {i (x1 , . . . , x n ) : i I }. Since there inno nite subcover every nite subset of is satisable. By compactness issatisable and this yields a type that is not contained in any B i .

Since S n (A) \ B = B , each B is open and closed. Thus S n (A) is totallydisconnected.

1 S n (A ) can be thought of as the set of ultralters on the Boolean algebra of A -denablesubsets of M n so it is in fact the Stone space of a Boolean algebra.


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Suppose K is an algebraically closed eld and F is a subeld of K . What are thecomplete n-types over F ? Suppose pS n (F ). Let I p = {f F [X 1 , . . . , X n ] :f (x1 , . . . , x n ) = 0 p}. Let Spec( F [X 1 , . . . , X n ]) be the set of prime ideals of F [X 1 , . . . , X n ]. We topologize this space by taking sub-basic open sets {P : f P }, for f F [X ].Proposition 5.4. p I p is a continuous bijection between S n (F ) and

Spec(F [X ]).

Proof. If fg I p, then f (x)g(x) = 0 p. Since p is complete eitherf (x) = 0 p or g(x) = 0 p. Thus I p is prime. It is just as easy to seethat it is an ideal.

If P is a prime ideal, then we can nd a prime ideal P 1 K [X ] such thatP 1 F [X ] = P . Let K 1 be the algebraic closure of K [X ]/P 1 and let a i = X i /P .For f

K [X ] f (a) = 0 if and only if f

P 1, thus I

tp( a/F )= P . Thus the map

is surjective.By quantier elimination if I p = I q , then p = q.Continuity is clear.

This shows that the Zariski topology on Spec( F [X ]) is compact.We can identify (as objects) S n (F ) and Spec( F [X ]), but the Stone topology

is much ner that the Zariski topology. The Stone topology corresponds to thetopology generated by the constructible sets.

If pS n (F ), let V = {xK n : f (x) = 0 for all f I p}. Then the type passerts that xV and xW for any W V dened over F . Thus realizationsof p are points of V generic over F .

What about real closed elds? If F is an ordered subeld of a real closedeld and p is an n-type, let I p be as above and let C p = {f /I p : f F [X ]and f (x) > 0p}. Then p (I p, C p) is a bijection onto the set of pairs of real prime ideals P (prime ideals where 1 = a i b2i where a i > 0, a i F [X ],biP ) and orderings of F [X ]/P . This is the real spectrum of F [X ].

In particular if R is a real closed eld, then elements of S 1(R) correspond toeither elements of R or cuts in the ordering of R.

6. Saturation

It is often useful to work in a very rich model of a theory. For example, it issometimes easier to prove things in an algebraically closed eld of innite tran-

scendence degree. Or when dealing with the reals it is useful to use nonstandardmethods by assuming there are innite elements. In model theory we make thisprecise in the following way.

Definition 6.1. Let be an innite cardinal. We say that a structure M is-saturated if for every AM with |A| < if pS 1(A), then there is b in M


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30 DAVID MARKER

such that b realizes p. An easy induction shows that in this case every n-typeover A is also realized in M .

We say that M is saturated if it is |M |-saturated.Lemma 6.2. If M is saturated , AM and |A| < |M |, then tp( a/A ) = tp( b/A)if and only if there is an automorphism of M xing A mapping a to b.

Proof. The argument from 5.1 can be done completely inside M .

Proposition 6.3. An algebraically closed eld K is saturated if and only if it has innite transcendence degree .

Proof. Suppose AK is nite and F is the eld generated by A. Let p be the1-type over A which says that x is transcendental over F . If K is0-saturated,then p must be realized in K . Thus every 0-saturated algebraically closed eldhas innite transcendence degree.

On the other hand suppose K has innite transcendence degree and F K is a eld generated by fewer that |K | elements. Let pS 1(F ) and let I p be asin Proposition 5.4. If I p = {0}, then p simply says x is transcendental over F ,and we can nd a realization in K . If I p is generated by f (X ), then any zero of f realizes p and we can nd a realization in K .

Unfortunately saturated models are not so easy to come by in general. In general

|S n (A)| can be as large as 2 | A | + |L | + 0 . For example, 1-types over Q in the theory

of real closed elds, correspond to cuts in the rationals so |S 1(Q )| = 20 . Thusset theoretic problems arise. Under assumptions like the generalized continuumhypothesis or the existence of inaccessible cardinals we can nd saturated models,but it is also possible that there are no saturated real closed elds.

Suppose |L | 0 and is an innite cardinal. We say that an L -theory T is -stable if and only for all M |= T and all AM , if |A| = , then |S n (A)| = .It is easy to see that algebraically closed elds are -stable for all innite .

Proposition 6.4. If T is -stable , then T has a saturated model of size + .

Proof. We build a saturated model of size + as a union of an elementarychain of models ( M : < + ) where each M has size . Let M 0 be any modelof size . For a limit let M be the union of the M , for < .

Given M . Let ( p : < ) list all 1-types over M . Build a chain of elementary extensions ( N , < ) where N 0 = M and where N contains arealization of p . Let M +1 be the union of the N .

Let M =


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7. Interpretability and Imaginaries

It is often very useful to study the structures which can be dened inside agive structure. For example, let K be a eld and let G be the group GL 2(K ).Let X = {(a,b,c,d)K 4 : ad bc = 0 }. Let f : X 2 X byf ((a1 , b1 , c1 , d1), (a2 , b2 , c2 , d2))

= ( a1a2 + b1c2 , a1b2 + b1c2 , c1a2 + d1c2 , c1b2 + d1d2).

Clearly X and f are denable and the set X with the operation f is isomorphicto GL 2(K ).

We say that an L 0-structure N is denable in an L -structure M if and only if we can nd a denable (in L ) subset X of M n for some n and we can interpretthe symbols of L 0 as denable subsets and functions on X so that the resultingL 0-structure is isomorphic to N .

The example above shows that GL n (K ) is denable in K . It is also easy tosee that any linear algebraic group is denable in K .

We give a more interesting example. Let F be a eld and let G be the groupof matricies of the form

a b0 1

where a, bK, a = 0. We will show that F is denable in the group G. Let

=1 10 1 and =

00 1

where = 0 , 1. Let

A = {gG : g = g}=1 x0 1 : xF

andB = {gG : g = g}=

x 00 1 : x = 0 .

Clearly A and B are denable.B acts on A by conjugation:

x 00 1

1 1 y0 1

x 00 1 =

1 y/x0 1 .

Clearly the action ( a, b) b 1ab is denable. We can dene the map i : A\{1} B by i(a) = b if b 1ab = 1, i.e.,

i 1 x0 1 =x 00 1 .

Dene an operation on A by

ab =i(b) 1ab if b = 11 if b = 1.


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32 DAVID MARKER

It is now easy to see that ( F, + , , 0, 1) is isomorphic to ( A, ,, 1, ). Thus theeld is denable in G.

This will not be true for all algebraic groups. For example, if E is an ellipticcurve and is the addition law on E then we cannot interpret a eld in thegroup ( E,).

Often we want to do more general constructions. For example, suppose wehave a denable group G and a denable normal subgroup H . We might wantto look at the group G/H . It is possible that G/H does not correspond toa denable group in our structure. But it does correspond to the cosets of adenable equivalence relation.

We say that an L 0-structure N is interpretable in an L -structure M if there isa denable set X , a denable equivalence relation E on X , and for each symbol of L

we can nd denable E -invariant sets on X , such that X/E with the inducedstructure is isomorphic to N .As an example let us show that we can interpret the additive group of integers

in the eld Q p. First note that we can dene Z p = {xQ p :y y2 = px2 + 1 }(at least for p = 2). Let U = {xZ p : yZ p : xy = 1}be the units of Z p.Then ( Z , +) is isomorphic to the multiplicative group Q p/U . We can dene theordering on Q p/U by

x/U y/U xy

Z p.

Quotient constructions are so useful that we often enrich our structure sothat we can deal with all quotients as elements of the structure. Let M be anL -structure. If E is a -denable equivalene relation on M n , let S E = M n /E

and let E : M n

M n

/E be the quotient map. Let Meq

be the structure whoseunderlying set is the disjoint union of M and all of the S E for E a -denableequivalence relation. In addition to the relations and functions of L , we addfunction symbols for each map E . We call the new elements of M eq imaginary elements.

If a structure N is interpretable in M , then N is denable in M eq . On otherhand, not much has changed, if X M

n is denable in M eq then X is alreadydenable in M .

An important property of many of the theories of elds that we will consideris that the passage from K to K eq is unnecessary.

Van den Dries showed that in real closed elds any denable equivalencerelation has a denable set of representatives. Nothing this strong could be truein algebraically closed elds as a set of represntatives for the equivalence relationxEy x2 = y2 would be innite and coinnite.

We say that M has elimination of imaginaries if, whenever E is a denableequivalence relation on M n, there is for some m a denable function f :M n M msuch that xEy f (x) = f (y).


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Theorem 7.1 (Poizat). Algebraically closed elds have elimination of imagi-naries .

In other words, if X is a constructible set and E is a constructible equivalencerelation on X , then X/E can be viewed as a constructible set.

The proof of Theorem 7.1 proceeds by rst showing that a theory has elimi-nation of imaginaries if and only if for any saturated model M and any X M

n

denable there is aM m for some m such that for all automorphisms of M ,

xes X setwise if and only if (a) = a . We call a canonical parameter for X .If X is dened by (x, b), we could dene an equivalence relation b1E b2 if andonly if (x, b1) (x, b2). Then b/E is a canonical parameter for X . In generalcanonical parameters will only be found in M eq .

Suppose K is an algebraically closed eld and X is an irreducible variety.There is a smallest subeld k K such that X is xed by an automorphism

if and only if k is xed pointwise ( k is called the eld of denition of X ). Thesubeld k must be nitely generated and if a generates k, then a is a canonicalparameter. From this observation and quantier elimination one can deriveelimination of imaginaries.

The eld of p-adics is a natural example where elimination of imaginariesfails. We saw above that we can interpret the integers in Q p. Analysis usingquantier elimination for the p-adics, shows that any denable set is either niteor uncountable, so the integers cannot be isomorphic to a denable set.

Here is another instructive example where elimination of imaginaries fails. LetK be an algebraically closed eld of characteristic zero. Let C be curve of genusat least one and let C be the structure with underlying set C and relation symbolsfor all constructible subsets of C n . Since there is a rational map : C

K , we

can intepret the eld on C using the equivalence relation xEy (x) = (y).If C/E was denably isomorphic to a denable set X C

n , this would give riseto a denable map f : K C . But (by Corollary 3.4) there is a rational mapg : K C which agrees with f on all but a nite set. By genus considerationsg is constant.

It is often very important to understand the groups and elds interpretablein a structure. For algebraically closed elds we get a very satisfying answer.It is easy to see that if K is an algebraically closed eld any algebraic groupdened over K is interpretable in K and hence, by elimination of imaginaries,isomorphic to a denable group. The following theorem is related to Weilstheorems on group chunks.

Theorem 7.2. (i) (van den Dries and Hrushovski) If a group G is denable in an algebraically closed eld K , then G is denably isomorphic to the K -rational points of an algebraic group dened over K .(ii) (Poizat) If F is an innite eld denable in an algebraically closed eld ,then F is denably isomorphic to K .


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34 DAVID MARKER

Appendix: Formulas

Here I give a precise denition of formulas. Let L be a language.

Definition. (i) The set of L -terms is the smallest set T such that all constantsymbols of L are in T ,(ii) all variables are in T , and(iii) if t1 , . . . , t n are in T and f is an n-ary function symbol of L , then

f (t1 , . . . , t n )T .

The set of atomic L -formulas is the smallest set A such that

(i) if t1 and t2 are terms, then t1 = t2 is in A , and(ii) if t1 , . . . , t n , are terms and R is an n-ary function symbol, then R(t1 , . . . , t n )is in A .

The set of L -formulas is the smallest set F such that(i) every atomic L -formula is in F ;(ii) if

F , then F ;(iii) if and are in F , then ( ), (), ( ) and ( ) are in F ;(iv) if is in F and x i is a variable, then x i and x i are in

F .

For example, x1 + ( x2 (x1 + 1)) is an L or -term, x1 (x2 + x3) = x1 + 1 andx1 < x 3 + x7 are atomic L or -formulas, and x1 (x1(x2 + x3) = x1 +1 x2 < x 1)is an L or -formula.

Bibliographical Notes

The following are good basic texts: [Chang and Keisler 1990; Hodges 1993;1997; Poizat 1985; Sacks 1972]. For an introduction to the model theory of algebraically closed, real closed, differentially closed and separably closed elds,see [Marker et al. 1996]. Proofs of Theorem 7.1 can be found in the same volumeor in [Poizat 1989]. A proof of 7.2 can be found in [Poizat 1987].

References

[Chang and Keisler 1990] C. C. Chang and H. J. Keisler, Model theory , Third ed.,Studies in Logic and the Foundations of Math. 73 , North-Holland, Amsterdam,1990.

[Chatzidakis 2000] Z. Chatzidakis, A survey on the model theory of difference elds,

pp. 6596 in Model theory, algebra and geometry , edited by D. Haskell et al., Math.Sci. Res. Inst. Publ. 39 , Cambridge Univ. Press, New York, 2000.

[Hodges 1993] W. Hodges, Model theory , Encyclopedia of Mathematics and itsApplications 42 , Cambridge University Press, Cambridge, 1993.

[Hodges 1997] W. Hodges, A shorter model theory , Cambridge University Press,Cambridge, 1997.


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[Lang 1984] S. Lang, Algebra , Second ed., Addison-Wesley, Reading, MA, 1984.

[Marker et al. 1996] D. Marker, M. Messmer, and A. Pillay, Model theory of elds ,

Lecture Notes in Logic 5 , Springer, Berlin, 1996.[Poizat 1985] B. Poizat, Cours de theorie des modeles: une introduction ` a la logique

mathematique contemporaine , Nur al-Mantiq wal Marifah 1 , Bruno Poizat, Ville-urbanne, 1985.

[Poizat 1987] B. Poizat, Groupes stables: une tentative de conciliation entre la geometrie algebrique et la logique mathematique , Nur al-Mantiq wal Marifah 2 ,Bruno Poizat, Lyon, 1987.

[Poizat 1989] B. Poizat, An introduction to algebraically closed elds and varieties,pp. 4167 in The model theory of groups (Notre Dame, IN, 19851987), edited byA. Nesin and A. Pillay, Univ. Notre Dame Press, Notre Dame, IN, 1989.

[Sacks 1972] G. E. Sacks, Saturated model theory , Math. Lecture Note Series, W. A.Benjamin, Reading, MA, 1972.

David MarkerUniversity of Illinois at ChicagoDepartment of Mathematics, Statistics, and Computer Science (M/C 249)851 S. Morgan St.Chicago, IL 60613United States

[email protected]

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Marker Introduction to Model Theory