# Lecture notes - Model Theory (Math 411) Autumn 2002.

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1 Lecture notes - Model Theory (Math 411) Autumn Anand Pillay December 9, Notation and review. Let me begin by briefly discussing many-sorted structures. Although in most of the course I will be working with the traditional 1-sorted structures, everything is valid in the more general context. By a many-sorted language L (or rather a language for many-sorted structures), we mean a set S of sorts, a set R of sorted relation symbols, and a set F of sorted function symbols. What this means is that each R R comes together with a certain finite sequence (S 1,.., S n ) of sorts where n 1, and each f F comes together with a sequence (S 1,.., S n, S) of sorts where n 0. By the cardinality L of L we mean S + R + F. By an L-structure M we mean a family (S(M) : S S) of nonempty sets, together with, for each R R of sort (S 1,.., S n ) a subset R(M) of S 1 (M)... S n (M), and for each f F of sort (S 1,.., S n, S) a function f(m) : S 1 (M).. S n (M) S(M). So an L-structure is just an interpretation of the formal language L. The S(M) are the sorts of the structure, the R(M) the basic relations of the structure and the f(m) the basic functions of the structure. By the cardinality M of M we mean the sum of the cardinalities of the sets S(M). A one-sorted language is a (many-sorted) language with only one sort, that is S is a singleton, say {S}. In that case the sort of a relation symbol is just a natural number n 1 and likewise for the sort of a function symbol. If M is an L-structure then S(M) is called the underlying set (or universe) of 1

2 M and we often notationally identify it with M (just as we often notationally identify a group G with its underlying set) if there is no ambiguity. The many-sorted point of view is quite natural, because a whole category of objects can be now viewed as a single structure. An example is the category of algebraic varieties over Q. By an algebraic variety over Q we mean (for now) a subset X of some C n which is the common zero-set of a finite number of polynomials P i (x 1,.., x n ) in Q[x 1,.., x n ]. By a morphism (over Q) between X C n and Y C m, we mean a map f : X Y given by m polynomials Q 1 (x 1,.., x n ),..., Q m (x 1,.., x n ) over Q. The many-sorted structure V ar Q has as sorts the algebraic varieties over Q, as basic relations the algebraic subvarieties (over Q) of products of the sorts, and as basic functions morphisms from products of sorts to sorts. Of course there is another natural associated one-sorted structure, the structure (C, +,,, 0, 1). This is the structure with universe C, two basic 2-ary functions, addition and multiplication, one basic unary function ( ) and two basic 0-ary functions (or constants), 0 and 1. Note that V ar Q and (C, +,,, 0, 1) are not only different structures, but structures for different languages. But we hope later to explain a sense in which they are the same. (They are bi-interpretable.) Given a many-sorted language L we build up in the usual manner terms and first order formulas of L from the relation and function symbols of L together with,,,,, and a countable set of variables for each sort S S, as well as the symbol = S for each S S. Note that every variable comes equipped with a sort. If φ is an L-formula we may write φ(x 1,.., x n ) to mean that the free variables in φ are among x 1,.., x n. An L-sentence is an L-formula with no free variables. If φ(x 1,.., x n ) is an L-formula where x i is of sort S i, M is an L-structure and a i S i (M) for i = 1,.., n we define (inductively) M = φ[x 1 /a 1,.., x n /a n ] in the usual way. ( (a 1,., a n ) satisfies φ(x 1,.., x n ) in M, or φ(x 1,.., x n ) is true of (a 1,.., a n ) in M.) When there is no ambiguity we just write M = φ[a 1,.., a n ]. An important fact is that exactly one of M = φ[a 1,.., a n ], M = ( φ)[a 1,.., a n ] holds. (By the way the = S symbols are interpreted as equality, namely if x, y are variables of sort S, M is an L-structure, and a, b S(M) we define M = x = S y[x/a, y/b] to hold if a = b.) If σ is an L-sentence we say M is a model of σ for M = σ and this is 2

3 where the expression model theory comes from. Definition 1.1 Let L be a language and M an L-structure, Σ a set of L- sentences, and σ an L-sentence. (i) We say that M = Σ (M is a model of Σ) if M = σ for all σ Σ. (ii) Σ = σ (Σ logically implies σ) means that every model of Σ is a model of σ. (iii) By an L-theory we mean a set of L-sentences closed under =. (Ltheories are often debtoed by T.) (iv) We say that Σ is consistent if Σ has a model. (v) We say that Σ is complete if Σ is consistent and for every σ, either σ Σ or σ Σ. (vi) Let K be a class of L-structures. By T h(k) we mean {σ : M = σ for all M K}. If K = {M} we write T h(m) in place of T h(k). (vii) Let K be a class of L-structures. We say that K is an elementary class if there is a set Σ of L-sentences such that K = {M : M = Σ}. (viii) We write M N (M is elementarily equivalent to N)if M and N are models of the same L-sentences, that is, if T h(m) = T h(n). (ix) Let T be an L-theory. We say that Σ axiomatizes T if Σ T and Σ = σ for all σ T. Exercise 1.2 (i) Let Σ be a set of L-sentences and Σ = {σ : Σ = σ}. Then Σ is a theory. (ii) T h(m) is a complete theory. (iii) Let K be a class of L-structures. Then K is an elementary class iff K = {M : M = T h(k)}. (iv) T is complete iff all models of T are elementarily equivalent Example 1.3 Let L be the 1-sorted language containing a single binary function symbol f say, and a single constant sybol e say. Any group can be considered to be an L-structure. (i) The class of groups is elementary. (ii) The class of torsion-free divisible abelian groups is an elementary class. (iii) (needs compactness theorem) The class of torsion abelian groups is NOT elementary. (iv) (needs more) The class of simple nonabelian groups is NOT an elementary class. 3

4 Example 1.4 Let L be the language of unitary rings that is L is again one-sorted, contains two binary function symbolds + and say, one unary funtion symbol, and two constant symbols 0, 1. Any ring can be naturally viewed as an L-structure. (i) The class of finite fields is NOT an elementary class. (Needs compactness.) (ii) Let K be the class of finite fields. By definition a pseudofinite field is an infinite model of T h(k). Then the class of pseudofinite fields IS an elementary class. (Why?) We now consider relations between L-structures. Let us fix a (many-sorted) language L. The notion of an isomorphism between two L-structures is pretty obvious and clearly we are only interested in L-structures up to isomorphism. We have already defined elementary equvalence of L-structures and it is important that this is generally weaker than isomorphism. However: Exercise 1.5 Let M and N be elementarily equivalent L-structures. Suppose that for every sort symbol S of L, S(M) is finite. Then M and N are isomorphic. Definition 1.6 (i) Let M, N be L-structures. We say that M is a substructure of N if for each sort symbol S, S(M) S(N), for each relation symbol R of L of sort S 1... S n, R(M) = R(N) (S 1 (M)... S n (M)) and for each function symbol f of L of sort (S 1,.., S n, S), and choice a i S i (M) for i = 1,.., n, f(m)(a 1,., a n ) = f(n)(a 1,.., a n ). (ii) We say that M is an elementary substructure of N if M is a substructure of N, and for all L-formulas φ(x 1,., x n ) and a 1,., a n in the sorts of M corresponding to variables x 1,.., x n respectively, we have M = φ[a 1,.., a n ] iff N = φ[a 1,.., a n ]. (iii) An (elementary) embedding of M into N is an isomorphism of M with a (elementary) substructure of N. Exercise 1.7 (i) Let L be the language of unitary rings mentioned earlier. Let F, K be fields. Then F is a subfield of K iff F is a substructure of K. (ii) Let L be 1-sorted and consist of a single unary function symbol. Let S be the succesor function on Z. Then (Z, S) and N, S) are naturally L- structures. The first is a substructure of the second, but not an elementary 4

5 substructure. (iii) Let M be a substructure of N. Then for any quantifier-free L-formula φ(x 1,.., x n ) and a i M of suitable sort, M = φ[a 1,.., a n ] iff N = φ[a 1,.., a n ]. Exercise 1.8 (Tarski-Vaught test.) Let M be a substructure of N. Then M is an elementary substructure if M if and only if for each L-formula φ(x 1,.., x n, x) and a 1,.., a n M (in the appropriate sorts), if N = ( x(φ))[a 1,.., a n ] then for some b of the appropriate sort in M, N = φ[a 1,.., a n, b]. Let us introduce a bit of notation. Let L be a language as usual, and M an L-structure. Let L M be L together with a set of new (sorted) constant symbols c m for m ranging over elements of M (i.e. of elements of sorts of M). Then we can expand canonically M to an L M -structure M say by defining c m (M ) = m. We often write M as (M, m) m M. For φ(x 1,.., x n ) an L-formula, and m 1,.., m n M of appropriate sorts, let φ(c m1,.., c mn ) be the L M -formula which results from substituting c mi for x i in φ. In this context we have: Remark 1.9 (i) M = φ[m 1,.., m n ] iff (M, m) m M = φ(c m1,.., c mn ). Note that we can do exactly the same thing, if we add constants for a subset A of M (that is each element of A is in some S(M)), to obtain a language L A and an expansion (M, a) a A of M to an L A -structure. With this notation we have: Lemma 1.10 Let M be a substructure of N. Then M is an elementary substructure of N iff and only if (M, m) m M and (N, m) m M are elementarily equivalent as L M -structures. Sometimes we completely abuse notation by writing M = φ(m 1,.., m n ) in place of the equivalent conditions in Remark 1.9. Here is a useful slight strengthening of 1.8 which we state for convenience in the 1-sorted context. Exercise 1.11 (Assume L to be 1-sorted.) Let M be an L-structure, and A a subset of the underlying set (or universe) of M. Then A is the universe of an elementary substructure of M if and only if for each L A -formula φ(x), if M = x(φ(x)) then there is b A such that M = φ(b). 5

6 By a chain of L-structures we mean a sequence (M i : i I) (of L- structures) where I = (I, <) is an ordered set and i < j implies that M i is a substructure of M j. Note that the union of the underlying sets (or sorts) of the M i is naturally equipped with an L-structure, which is an extension of each M i. We call this L-structure i M i. An elelementary chain of L- structures is a chain as above but with the additional feature that i < j implies M i is an elementary substructure of M j. Lemma 1.12 Let (M i : i I) be an elementary chain, and M = i M i. Then M i is an elementary substructure of M for all i. Proof. We prove by induction on the complexity of the L-formula φ(x 1,.., x n ) that for all i and a 1,.., a n M i, M i = φ(a 1,., a n ) iff M = φ(a 1,.., a n ). For φ quantifier-free this is Exercise 1.7 (iii). If φ is built from formulas we already dealt with, by,,, or, it is easy. Suppose φ is x(ψ(x, x 1,.., x n )). and suppose that M i = φ(a 1,.., a n ), then M i ψ(b, a 1,.., a n ) for some b M i, so M = ψ(b, a 1,.., a n ) by induction hypothesis and so M = φ(a 1,.., a n ). Conversely, suppse M = φ(a 1,.., a n ), so M = ψ(b, a 1,.., a n ) for some b M. Then b M j for some i < j. By induction hypothesis M j = ψ(b, a 1,.., a n ), so M j = φ(a 1,.., a n ) so by our assumptions M i = φ(a 1,.., a n ). Definition 1.13 Let Σ be a set of L-sentences. We say that Σ is finitely consistent, if every finite subset of Σ is consistent (namely every finite subset of Σ has a model). From now on we assume L to be one-sorted for convenience. Exercise 1.14 Suppose Σ is a finitely consistent set of L-sentences. Then (i) For any L-sentence σ either Σ σ or Σ σ is finitely consustent. (ii) Suppose xφ(x) Σ and c is a constant symbol not appearing in Σ. Then Σ φ(c) is finitely consistent. Theorem 1.15 (Compactness) Let Σ be a set of L-sentences. Then Σ is finitely consistent if and only if Σ is consistent. Note that right to left is trivial. We will give a couple of sketch proofs of left to right. Proof A. (essentially Henkin.) Assume for simplicity that the language L is 6

7 countable and has no function symbols. Assume Σ to be finitely consistent. Let {c i : i < ω} be a countable set of new constant symbols. Let L be L together with these new symbols. Let (σ i : i < ω) be an enumeration of all L -sentences. We build an increasing sequence Σ 0 Σ 1 Σ 2... of sets of L-sentences with the following properties: (i) Σ 0 = Σ, (ii) Each Σ i is finitely consistent and consists of Σ together with at most finitely many additional sentences, (iii) If (a) Σ i {σ i } is finitely consustent then σ i Σ i+1. Otherwise σ i Σ i+1. (iv) If (a) holds in (iii) and σ i has the form xφ(x) then for some new constant c j, φ(c j ) Σ i+1. The construction of the Σ i is by induction, using Exercise Let Σ be the union of the Σ i. Then we have: (v) Σ is a finitely consistent set of L -sentences, (vi) For each L -sentence σ, exactly one of σ, σ is in Σ, and (vii) if xφ(x) Σ then φ(c) Σ for some constant symbol c of L. Now we build an L -structure. Let be the following relation on {c i : i < ω}: c i c j if c i = c j Σ. Check that is an equivalence relation. Let M be the L -structure whose elements are the classes, and such that for an n-place relation symbol R of L, ((c i1 / ),..., (c in / )) R(M) if R(c i1,.., c in ) Σ. Check this is well-defined. Finally check that for each L -sentence σ, M = σ iff σ Σ. (By induction on the complexity of σ.) In particular M = Σ. Let M 0 be the L-reduct of M, that is M considered as an L-structure. So M 0 is a model of Σ, and Σ is consistent. (Where in the proof do we use finite consistency of Σ?) Proof B. This proof is by ultraproducts which we will define subsequently. Let I be the set of finite subsets of Σ. For each i I let M i be a model of i. For each σ Σ, let X σ = {i I : σ i}. Let F 0 = {X σ : σ Σ}. Then F 0 is a collection of subsets of I, i.e. a subset of P (I). Let F be the filter on I generated by F 0, namely the closure of F 0 under supersets and finite intersections. 7

8 Claim. F is a nontrivial filter on I, that is / F. Proof. It is enough to show that for σ 1,.., σ n Σ, X σ1... X σn, which is clear since it contains {σ 1,.., σ n }. By Zorn s Lemma we can extend F to an ultrafilter U on I. This means that U is a maximal nontrivial filter on I, equivalent U is a nontrivial filter on I such that for any X I, either X or the complement of X is in U. Now let M be the ultraproduct ( M i )/U (to be expained below). Los Theorem states that for any L-sentence σ, M = σ if and only if {i I : M i = σ} U. Claim. M is a model of Σ. Proof. Let σ Σ. Then for each i X σ, M i = σ (by choice), so {i I : M i = σ} X σ so is in F so also in U. By Los Theorem, M = σ. Let us define ultraproducts and state Los Theorem. Let I be a set, and U an ultrafilter on I. We will say that a subset X of I is large if X U. By i I M i we mean for now just the set of sequences s = (s(i) : i I) such that s(i) M i for each i. Define on this set by s t if {i : s(i) = t(i)} is large. M = ( i M i )/U will be the L-structure whose underlying set is {s/ : s i M i ), and such that for R an n-ary relation symbol (s 1 /,.., s n / ) R(M) if {i I : (s 1 (i),.., s n (i)) R(M i )} is large, and for an n-ary function symbol f, f(m)(s 1 /,.., s n / ) = t/ where t(i) = f(m i )(s 1 (i),.., s n (i)). One must prove that this is well-defined. Theorem 1.16 (Los.) Let φ(x 1,.., x n ) be an L-formula, and s 1 /,., s n / elements of ( i M i )/U. Then ( i M i )/U = φ(s 1 /,.., s n / ) if and only if {i I : M i = φ(s 1 (i),.., s n (i))} is large. The theorem is proved by induction on the complexity of φ (with maybe a prior lemma on terms). In the proof B above of the compactness theorem we used the special case of the Theorem where σ is a sentence. Lemma 1.17 Let K be a class of L-structures. Then K is an elementary class if and only if K is closed under elementary equivalence and ultraproducts. Proof. Left to right is immediate (using Theorem 1.16). For right to left. Let T = T h(k). We must prove that every model M of T is in K. Let M 8

9 be a model of T. Let Σ = T h(m). Let I be the set of finite subsets of Σ. Note that for each i I there is M i K which is a model of i. (Otherwise i is a sentence in T so in Σ, contradiction.) Let U be the ultrafilter on I produced as in the proof B of the compactness theorem. Then ( i M i )/U is a model of Σ so elementarily equivelent to M. By the hypotheses, M K. Let us give some corollaries and reformulations of the compactness theorem. Proposition 1.18 Let Σ be a set of L-sentences and suppose that for every n < ω, every finite subset of Σ has a model of cardinality at least n. Then Σ has an infinite model Proof. Let σ n be the L-sentence expressing that there are at least n elements (for n < ω). Then By hypothesis Σ {σ n : n < ω} is finitely consistent, so consistent by the compactness theorem. Proposition 1.19 Let Σ and Γ be sets of L-sentences. Suppose that for every model M of Σ there is γ Γ such that M = γ. (We could write here Σ = Γ.) Then there is are finite subsets Σ of Σ and Γ of Γ such that Σ = Γ. Proof. The hypothesis says that Σ { γ : γ Γ} is inconsistent. By the compactness theorem a finite subset is inconsistent, yielding the desired conclusion. Proposition 1.20 Let T be the set of complete (consistent) L-theories. Define X T to be closed if X = {T : Σ T } for some set Σ of L-sentences. Then this makes T into a topological space which is compact (Hausdorff) and totally disconnected. Proof. First as a word of explanation, a topological space is said to be totally disconnected if it has a basis of clopens. First we ll show that T is a topological space. Clearly an intersection of closed sets is closed. The whole space is closed (by taking Σ = ) and the empty set is closed (by taking Σ to be {σ σ} for some sentence σ). T is Hausdorff because if T 1 and T 2 are distinct complete L-theories then there is σ such that σ T 1 and σ T 2. σ and σ yield disjoint clopen subsets of T. For σ a sentence X σ = def {T : σ T } is clopen, and the X σ form a basis 9

10 for T so T is totally disconnected. Finally compactness of T is precisely the compactness theorem: Suppose that T is covered by open sets U i so by basic open sets X σi, for i I. This means that every complete theory T contains some σ i, and thus no complete theory contains σ i for all i. So { σ i : i I} is inconsistent (has no model) thus by the compactness theorem sme finite subset, say { σ 1,.., σ n } is inconsistent. But then every complete theory T contains one of the σ 1,.., σ n so X σ1,.., X σn cover T. Remark 1.21 Compact (Hausdorff) totally disconnected spaces are precisely the Boolean spaces. The set of clopen subsets of such a space S form (under the natural operations) a Boolean algebra B. The set of ultrafilters on B form a Boolean space homeomorphic to S. There are other interpretations of the compactness theorem, including Deligne s Theorem : A coherent topos has enough points. I will point out now how the compactness theorem for sets of sentences also holds for sets of formulas. Let L be a language as before. We defined the cardinality of L to be the number of (nonlogical) symbols in L. We built up L-formulas from these symbols together with the logical symbols (including a countably infinite supply of variables). Note that the cardinality of the set of L-formulas is L + ω. Let us now allow ourselves as many variables as we want. We then get possibly more L-formulas, but still the same number of L-sentences up to logical equivalence. (Why?) Let us write Σ(x i ) i I to mean that Σ is a set of L-formulas with free variables among {x i : i I}. By a model of Σ(x i ) i I we mean an L-structure M together with a set {a i : i I} of elements of M such that for each formula φ(x i1,.., x in ) Σ, M = φ(a i1,.., a in ). We ll say that Σ(x i ) i I is consistent if it has a model, and finitely consistent if every finite subset has a model. Remark 1.22 Σ(x i ) i I is finitely consistent, if for each finite subset Σ of Σ, and finite sequence x of variables from among the x i which includes the free varoables occuring in Σ, the L-sentence x( Σ ( x)) is consistent. Exercise 1.23 Let Σ(x i ) i I be a set of L-formulas. Let {c i : i I} be a set of new constant symbols, and let L be the language got by adjoining these to L. Let Σ(c i ) i I be the set of L -sentences obtained by substituting the c i for the x i in Σ. Then σ(x i ) i I is consustent if and only if Σ(c i ) i I is consistent (as a set of L -sentences. 10

11 This we get: Proposition 1.24 Σ(x i ) i I is consistent if and only if it is finitely consistent. Definition 1.25 Let T be an L-theory. Let (x i : i I) be a sequence of variables. By an I-type of T we mean a set Σ(x i ) i I of L-formulas such that T Σ is consistent. Lemma 1.26 Suppose T = T h(m). Let Σ(x i ) i I be a set of L-formulas. Then Σ(x i ) i I is an I-type of T if and only if for every finite subset Σ ( x) of Σ, M = x( Σ ( x)). Proof. By 1.24, Σ is an I-type of T if and only if T Σ is finitely consistent if and only if for each finite subset Σ ( x) of Σ, T Σ ( x) is consistent if and only if for each such Σ, T x( Σ ( x)) is consistent if and only for each such Σ, M = x( Σ ( x)). We will come back to types in the next section. We complete this section with the Lowenheim-Skolem theorems and some applications. Proposition 1.27 (Downward L-S.) Let M be an L-structure of cardinality κ. Let A be a subset of the universe of M and let λ be a cardinal such that L + ω + A λ κ. Then There is an elementary substructure N of M such that A is a subset of the universe of N and N has cardinality λ. Proof. Without loss of generality A = λ. Note that the cardinality of the set of L A -formulas is λ. For each L A -formula φ(x) such that M = x(φ(x)), pick b φ M such that M = φ(b φ ) and let A 1 = A {b φ } φ. Then A 1 = λ. Now construct A 2 from A 1 in the same way that A 1 was constructed from A. Continue to get A A 1 A 2 A 3..., and let B be the union. Then B is a subset of the universe of M of cardinality λ and for each L B -formula φ(x) such that M = xφ(x), there is b B such that M = φ(b). By 1.11, B is the universe of an elementary substructure of M. Definition 1.28 Let L L be say one-sorted languages. Let M be an L -structure. By M L (the L-reduct of M ) we mean the L-structure whose universe is the same as that of M and such that for all relation symbols 11

12 R L, R(M L) = R(M ) and for all functions symbols f L, f(m L) = f(m ). We call M an expansion of M L to an L -structure (If L and L are many sorted languages and L L then maybe sort sort symbols of L are not in L. It is then natural to consider the L-reduct of an L -structure, to be the L-structure M such that S(M) = S(M ) for all sort symbols in L, etc.) Definition 1.29 Let M be an L-stucture. Let D c (M), the complete diagram of M be T h(m, m) m M in the language L M. Lemma 1.30 Let M, N be L-structures. Then M can be elementarily embedded in N if and only if N can be expanded to a model of D c (M). Proof. Suppose first that there is an elementary embedding f of M into N. Expand N to an L M -structure by interpreting c m as f(m). Check that this expansion N say is a model of D c (M). Conversely, if n can be expanded to a model N say of D c (M), let us denote f(m) as the interpretation of c m in M, and then check that f is an elemetary embedding of M into N. Proposition 1.31 (Upward L-S.) Let M be an infinite L-structure. λ M + L. Then M has an elementary extension of cardinality λ. Let Proof. Let (c i : i < λ) be a sequence of new constants. Let L be L M together with these new constant symbols. Let Σ = D c (M) {c i c j : i < j < λ}. Σ is a set of L -sentences. We claim that Σ is finitely consistent. This is immediate as any finite subset Σ of Σ involves a finite part of D c (M) and only finitely many of the constant symbols c i. So (M, m) m M together with distinct interpretations of these finitely many c i will be a model of Σ. By the compactness theorem, Σ has a model N say. N has cardinality at least λ. Let N the be L-reduct of N. By 1.30 we may assume that N is an elementary extension of M. By 1.27 we may find an elementary substructure of N of cardinality exactly λ which contais the universe of M and so is also an elementary extension of M. Corollary 1.32 Let T be an L-theory, and suppose that T has an infinite model. Then for any infinite cardinal λ L, T has a model of cardinality λ. 12

13 Here is another important application of Proposition 1.33 Suppose that M and N are elementarily equivalent L- structures. Then there is an L-structure M 1 and elementary embedding f of M into M 1 and g of N into M 1. (Also we could choose one of f, g to be the identity map.) Proof. There is no harm in assuming the universes of M and N to be disjoint. Let L = L M L N. (So the new constants for elements of M are assumed disjoint from the new constants for elements of N. By 1.30 it suffices to prove that D c (M) D c (N) is consistent (as a set of L -sentences). We ll show it to be finitely consistent (and use compactness). Let Σ 1 be a finite subset of D c (M) and Σ 2 a finite subset of D c (M). Let c n1,.., c nk be the new constants appearing in Σ 2. So we can write Σ 2 as Σ 3 (c n1,.., c nk ) where Σ 3 (x 1,.., x k ) is a finite set of L-formulas. Note that N = Σ 3 (n 1,.., n k ) and thus N = x 1,.., x k ( Σ 3 (x 1,.., x k )), and so also M = x 1...x k ( Σ 3 (x 1,.., x k )). Let a 1,.., a k M be such that M = Σ 3 (a 1,.., a k ). Then clearly, if we interpret c ni as a i (and the other c n as anything you want) we expand M to an L N -structure which is a model of Σ 2. Of course M can also be expanded (tautologically) to an L M -structure which is a mdel of Σ 1. So Σ 1 Σ 2 is consistent. Definition 1.34 Let T be an L-theory and κ a (possibly finite) cardinal. T is said to be κ-categorical if T has exactly one model of cardinality κ (up to isomorphism). Proposition 1.35 Suppose that T has only infinite models, and is κ-categorical for some cardinal κ L. Then T is complete. Proof. Let M, N be models of T. Let T 1 = T h(m) and T 2 = T h(n). By Corollary 1.31, T 1 has a model M 1 of cardinality κ and T 2 has a model N 2 of cardinality κ. Then M 1 and N 1 are models of T of cardinality κ so are isomorphic by assumption. It follows that T 1 = T 2 and M N. Thus T is complete. Example 1.36 Let L be the empty language and T the theory axiomatized by {σ n : n < ω} where σ n says there are at least n elements. Then T is totally categorical (categorical in all infinite cardinalities) and so complete. 13

14 Proof. A model of T is just an infinite set. An isomorphism between models is simply a bijection between their universes. So T is κ-categorical for any infinite cardinal. Example 1.37 Let L = {+, 0}. Let T be the theory of torsion-free divisible nontrivial abelian groups. Then T is categorical in every uncountable cardinality, and complete. Proof. I leave it to you to axiomatize T. The models of T are precisely the nontrivial torsion-free divisible abelian groups. Any such model is a Q- vector space of dimension 1. An isomorphism between two such models is precisely an isomorphism of Q-vector spaces. Any nontrivial Q-vector space V is infinite, and its isomorphism type is determined by its dimension (the cardinality of a Q-basis. If V has cardinality λ > ω, then clearly dim(v ) = κ. (Why?) Thus T is κ-categorical for any uncountable κ. Example 1.38 Let G be an infinite group of cardinality κ say. Let L = {λ g : g G} where the λ g are unary function symbols. Let T be the theory of free G-sets. Then T is categorical in every cardinality > κ, and complete. Proof. First what do we mean by a free G-set? First of all, a G-set is a set X together with mapping from G X to X ((g, x) g x) such that 1.x = x for all x X and (gh) x = g (h x) for all g, h G and x X. X is said to be a free G-set if it is a G-set and for g 1 in G and x X, g x x. Any G-set X can be viewed naturally as an L-structure, and we can write axioms for free G-sets in the language L. If X is a free G-set then X is the disjoint union of G-orbits where a G-orbit is G a = {g a : g G}. Any such G-orbit is in bijection with G via g g a (for a fixed a in the G-orbit). If X, Y are free G-sets with the same number of orbits then X and Y are isomorphic (as G-sets and as L-structures) as follows: Let (a i : i I) and (b i : i I) be representatives of the G-orbits in X, Y respectively. Define a bijection between X and Y by (g a i ) goes to g b i for each g G and i I. Then this is an isomorphism. Now if X is a G-set of cardinality λ > κ then X has precisely λ-many G-orbits. It follows that T is λ-categorical. Example 1.39 Let L = {<} and let T be the theory of dense linear orderings without endpoints. Then T ω-categorical and complete. 14

15 Proof. The axioms are obvious. Any model is infinite. A well-known backand forth argument shows that any two countable models of T are isomorphic so T is ω-categorical, so complete. T has 2 κ models of cardinality κ for every uncountable κ. Example 1.40 Let L = {R} where R is a binary relation symbol. T says (informally) that R is irreflexive, symmetric and that for any distinct x 1,.., x n, y 1,., y n there is z such that R(z, x i ) and R(z, y i ) for all i = 1,.., n. Then T is ω-categorical and complete. Proof. Again the axiomatization of T is clear. Clearly T has only infinite models. One issue is why T has a model. An important general construction of Fraisse produces a countable model. For now we just assume T to be consistent. Now if M, N are countable models we can show by a back-and-forth argument that M and N are isomorphic. T is ω-categorical, so complete. However again T has 2 κ models of cardinality κ for any uncountable κ. The countable model of T is called the random graph. 2 Types, ω-saturation and quantifier elimination We have seen in the previous section that categoricity can be used to prove completeness of a theory. However this method does not always work as there are complete theories which are not categorical in any uncountable cardinality (such as what). The method of (relative) quantifier elimination often enables one to prove completeness of particular theories and we ll talk about this now. Moreover quantifier elimination tells one about the definable sets in models of a theory, and that is also very important. We continue our convention of working with one-sorted structures although everything holds in the many-sorted case. Definition 2.1 Let M be an L-structure, and X M n. We say that X is -definable in M if there is an L-formula φ(x 1,.., x n ) such that X = {(a 1,.., a n ) M n : M = φ(a 1,.., a n )} ( = φ(m).) If A is a subset of the universe of M we say that X is A-definable in M if we can find an L A - formula (or if you want an L M -formula where only constants for elements of 15

16 A appear) such that X = φ(m) again. A definable set in M is by definition an L M -definable set in M. Understanding a structure M from the model-theoretic point of view involves at least understanding its definable sets. Definition 2.2 Let T be a theory in language L. We say that T has quantifier elimination if for any L-formula φ(x 1,.., x n ) (n 1) there is a quantifierfree L-formula ψ(x 1,.., x n ) such that T = x(φ( x) ψ( x)). Remark 2.3 (i) Suppose that T has quantifier-elimination and L contains at least one constant symbol. Then for any L-sentence σ there is a quantifierfree L-sentence τ such that T = σ τ. (ii) Suppose T has quantifier-elimination and M is a model of T. Then every definable set in M is defined by a quantifier-free L M -formula. Every -definable set in M is defined by a quantifier-free L-formula. Definition 2.4 Let T be an L-theory. Let L = L together with some new relation symbols and function symbols. For each new n-ary relation symbol R, let φ R (x 1,.., x n ) be an L-formula. For each new nary function symbol f, let ψ f ( x, y) be an L-formula such that T = ( x)( =1 y)(ψ f ( x, y)). Let T be axiomatized by T together with x(r(x 1,., x n ) φ R (x 1,.., x n )), and x y(ψ f ( x, y) f( x) = y)), for all new R and f. We call any T obtained this way a definitional expansion of T. Remark 2.5 Note that a definitional expansion T of T is to all intents and purposes the same as T : any model of T ca be expanded to a unique model of T, etc. On the other hand for any theory T we can find a definitional expansion of T which has quantifier-elimination: for each L-formula φ(x 1,.., x n ) adjoin a new n-ary relation symbol R φ and the axiom x(r φ ( x) φ( x)). This definitional expansion is called the Morleyization of T. We will give a useful method for proving quantifier elimination (when it s true) using back-and-forth systems of partial isomorphisms in ω-saturated models. Let us first recall types. We saw in 1.25 the definition of an I-type of a theory. By an n-type we mean an I-type where I = {1,.., n}. x will usually denote an n-tuple of variables (x 1,.., x n ) for some 1 n < ω. 16

17 Definition 2.6 (i) Let Σ(x i ) i I be a collection of L-formulas and M an L- structure. We say that Σ is realized in M if there is a tuple (a i ) i I of elements of M such that M = Σ(a i ) i I (and we also say then that (a i ) i I realizes Σ in M). We say that Σ is omitted in M if its is not realized in M. (ii) By a complete n-type of T we mean an n-type Σ( x) of T such that for each L-formula φ( x) either φ( x) or φ( x) is in Σ. Complete types are often denoted by p, q,... (iii) Let M be an L-structure and ā be an n-tuple from M. By tp M (ā) (the type of ā in M) we mean {φ( x) L : M = φ(ā)}. Remark 2.7 (i) By definition, an I-type of T is realized in some model of T. (ii) Let Σ( x) be a set of L-formulas. Then Σ is a complete n-type of T if and only if Σ( x) = tp M (ā) for some model M of T and n-tuple ā from M. Definition 2.8 The L-structure M is said to be ω-saturated if whenever A is a finite subset of (the universe of) M and Σ( x) is a set of L A -formulas which is finitely satisfiable in M (that is every finite subset of Σ is realized in M), then Σ is realized in M. (ii) For κ an arbitrary infinite cardinal, we define κ-saturated in the same way except that A has cardinality < κ. Lemma 2.9 The following are equuivalent: (i) M is ω-saturated. (ii) For any finite subset A of M and n-type Σ of T h(m, a) a A, Σ is realized in M. (iii) For any finite subset A of M and complete n-type p( x) of T h(m, a) a A, p is realized in M. Proof. The lemma follows from a couple of observations. First, a set Σ( x) of L A -formulas is a n-type of T h(m, a) a A if and only if Σ is finitely satisfiable in M (by 1.26). Second, any n-type of a theory extends to a complete m-type of that theory. Remark 2.10 (i) If the structure M is understood and A is a subset of M then by a type over A we mean a type of T h(m, a) a A. To make things more explicit we may say : a type over A in the sense of M. Note that if N is an elementary extension of M then a type over A in the sense of M is the same 17

18 thing as a type over A in the sense of N. (ii) With the notation of (i), any type over A is realized in an elementary extension of M. Proof of (ii). If Σ( x) is a type over A then it is easy to see that it is also an n-type of D c (M), and so is realized in a model of D c (M) and thus in an elementary extension of M. A related notion to ω-saturation is ω-homogeneity. Definition 2.11 M is ω-homogeneous if for any n, whenever ā, b are n- tuples from M and tp M (ā) = tp M ( b) then for any c M there is d M such that tp M (āc) = tp M ( bd). Lemma 2.12 Any L-structure M has an ω-saturated elementary extension N. Moreover if L and M are at most countable then we can choose N to be of cardinality at most 2 ω. Proof. By compactness (a variant of Remark 2.10(ii)) we can find an elementary extension M 1 of M realizing all types over finite subsets of M. Continue with M 1 in place of M. We build an elementary chain M M 1 M 2... and let N be the union, an elementary extension of M. N is ω-saturated, as any finite subset of N is contained in some M i. Under the additional cardinality hypotheses, note that there are at most continuum many types over finite subsets of M so we can choose M 1 to have cardiality at most 2 ω (by downward Lowenheim-Skolem). Likewise there are at most continuum many types over finite subsets of M 1 etc. Although it is not easy to recogize when a model of a theory is ω-saturated, we can nevertheless often deduce properties from ω-saturation: Example 2.13 Let T be the theory of torsion-free divisible abelian groups (from 1.35). Let (V, +, 0) be an ω-saturated model of T. Then dim Q (V ) is infinite. Proof. Without loss of generality we will assume that our language also contains symbols for scalar multiplication by each element of Q. (Why can we assume this?). For each n consider the set Σ n (x 1,.., x n ) of L-formulas: {r 1 x r n x n 0 : r 1,.., r n Q not all zero}. Then Σ n is finitely satisfiable in V (why?), so realized, by ω-saturation. 18

19 Lemma 2.14 Let M be a structure and ā, b two k-tuples in M such that tp M (ā) = tp M ( b). Let Σ( x) be an n-type over a. Let Σ ( x) = {φ(x 1,.., x n, b 1,.., b k ) : φ(x 1,.., x n ) L and φ(x 1,.., x n, a 1,.., a k ) Σ}. Then Σ ( x) is an n-type over b. Moreover Σ is complete (as an n-type over b) if Σ( x) is complete as an n-type over ā. Proof. We have to show that Σ is finitely satisfiable in M. Choose a finite subset, without loss of generality a single formula φ(x 1,.., x n, b 1,.., b k ). So φ(x 1,.., x n, a 1,.., a k ) Σ so M = x 1,.., x n φ(x 1,.., x n, a 1,.., a k ). Let ψ(y 1,.., y k ) be the L-formula x 1,..., x n φ(x 1,., x n, y 1,.., y k ). So ψ(y 1,.., y k ) tp M (ā) so also in tp M ( b). Thus M = ψ(b 1,.., b n ). Remark 2.15 Here are some useful bits of notation. (i)in the above lemma let us write Σ as Σā to emphasize the dependence on ā. Then we write Σ as Σ b. (So Σ is just the result of replacing the parameters for ā by b.) (ii) Let A be a subset of M and b a tuple from M. By tp M ( b/a) we mean the type of b in the structure (M, a) a A. Lemma 2.16 Any ω-saturated structure is ω-homogeneous. Proof. Let M be ω-saturated. Let ā, b be tuples from M with the same (complete) type in M. Let c M. Let p a (x) = tp M (c/a). By Lemma 2.14, p b (x) is a complete type (over b in M). By ω-saturation of M we can realize p b (x) by some d. Claim. tp M (ā, c) = tp M ( b, d). Proof. Let φ(x, ȳ) be an L-formula. Then M = φ(c, ā) iff φ(x, ā) p a (x) iff φ(x, b) p b (x) iff M = φ(d, b). Corollary 2.17 Any structure has an ω-homogeneous elementary extension. We can do better if L is countable. Proposition 2.18 Let L be countable and M a countable L-structure. Then M has a countable ω-homogeneous elementary extension. Proof. Let P be the set of all types of the form p b(x) where ā and b are finite tuples from M with the same type in M and where pā(x) = tp M (c/ā) for 19

20 some c M. Note that P is a countable set of types, so there is a countable elementary extension M 1 of M which realizes all types in P. Now do the same thing, with M 1 in place of M. Continue to get a countable elementary chain M i for i < ω. If N is the union, then N is ω-homogenous by construction. Lemma 2.18 can be used to relate types and automorphisms. Lemma 2.19 Let M be a countable ω-homogeneous structure. Let ā, b be finite tuples from M with the same type in M. Then there is an automorphism f of M such that f(ā) = b. Proof. (Back-and-forth.) List the elements of M as (e i : i < ω). Let us define c i and d i from M by induction, such that tp M (ā, c 1,.., c n ) = tp M ( b, d 1,.., d n ) for all n. At stage 2n, let c 2n be e n, and let d 2n be such that tp M (ā, c 0,.., c 2n 1, c 2n ) = tp M ( b, d 0,.., d 2n 1, d 2n ) (by induction hypothesis and homogeneity of M). At stage 2n+1 let d 2n+1 = e n and let c 2n+1 be such that tp M (ā, c 0,.., c 2n, c 2n+1 ) = tp M ( b, d 0,.., d 2n, d 2n+1 ). Then the map f which takes c n to d n is by construction well-defined (why?), a permutation of the universe of M (why?) and as for each formula φ(x 1,.., x m ) of L, M = φ(c 1,.., c m ) iff M = φ(d 1,.., d m ), f is an automorphism of M. Finally f(ā) = b (why?) Corollary 2.20 Let M be an L-structure, and ā, b n-tuples from M. Then tp M (ā) = tp M ( b) iff there is an elementary extension N of M and an automorphism f of N such that f(ā) = b. Proof. Right to left is immediate (???). Left to right: Add a new function symbol f. Note that we can express f is an L-automorphism by set of sentences in L {f}. Let L = L M {f}, and let Σ be D c (M) f is an L-automorphism {f(ā) = b}. It is enough to show that Σ is finitely consistent. Let Σ 0 be a finite subset of Σ. Σ 0 involves only symbols from a finite sublanguage L 0 of L and finitely many elements of M. Let M 0 be a countable elementary substructure of M L 0 containing those elements as well as ā and b. Let N 0 be a countable ω-homogeneous elementary extension of M 0. Then by Lemma 2.19 there is an automorphism of N 0 taking ā to b. So Σ 0 has a model. Definition 2.21 (i) By a quantifier-free n-type of T we mean a set Σ(x 1,.., x n ) of quantifier-free L-formulas such that T Σ is consistent. (ii) A complete quantifier-free n-type of T is a quantifier-free n-type Σ of T 20

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