tp(c/a) tp(c/ab) T h(m M ) is assumed in the background.

Model Theory II. 80824 22.10.2006-22.01-2007 (not: 17.12) Time: The first meeting will be on SUNDAY, OCT. 22, 10-12, room 209. We will try to make this time change permanent. Please write ehud@math.huji.ac.il if you are interested but cannot come to the first meeting. The aim is to cover general model theoretic methods, initially developed within stability theory, that are of use in studying theories of geometric significance. In the last weeks (if time permits) and in the following semeseter (seminar in model theoretic algebra) we will concentrate on fields with automorphisms, and valued fields. 1) a) Review of saturated models: definition, existence, uniqueness, homogeneity. Applications: quantifier elimination criteria. b) Induced structure. Stable embeddedness of definable sets. c) Imaginary elements. Definable Galois theory. Definable and algebraic closure. The compact Lascar group. 2) Internality and definable automorphism groups. Covers and definable groupoids. 3) Definable types. Fundamental theorem of stability: existence and uniqueness of a definable type extending a given type over an algebraically closed set. Groups with generic types. 4) Domination and orthogonality. Superstability: decomosition into regular types. 5) Modularity. 6) Simple theories and higher independence. 7) Grothendieck rings and related structures. 8) Model theory of valued fields. Prerequisites: structures, formulas and the compactness theorem; types and saturated models. The latter will be reviewed in the first week. Occasionally, other standard notions from Model Theory I may be mentioned, but they will not be used extensively. All prerequisites can be found in: Model Theory books by Chang and Keisler, David Marker, in library. Model theory book by Poizat, not in library but available in Russian translation at http://www.maths.ox.ac.uk/ zilber/ Model Theory notes in http://www.math.uiuc.edu/people/pillay.html Grades will be based mostly on homework assignments. Some excercises will depend on more knowledge. These are optional. 1. Languages, theories, types, saturated models 1.1. Many-sorted languages. A (many-sorted) language L consists of: 1) A number of sorts. For each sort S, infinitely many variables of sort S. 2) Function symbols F. Each function symbol comes along with a list (S 1,... S m ) of domain sorts, and a range sort S. We write F : S 1... S m S though F is only a symbol. In case m = 0, we view F as a constant symbol of sort S. 3) Relation symbols R. Each comes with a list of sorts (S 1,... S m ), and we write R S 1... S m The definition of terms and formulas proceeds as in e.g. [?] in the one-sorted case, except that a composed term F (t 1,..., t k ) : S 1... S m S is defined only when t i : S 1... S m T i and F : T 1... T k S. Excercise 1.2. (Requires the definition of a category.) Let L be a language. Define a category L as follows: Ob L is the set of n-tuples (S 1,..., S n ) of sorts of L. Given a = (S 1,..., S n ), b = (T 1,..., T m ) Ob L, let Mor(a, b) be the set of m-tuples (t 1,..., t m ) where t i is a term S 1... S n T i. 1) Define composition. 1

2 2) Assume L has no relation symbols (function symbols only.) Show that an L-structure is the same thing as a functor L Sets. 3) Conversely, given a small category with finite products, show that it comes from a language L (with no relation symbols) iff it is freely generated as a category with finite products by some set of objects X and some set of morphisms (x 1... x n ) y, with x 1,..., x n, y X. Let L be a language with no relation symbols. An equational theory is a theory consisting of sentences of the form ( x 1 )... ( x n )(t = s) where t, s : S(x 1 )... S(x n ) T are terms of L. Excercise 1.3. Let S, S 1, S 2 be sorts, f i : S S i, g : S 1 S 2 S function symbols. Write down equational sentences saying that S is the direct product S 1 S 2. Since we will work with languages along with theories, it is convenient to always allow at least sentences of the form above, and to assume that a finite product of sorts is identified with another sort. Thus an n-tuple of a structure can be identified with an element of another sort. Excercise 1.4. Let L be a language with no relation symbols, and T an equational theory. Define a category whose objects are the finite tuples of sorts of L, and whose morphisms are the terms of L up to T -equivalence, i.e. s, t are identified if T = ( x 1 )... ( x n )(t = s). Show any small category with finite products can be obtained in this way (up to equivalence.) Definition 1.5. A formula is called universal if it has the form ( x 1 )... ( x n )ψ, with ψ quantifier - free. T is the theory generated by all universal sentences of T. 1.6. Structures. An L-structure A consists of a set S A for each sort S of L, a function f A : S1 A... Sm A T A for each function symbol f : S 1... S m T of L, and a set R A S1 A... Sm A for each relation symbol R S 1... S m of L. The many-sorted cardinality of A is the function S S A. The cardinality of A is the supremum. However a finite structure means: S A < for each S. If X is a set of variables, X A denotes the set of functions c : X A such that if x is a variable of sort S, then c(x) S A. N.B.: The main case is X = {x 1,..., x n }, so that X A can be identified with the set of n-tuples (of appropriate sorts.) But it is convenient to allow X to be arbitrary. We write φ(x) to mean: φ is a formula, and X is a set of variables containing all the free variables of φ. Define φ(x) A X A. This really depends on X and not only on φ, but we write φ A when X is clear from the context (the default is the set of free variables of φ.). Also write M = φ(a) for a φ M. A theory is a set of sentences, closed under consequences. If M is a class of structures, T h(m is the set of L-sentences true in all of them. If M is an L-structure, we write: and A M to say: A is a substructure. Let L A be the language L expanded by a constant c a for each a A; and M A the structure M with the addional interpretations c M A a = a. T qf A is T augmented with the set of quantifier-free sentences of L A true in M A. T eliminates quantifiers (QE) if every formula φ is T -equivalent to a quantifier-free formula ψ with the same free variables x = {x 1,..., x n }; i.e. T ( x)(φ ψ). Excercise 1.7. (Morley-zation). Given L, T find a language Mor L L and a theory Mor T T such that: i) Mor T eliminates quantifiers. ii) The restriction map A A L is a bijection from models of Mor T to models of T.

3 iii) Every formula of Mor L is Mor T -equivalent to a formula of L. (Hint: add a relation symbol R φ for any formula φ of L. This excercise will allow us to assume without loss of generality, in sections (1-7) of this class, that T eliminates quantifiers. In section (8), quantifier elimination becomes a theorem (e.g. for algebraically closed valued fields) rather than an assumption. Notation: Inside expressions such as tp(c/a {b}), we often shorten A {b} to Ab. Let a X M, and A M. The type of a over A, tp(a/a), is the set of formulas φ(x) of L A such that M A = φ(a). S X (A) is the set of types (in variables X) over A. = maximal consistent sets of formulas φ(x). Important: Excercise 1.8. tp qf (c/a) T qf Ab tpqf (c/ab) defines a symmetric relation in b, c. Hence, so is the relation: tp(c/a) T h(m Ab ) tp(c/ab) N.B. Part of the problem is explaining the hence. By a partial type over M one means: a set of formulas of L M, containing T h(m M ). In the literature one often sees the latter relation written as: T h(m M ) is assumed in the background. tp(c/a) tp(c/ab) 1.9. Set theoretic discussion. We will fix a cardinal λ greater than the size of any structure we are interested in, and assume 2 κ λ for all cardinals κ < λ. The purpose of this paragraph is to explain why this assumption is harmless. If you have not learned set theory, you may accept the assumption and skip the paragraph. A number of set theoretic methods can be used to avoid the assumption it or to show a posteriori that it is inessential. These methods include: (1) Forcing: let λ 0 be greater than any cardinal of interest; so that all our model theoretic discussions will take place in H(λ 0 ), the universe of sets hereditarily of size < λ 0. Change the set theoretic universe so as to obtain 2 λ0 = λ + 0 withough changing H(λ 0). Set λ = λ + 0. (2) Similarly using Gödel s L[X] where X is a set of ordinals coding H(λ 0 ). (3) Use singular cardinals and special models instead, cf. [?]. (4) Recursively saturated models. He we assume the language is countable and take λ = ℵ 0 but the notion of saturated is modified, so that only recursive types are used, and the level of saturation becomes context-dependent. (5) Using class models. We do not take λ to be a cardinal; rather we use the size distinction: proper class vs. set. (So λ = ORD.) Excercise 1.10. Let T be a consistent theory. For φ a formula, let F V (φ) be the (finite) set of free variables of φ. Recall that in set theory a class is a formula θ(u; z) in the language of set theory, together with a set b; the class is though of as {u : θ(u, b)}. Let Q be the class of pairs (φ, s) where φ is a formula of T, and s : dom(s) ORD is a function with finite domain dom(s) containing F V (φ). (Thus s assigns ordinals to the free variables of φ, and possibly to some other dummy variables ). By a class model of T (with universe ORD) we mean a class function T : Q 2 (where 2 = {0, 1} is the 2-element Boolean algebra) such that T is a Boolean homomorphism, T(φ, s) = T(φ, s F V (φ)), and = (( x)φ, s) = T iff for some s extending s dom(s) \ x, = (φ, s ) is defined and equal to T.

4 In particular, for any definable n-place relation R of L we obtain a class relation R T = {a ORD n : = ((R, a)) = 1}. (Here R is identified with the formula R(x 1,..., x n ) and a = (a 1,..., a n ) is identified with the assignment s(x i ) = a i.) Similarly, for each definable n-place function symbol we obtain a class function (called a basic function.) By a substructure we mean a set E of ordinals, closed under the basic functions. A class model is saturated if any type over any substructure is realized. Show existence of a saturated model U with universe ORD. Given two such saturated models, show the existence of an isomorphism between them. Remarks: (5) is most suited philosphically to both the way we think about a universal domain, and to the usual cut-off in set theory above all cardinals of interest.. It is inconvenient in that we cannot freely discuss higher level constructs such as Aut(U). We can talk about the set of all definable types over U, or Aut(U/C)-invariant ones (see below.) The device of determining the truth value of all formulas at once, instead of the usual definition of a structure, can be avoided if one assumes QE in advance. 1.11. Saturated structures. Assume QE. Also, at this level of generality, one can assume L has no function symbols. In this case for L-structure M, any subset of M is the universe of a unique substructure of M; and a substructure is κ-generated iff it has size κ. We say A is < κ-generated if it has a set of generators of size < κ. A 1-type is a type in one variable. Definition 1.12. Let λ be a cardinal. M is λ-saturated if for any < λ-generated substructure A of M, and 1-type over A is realized in M. Excercise 1.13. Let M be λ-saturated. Then either M is finite, or M λ. Lemma 1.14. Let M be λ-saturated, and let A be a < λ-generated substructure. Let q be a type in λ variables over A. Then q is realized in M. Proof. Enumerate the variables of q as {x i : i < λ}. Here we use a well ordering < on λ, such that every initial section has size < λ. Find a i by transfinite recursion; if a j has been defined for j < i, let A i = A {a j : j < i}. Let q i be the set of all formulas of q whose variables are among {x j : j < i}; assume inductively that (a j : j < i) realizes q i. Excercise 1.15. When i is a limit ordinal, if the inductive hypothesis holds for all j < i, then it holds for i Excercise 1.16. There exists a type r over A i such that for any b, (a j : j < i, b) = q i+1 iff b = r. Let a i be any realization of r (using the definition of saturation.) The induction hypothesis is preserved. Thus a j is defined for j < λ. So (a j : j < λ) realizes q. Assume QE. Corollary 1.17. If M = T is λ-saturated, it is λ-universal: any model of T of size λ admits an embedding into M. Proof. Let A = T. Let q = tp qf (A): i.e. let x a be a variable for a A, and let q be the set of formulas in these variables that are true in A under the assignment x a a. Claim. q is consistent with T : otherwise by compactness q is inconsistent with a finite conjunction φ of formulas in q. These formulas involve variables y = (y 1,..., y m ) say. So T = ( y) φ. But this is a universal formula, so T = ( y) φ. This contradicts the assumption: A = T.

5 Claim. q is a complete type. More precisely, the set of consequences of T {q} is complete. This is immediate from quantifier elimination. By Lemma 1.17, q is realized in M, by some assignment s. Then clearly a s(x a ) is an embedding of A into M. Lemma 1.18. Assume QE. Let M, N = T, and f : A N an embedding of L-structures. If A < λ and N is λ-saturated, and c M, then f extends to an embedding of A(c). Theorem 1.19. Assume QE. Let M, N = T be λ-saturated, with M, N λ. Let f : A B be an isomorphism of substructures of M, N respectively, with A < λ. Then f extends to an isomorphism F : M N. Corollary 1.20. Let M, N = T be λ-saturated, with T h(m) = T h(n) then M = N. Corollary 1.21. Let M = T be λ-saturated. Then M is λ-homogeneous: if q is any type in < λ variables, then Aut(M) acts transitively on the set of realizations q(m) of q in M. (Or else this set is empty.) Excercise 1.22. Conversely, if M is λ-homogeneous and λ-universal, it is saturated. Excercise 1.23. Let E be a binary relation. Let EQ be the (universal) theory such that (A, E) = EQ iff E A is an equivalence relation on A. Let ẼQ be the theory of models of EQ with infinitely many classes, each infinite. 1) Explain how to write down EQ, ẼQ. Show ẼQ is not universal, and in fact ẼQ = EQ. 2) An existentially closed model of EQ is precisely a model of ẼQ. 3) Let M be a saturated model of ẼQ. Show that M has λ classes, each of size λ. 4) Show that a model M of ẼQ is homogeneous iff all classes have the same size. 5) Let M have one class of size ℵ 0, and λ classes of size λ. Show M is universal, but not saturated. Proposition 1.24. Assume L < λ = λ <λ. Then T has a λ-saturated model of size λ (i.e. finite, or of size λ.) Proof. See [1]. Here, to avoid book-keeping we sketch the proof in case λ is inaccessible, i.e. regular with 2 κ < λ for κ < λ. Claim. Let A = T, A < λ. Then there exists B = T A realizing all 1-types over A, and with B < λ. Proof of Claim: for each type p over A, let c p be a constant symbol; let T be the theory in the language L A + {c p }, stating that p(c p ). This theory is consistent, hence has a model B, and the reduct of B to L A satisfies the requirements. Now construct a chain of models (A i : i < λ), such that A i is an elementary submodel of A j if i < j, and all 1-types of A i are realized in A j. Let M be the limit of the A i. By regularity of λ, any < λ-generated subset of M is contained in some A i. It follows that M is λ-saturated. Corollary 1.25. Assume L < λ = λ <λ. Let M be λ-saturated, with M λ. Let L L with L < λ, and let T be a consistent extension of T in the language L. Then M expands to a model of T ; in fact to a λ-saturated one. Proof. Let N be a λ-saturated model of T. Let M be the reduct of N to L. Excercise 1.26. M is saturated. Hence by uniqueness, there exists an isomorphism g : M M. Pulling back the relations of N, we find an expansion N of M with g : N N an isomorphism.

6 1.27. The universal domain. From now on we fix a cardinal λ as above, and greater than any language or structure of interest. For each complete theory T we consider, we fix a λ-saturated model U T of size λ. Note U T is unique up to isomorphism. By a structure, we will mean a substructure of T Mor, generated by < λ elements.. Or assuming quantifier elimination, simply a substructure generated by < λ elements. Note that any < λ-generated model of T embeds into U T uniquely up to conjugacy. 1.28. More on quantifier elimination. We show how T, T determine each other. An existentially closed model of T is a model N = T such that if N N = T, and φ L(N) is quantifier-free and realized in N, then φ has a solution in N. Excercise 1.29. T is the theory of the class of all substructures of models of T. Hint: see Corollary 1.17. Excercise 1.30. Assume T is complete, with quantifier elimination. Then T is the theory of the class of all existentially closed models of T. Hint: If M = T, show that M is an existentially closed model of T : if M N = T by Corollary 1.17 there exists N = T with M N. Now the fact that existential formulas are T -equivalent to quantifier-free ones. Conversely, let T be the set of all sentences of T of the form ( x 1 )... ( x n )( y 1 )... ( y m )ψ, with ψ quantifier free. Show that T is realized in any existentially closed model of T. Show that T axiomatizes T. [1] Chang and Keisler, Model Theory References