Model Theory - Introduction to the Course Advanced Topics in Logic, Spring 2009

Model Theory - Introduction to the course Advanced Topics in Logic, Spring 2009

Jaap van Oosten February 2009

Introduction

Prerequisite for the course is the course Foundations of Mathematics, see the lecture notes “Sets, Modela, Proofs” by I. Moerdijk and J. van Oosten: (http://www.math.uu.nl/people/jvoosten/syllabi/logicasyllmoeder.pdf) We shall work from the book by D. Marker, Model Theory: An Introduction, Springer Graduate Texts in Mathematics 217, 2002. Model Theory has many faces, but I limit myself to the following rough division: Classical Model Theory and Modern Model Theory. The dividing year is 1965, when Morley published the proof of his theorem.

1 Classical Model Theory

Classical Model Theory studies theories by their classes of models (recall that a theory is just a set of sentences in a given language). A famous and simple example is the following (see also Marker, 2.3.9). Call a sentence universal if it has a string of universal quantifiers followed by a quantifier-free formula: ∀x1 · · · ∀xnφ with φ quantifier-free. We say that a theory T has a universal axiomatization if there is a set T 0 of universal sentences, such that T and T 0 have the same consequences: T |= φ if and only if T 0 |= φ, for every sentence φ in the language. Clearly, if a structure M satisfies a universal sentence, then every substructure of M also satisfies that sentence; so if T has a universal axiomatization, every substructure of a model of T is also a model of T . The theorem of Tarski-L os says that the converse also holds. This gives a nice relation between a syntactical property of a theory (having a universal axiomatization) and a property of its class of models (being closed under substructures). Theories with a universal axiomatization include: groups (in the language (·, 1, (−)−1)), linear orders, vector spaces, rings.

1 There is quite a number of theorems of this sort, for example also a theorem (Exercise 2.5.15 in Marker) about theories which have an axiomatization of sentences of the form ∀x1 · · · ∀xn∃y1 · · · ∃ykφ, φ quantiﬁer-free. A couple of important names in Classical Model Theory: L os, Tarski, Robinson.

Important is, that Model Theory works independently from any notion of formal proof. Yet, we constantly use the Compactness Theorem: if Γ |= φ, there is a finite subset Γ0 ⊂ Γ such that Γ0 |= φ. Now in the Foundations course, we proved this theorem using the Complete- ness Theorem. This is undesirable from the point of view of Model Theory, which needs this theorem but doesn’t want to have anything to do with proof trees. In fact, Los gave a proof of the Compactness Theorem independent of formal proofs, using Ultraproducts (Exercises 2.5.19 and 2.5.20 in Marker). Robinson invented the method of diagrams, which we have already (tacitly) used in the Foundations course. Recall that if M is an L-structure, we can consider the language LM of the structure M (constants for elements of M added). The diagram of M, Diag(M), is the set of quantifier-free LM -sentences which are true in M (the definition in Marker, 2.3.2., is a bit different: there it is the set of atomic LM -sentences and negations of atomic LM -sentences which hold in M). If L has no relation symbols, Diag(M) is determined by all equalities F (a1, . . . , an) = a for a1, . . . , an, a ∈ M; that is: the diagram of a structure is like the multiplication table of a group. Exercise: M is isomorphic to a substructure of N if and only if N can be made into an LM -structure which is a model of Diag(M).

The elementary diagram of M, Diagel(M), is the set of all LM -sentences which are true in M. Recall that an elementary embedding from M into N is a function f : M → N such that for every L-formula φ(v1, . . . , vn) and every n-tuple a1, . . . an from M, we have M |= φ(a1, . . . , an) if and only if N |= φ(f(a1), . . . , f(an)). Exercise: There is an elementary embedding M → N if and only if N can be made into an LM -structure which is a model of Diagel(M) The following result (2.3.11 in Marker) is the Elementary Chain Theorem by Tarski and Vaught. Suppose that (I, <) is a linear order, and for every i ∈ I an L-structure Mi is given, such that whenever i < j, Mi is an elementary substructure of Mj. Then the union Si∈I Mi is an L-structure, and every Mi is an elementary substructure of Si∈I Mi.

Exercise: Prove the Elementary Amalgamation Theorem: suppose M, M1, M2 are L-structures and f : M → M1 and g : M → M2 are elementary embeddings. Show that there is an L-structure N and elementary embeddings h : M1 → N and k : M2 → N such that hf = kg.

2 Another element of Classical Model Theory that I want to recall, is what is known as the canonical model or Henkin construction (section 2.1 in Marker). Suppose T is a complete L-theory with enough constants. Recall that this means: for every L-formula φ(v) with one free variable v, there is a constant c in L such that T |= (∃xφ(x)) → φ(c) Then we can construct a model for T ‘out of the language’: let C the set of constants of L. Deﬁne c ∼ c0 if T |= c = c0. Let M = C/ ∼. For a constant c of L, we set cM = [c]. For an n-place relation symbol R of L, we set

M R = {([c1], . . . , [cn]) | T |= R(c1, . . . , cn)} and for an n-ary function symbol F of L, we deﬁne F ([c1], . . . , [cn]) by picking c such that T |= F (c1, . . . , cn) = c, and taking [c]. We have a well-deﬁned L-structure, and for every L-formula φ(v1, . . . , vn) and elements [c1], . . . , [cn] of M, we have M |= φ([c1], . . . , [cn]) precisely if T |= φ(c1, . . . , cn).

2 Modern Model Theory

First of all I recall the notion ‘κ-categorical’ for a cardinal number κ. A theory is κ-categorical if whenever M and N are models of T , both of cardinality κ, then M and N are isomorphic. Examples: 1) The theory DLO of dense linear orders without end-points is ω-categorical by Cantor’s back-and-forth method as we saw in the Foundations course, ω 1 but it is not 2 -categorical, as the models (0, 1) and (0, 1) − { 2 } are not isomorphic. 2) The theory of ‘sets with a free Z-action’ (i.e., sets together with a bijective function f such that f n(x) 6= x for all n > 0 and x). This theory is not ω- categorical (consider (Z, +1) and (Q−{0}, x 7→ 2x)), but it is κ-categorical for every uncountable κ. 3) The theory of ‘divisible torsion-free abelian groups’ (abelian groups in which no nonzero element has ﬁnite order, and in which every element can be divided by a positive integer). Such a group is nothing but a Q-vector space. Hence, again the theory is not ω-categorical but it is κ-categorical for every uncountable κ. In 1965 the following theorem was proved: Theorem 2.1 (Morley) Let L be a countable language and T a complete L- theory which has inﬁnite models. If T is κ-categorical for some uncountable κ, then T is κ-categorical for every uncountable κ.

3 Modern Model Theory starts with Morley’s Theorem. Not so much because the theorem itself is important (although it certainly is), but because of the techniques is requires for analyzing models. Convention From now on, we shall often use the term ‘model’ instead of ‘L- structure’. Modern Model Theory uses techniques known from Algebra and Geometry, and generalizes these in the study of arbitrary models. Later, we shall see that in models of ‘good’ theories (stable theories), one can mimick a good deal of (abstract) geometry. Examples

1) There is a notion of ‘independence’ between elements of a model. This generalizes linear independence in vector spaces, and algebraic independence in rings. 2) Hence, a theory of dimension of definable subsets of a model. 3) Also, we shall see the notion of algebraic closure of a subset of a model. We shall start by considering what can be seen as a generalization of the con- cept of ideal. If K is a field, then every element a of a ring which extends K determines an ideal Ia of K[X]: Ia is the set of those polynomials f for which f(a) = 0. Conversely, if I is an ideal of K[X] then the element a = [X] of the ring K[X]/I is such that Ia = I. So there is a connection between ideals of K[X] and elements of extensions of K. If M is a model, we shall consider not only elements of M but also ‘ideal elements of M’, that is: elements which exist in some elementary extension of M. If x ∈ N M and x0 ∈ N 0 M are such elements, we call them equivalent if there is a common elementary extension N 00 of N and N 0, and an automorphism τ of N 00 such that τ(x) = x0. The equivalence classes then form a set and, more importantly, a topological space. This space has features in common with the Zariski topology on a spectrum of a commutative ring (but there are also differences). We shall study these topological spaces quite a bit. Modern Model Theory has applications to ordinary Mathematics. The most striking application is E. Hrushovski’s proof of the Mordell-Lang Conjecture in Algebraic Geometry (1996). An exposition of this proof is in the book Model Theory and Algebraic Geometry, Elisabeth Bouscaren (ed.), Springer Lecture Notes in Mathematics 1696, 1998. For a student who has followed both this Model Theory course and a course in Algebraic Geometry, this might be a good topic for a Master thesis. Some important names in Modern Model Theory are: S. Shelah, B. Zil’ber, E. Hrushovski, A. Pillay.

4 3 Literature

A good book on Classical Model Theory is Model Theory by Chang and Keisler, Studies in Logic 73, Elsevier 1989 (3rd edition). A more encyclopedic source is Model Theory by Wilfrid Hodges, Encyclopedia of Mathematics 42, Cambridge University Press 1993. Good books on stability theory (the heart of Modern Model Theory) are Essential Stability Theory by Stefan Buechler (Perspec- tives in Mathematical Logic, Springer 1996) and Geometric Stability Theory by Anand Pillay (Oxford Logic Guides 32, Clarendon Press 1996). I also recommend a visit to A. Pillay’s web page: http://www.amsta.leeds.ac.uk/~pillay/ Here you can ﬁnd lecture course material on Model Theory, Stability Theory and Applied Stability Theory.

4 Plan of the course (provisional)

We shall start with Chapter 4 of Marker. I shall go through Chapter 5 in part, and we shall certainly see Chapter 6. Hopefully, there is time to do bits of Chapters 7 and 8 as well.

5 Preliminaries on Ordinals and Cardinals

Since we shall do a lot of transfinite inductions, it is good to recall a bit of what we saw in the Foundations course on ordinal numbers, and expand a little on that. An ordinal number or ordinal is a well-ordered set α such that for every β ∈ α, the set {γ ∈ α | γ < β} equals β. Hence, every element of an ordinal is again an ordinal. If an ordinal has a least element, it is ∅ which we also denote by 0; its successor is {∅} = 1; then 2 = {0, 1} = {∅, {∅}}, etc. The least infinite ordinal is ω = {0, 1, 2, . . .}. We can continue: ω + 1, ω + 2, . . . , ω + ω,. . . If two ordinals are isomorphic as well-ordered sets, they are equal. Every well-ordered set is isomorphic to a unique ordinal. The class of all ordinals is ordered by the ∈-relation; we shall treat the expressions α < β and α ∈ β as synonymous (for ordinals α, β). If α is an ordinal, there is a least ordinal > α, which we call α + 1. It is α ∪ {α}, and has a greatest element α. Such ordinals are successor ordinals. Ordinals with no greatest element are limit ordinals. The least limit ordinal is 0; the next one is ω. Every nonempty class of ordinals has a least element, so we can do induction over ordinals, and define functions on the class of ordinals by recursion. As an example, we can define addition and multiplication of ordinals:

α + 0 = α α + (β + 1) = (α + β) + 1 α + γ = S{α + β | β ∈ γ} if γ is a limit ordinal

5 As a well-ordered set, α + β is the sum of α and β, with every element of α less than every element of β. Note: this addition is associative but not commutative: 1 + ω = ω 6= ω + 1.

α·0 = 0 α·(β + 1) = (α·β) + α α·γ = S{α·β | β ∈ γ} if γ is a limit ordinal

As a well-ordered set, α·β can be seen as: every element of β replaced by a copy of α. Again, this multiplication is associative but not commutative: 2·ω = ω 6= ω·2 = ω + ω. It is possible for different ordinals to be in bijective correspondence: there is a bijective function ω → ω + ω for example. An ordinal α with the property that for no β < α there is a bijective function β → α, is called a cardinal number or cardinal. Note that α is a cardinal if and only if for no β < α there is an injective function α → β. It follows that if I is a chain of cardinals, then the least upper bound of I (qua ordinal) is also a cardinal. Cardinal numbers are denoted κ, λ, µ, ν. Every finite ordinal is a cardinal. The first infinite cardinal is ω; we also write ℵ0 (‘aleph-null’). If α is any ordinal, by Hartogs’ Lemma there is a well-ordered set X, hence an ordinal β, which does not inject into α; so there is a least β with this property. This β is a cardinal, which we denote by (α)+. We can now define a sequence of infinite cardinals, indexed by ordinal numbers: ℵ0 = ω + ℵα+1 = (ℵα) ℵγ = S{ℵβ | β ∈ γ} if γ is a limit ordinal

So the least uncountable ordinal is ℵ1 (or ω1); it is the set of all countable ordinals. Note, that we always have α ≤ ℵα but the inequality is not always strict! From this it follows that every inﬁnite cardinal is of the form ℵα for a unique α: let κ be such a cardinal. Then κ ≤ ℵκ so there is a least ordinal α such that κ ≤ ℵα; for this α, κ = ℵα must hold. A cardinal is called a successor cardinal if it is of the form ℵα+1; it is a limit cardinal if it is of the form ℵγ with γ a limit ordinal. The coﬁnality of a cardinal κ, cof(κ), is the least cardinal λ such that there is a function f : λ → κ with S{f(α) | α ∈ λ} = κ. Clearly, cof(κ) ≤ κ since we can take the idetity function on κ for f. A cardinal κ is called regular if cof(κ) = κ. The antonym is singular. Example: consider ℵω. We have f : ω → ℵω by f(n) = ℵn; then ℵω = S{f(n) | n ∈ ω} so cof(ℵω) = ω and ℵω is singular. Exercise. Use AC to show that every successor cardinal is regular. Exercise. Suppose κ is a regular limit cardinal. Prove: κ = ℵκ.