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: 8x1 · · · 8xnφ 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 j= φ if and only if T 0 j= φ, for every sentence φ in the language. Clearly, if a structure M satisfies a universal sentence, then every substruc- ture 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 8x1 · · · 8xn9y1 · · · 9ykφ, 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 j= φ, there is a finite subset Γ0 ⊂ Γ such that Γ0 j= 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 2 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 j= φ(a1; : : : ; an) if and only if N j= 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 2 I an L-structure Mi is given, such that whenever i < j, Mi is an elementary substructure of Mj. Then the union Si2I Mi is an L-structure, and every Mi is an elementary substructure of Si2I 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 j= (9xφ(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 j= 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 = f([c1]; : : : ; [cn]) j T j= R(c1; : : : ; cn)g and for an n-ary function symbol F of L, we define F ([c1]; : : : ; [cn]) by picking c such that T j= 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 j= φ([c1]; : : : ; [cn]) precisely if T j= 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 !-categorical by Cantor's back-and-forth method as we saw in the Foundations course, ! 1 but it is not 2 -categorical, as the models (0; 1) and (0; 1) − f 2 g 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 !- categorical (consider (Z; +1) and (Q−f0g; 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 !-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 indepen- dence 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 2 N M and x0 2 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.