Model Theory - Broadening Assessment Adam Keilthy December 11, 2019 1 Introduction The goal of this note is to develop a completeness criterion for a theory T , analogous to theLos-Vaught test for completeness, by considering saturated models of cardinality @1. However, there are issues with this approach; in particular, within standard set theory, there is no guarentee of the existence of such models. As such, we will assume the generalised continuum hypothesis (GCH) @α 2 = @α+1 in order to ensure our constructions work. This note will proceed as follows. We will first establish common notions, recalling various core definitions and notations that will be used in our discus- sion. We will then introduce the idea of ultrafilters and ultraproducts, which we use to establish the existence of saturated models. We finally provide an analogue of theLos-Vaught test for completeness. 2 Preliminary Definitions We assume the reader is familiar with the definitions of a first order language, a model, a model of a theory,completeness of a theory, along with the concepts of elementary equivalence and isomorphism of models. Definition 2.1. We define 1. M φ for a formula φ, if and only if φ is true in the realisation of M. 2. M T for a theory T if and only if M is a model of T . 3. T ` φ if and only if every model of T is a model of φ. 4. M ≡ N if and only if M and N are elementarily equivalent. 5. M =∼ N if and only if M and N are isomorphic. 1 Definition 2.2. A theory T is called κ-categorical if it has a unique model of size κ, up to isomorphism The theorem we want to emulate is the following. Theorem 2.3 (Los-Vaught test for completeness). Suppose a theory T is κ- categorical for some infinite κ and has no finite models. Then T is complete. However, we want a test involving saturated models. Definition 2.4. Let M be an L-structure for a language L. For a subset A ⊆ M, let LA := L [ fca : a 2 Ag be the expansion of L by constant symbols corresponding to elements of A. We can expand M to an LA-structure by interpreting ca as the element a 2 A. In a slight abuse of notation, we still denote the expansion by M. Definition 2.5. An n-type of M over A is a set p(x) of formulae in LA with free variables x1; : : : ; xn such that, for every finite subset p0(x) ⊆ p(x), there n exists b 2 M with M p0(b). We call an n-type p(x) complete if, for every φ an LA formula with n free variables, either φ 2 p(x) or :φ 2 p(x). Definition 2.6. For p a complete n-type over A ⊆ M, we say the model M n realises p if there exists a 2 M such that M φ(a) for all φ 2 p. We call M κ-saturated, for any cardinal κ, if for all A ⊆ M with jAj < κ, M realises all complete types over A. If jMj = κ, we call M saturated. Finally, we recall the L¨owenheim-Skolem theorems, here compiled into one statement Theorem 2.7 (L¨owenheim-Skolem). For any infinite L-structure M and any cardinal κ ≥ jLj, there exists an L-structure N such that jNj = κ and • if κ < jMj, then N is an elementary substructure of M. • if κ > jMj, then N is an elementary extension of M. 3 Ultrafilters and Ultrapowers Definition 3.1. For a set I, an ultrafilter U over I is a subset of the power set P(I) satisfying 1. I 2 U. 2. If V; W 2 U, then V \ W 2 U. 3. If V 2 U and V ⊆ W ⊆ I, then W 2 U. 4. For any V ⊆ I, either V 2 U or I n V 2 U. 2 We call an ultrafilter principal if there exists i 2 I such that V 2 U if and only if i 2 V . Definition 3.2. Given a family of sets Ai, with i 2 I an index set, and an Q Q ultrafilter U over I, we define the ultraproduct Ai = Ai=U to be the Q U i2I quotient of the cartesian product i2I Ai by the equivalence relation faigi2I ∼ fbigi2I if and only if fi 2 I : ai = big 2 U Definition 3.3. Given an ultrafilter U over an index set I and a collection of Q models Mi for a language L, define the ultraproduct Mi to the be L-structure Q U with underlying set U Mi and realisation defined for a relation symbol R by Q 1 n U Mi 1 n Mi (a ; : : : ; a ) 2 R if and only if fi 2 I :(ai ; : : : ; ai ) 2 R g 2 U and similarly for functions and constants. In the case where there exists M such that Mi = M for every i 2 I, we call this the ultrapower of M and denote it by MjIj=U. We have the following theorem Theorem 3.4 (Los'sTheorem) . For all sentences φ of L, Y Mi φ if and only if fi 2 I : Mi φg 2 U U In the case of ultrapowers, we have M ≡ MjIj=U. Definition 3.5. We call an ultrafilter U countably incomplete if there exists a countable V ⊆ U such that \W 2V W = ;. Fact 3.6. There exist countably incomplete ultrafilters Remark 3.7. There also exist countably complete ultrafilters, on N. That is, an ultrafilter on N such that every countable intersection is non-empty. Indeed, every principal ultrafilter on N is countably complete. In the case of ultrafilters over countable sets, this is an exact criterion: countably complete ultrafilters are precisely principal ultrafilters. We will demonstrate this in the case over ultrafitlers over N. Let U be a non-principal ultrafilter. Then U cannot contain any finite subsets of N, and so the set fN n f1; : : : ; ng : n 2 Ng is a countable subset of U with empty intersection. Thus non-principal ultra- filters are countably incomplete. However, the existence of cardinals admitting non-principal countably complete ultrafilters is not provable within ZFC, so this criterion may not hold for higher cardinals. Lemma 3.8. Every countably incomplete ultraproduct over a countable language is @1-saturated. 3 Proof. We need to show that, given a language jLj ≤ @0, a cardinal κ ≥ @1 so that there exists a countably incomplete ultrafilter U over I (jIj = κ), and an Q ultraproduct M = U Mi, then M realises all complete types over any A ⊆ M with jAj < @1. Since A is countable, LA. Thus, we can take p(x), a complete n-type, and enumerate p(x) = fφi(x): i 2 Ng where φi(x) are LA formulae. Defining ^ i(x) = φi(x) j≤i we see that, byLos'stheorem, since 9x i(x) is true in M, we have that the set of Mt realising 9x i(x) must be an element of the ultrafilter, i.e. ft < @0 : Mt 9x i(x)g 2 U where we restrict constants, etc, in i(x) as necessary. Here, we have technically pulled the wool over the reader's eyes, as we have been identifying our models over L with their expansions to models over LA. Fortunately for us, Theorem 3.9 allows us to get away with this, performing the following argument on the expansions, then restricting to our original language. Now , as U is countably incomplete, we can construct a descending chain I = I0 ⊃ I1 ⊃ · · · 1 with each In 2 U, but \n = 0 In = ;. Now define X0 = I0, and for all n ≥ 0, define Xn = In \ ft < @0 : Mt 9x n(x)g Each Xn 2 U, and there form a descending chain with empty intersection. Now, for each i 2 I,where exists ni, the greatest integer for which i 2 Xni . Define a sequence fligi2I by the following rules If ni = 0, then li can be any element of Mi. Otherwise, choose li in the projection of A onto Mi so that Mi ni (li). Thus, for any n > 0, and i 2 Xn, we have n ≤ ni, so Mi φn(li). We then applyLos's Theorem to conclude M φn(l) for all n > 0, and hence, l realises p(x) in M, proving the result. Theorem 3.9. Given a language L, and an expansion L0 ⊃ L, let I be a non-empty set, U an ultrafilter over I, so that, for each i 2 I, we have an L- 0 0 Q 0 structure Mi and an elementary extension Mi to an L -structure. Then U Mi Q 0 is an elementary extension of U Mi to an L -structure. Theorem 3.10. Given elementarily equivalent saturated models M; N of the same cardinality, we must have M =∼ N A full proof of this statement can be found in Chang and Keisler's Model Theory, however we will only sketch the main idea, and contrast to the countable case. We first require two lemmas, both proven by transfinite induction. 4 Definition 3.11. For two sets, X, Y , denote by X Y the set of functions f : X ! Y . Definition 3.12. For a cardinal α, denote by α+ the least cardinal greater than α. Lemma 3.13. Suppose the M is α-saturated, M =∼ N and b 2 αN.