Completeness Criterion for Saturated Models

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 ℵ1. 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)

ℵα 2 = ℵα+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 ∪ {ca : a ∈ A} 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 ∈ 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 ∈ 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 ∈ p(x) or ¬φ ∈ 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 ∈ M such that M φ(a) for all φ ∈ p. We call M κ-saturated, for any cardinal κ, if for all A ⊆ M with |A| < κ, M realises all complete types over A. If |M| = κ, we call M saturated. Finally, we recall the Löwenheim-Skolem theorems, here compiled into one statement Theorem 2.7 (Löwenheim-Skolem). For any infinite L-structure M and any cardinal κ ≥ |L|, there exists an L-structure N such that |N| = κ and • if κ < |M|, then N is an elementary substructure of M. • if κ > |M|, 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 ∈ U. 2. If V,W ∈ U, then V ∩ W ∈ U. 3. If V ∈ U and V ⊆ W ⊆ I, then W ∈ U. 4. For any V ⊆ I, either V ∈ U or I \ V ∈ U.

2 We call an ultraﬁlter principal if there exists i ∈ I such that V ∈ U if and only if i ∈ V .

Definition 3.2. Given a family of sets Ai, with i ∈ I an index set, and an Q Q ultrafilter U over I, we define the ultraproduct Ai = Ai/U to be the Q U i∈I quotient of the cartesian product i∈I Ai by the equivalence relation

{ai}i∈I ∼ {bi}i∈I if and only if {i ∈ I : ai = bi} ∈ 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 ) ∈ R if and only if {i ∈ I :(ai , . . . , ai ) ∈ R } ∈ U and similarly for functions and constants. In the case where there exists M such that Mi = M for every i ∈ I, we call this the ultrapower of M and denote it by M|I|/U. We have the following theorem Theorem 3.4 (Los’sTheorem) . For all sentences φ of L, Y Mi φ if and only if {i ∈ I : Mi φ} ∈ U U

In the case of ultrapowers, we have M ≡ M|I|/U. Definition 3.5. We call an ultrafilter U countably incomplete if there exists a countable V ⊆ U such that ∩W ∈V 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

{N \{1, . . . , n} : n ∈ N} 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 ℵ1-saturated.

3 Proof. We need to show that, given a language |L| ≤ ℵ0, a cardinal κ ≥ ℵ1 so that there exists a countably incomplete ultrafilter U over I (|I| = κ), and an Q ultraproduct M = U Mi, then M realises all complete types over any A ⊆ M with |A| < ℵ1. Since A is countable, LA. Thus, we can take p(x), a complete n-type, and enumerate p(x) = {φi(x): i ∈ N} where φi(x) are LA formulae. Defining ^ ψi(x) = φi(x) j≤i we see that, byLos’stheorem, since ∃xψi(x) is true in M, we have that the set of Mt realising ∃xψi(x) must be an element of the ultrafilter, i.e.

{t < ℵ0 : Mt ∃xψi(x)} ∈ U where we restrict constants, etc, in ψi(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 ⊃ · · ·

∞ with each In ∈ U, but ∩n = 0 In = ∅. Now deﬁne X0 = I0, and for all n ≥ 0, deﬁne Xn = In ∩ {t < ℵ0 : Mt ∃xψn(x)}

Each Xn ∈ U, and there form a descending chain with empty intersection.

Now, for each i ∈ I,where exists ni, the greatest integer for which i ∈ Xni . Deﬁne a sequence {li}i∈I 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 ψni (li). Thus, for any n > 0, and i ∈ 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 ∈ 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 ∈ αN. Then there exists a ∈ αM such that

M ∪ {a(η): η < α} ≡ N ∪ {b(η): η < α}

Lemma 3.14. Suppose α is inﬁnite, M, N are both α-saturated and M ≡ N. Let a ∈ αM, b ∈ αN. Then there are a¯ ∈ αM, ¯b ∈ αN such that

range(a) ⊂ range(¯a) range(b) ⊂ range(¯b) M ∪ {a¯(η): η < α} ≡ N ∪ {¯b(η): η < α}

Using Lemma 3.13, one can easily show

Theorem 3.15. An α-saturated model M have the property that, for every model N ≡ M of size less than α+, there is an elementary embedding of N into M. We say M is α+-universal. Theorem 3.10 then follows by application of Lemma 3.14 and Theorem 3.15 to construct inverse elementary embeddings. To be precise, for |M| = |N| = α, one constructs a ∈ αM, b ∈ αN such that

M ∪ {a¯(η): η < α} ≡ N ∪ {¯b(η): η < α}

Then the map ai 7→ bi is an invertible map, which from universality must be an isomorphism. Following Theorem 3.10, we get the following result.

Lemma 3.16. Assuming GCH, given a model M of a countable language L ℵ0 with |M| ≤ ℵ1, then N := M /U is saturated of cardinality ℵ1.

Proof. We have, from Lemma 3.8 that N is ℵ1-saturated, so it suffices to show that |N| ≤ ℵ1. But ℵ0 |M| ≤ ℵ1 = 2 by GCH, so our ultraproduct has cardinality at most ℵ1. Remark 3.17. Theorem 3.10 is a generalisation of a uniqueness theorem for countably saturated models, of which the proof is quite similar, using the back and forth method to construct an enumeration of M,N, which in turn provides an isomorphism. Indeed, the key difference is that in the countable case, one can simply define these enumerations, whereas here transfinite methods were needed, and we must definte an idea of parity for ordinals. Furthermore, GCH is implicitly used via α+, complicating matters further.

5 4 Saturated Models and the Completeness Cri- terion

Theorem 4.1. Suppose T is a theory in a countable language L with no finite models, and a unique saturated model M of cardinality ℵ1, up to elementary equivalence, i.e all saturated models of T of cardinality ℵ1 are elementary equivalent to M. Then T is complete, assuming GCH. Proof. First note that Theorem 3.10 implies M is the unique saturated model of T of cardinality ℵ1, up to isomorphism. Let φ be an L-sentence and assume M φ. Thus, there cannot exist a saturated model of cardinality ℵ1 of T ∪ {¬φ}. Suppose there exists a model N, of cardinality strictly less than ℵ1, such that N T ∪ {¬φ}. But then, by Lemma 3.16 andLos’s Theorem, taking the ultrapower of this over a countably incomplete ultrafilter of ℵ0 will give a saturated model of cardinality ℵ1, isomorphic to M, but elementarily equivalent to N, and hence realising both φ and ¬φ. Thus no such model can exist, and so, by Löwerheim Skolem, no infinite models of T ∪ {¬φ} can exist. As we assumed no finite models of T exist, we have that all models of T are models of φ, and so T ` φ. The argument proceeds similarly if M ¬φ. Hence T is complete.

5 A note on the existence of saturated models

In Theorem 4.1, we have assumed the existence of a saturated model of the correct cardinality. However, the existence of such a model is not provable within ZFC. We can construct one by assuming GCH: find any infinite model, use Löwenheim-Skolem to find a countable model, and then take its ultrapower with respect to a countably incomplete ultrafilter of ℵ0. Furthermore, we require CH in the proof of Theorem 3.10. However, the author cannot say that Theorem 4.1 is not independent of GCH, so it may hold without that assumption. In this scenario, one may with for a criterion for the existence of saturated models. Analoguous to the theory of 0-stable theories and ℵ0-saturation, we can define a higher stability, which gives the existence of saturated models for complete theories. For the definitions of 0-stability, and models with parameters, we refer the reader to Kirby’s An Invitation to Model Theory via definable sets. Definition 5.1. Let κ be an infinite cardinal. A complete L theory T is called κ-stable if, over any set of at most κ parameters in any model of T , there are at most κ complete types.

Theorem 5.2. If T is κ-stable, then there exists a saturated model of T of cardinality κ. For a proof of this theorem, see Harnik’s On the Existence of Saturated Models of Stable Theories In his proof, he assumes the existence of a large saturated model of T , of which every other model of T is an elementary substructure. This is quite a

6 common approach in modern model theory, and can be established without use of GCH, or similar set theoretic assumptions.