Contents 1. Introduction 1 2. Background 2 3. Examples of Complete Theories with Diﬀerent I(T, ℵ0) 6 4

Home , Categorical theory, Complete theory, Prime model

HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE?

ALEXANDER KASTNER

Abstract. The purpose of this paper is expository. We investigate the number of non-isomorphic countable models of a complete, countable theory. Ex- ℵ amples of complete theories are given that either have exactly ℵ0, 2 0 or n countable models, where n is any natural number diﬀerent from 2. We then prove Vaught’s famous No-Two Theorem, which states that a complete countable theory cannot have exactly two countable models. Along the way, we prove the Ryll-Nardzewski–Svenonius–Engeler characterization of ℵ0-categoricity. We also discuss Morley’s result on countable models and the current state of Vaught’s conjecture.

Contents 1. Introduction 1 2. Background 2 3. Examples of complete theories with diﬀerent I(T, ℵ0) 6 4. Countable atomic and saturated models 9 Acknowledgments 15 References 15

1. Introduction One of the main goals of model theory is to understand the different models of a formal theory. In particular, given a theory T , it is natural to ask about the number of non-isomorphic models of T of any cardinality κ. This function of κ is usually denoted by I(T, κ) and called the spectrum of T . If I(T, κ) = 1, then we say that T is κ-categorical. Complete countable theories, which are our focus here, are a natural object of study. If such a theory T has a finite model with n elements, then I(T, κ) = 0 except when κ = n. On the other hand, if T does have an infinite model, then by the Löwenheim-Skolem-Tarski Theorem, we have I(T, κ) ≥ 1 for every infinite cardinal κ. A simple computation establishes that we always have I(T, κ) ≤ 2κ. The above facts are all that can easily be said in general about the spectrum of a theory. The investigation of I(T, κ) has yielded a horde of fascinating – and

Date: October 10, 2017. 1 2 ALEXANDER KASTNER often quite surprising – results. One of the most famous results is Morley’s Cate- goricity Theorem (see [3]): a complete, countable theory that is categorical in some uncountable cardinal must be categorical in every uncountable cardinal. The main focus of this paper, however, is I(T, ℵ0). Why should we care about the number of countable models of a complete (countable) theory? One reason is that the countable models – being small – constitute many of our examples and are thus quite familiar objects to us. Furthermore, one of most notorious questions in model theory, known as Vaught’s conjecture, concerns the behavior of I(T, ℵ0). ℵ0 It was proved by Morley in 1970 in [4] that if I(T, ℵ0) > ℵ1, then I(T, ℵ0) = 2 . ℵ0 Vaught conjectured in the 1960’s that if I(T, ℵ0) > ℵ0, then I(T, ℵ0) = 2 . Of course, this question disappears if one assumes the Continuum Hypothesis (CH), so work on Vaught’s Conjecture assumes the negation of CH. A counterexample to Vaught’s Conjecture was announced by R. Knight [2] in 2002. As of August 2017, the validity of Knight’s counterexample has not yet been veriﬁed. What follows is an outline of the paper. In Section 2, we review background material and establish notation. The section is structured in such a way as to make the paper nearly self-contained. In Section 3, we consider examples of complete theories T with diﬀerent values for I(T, ℵ0). In Section 4, we develop the theory of countable atomic and saturated models. In particular, this section contains a proof of the Ryll-Nardzewski–Svenonius–Engeler characterization of ℵ0-categoricity, and culminates with a proof of Vaught’s No-Two Theorem. The latter result states that a complete, countable theory cannot have exactly two non-isomorphic countable models, a rather surprising fact.

2. Background The set theory background required to understand this paper is minimal. We employ the Fundamental Theorem of Cardinal Arithmetic, but not much else. We state it here for completeness. Theorem 2.1 (Fundamental Theorem of Cardinal Arithmetic). If κ and λ are nonzero cardinals and at least one of them is infinite, then κ + λ = κ · λ = max(κ, λ). We use the letters κ, λ and µ to denote cardinals, and the letters α, β and γ to denote ordinals. The cardinality of a set X will be denoted by |X|. We use ℵβ to denote the β-th largest infinite cardinal. In particular, ℵ0 is the cardinality of the set ω of natural numbers. By a (first-order) language L , we mean a set of constant symbols, relation symbols and function symbols. For example, L = {0, <, S, +, ·} is a language consisting of one constant symbol 0, one 2-placed relation symbol <, one 1-placed function symbol S, and two 2-placed function symbols + and ·. If Y is a set of constant symbols not in L , we use LY denote the language L ∪ Y . A sequence of symbols τ from L ∪ {(, ), v0, v1,... } is called a term if τ has one of the following forms:

(1) vi, where vi is a variable symbol; (2) c, where c is a constant symbol in L ; (3) F (τ1, . . . , τn), where F is an n-placed function symbol in L and τ1, . . . , τn are terms. HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 3

A sequence of symbols φ from L ∪ {(, ), =, ¬, ∧, ∨, →, ↔, ∃, ∀, v0, v1,... } is called a formula if φ has one of the following forms:

(1) τ1 = τ2, where τ1 and τ2 are terms; (2) R(τ1, . . . , τn), where R is an n-placed relation symbol and τ1, . . . , τn are terms. (3) ¬φ1,(φ1 ∧ φ2), (φ1 ∨ φ2), (φ1 → φ2), (φ1 ↔ φ2), ∃φ1, or ∀φ1, where φ1 and φ2 are formulas. Formulas of the form (1) and (2) are called atomic formulas. Informally, we could say that terms are strings of symbols obtained from variables and constants via the application of finitely many function symbols. Similarly we could say that formulas are strings of symbols obtained from atomic formulas via the application of finitely many connectives and quantifiers. An occurrence of a variable vi in a formula φ is said to be free if it is not bound by a quantifier. We write φ(v1 . . . vn) to mean that the free variables in φ are among v1, . . . , vn. A formula φ is a sentence if it does not contain any free variables. If there is no danger of confusion, we will omit the parentheses in (φ1 ∧ φ2), (φ1, ∨φ2), etc. A model A for a language L consists of a nonempty set (called the universe) and interpretations of all the constant, function and relation symbols in L . For example, if L = {0, <, S, +, ·} as above, then a model for L is the set of natural numbers ω together with the interpretations of the symbols as the number 0, the usual order, the successor function, addition and multiplication, respectively. We write cA, RA and FA for the interpretations in a model A of the constant symbol c, the relation symbol R and the function symbol F . We denote models by A, B and C, and their corresponding universes by A, B and C. By the cardinality of a model A, we mean the cardinality of its universe A. If L ⊆ L 0 and A0 is a model for L 0, the reduct A of A0 to L is the model we obtain from A0 by forgetting about the interpretations of the symbols in L 0 \ L . In this case, we also say that A0 is an expansion of A. If A is a model and Y ⊆ A, we write AY or (A, a)a∈Y for the expansion of the model A to the language LY where the new constant symbols are interpreted by the corresponding elements in A. We assume the reader is familiar with the definition of a formula φ(v1 . . . vn) being satisfied by a tuple (a1, . . . , an) in a model A. In this case, we write

A |= φ[a1 . . . an]. In the special case where φ is a sentence, we simply write

A |= φ.

Of course, in the deﬁnition of satisfaction we assume that the formula and model are over the same language. We say that two models A and B (for the same language L ) are isomorphic, written A =∼ B, if there exists a bijection f : A → B such that

(1) f(cA) = cB for all constant symbols c; (2) f(FA(a1, . . . , an)) = FB(f(a1), . . . , f(an)) for all n-placed functions symbols F and a1, . . . , an ∈ A; (3) (a1, . . . , an) ∈ RA iﬀ (f(a1), . . . , f(an)) ∈ RB for all n-placed relation symbols R and a1, . . . , an ∈ A. 4 ALEXANDER KASTNER

If f is injective, but not necessarily onto, we call f an isomorphic embedding of A into B. A model A is a submodel of a model B, written A ⊆ B, if A ⊆ B contains all the constant elements cB, is closed under the functions FB, and the interpretations in A are the restrictions of the interpretations in B. In this case, we also say that B is an extension of A. If X ⊆ A, the submodel of A generated by X is the intersection of all submodels of A which include X. We say that two models A and B (for the same language L ) are elementarily equivalent, written A ≡ B, if they satisfy exactly the same sentences. Clearly, A =∼ B implies A ≡ B but the converse need not hold. Given models A and B, an injective map f : A → B is called an elementary embedding, written f : A ≺ B, if for all formulas φ(v1 . . . vn) and a1, . . . , an ∈ A, we have

A |= φ[a1 . . . an] implies B |= φ[f(a1) . . . f(an)]. In this case, f is an isomorphic embedding and A ≡ B. The expressions “elementary submodel” and “elementary extension” are deﬁned in the obvious way. A set of sentences T for a language L is called a theory. We say that A is a model of T , and write A |= T, if every sentence in T is satisﬁed in A. A theory T is called consistent if it has a model. We say that φ is a consequence of a theory T , written T |= φ, if every model of T is a model of φ. A subset Σ of a theory T is called a set of axioms for T if Σ and T have the same consequences. If φ(v1 . . . vn) is a formula with free variables, we will sometimes also write

T |= φ(v1 . . . vn) to mean that for all models A of T and for all a1, . . . , an ∈ A,

A |= φ[a1 . . . an]. Note that T |= φ(v1 . . . vn) iff T |= ∀v1 . . . vnφ(v1 . . . vn). The following two theorems are fundamental to model theory. Proofs can be found in the early chapters of any model theory textbook (e.g. Chang and Keisler [1]). We will invoke these theorems repeatedly in the course of this paper. Theorem 2.2 (Compactness Theorem). If every finite subset of a theory T has a model, then T has a model. Theorem 2.3 (Löwenheim-Skolem-Tarski Theorem). If a theory T has an infinite model, then T must have models of any cardinality κ ≥ max(ℵ0, |L |). In this paper, we will be particularly interested in complete theories T , whose defining property is that for every sentence φ, exactly one of T |= φ and T |= ¬φ holds. Complete theories can be viewed as the “irreducible” objects among theories as they have the “fewest” models. If we understand the set of complete extensions of a theory, it seems natural to assert that we also understand the original theory. An easy way to obtain complete theories is to consider the set of all sentences satisfied in a model A, denoted by Th(A) and called the theory of A. It is clear that A ≡ B iff Th(A) = Th(B), and that all models of a complete theory are elementarily equivalent. Given a model A, the diagram of A, denoted by ∆A, is defined to HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 5 be the set of all atomic sentences and negations of atomic sentences satisfied in (A, a)a∈A. It is not hard to show that A can be isomorphically embedded in B iff B can be expanded to a model of ∆A. Similarly, we define the elementary diagram of A, denoted by ΓA, to be Th((A, a)a∈A). Again, it is an easy fact that A can be elementarily embedded in B iff B can be expanded to a model of ΓA. Showing that a given theory is complete can prove quite challenging. The next result provides a sufficient condition for completeness, which can be applied in a few important cases. Theorem 2.4 (Lo´s-Vaught Test for Completeness). Suppose that T is a consistent theory with no finite models, and that for some cardinal κ ≥ max(ℵ0, |L |), all models of T of cardinality κ are elementarily equivalent. Then T is a complete theory. Proof. Assume, for contradiction, that T is not complete. Then for some sentence φ, both T ∪ {φ} and T ∪ {¬φ} must be consistent. Since T has no finite models, the Löwenheim-Skolem-Tarski Theorem implies that there exist models A and B of cardinality κ of T ∪{φ} and T ∪{¬φ}, respectively. Since A and B are both models of T , we must have Th(A) = Th(B), which contradicts the fact that φ ∈ Th(A) and ¬φ ∈ Th(B). While the notion of a theory captures global properties of a model, the next notion we introduce captures properties of specific elements or tuples within a model. A type Σ(v1 . . . vn) is defined to be a set of formulas whose free variables are among v1, . . . , vn. A model A is said to realize a type Σ(v1 . . . vn) if there exist a1, . . . , an ∈ A such that for all formulas φ in Σ,

A |= φ[a1 . . . an]. In this case, we write

A |= Σ[a1 . . . an].

We say that A omits Σ if it does not realize Σ. A type Σ(v1 . . . vn) is said to be consistent if it is realized in some model; Σ is said to be consistent with a theory T , or to be a type of T , if it is realized by some model of T . A type Σ(v1 . . . vn) is said to be consistent with a model A if it is consistent with Th(A). An application of the Compactness Theorem shows that Σ is consistent with A iff every finite subset of Σ is realized in A. When we introduce “saturated models”, we will need to consider types over parameter sets. If A is a model and Y ⊆ A (Y is the set of parameters), then a type Σ(v1 . . . vn) of A over Y is defined to be a type consistent with Th(AY ). We recall that AY = (A, a)a∈Y is the model in the expanded language LY where the new constants in Y are interpreted by the corresponding elements in A. A complete type is a type Σ(v1 . . . vn) which is maximal consistent, i.e. contains exactly one of φ(v1 . . . vn) and ¬φ(v1 . . . vn) for each formula φ(v1 . . . vn). Every type of a (consistent) theory can be enlarged to a complete type of that theory. Given a model A and a1, . . . , an ∈ A, the type determined by a1, . . . , an in A, denoted by tp(a1, . . . , an, A), is the complete type which consists of all formulas satisfied by (a1, . . . , an) in A. We note that a complete type Σ(v1 . . . vn) is consistent with a theory T iff

Σ = tp(a1, . . . , an, A) for some model A of T and a1, . . . , an ∈ A. 6 ALEXANDER KASTNER

The Omitting Types Theorem, the last theorem we assume as a prerequisite, provides a necessary and sufficient condition for a complete theory in a countable language to omit a type. In order to state the result, we need to introduce one more notion. A theory T is said to locally realize a type Σ(v1 . . . vn) if there exists a formula φ(v1 . . . vn) consistent with T such that for all σ ∈ Σ, T |= φ → σ. In other words, every n-tuple in a model of T which satisfies φ realizes Σ. A theory T is said to locally omit Σ(v1 . . . vn) if T does not locally realize Σ. In other words, T locally omits Σ(v1 . . . vn) iff for every formula φ(v1 . . . vn) there exists σ ∈ Σ such that φ ∧ ¬σ is consistent with T . Theorem 2.5 (Omitting Types Theorem). Let L be a countable language, T a complete theory in L , and Σ(v1 . . . vn) a type consistent with T . Then T has a countable model that omits Σ iff T locally omits Σ. A slightly stronger statement, which we also need, is the following. Theorem 2.6 (Extended Omitting Types Theorem). Let L be a countable language and let T be a complete theory in L . For each r < ω, let Σr(v1 . . . vnr ) be a type consistent with T . Then T has a countable model that omits each Σr iff T locally omits each Σr.

3. Examples of complete theories with different I(T, ℵ0)

In this section, we provide examples of complete theories T such that I(T, ℵ0) takes on every value that we currently know is possible. Thus, we provide examples where I(T, ℵ0) = n for every natural number n 6= 2, as well as examples where ℵ0 I(T, ℵ0) = ℵ0 and where I(T, ℵ0) = 2 . Our first example is due to Cantor. It is one of the most interesting examples of an ℵ0-categorical theory, i.e. of a theory T with I(T, ℵ0) = ℵ0. Example 3.1 (Dense linear order without endpoints). Let L = {<}, and let T be the theory of dense linear orders without endpoints. Two classical models of T are (Q, <) and (R, <), and it is clear that T has no finite models. We argue that (Q, <) is the unique countable model of T up to isomorphism. The standard way of proving this fact is by a “back-and-forth” construction. Let q0, q1,... be a fixed enumeration of Q and let a0, a1,... be a fixed enumeration of another countable dense linear order A = (A, <) without endpoints. We define an isomorphism f :(Q, <) → (A, <) in stages. Define f(q0) = a0. Now suppose that we have partially defined f so that its partial domain domn(f) is finite and contains q0, q1, . . . , qn, and so that its partial range rann(f) is finite and contains a0, a1, . . . , an. We wish to extend f so that (1) the domain will contain qn+1 and (2) the range will contain an+1. We first take care of (1). If qn+1 is already in the domain of f, then we are done. If not, then for each p ∈ domn(f), one of the following holds in (Q, <): qn+1 < p or p < qn+1. Using the defining properties of dense linear orders without endpoints, we can find a b ∈ A such that for each p ∈ domn(f), if (Q, <) |= qn+1 < p, then A |= b < f(p) and if (Q, <) |= p < qn+1, then A |= f(p) < b. We define f(qn+1) = b. We take care of (2) in a similar fashion. If an+1 is already in the range of f, then we are done. If not, we can find p ∈ Q with the same “order properties” as an+1 and define f(p) = an+1. HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 7

Repeating this process yields the desired isomorphism f :(Q, <) → (A, <). The back-and-forth argument described above appears frequently in model theory. An alternate argument, which is perhaps a bit simpler, is to define f :(Q, <) → (A, <) inductively as follows. Define f(q0) = a0. Then assuming f is defined appropriately on q0, . . . , qn, define f(qn+1) to be the first element in A (in the chosen enumeration) with the same order properties as qn+1. By theLo´s-Vaught Test for Completeness, we can also conclude that T is a complete theory.

Finding examples of complete theories with I(T, ℵ0) = n for natural numbers n > 1 is more diﬃcult. The two examples below are “built on” the theory of dense linear orders without endpoints. To the author’s knowledge, these are the only known examples with 1 < I(T, ℵ0) < ℵ0. It would be interesting to construct examples of a diﬀerent nature, which do not rely on the theory of dense linear orders without endpoints. Example 3.2 (Ehrenfeucht). We describe an example of a complete theory T with I(T, ℵ0) = 3. The language is

L = {<, c0, c1, c2,... }, and the theory T consists of the axioms for dense linear orders without endpoints, together with the axioms

c0 < c1, c1 < c2,... Any countable model of T must be isomorphic to (Q, <) together with some spec- iﬁed increasing sequence (cn). Depending on whether

lim cn = ∞, lim cn ∈ , or lim cn ∈ \ , n→∞ n→∞ Q n→∞ R Q we obtain three non-isomorphic countable models. From the proof that (Q, <) is the unique countable dense linear order without endpoints, it is easy to see that there are no other countable models. We now argue that T is complete. If we consider the restriction Tn of T to a language Ln = {<, c0, . . . , cn} with only ﬁnitely many constant symbols, then a back-and-forth argument establishes that Tn is ℵ0-categorical, and hence complete. If φ is any sentence in the original language L , it can only contain ﬁnitely many constant symbols so must be a sentence in Ln for some n < ω. Since Tn is complete, either Tn |= φ, or Tn |= ¬φ. It follows that either T |= φ, or T |= ¬φ, whence T is complete. Example 3.3 (Ehrenfeucht). We modify Example 3.2 to obtain complete theories Tn with I(Tn, ℵ0) = n for 3 ≤ n < ℵ0. The language for Tn is

Ln = {<, P1,P2,...,Pn−2, c0, c1, c2 ... }, where each Pi is a unary predicate symbol. In addition to the axioms for T in Example 3.2, the theory Tn contains ﬁrst-order axioms which say the following:

(1) the sets Pi partition the universe; (2) all the constants cj belong to P1; (3) each set Pi is dense in the order.

The resulting theory Tn is complete, as can be established by a modiﬁcation of the argument in Example 3.2. Moreover, the situation when limn→∞ cn ∈ Q splits into 8 ALEXANDER KASTNER n − 2 cases depending on which set Pi contains the limit. Thus, Tn has exactly n non-isomorphic countable models.

In the next example, we consider a few complete theories T with I(T, ℵ0) = ℵ0.

Example 3.4. Let L = {c0, c1, c2,... }, and let T be the theory consisting of the sentences ci 6= cj for all 0 ≤ i < j < ω. The theory T has different countable models depending on whether there are 0, 1, 2, ... , or ℵ0 non-constant elements, so that I(T, ℵ0) = ℵ0. We can view the number of non-constant elements as an invariant of the model. The completeness of T follows from theLo´s-Vaught Test and the fact that T has a single model for any cardinality κ > ℵ0 (up to isomorphism). Similar examples of complete theories with I(T, ℵ0) = ℵ0 can be constructed. Consider the theory of a permutation F : A → A with no finite cycles, and the theory of an equivalence relation R with a single equivalence class of each size n < ω. In the first case, the invariant is the number of orbits; in the second case, the invariant is the number of infinite equivalence classes.

ℵ0 Finally, we give an example of a complete theory T with I(T, ℵ0) = 2 .

Example 3.5. Let L consist of countably many unary predicate symbols P0,P1,... . The theory T consists of the sentences

∃x(Pi1 (x) ∧ · · · ∧ Pim (x) ∧ ¬Pj1 (x) ∧ · · · ∧ ¬Pjn (x)) for each choice of disjoint ﬁnite subsets {i1, . . . , im} and {j1, . . . , jn} of ω. The theory T is sometimes called “the theory of countably many independent unary predicates”. We ﬁrst argue that T has 2ℵ0 countable models. If A is a model of T , then each element a ∈ A corresponds to a subset of ω, namely the subset of all indices i such that a ∈ Pi. Any countable model thus corresponds to ℵ0 many subsets of ω. Combined with the observation that for each subset of ω, there must be a model of T with an element corresponding to that subset, we see that T must have 2ℵ0 many countable models. To prove that T is complete, we show that any countable model A of T elementarily embeds in a model B where for each X ⊆ ω, there are 2ℵ0 many elements b ∈ B such that

B |= Pi[b] iﬀ i ∈ X. Now there is exactly one such model B up to isomorphism. This would thus show that all countable models of T are elementarily equivalent, whence by theLo´s- Vaught test T would be complete. 0 Consider an expanded language L with constant symbols cX,α, where X ranges over the subsets of ω and α < 2ℵ0 . Deﬁne T 0 to be the theory consisting of

(1) the elementary diagram Th((A, a)a∈A); (2) the sentences cX,α 6= cX,β where α < β; (3) the sentences Pi(cX,α) for all i ∈ X; (4) the sentences ¬Pi(cX,α) for all i 6∈ X. By the Compactness Theorem and the L¨owenheim-Skolem-Tarski Theorem, T 0 has 0 ℵ 0 a model B of cardinality 20 . The reduct B of B to L is the desired elementary extension. HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 9

4. Countable atomic and saturated models Important: In this section, the language L is always assumed to be countable. We analyze the behavior of the countable models of a complete theory. For this analysis, the notions of “atomic models” and “saturated models” are fundamental. The atomic models are thought of as “small”, and the saturated models are thought of as “large”. We ﬁrst consider the atomic models. Let T be a complete theory. A formula φ(v1 . . . vn) is said to be complete (in T ) if for all ψ(v1 . . . vn), exactly one of T |= φ → ψ and T |= φ → ¬ψ holds. A formula ψ(v1 . . . vn) is said to be completable (in T ) if there exists a complete formula φ(v1 . . . vn) such that T |= φ → ψ.

Deﬁnition 4.1. A model A is said to be atomic if every tuple (a1, . . . , an) of elements of A satisﬁes a complete formula in Th(A).

In other words, A is atomic iff for all a1, . . . , an ∈ A, the complete type tp(a1, . . . , an, A) is finitely axiomatized with respect to Th(A). The following theorem explains why we think of atomic models as “small”. A model A is called prime if it can be elementarily embedded in all models B ≡ A. Obviously, a prime model (for a countable language) must be countable. Theorem 4.2. A countable model is atomic iff it is prime. We omit the proof of this theorem, as we will not need it. A proof can be found in Chang and Keisler [1]. Theorem 4.3 (Existence Theorem for Atomic Models). A complete theory T has a countable atomic model iff every formula ψ(v1 . . . vn) consistent with T is completable in T . Proof. Suppose A is a countable atomic model of the complete theory T , and let ψ(v1 . . . vn) be consistent with T . Since T is complete we must have

T |= ∃v1 . . . vnψ(v1 . . . vn).

Hence, there exist a1, . . . , an ∈ A such that

A |= ψ[a1 . . . an].

Since A is atomic, (a1, . . . , an) must satisfy a complete formula φ(v1, . . . , vn). It is clear that T |= φ → ψ. For the backward direction, we need to use the Extended Omitting Types The- orem. Suppose every formula ψ(v1 . . . vn) consistent with T is completable in T . For each n < ω, let Σn(v1 . . . vn) denote the set of all negations of complete formulas in n variables. We claim that T locally omits each Σn(v1 . . . vn). Indeed, for any formula ψ(v1 . . . vn) consistent with T , there must exist a complete formula φ(v1 . . . vn) such that T |= φ → ψ.

In particular, ψ ∧ φ is consistent with T , and by construction ¬φ ∈ Σn(v1 . . . vn). It follows that T locally omits Σn(v1 . . . vn). By the Extended Omitting Types Theorem, T has a countable model A that omits each type Σn(v1 . . . vn). Now if a1, . . . , an ∈ A, then there must be some ¬φ ∈ Σn(v1 . . . vn) such that

A |= φ[a1 . . . an]. 10 ALEXANDER KASTNER

Thus, A is an atomic model. Theorem 4.4 (Uniqueness Theorem for Atomic Models). If A and B are two countable atomic models such that A ≡ B, then A =∼ B. Proof. The proof uses a back-and-forth argument which is similar to the one used to show that all countable dense linear orders without endpoints are isomorphic. Pick an enumeration of the sets A and B so that it is meaningful to write min X if X ⊆ A or X ⊆ B. Write T for Th(A) = Th(B). Let a0 = min A. Since A is atomic, a0 must satisfy a complete formula φ0(v0) in A. Since

T |= ∃v0φ0(v0), we can choose b0 ∈ B such that b0 satisﬁes φ0(v0) in B. Now let b1 = min B \{b0}. Since B is atomic, (b0, b1) must satisfy a complete formula φ1(v0v1) in B. Using that φ0 is complete (this is where we really need the atomic property!), we have

T |= ∀v0(φ0(v0) → ∃v1φ1(v0v1)).

Hence, we can choose a1 ∈ A such that (a0, a1) satisfies φ1(v0v1) in A. Repeating this process, we obtain enumerations a0, a1,... of A and b0, b1,... of B such that for each n < ω,(a0, . . . , an) and (b0, . . . , bn) satisfy the same complete formula φn(v0 . . . vn). It follows readily that the map an 7→ bn is an isomorphism from A onto B. We next study the countable saturated models of a complete theory. Recall that if A is a model and Y ⊆ A, then a type Σ(v1 . . . vn) of A over the parameter set Y is simply a type consistent with Th(AY ). By the Compactness Theorem, Σ(v1 . . . vn) being consistent with Th(AY ) is equivalent to every finite subset of Σ being realizable in AY . Definition 4.5. A model A is said to be κ-saturated if for all parameter sets Y ⊆ A, |Y | < κ, all types Σ(x) of A over Y are realized in A. A is called saturated if A is |A|-saturated. Intuitively, a saturated model is one that realizes as many types as possible (if we allowed |Y | = |A|, then we could take the entire universe A as our parameter set Y , and the type consisting of the formulas x 6= a, for each a ∈ A, would always be consistent with Th(AY ) but not realizable in AY ). In this section we will only be concerned with countable saturated models, i.e. countable models A that realize all of their types over finite parameter sets. In contrast with the countable atomic models, the countable saturated models are thought of as “large”. We say that a countable model A is countably universal if for all countable models B ≡ A, B can be elementarily embedded in A. Theorem 4.6. All countable saturated models are countably universal. Again, we omit the proof of this theorem as we will not need the result. The interested reader can refer once more to Chang and Keisler [1]. As an aside, we remark that the converse of Theorem 4.6 does not hold. Theorem 4.7 (Existence Theorem for Saturated Models). A complete theory T has a countable saturated model iff T has countably many complete types Σ(v1 . . . vn). HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 11

Proof. If A is a countable saturated model of T , then each complete type Σ(v1 . . . vn) of T must be realized in A. Since there are only countably many complete types realized in a countable model, T must only have countably many complete types. We now prove the more difficult backward direction. The proof is an example of the Henkin method of adding constants, which has become the standard method for proving both the Completeness Theorem for First-Order Logic and the Omitting Types Theorem. Suppose that T has only countably many complete types Σ(v1 . . . vn). We construct a countable saturated model A of T . Add a countable set of constant symbols 0 C = {c0, c1, c2,... } to the language L to obtain a new language L . For each finite subset Y = {d1, . . . , dn} ⊆ C, the complete types Γ(x) with one variable of T in LY are in one-to-one correspon- dence with the complete types Σ(xv1 . . . vn) with n + 1 variables of T in L . Since there are countably many finite subsets of C, our assumption implies that there are countably many complete types Γ(x) of T in finite expansions LY . Enumerate them as Γ0(x), Γ1(x), Γ2(x),... Enumerate the sentences in L 0 as

φ0, φ1, φ2,... We form an increasing chain of theories

T = T0 ⊆ T1 ⊆ T2 ⊆ ... satisfying the following conditions:

(1) Each theory Tm is consistent and uses finitely many constant symbols from C; (2) Either φm ∈ Tm+1 or ¬φm ∈ Tm+1; (3) If φm = ∃xψ(x) is in Tm+1, then ψ(c) ∈ Tm+1 for some c ∈ C; (4) If Γm(x) is consistent with Tm+1, then Γm(d) ⊆ Tm+1 for some d ∈ C. That such a construction can be carried out is clear. By (2), the theory [ Tω = Tm m<ω 0 is maximal consistent. Let B = (B, a0, a1,... ) be a countable model of Tω (where as usual a0, a1,... are the interpretations of the constant symbols c0, c1,... ). Let 0 0 A = (A, a0, a1,... ) be the submodel of B generated by the constant elements 0 a0, a1,... By (2) and (3), A = {a0, a1,... }. Condition (3) also shows that A is an 0 0 elementary submodel of B , and that therefore A is a model of Tω. We claim that A is a countable saturated model of T . Given a complete type Γ(x) of Th(AY ) for some finite Y ⊆ A, we have Γ(x) = Γm(x) for some m. Since Γm(x) is consistent with Tω,Γm(x) must have been consistent with Tm+1. By (4), Γm(ci) ⊆ Tm+1 for some ci ∈ C. It follows that Γm(x) is realized by ai in A. Corollary 4.8. A complete theory with countably many non-isomorphic countable models must have a countable saturated model. Proof. We claim that if T is a complete theory with countably many non-isomorphic countable models, then T must have countably many complete types. Indeed, any 12 ALEXANDER KASTNER type Σ(v1 . . . vn) of T must be realized in a countable model of T by the Löwenheim- Skolem-Tarski Theorem. Since a countable model can only realize countably many complete types, and T has countably many countable models, it follows that T has countably many complete types. Now apply Theorem 4.7. As for atomic models, there is a uniqueness theorem for countable saturated models. It is proved via a back-and-forth argument that is very similar to the proof of Theorem 4.4. Theorem 4.9 (Uniqueness Theorem for Saturated Models). If A and B are two countable saturated models such that A ≡ B, then A =∼ B. Theorem 4.10. If a complete theory T has a countable saturated model, then T also has a countable atomic model. Proof. Suppose T is a complete theory that does not have a countable atomic ℵ0 model. We prove that for some n < ω, T has 2 complete types Σ(v1 . . . vn). By Theorem 4.7, this would show that T has no countable saturated model. By Theorem 4.3, there must exist a formula φ(v1 . . . vn) that is consistent with T but not completable in T . In particular, φ is not complete, so we can find a formula ψ(v1 . . . vn) consistent with T such that neither T |= φ → ψ nor T |= φ → ¬ψ. Thus if we let φ0 = ψ ∧ φ and φ1 = ¬ψ ∧ φ, we have

T |= φ0 → φ, T |= φ1 → φ, T |= ¬(φ0 ∧ φ1).

Since φ0 and φ1 are not completable (otherwise φ would be completable), we can repeat the process to obtain the following tree: φ

φ0 φ1

φ00 φ01 φ10 φ11 . .

Each of the 2ℵ0 infinite branches of the tree corresponds to a type of T (this follows from the Compactness Theorem and the fact that finite branches are consistent with T ). The tree was defined in such a way that the types corresponding to the branches are mutually inconsistent. Now enlarge the types to complete types ℵ in order to obtain 2 0 complete types of T . The following theorem was established independently by Ryll-Nardzewski, Sveno- nius and Engeler in 1959. Recall that we defined a theory T to be κ-categorical if I(T, κ) = 1.

Theorem 4.11 (Characterization of ℵ0-categoricity). Let T be a complete theory. The following are equivalent:

(a) T is ℵ0-categorical; (b) T has a countable model that is both atomic and saturated; (c) Each complete type of T contains a complete formula; (d) For each n < ω, there are only ﬁnitely many complete types Σ(v1 . . . vn) consistent with T . HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 13

(e) For each n < ω, there are only ﬁnitely many formulas φ(v1 . . . vn) up to equivalence with respect to T . (f) Every model of T is atomic. We repeat here the wise words of Chang and Keisler [1]: “The reader is advised to sit down before beginning this proof.”

Proof. (a) ⇒ (b): Suppose that T is ℵ0-categorical. By the L¨owenheim-Skolem- Tarski theorem, every type of T must be realized in a countable model of T . Since T is ℵ0-categorical, it follows that T has countably many complete types Σ(v1 . . . vn). By Theorem 4.7, this implies that T has a countable saturated model. Now by Theorem 4.10, T also has a countable atomic model. Since T is ℵ0-categorical, the unique countable model of T must be both atomic and saturated. (b) ⇒ (c): Let A be a countable model of T that is both atomic and saturated. Consider a complete type Σ(v1 . . . vn) of T . Since A is saturated, there exist a1, . . . , an such that A |= Σ[a1, . . . , an].

Further, the fact that A is atomic implies that (a1, . . . , an) satisﬁes a complete formula φ(v1 . . . vn) in T , so that φ ∈ Σ(v1 . . . vn). (c) ⇒ (d): Suppose that each complete type of T contains a complete formula. Let n < ω, and let Γ(v1 . . . vn) denote the set of all negations of complete formulas (in T ) with n free variables. By assumption, Γ(v1 . . . vn) cannot be extended to a complete type consistent with T . In particular, this means that Γ(v1 . . . vn) is inconsistent with T . By the Compactness Theorem, there exist ¬φ1,..., ¬φm ∈ Γ(v1 . . . vn) such that ¬φ1 ∧ · · · ∧ ¬φm is inconsistent with T . Thus,

T |= φ1 ∨ · · · ∨ φm.

Recalling that each φi(v1 . . . vn) is a complete formula, it follows that T has only finitely many types Σ(v1 . . . vn). (d) ⇒ (e): Suppose that for each n < ω, there are only finitely many complete types Σ(v1 . . . vn) consistent with T . Fix n < ω. Given a formula φ(v1 . . . vn), ∗ define φ to be the set of all complete types Σ(v1 . . . vn) of T that contain φ. We claim that φ∗ = ψ∗ ⇐⇒ T |= φ ↔ ψ. The backward direction is clear. Now suppose that φ∗ = ψ∗, and suppose that A |= T and (a1, . . . , an) realize φ in A. The complete type tp(a1 . . . an, A) contains φ so must be a member of φ∗ = ψ∗. It follows that this type contains ψ and that therefore A |= ψ[a1 . . . an]. This proves that T |= φ → ψ, and by an analogous argument we obtain T |= φ ↔ ψ. Finally consider the map φ 7→ φ∗.

This induces an injection of the collection of sets of T -equivalent formulas φ(v1 . . . vn) into the collection of sets of complete types Σ(v1 . . . vn). Since we assumed that there are only finitely many complete types Σ(v1 . . . vn), this proves that there are only finitely many formulas φ(v1 . . . vn) up to equivalence with respect to T . (e) ⇒ (f): Suppose that for each n < ω, there are only finitely many formulas φ(v1 . . . vn) up to equivalence with respect to T . Let A be a model of T , and let a1, . . . , an ∈ A. Let φ1, . . . , φm be all the formulas that are satisfied by (a1, . . . , an) 14 ALEXANDER KASTNER in A up to equivalence (there must be finitely many of them by our assumption). Then φ1 ∧ · · · ∧ φm is a complete formula which is satisfied by (a1, . . . , an) in A. (f) ⇒ (a): If all models of T are atomic, then by the Uniqueness Theorem for Atomic Models there must be at most one countable model. Theorem 4.12 (Vaught’s No-Two Theorem). No complete theory has exactly two non-isomorphic countable models.

Proof. We prove that if T is a complete theory with 2 ≤ I(T, ℵ0) ≤ ℵ0, then T must have at least three non-isomorphic countable models. Since T has countably many countable models, T must have a countable saturated model B by Corollary 4.8. By Theorem 4.10, this implies the existence of a countable atomic model A. Since ∼ T is not ℵ0-categorical, we must have A =6 B (as A cannot be saturated and B cannot be atomic). We prove that there exists a model C of T that is neither atomic nor saturated, which would give us a third non-isomorphic model. Since B is not atomic, there exists a tuple (b1, . . . , bn) that does not satisfy a complete formula φ(v1, . . . , vn) in T = Th(B). The model (B, b1, . . . , bn) must be saturated since B is saturated. By Theorem 4.10, there must exist a countable atomic model (C, c1, . . . , cn) ≡ (B, b1, . . . , bn). We claim that the reduct C is neither atomic nor saturated. To see that C is not atomic, observe that the tuple (c1, . . . , cn) does not satisfy a complete formula in T = Th(C) = Th(B) (since (c1, . . . , cn) satisfies the same formulas in C as (b1, . . . , bn) satisfies in B). Finally, we prove that C is not saturated by showing that (C, c1, . . . , cn) is not saturated. Since T is not ℵ0-categorical, for each n < ω there are infinitely many formulas φ(v1, . . . , vn) up to equivalence with respect to T . We claim that there must also be infinitely many formulas φ(v1, . . . , vn) up to equivalence with respect to Th((C, c1, . . . , cn)). Indeed, if φ(v1, . . . , vn) and ψ(v1, . . . , vn) are not equivalent with respect to T , then there exists a model of T that realizes φ∧¬ψ (or alternatively ψ ∧ ¬φ). Since T is complete,

T |= ∃v1 . . . vn(φ ∧ ¬ψ), and therefore

Th((C, c1, . . . , cn)) |= ∃v1 . . . vn(φ ∧ ¬ψ).

Thus, φ and ψ are still not equivalent with respect to Th((C, c1, . . . , cn)). Hence Th((C, c1, . . . , cn)) is not ℵ0-categorical. Since (C, c1, . . . , cn) is atomic, it therefore cannot be saturated as well. Remark 4.13. The above proof shows that a complete theory T with

2 ≤ I(T, ℵ0) ≤ ℵ0 must have (1) an atomic model that is not saturated. (2) a saturated model that is not atomic. (3) a model that is neither atomic nor saturated.

Moreover, if I(T, ℵ0) ≤ ℵ0, then there is always a single countable atomic model and a single countable saturated model; all other countable models are neither atomic nor saturated. HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? 15

Example 4.14. In Example 3.2 we considered the theory T of dense linear orders without endpoints together with an increasing sequence c0 < c1 < c2 < . . . . We showed that there are three countable models of T up to isomorphism characterized by the limiting behavior of the sequence (cn). In the spirit of Remark 4.13, we can identify among the countable models of T the atomic model, the saturated model, and the model that is neither atomic nor saturated:

(1) limn→∞ cn = ∞ corresponds to the atomic model. (2) limn→∞ cn ∈ R \ Q corresponds to the saturated model. (3) limn→∞ cn ∈ Q corresponds to the model that is neither atomic nor saturated. A similar analysis holds for Example 3.3. Acknowledgments. This project would not have been possible without the help of a few individuals. I ﬁrst want to thank Professor Maryanthe Malliaris. Professor Malliaris was great at conveying intuition about model theory during our weekly meetings. Her enthusiasm for her work is clear to anyone who knows her. Second I’d like to thank my mentor Sarah Reitzes. Sarah led the group of logic students and was excellent at keeping us all on track. Her comments and criticisms of this paper have improved it considerably. Third, I’d like to thank my fellow logic students Alex Johnson and Diego Bejarano Rayo. The numerous conversations I had with both Alex and Diego really helped to reinforce my understanding of the subject. Finally, I’d like to thank Professor Peter May for organizing the UChicago REU program. It was a thoroughly enjoyable experience.

References [1] C. C. Chang and H. Jerome Keisler. Model Theory, Third Edition, Dover (2012). [2] R. W. Knight. The Vaught conjecture: a counterexample, Manuscript (2002) [3] M. Morley Categoricity in power, Trans. Am. Math. Soc. 114, 514-538. [4] M. Morley. The number of countable models, J. Symb. Logic 35, 14-18 (1970) [5] R. Vaught. Denumerable models of complete theories, Inﬁnitistic Methods (Pergamon, Lon- don), 303-321 (1961)