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

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Contents 1. Introduction 1 2. Background 2 3. Examples of Complete Theories with Different I(T, ℵ0) 6 4 HOW MANY COUNTABLE MODELS CAN A COMPLETE THEORY HAVE? ALEXANDER KASTNER Abstract. The purpose of this paper is expository. We investigate the num- ber of non-isomorphic countable models of a complete, countable theory. Ex- @ amples of complete theories are given that either have exactly @0, 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 count- able theory cannot have exactly two countable models. Along the way, we prove the Ryll-Nardzewski{Svenonius{Engeler characterization of @0-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; @0) 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 L¨owenheim-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; @0). 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; @0). @0 It was proved by Morley in 1970 in [4] that if I(T; @0) > @1, then I(T; @0) = 2 . @0 Vaught conjectured in the 1960's that if I(T; @0) > @0, then I(T; @0) = 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; @0). 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 @0-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 jXj. We use @β to denote the β-th largest infinite cardinal. In particular, @0 is the cardinality of the set ! of natural numbers. By a (first-order) language L , we mean a set of constant symbols, relation sym- bols and function symbols. For example, L = f0; <; 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 [ f(; ); v0; v1;::: g 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 [ f(; ); =; :; ^; _; !; $; 9; 8; v0; v1;::: g 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), 9φ1, or 8φ1, 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 = f0; <; S; +; ·} as above, then a model for L is the set of natural numbers ! 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 n 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)a2Y 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 j= φ[a1 : : : an]: In the special case where φ is a sentence, we simply write A j= 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 sym- bols F and a1; : : : ; an 2 A; (3) (a1; : : : ; an) 2 RA iff (f(a1); : : : ; f(an)) 2 RB for all n-placed relation sym- bols R and a1; : : : ; an 2 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.
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