Number of countable models

1978 ◽  
Vol 43 (3) ◽  
pp. 492-496 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Complete Theory ◽  
Prime Model ◽  
Categorical Theory ◽  
Definable Subset ◽  
Algebraic Elements ◽  
Minimal Prime ◽  
Countable Theory ◽  
The Universe ◽  
Countable Models

We prove that a countable complete theory whose prime model has an infinite definable subset, all of whose elements are named, has at least four countable models up to isomorphism. The motivation for this is the conjecture that a countable theory with a minimal model has infinitely many countable models. In this connection we first prove that a minimal prime model A has an expansion by a finite number of constants A′ such that the set of algebraic elements of A′ contains an infinite definable subset.We note that our main conjecture strengthens the Baldwin–Lachlan theorem. We also note that due to Vaught's result that a countable theory cannot have exactly two countable models, the weakest possible nontrivial result for a non-ℵ0-categorical theory is that it has at least four countable models.§1. Notation and preliminaries. Our notation follows Chang and Keisler [1], except that we denote models by A, B, etc. We use the same symbol to refer to the universe of a model. Models we refer to are always in a countable language. For T a countable complete theory we let n(T) be the number of countable models of T up to isomorphism. ∃n means ‘there are exactly n’.

Why some people are excited by Vaught's conjecture

1985 ◽  
Vol 50 (4) ◽  
pp. 973-982 ◽  
Author(s):  
Daniel Lascar
Keyword(s):  
Model Theory ◽  
Continuum Hypothesis ◽  
Complete Theory ◽  
Scott Sentences ◽  
Countable Theory ◽  
The Continuum ◽  
Countable Models

§I. In 1961, R. L. Vaught ([V]) asked if one could prove, without the continuum hypothesis, that there exists a countable complete theory with exactly ℵ1 isomorphism types of countable models. The following statement is known as Vaught conjecture:Let T be a countable theory. If T has uncountably many countable models, then T hascountable models.More than twenty years later, this question is still open. Many papers have been written on the question: see for example [HM], [M1], [M2] and [St]. In the opinion of many people, it is a major problem in model theory.Of course, I cannot say what Vaught had in mind when he asked the question. I just want to explain here what meaning I personally see to this problem. In particular, I will not speak about the topological Vaught conjecture, which is quite another issue.I suppose that the first question I shall have to face is the following: “Why on earth are you interested in the number of countable models—particularly since the whole question disappears if we assume the continuum hypothesis?” The answer is simply that I am not interested in the number of countable models, nor in the number of models in any cardinality, as a matter of fact. An explanation is due here; it will be a little technical and it will rest upon two names: Scott (sentences) and Morley (theorem).

Recursively presentable prime models

1974 ◽  
Vol 39 (2) ◽  
pp. 305-309 ◽  
Author(s):  
Leo Harrington
Keyword(s):  
Decision Procedure ◽  
Positive Answer ◽  
Prime Model ◽  
Property A ◽  
Categorical Theory ◽  
Decidable Theory ◽  
Free Variables ◽  
Closed Fields ◽  
Countable Theory ◽  
Algebraically Closed Fields

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ1-categorical theory.The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula φ(x) corresponds to is generally called the Gödel number of φ(x) and is denoted by ⌜φ(x)⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type Γ is recursive with index e if {⌜φ⌝.; φ ∈ Γ} is a recursive set of integers with index e.

On theories having a finite number of nonisomorphic countable models

1985 ◽  
Vol 50 (3) ◽  
pp. 806-808 ◽  
Author(s):  
Akito Tsuboi
Keyword(s):  
Finite Number ◽  
Positive Solutions ◽  
Linear Order ◽  
Order Property ◽  
Categorical Theory ◽  
Base Theory ◽  
Almost All ◽  
Countable Models

In this paper we shall state some interesting facts concerning non-ω-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from ω-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory.Many papers including [4] and [5] are motivated by the conjecture that every non-ω-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory T0 of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory T0 is stable and ω-categorical, then T0 cannot be extended to a theory T which has n countable models (1 < n < ω) by adding axioms for constant symbols.

Theories with exactly three countable models and theories with algebraic prime models

1980 ◽  
Vol 45 (2) ◽  
pp. 302-310 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Complete Theory ◽  
Uniform Bound ◽  
Bounded Degree ◽  
Definable Set ◽  
Countable Theory ◽  
Uniformly Bounded ◽  
Pathological Case ◽  
Countable Models

We prove first that if T is a countable complete theory with n(T), the number of countable models of T, equal to three, then T is similar to the Ehrenfeucht example of such a theory. Woodrow [4] showed that if T is in the same language as the Ehrenfeucht example, T has elimination of quantifiers, and n(T) = 3 then T is very much like this example. All known examples of theories T with n(T) finite and greater than one are based on the Ehrenfeucht example. We feel that such theories are a pathological case. Our second theorem strengthens the main result of [2]. The theorem in the present paper says that if T is a countable theory which has a model in which all the elements of some infinite definable set are algebraic of uniformly bounded degree, then n(T) ≥ 4. It is known [3] that if n(T) > 1, then n(T) > 3, so our result is the first nontrivial step towards proving that n(T) ≥ ℵ0. We would also like, of course, to prove the result without the uniform bound on the finite degrees of the elements in the subset.Theorem 2.1 is included in the author's Ph. D. thesis, as is a weaker version of Theorem 3.7. Thanks are due to Harry Simmons for his suggestions concerning the presentation of the material, and to Wilfrid Hodges for his advice while I was a Ph. D. student.

Omitting models

1977 ◽  
Vol 42 (1) ◽  
pp. 29-32
Author(s):  
Ernest Snapper
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
Complete Theory ◽  
Finite Set ◽  
Homogeneous Models ◽  
Definition Of ◽  
General Aspects ◽  
The Universe ◽  
Main Definition ◽  
Countable Models

The purpose of this paper is to introduce the notion of “omitting models” and to derive a very natural theorem concerning it (Theorem 1). A corollary of this theorem is the remarkable theorem of Vaught [3] which states that a countable complete theory cannot have precisely two nonisomorphic countable models. In fact, we show that our theorem implies Rosenstein's theorem [2] which, in turn, implies Vaught's theorem.T stands for a countable complete theory whose (countable) language is denoted by L. Following [1], a countably homogeneous model of T is a countable model of T with the property that, for any two n-tuples a1, …, an and b1,…,bn of the universe of whose types are the same, there is an automorphism of which maps ai, on bi, for i = 1, …, n [1, p. 129 and Proposition 3.2.9, p. 131]. “Homogeneous model” always means “countably homogeneous model.” “Type of T” always stands for “n-type of T” where n ≥ s 0, i.e., for the type of some n-tuple of individuals of the universe of some model of T. We often use that two homogeneous models which realize the same types are isomorphic [1, Proposition 3.2.9, p. 131].It is well known that every type of T is realized by at least one countable model of T. The main definition of this paper is:Definition 1. A set of countable models of T is omissible or “may be omitted” if every type of T is realized by at least one countable model of T which is not isomorphic to a model in the set.The main theorem of the paper is:Theorem 1. If a countable complete theory is not ω-categorical, every finite set of its homogeneous models may be omitted.The theorem is proved in §1 and in §2 it is shown how Vaught's and Rosenstein's theorems follow from it. §3 discusses some general aspects of omitting models.

Vaught's conjecture for o-minimal theories

1988 ◽  
Vol 53 (1) ◽  
pp. 146-159 ◽  
Author(s):  
Laura L. Mayer
Keyword(s):  
Structure Theorem ◽  
Classification Theory ◽  
Real Closed Fields ◽  
Minimal Theory ◽  
Definable Subset ◽  
Closed Fields ◽  
The Universe ◽  
Countable Models

The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory.We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2ω countable models. This proviso (viz. that T has fewer than 2ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2α, then no structure theorem is expected (cf. [Sh1]).O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set.In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension.In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2ω countable models.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (04) ◽  
pp. 922-936
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

scholarly journals Generalized prime models

1971 ◽  
Vol 36 (4) ◽  
pp. 593-606 ◽  
Author(s):  
Robert Fittler
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
General Conditions

A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

Asymptotic behavior of Hill's estimate and applications

1986 ◽  
Vol 23 (4) ◽  
pp. 922-936 ◽  
Author(s):  
Gane Samb Lo
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Finite Number ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Complete Theory ◽  
General Distribution ◽  
Stable Law ◽  
Biological Phenomena ◽  
Pareto’S Law

The problem of estimating the exponent of a stable law is receiving an increasing amount of attention because Pareto's law (or Zipf's law) describes many biological phenomena very well (see e.g. Hill (1974)). This problem was first solved by Hill (1975), who proposed an estimate, and the convergence of that estimate to some positive and finite number was shown to be a characteristic of distribution functions belonging to the Fréchet domain of attraction (Mason (1982)). As a contribution to a complete theory of inference for the upper tail of a general distribution function, we give the asymptotic behavior (weak and strong) of Hill's estimate when the associated distribution function belongs to the Gumbel domain of attraction. Examples, applications and simulations are given.

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