Finite extensions and the number of countable models

A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive.It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy.Recursive model theory considers model-theoretic objects and constructions from the point of view of recursion-theoretic complexity.For instance, must isomorphism types of countable models that are structurally simple have models of low recursion-theoretic complexity?This paper considers one such question involving countable homogeneous models.If is a countable homogeneous model of a complete theory T, then is necessarily decidable in the degree of Tl Since there are complete theories with 2m complete types, the answer is obviously "no".On the other hand, by results in [1], if T has only finitely many countable models and all of the complete types of the theory are recursive, then the answer is "yes".This paper considers decidable theories T that have at most countably many countable models and only recursive complete types.Call such a theory tame.The notational conventions in the paper are standard.Fact.Every countable homogeneous model of a tame theory T is hyperarithmetic.Proof.Let {p^i < co} be an effective enumeration of all the partial recursive functions : N - TV, and let {B(\i < co} be an effective enumeration of all formulas in L(T).Since all of the types of T are recursive, we can fix a hyperarithmetic set 7 C TV such that (i) Vi I[{0j\i(j) = 17 < <y} is a complete type of T]; and(ii) vr complete type of T miel Vj[8j G T iff .(J) = I].For a countable model il I-T, say that W CI is a witness set for il iff there is some enumeration {jS.\i < co} of |il|<cu such that (1) Ve W [n is total];

Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in 0 ′ 0’ . The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.

Let T be a complete theory over a relational language which has an axiomatization by 3V-sentences.The properties of models of T are studied.It is shown that quantifier-free formulas are stable.This limited stability is used to show that in 3V-saturated models the elementary types of tuples are determined by their 3-types and algebraicity is determined by existential formulas.As an application, under the additional assumption that no quantifier-free formula has the FCP, the models L of T are completely characterized in terms of certain O-definable equivalence relations on cartesian powers of M. This characterization yields a result similar to that of Schmerl for the case in whichThis is the first of two papers about complete first-order theories which have axiomatizations consisting entirely of 3V-sentences over a relational language.We call theories with such axiomatizations 3V-theories.Schmerl has called them coinductive.This paper is a sequel to [7] which characterized complete theories over a relational language which have axiomatizations consisting of 3-sentences and V-sentences.(From now on the restriction to relational languages which obtains throughout will be tacit.)Such theories are called primitive.Following Hodkinson and Macpherson [5] a structure L is said to be finitely partitioned if there exists a partition M = FUX 1 U ... UXn of the universe of L invariant underand the pointwise stabilizer of M\X i in Aut(L) induces Sym(X i ).In [7] it was shown that a complete theory is primitive if and only if all its models are finitely partitioned.Some acquaintance with [7] would be very helpful to readers of the present work.Schmerl [10] studied No-categorical 3V-theories over a finite relational language and proved that the unique countable model of such a theory is cellular in the following sense.L is cellular if there exist a finite subset X of M and equivalence relations E, F on M\X such that(1) (M\X)jE is finite.(2) If C is an E-class and D an F -class, then IC n DI = 1.

Stable theories, pseudoplanes and the number of countable models

Let $T$ be a complete, superstable theory with fewer than $2^{\aleph_{0}}$ countable models. Assuming that generic types of infinite, simple groups definable in $T^{eq}$ are sufficiently non-isolated we prove that $\omega^{\omega}$ is the strict upper bound for the Lascar rank of $T$.

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the

Let= {0,1, +,·,&lt;} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order-theory containingIΔ0+ exp such that every complete extensionTof it has a countable modelKsatisfying(i)Khas no proper elementary substructures, and(ii) wheneverL≻Kis a countable elementary extension there isandsuch that.Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

The paper is aimed at studying the countable spectrum of small linearly ordered theories. The objectives of the research are to study the structural properties of countable linearly ordered theories, as well as to promote the solution to the well-known open problem of model theory, Vaught's conjecture, which assumes that the number of countable models of a countable complete first-order theory cannot be equal to ℵ1. An important step in solving Vaught's conjecture is the search for conditions under which the theory has the maximal number of countable pairwise non-isomorphic models. By limiting ourselves to linearly ordered theories we do not get special advantages from the viewpoint of studying their countable spectrum. Therefore, in the article, a restriction on 1-types and 1-formulas of the theory is introduced. It is proved that a small countable linearly ordered theory that satisfies the restriction and has an infinite discrete chain has the maximal number of countable non-isomorphic models. To build models, the authors use the method of constructing countable models over countable sets, based on the Tarski-Vaught criterion. It is shown that it is possible to carry out the construction in such a way that the types of unnecessary elements in the resulting model are omitted, what guarantees non-isomorphism of the models and their maximal number.

We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$ . We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

We give effectiveness conditions on a strongly minimal theory T guaranteeing that all countable models have computable copies. In particular, we show that if T is strongly minimal and for all n\geq 1 , T\cap\exists_{n+2} is \Delta^0_n , uniformly in n , then every countable model has a computable copy. A longstanding question of computable model theory asked whether for a strongly minimal theory with one computable model, every countable model has an arithmetical copy. Relativizing our main result, we get the fact that if there is one computable model, then every countable model has a \Delta^0_4 copy.

Hypergraphs of prime models over realizations of types in small theories are defined. On the basis of graph structures of models of small theories, hierarchies of sets in these hypergraphs, revealing structural connections in countable models of small theories, are established. The key role of graph-theoretic objects in constructions of Ehrenfeucht theories is proved. Using hypergraph constructions, a classification of complete first-order theories with finite Rudin–Keisler preorders is generalized to the class of all small theories.

Model theory for universal classes with the amalgamation property: A study in the foundations of model theory and algebra

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.

The model companions of the theories of n-dimensional partial orderings and n-dimensional distributive lattices are found for each finite n. Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n-dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n-dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory.In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T* theories in a common language, T* is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T* are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T* is model complete, which means that if and are both models of T* and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T* admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ0-categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.