Infinite discrete chains and the maximal number of countable models

Abstract

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.