Complete Theories with Finitely Many Countable Models. I

Abstract

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin-Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin-Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories.