Gaifman operations, minimal models and the number of countable models

Abstract

We deal with questions and problems in first order countable model theory. Chapter 1 examines countable first order Gaifman operations, which are theories whose models are determined, up to isomorphism, by their relativised reducta. We first prove some reduction and preservation results. Then we prove that the class of relativised reducts of a Gaifman operation is generalised elementary. Finally, we examine the degree of 1-cardinality of such theories. Chapter 2 is basically concerned with trying to get lots of pairwise elementarily equivalent countable models, or to begin with, at least four models. We first show that a minimal prime model is "fairly" algebraic. Then, under various conditions on the algebraicity of the countable models of a theory, we prove results concerning the number of its countable models. The main result is that a countable complete theory whlch has a model with an infinite definable subset all of whose elements are aliebraic or degree at most two, has at least four countable models, up to isomorphism. Chapters 1 and 2 are fermally independent and self-contained. However there are certain common themes. The notion of a minimal model is important in both chapters. More generally, both chapters are concerned with a question at the centre of model theory - the number of models of a theory. In Chapter 1, it is the number of models over a predicate, in particular the case where the number is one. In Chapter 2 it is the number of countable models.