Classification Of Countable Models Of Complete Theories Art 1

Classification of countable models of complete theories. Рart 1

The book is the first part of the monograph “Classification of countable models of complete theories” consisting of two parts. In the monograph, a classification of countable models of complete theories with respect to two basic characteristics (Rudin–Keisler preorders and distribution functions for numbers of limit models) is presented and applied to the most important classes of countable theories such as the class of Ehrenfeucht theories (i. e., complete first-order theories with finitely many but more than one pairwise non-isomorphic countable models), the class of small theories (i. e., complete first-order theories with countably many types), and the class of countable first-order theories with continuum many types. For realizations of basic characteristics of countable complete theories, syntactic generic constructions, generalizing the Jonsson–Fraïssé construction and the Hrushovski construction, are presented. Using these constructions a solution of the Goncharov–Millar problem (on the existence of Ehrenfeucht theories with countable models which are not almost homogeneous) is described. Modifying the Hrushovski–Herwig generic construction, a solution of the Lachlan problem on the existence of stable Ehrenfeucht theories is shown. In the first part, a characterization of Ehrenfeuchtness, properties of Ehrenfeucht theories, generic constructions, and algebras for distributions of binary semi-isolating formulas of a complete theory are considered.The book is intended for specialists interested in Mathematical Logic.

Classification Theory

Model Theory and Modules

In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction to the subject and introduces the requisite model theory and module theory as it is needed. Dr Prest develops the basic ideas concerning what can be said about modules using the information which may be expressed in a first-order language. Later chapters discuss stability-theoretic aspects of modules, and structure and classification theorems over various types of rings and for certain classes of modules. Both algebraists and logicians will enjoy this account of an area in which algebra and model theory interact in a significant way. The book includes numerous examples and exercises and consequently will make an ideal introduction for graduate students coming to this subject for the first time.