Fundamentals of Stability Theory (Perspectives in Mathematical Logic)

This new volume of the O-series is written as an introduction to first order stability theory. It is organized around the the spectrum problem: calculate the number of models a first order theory T has in each uncountable cardinal. To solve this problem a generalization of the notion of algebraic independence "nonforking" was developed. In this text the abstract properties of this relation (in contrast to other books which begin with the technical description). The important notions of orthogonality and regularity are carefully developed: this machinery is then applied to the spectrum problem. Complete proofs of the Vaught conjecture for omega-stable theories are presented here for the first time in book form. Considerable effort has been made by the author to provide much needed examples. In particular, the book contains the first publication of Shelah's infamous example showing the necessity of his methods to solve Vaught's conjecture for omega-stable theories. The connections of abstract stability theory with algebra particularly with the theory of modules are emphasized.