Essential Stability Theory

This book grew out of lectures to graduate students in logic at the University Dame in the academic year 1992-93. The purpose of the course was to of Notre bridge the gap between the model theory in a first year graduate logic course (say, the first two chapters in Chang-Keisler) and research papers in stability theory. While the most basic definitions in model theory are repeated in Chapter 1, realistically, I expect the reader to have completed an introductory course in mathematical logic. My intention in writing this book was not to give a comprehensive treatment of elementary stability theory, but to get the student through the basics as quickly as possible. It was also written (hopefully) so that a well-prepared student can begin the book at a chapter appropriate to his or her needs. Stability theory began in the early '60's with Morley's Categoricity The­ orem. (See Section 3. 1. ) In the period 1965 to 1982 (or so) virtually all atten­ tion focused on Shelah's work on (and eventual proof of) Morley's Conjecture. The spectrum function of a complete first-order theory T is a map I( -, T) such that for any cardinal A, I(A, T) is the number of models of T of cardi­ nality A. In the late '60's Morley conjectured that the spectrum function of a complete countable first-order theory T is nondecreasing on uncountable car­ dinals; i. e. , for all uncountable cardinals A < K, I(A, T) ~ I(K, T). Content: Front Matter....Pages I-XIV The Basics....Pages 1-9 Constructing Models with Special Properties....Pages 11-47 Uncountably Categorical and ?0-stable Theories....Pages 49-124 Fine Structure of Uncountably Categorical Theories....Pages 125-211 Stability....Pages 213-291 Superstable Theories....Pages 293-322 Selected Topics....Pages 323-343 Back Matter....Pages 345-358