Classification Theory. Second Edition with a New Introduction
Classification Theory. Second Edition with a New Introduction by Saharon Shelah, published by College Publications on January 13, 2023, is a comprehensive exploration of model theory aimed at readers familiar with first-order logic and related concepts. This edition spans 748 pages and is presented in English, offering an in-depth examination of the dividing lines among complete first-order theories, particularly those that are countable.
Readers will find a detailed discussion on the main test problem concerning the number of models of a theory as a function of infinite cardinality. The book delves into significant properties such as stability and super stability, and introduces concepts like dimension and weight, especially in the context of super stable theories. This work is intended for those who have a foundational understanding of logic, set theory, and model theory, making it a valuable resource for advanced study in mathematics, logic, and philosophy.
Official synopsis Publisher
This book is for readers who have learned about first order logic; Gödel’s completeness theorem; the Löwenheim-Skolem theorem; the Tarski-Vaught criterion for being elementary sub-model; and who know naive set theory. A graduate course in model theory will be helpful. The thesis of the book is that we can find worthwhile dividing lines among complete first order theories T; mainly countable. That is, properties dividing them in some sense between understandable and complicated ones.
The main test problem is the number of models of T of the infinite cardinal l as a function of l. This culminates in the so-called main gap theorem saying the number is either maximal or quite small in suitable sense. Toward this, other properties are introduced and investigated, such as being stable or being super stable, where can we define dimension and weight, particularly for super stable theories.
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