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  • Print publication year: 2017
  • Online publication date: March 2017

Author's Preface

Summary

This book is intended to give a fairly comprehensive account of the theory of constructible sets at an advanced level. The intended reader is a graduate mathematician with some knowledge of mathematical logic. In particular, we assume familiarity with the notions of formal languages, axiomatic theories in formal languages, logical deductions in such theories, and the interpretation of languages in structures. Practically any introductory text on mathematical logic will supply the necessary material. We also assume some familiarity with Zermelo-Fraenkel set theory up to the development or ordinal and cardinal numbers. Any number of texts would suffice here, for instance Devlin (1979) or Levy (1979).

The book is not intended to provide a complete coverage of the many and diverse applications of the methods of constructibility theory, rather the theory itself. Such applications as are given are there to motivate and to exemplify the theory.

The book is divided into two parts. Part A (“Elementary Theory”) deals with the classical definition of the hierarchy of constructible sets. With some pruning, this part could be used as the basis of a graduate course on constructibility theory. Part B (“Advanced Theory”) deals with the -hierarchy and the Jensen “fine-structure theory”.

Chapter I is basic to the entire book. The first seven or eight sections of this chapter should be familiar to the reader, and they are included primarily for completeness, and to fix the notation for the rest of the book. Sections 9 through 11 may well be new to the reader, and are fundamental to the entire development. Thus a typical lecture course based on the book would essentially commence with section 9 of Chapter I. After Chapter II, where the basic development of constructibility theory is given, the remaining chapters of Part A are largely independent, though it would be most unnatural to cover Chapter IV without first looking at Chapter III. Likewise, in Part B, after the initial chapter (Chapter VI) there is a large degree of independence between the chapters. (Indeed, given suitable introduction by an instructor, Chapter IX could be read directly after Chapter IV.)