Set Theory and the Continuum Problem

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Set Theory and the Continuum Problem is a novel introduction to set theory, including axiomatic development, consistency, and independence results. It is self-contained and covers all the set theory that a mathematician should know. Part I introduces set theory, including basic axioms, development of the natural number system, Zorn's Lemma and other maximal principles. Part II proves the consistency of the continuum hypothesis and the axiom of choice, with material on collapsing mappings, model-theoretic results, and constructible sets. Part III presents a version of Cohen's proofs of the independence of the continuum hypothesis and the axiom of choice. It also presents, for the first time in a textbook, the double induction and superinduction principles, and Cowen's theorem. The book will interest students and researchers in logic and set theory.
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Editorial Reviews

From the Publisher

"Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice

"This text is a general introduction to NBG (von Neumann-Bernays-Gödel class-set theory), and to Gödel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews

"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory and its relationship to logic." --Computing Reviews

"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books

"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuum hypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic

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Product Details

  • ISBN-13: 9780198523956
  • Publisher: Oxford University Press, USA
  • Publication date: 11/14/1996
  • Series: Oxford Logic Guides Series , #34
  • Pages: 304
  • Product dimensions: 6.31 (w) x 9.43 (h) x 0.85 (d)

Meet the Author

Indiana University

City University of New York

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Table of Contents

1 General background 3
2 Some basics of class-set theory 14
3 The natural numbers 27
4 Superinduction, well ordering and choice 43
5 Ordinal numbers 64
6 Order isomorphism and transfinite recursion 70
7 Rank 81
8 Foundation, [epsilon]-induction, and rank 88
9 Cardinals 96
10 Mostowski-Shepherdson mappings 115
11 Reflection principles 128
12 Constructible sets 141
13 L is a well founded first-order universe 153
14 Constructibility is absolute over L 162
15 Constructibility and the continuum hypothesis 176
16 Forcing, the very idea 189
17 The construction of S4 models for ZF 203
18 The axiom of constructibility is independent 226
19 Independence of the continuum hypothesis 235
20 Independence of the axiom of choice 243
21 Constructing classical models 259
22 Forcing background 271
References 277
Subject index 281
Notation index 287
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