Set Theory and the Continuum Problem

Overview

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,...
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Overview

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: 9780486474847
  • Publisher: Dover Publications
  • Publication date: 4/21/2010
  • Series: Dover Books on Mathematics Series
  • Edition description: Revised
  • Pages: 336
  • Sales rank: 1,413,859
  • Product dimensions: 6.44 (w) x 9.06 (h) x 0.67 (d)

Meet the Author

Indiana University

City University of New York

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Table of Contents


Preface to the Revised 2010 Edition
Preface
I Axiomatic Set Theory
1. General Background
2. Some Basics of Class-Set Theory
3. The Natural Numbers
4. Superinduction, Well Ordering and Choice
5. Ordinal Numbers
6. Order Isomorphism and Transfinite Recursion
7. Rank
8. Foundation, Induction and Rank
9. Cardinals
II Consistency of the Continuum Hypothesis
10. Mostowski-Shepherdson Mappings
11. Reflection Principles
12. Constructible Sets
13. L is a Well-Founded First-Order Universe
14. Constructibility is Absolute Over L
15. Constructibility and the Continuum Hypothesis
III Forcing and Independence Results
16. Forcing, the Very Idea
17. The Construction of S 4 Models for ZF
18. The Axiom of Constructibility is Independent
19. Independence in the Continuum Hypothesis
20. Independence of the Axiom of Choice
21. Constructing Classical Models
22. Forcing Backward
Bibliography
Index
List of Notation
  
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