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.
Author(s): Raymond M. Smullyan, Melvin Fitting
Series: Oxford Logic Guides 34
Publisher: Oxford University Press
Year: 1996
Language: English
Pages: 302
Preface ......Page 5
Contents ......Page 9
Part 1. Axiomatic set theory ......Page 15
01 General Background ......Page 17
02 Some Basics of Class-Set Theory ......Page 28
03 The Natural Numbers ......Page 41
04 Superinduction, Well Ordering and Choice ......Page 57
05 Ordinal Numbers ......Page 78
06 Order Isomorphism and Transfinite Recursion ......Page 84
07 Rank ......Page 95
08 Foundation, $in$-Induction, and Rank ......Page 102
09 Cardinals ......Page 110
Part 2. Consistency of the continuum hypothesis ......Page 127
10 Mostowski-Shepherdson Mappings ......Page 129
11 Reflection Principles ......Page 142
12 Constructible Sets ......Page 155
13 L is a Well Founded First-Order Universe ......Page 167
14 Constructibility is Absolute Over L ......Page 176
15 Constructibility and the Continuum Hypothesis ......Page 190
Part 3. Forcing and independence results ......Page 201
16 Forcing, The Very Idea ......Page 203
17 The Construction of S4 Models for ZF ......Page 217
18 The Axiom of Constructibility is Independent ......Page 240
19 Independence of the Continuum Hypothesis ......Page 249
20 Independence of the Axiom of Choice ......Page 257
21 Constructing Classical Models ......Page 273
22 Forcing Background ......Page 285
References ......Page 291
Subject Index ......Page 295
Notation Index ......Page 301
