Basic Set Theory

By: Azriel Levy

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Paperback | 13 August 2002

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416 Pages

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23 x 15.5 x 2.5

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Although this book deals with basic set theory (in general, it stops short of areas where model-theoretic methods are used) on a rather advanced level, it does it at an unhurried pace. This enables the author to pay close attention to interesting and important aspects of the topic that might otherwise be skipped over. Written for upper-level undergraduate and graduate students, the book is divided into two parts. The first covers pure set theory, including the basic notions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of its consequences. The second part deals with applications and advanced topics, among them a review of point set topology, the real spaces, Boolean algebras, and infinite combinatorics and large cardinals. A helpful appendix deals with eliminability and conservation theorems, while numerous exercises supply additional information on the subject matter and help students test their grasp of the material. Slightly revised republication of the edition published by Springer-Verlag, Berlin and New York, 1979. New preface. Introduction. Appendix. Bibliography. Index of Notation. Index. 20 figures.

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Pure Set Theory	p. 1
The Basic Notions	p. 3
The Basic Language of Set Theory	p. 3
The Axioms of Extensionality and Comprehension	p. 5
Classes, Why and How	p. 7
Classes, the Formal Introduction	p. 12
The Axioms of Set Theory	p. 18
Relations and Functions	p. 24
Order and Well-Foundedness	p. 32
Order	p. 32
Well-Order	p. 37
Ordinals	p. 49
Natural Numbers and Finite Sequences	p. 56
Well-Founded Relations	p. 63
Well-Founded Sets	p. 68
The Axiom of Foundation	p. 72
Cardinal Numbers	p. 76
Finite Sets	p. 76
The Partial Order of the Cardinals	p. 83
The Finite Arithmetic of the Cardinals	p. 92
The Infinite Arithmetic of the Well Ordered Cardinals	p. 100
The Ordinals	p. 112
Ordinal Addition and Multiplication	p. 112
Ordinal Exponentiation	p. 122
Cofinality and Regular Ordinals	p. 131
Closed Unbounded Classes and Stationary Classes	p. 141
The Axiom of Choice and Some of its Consequences	p. 158
The Axiom of Choice and Equivalent Statements	p. 158
Some Weaker Versions of the Axiom of Choice	p. 166
Definable Sets	p. 171
Set Theory with Global Choice	p. 176
Cardinal Exponentiation	p. 180
Applications and Advanced Topics	p. 197
A Review of Point Set Topology	p. 199
Basic Concepts	p. 199
Useful Properties and Operations	p. 205
Category, Baire and Borel Sets	p. 212
The Real Spaces	p. 216
The Real Numbers	p. 217
The Separable Complete Metric Spaces	p. 222
The Close Relationship Between the Real Numbers, the Cantor Space and the Baire Space	p. 234
Boolean Algebras	p. 244
The Basic Theory	p. 245
Prime Ideals and Representation	p. 251
Complete Boolean Algebras	p. 263
Martin's Axiom	p. 276
Infinite Combinatorics and Large Cardinals	p. 289
The Axiom of Constructibility	p. 289
Trees	p. 292
Partition Properties	p. 320
Measurable Cardinals	p. 342
The Eliminability and Conservation Theorems	p. 357
Bibliography	p. 367
Additional Bibliography	p. 376
Index of Notation	p. 377
Index	p. 383
Corrections and Additions	p. 393
Table of Contents provided by Syndetics. All Rights Reserved.

Basic Set Theory

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Paperback

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