Set Theory

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Paperback | 10 January 2000

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328 Pages

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This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

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'This is a classic introduction to set theory ...'. L'Enseignement Mathematique 'Give this book fifteen minutes attention - you will find that you must buy it! European Maths Society Journal

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Non-Fiction Mathematics Mathematical Foundation Set Theory Mathematical Logic

Preface	p. vii
Introduction to set theory	p. 1
Introduction	p. 3
Notation, conventions	p. 5
Definition of equivalence. The concept of cardinality. The Axiom of Choice	p. 11
Countable cardinal, continuum cardinal	p. 15
Comparison of cardinals	p. 21
Operations with sets and cardinals	p. 28
Examples	p. 36
Ordered sets. Order types. Ordinals	p. 41
Properties of wellordered sets. Good sets. The ordinal operation	p. 54
Transfinite induction and recursion. Some consequences of the Axiom of Choice, the Wellordering Theorem	p. 66
Definition of the cardinality operation. Properties of cardinalities. The cofinality operation	p. 77
Properties of the power operation	p. 93
Hints for solving problems marked with * in Part I	p. 101
An axiomatic development of set theory	p. 107
Introduction	p. 109
The Zermelo-Fraenkel axiom system of set theory	p. 111
Definition of concepts; extension of the language	p. 114
A sketch of the development. Metatheorems	p. 117
A sketch of the development. Definitions of simple operations and properties (continued)	p. 122
A sketch of the development. Basic theorems, the introduction of [omega] and R (continued)	p. 124
The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2-7	p. 128
The role of the Axiom of Regularity	p. 130
Proofs of relative consistency. The method of interpretation	p. 133
Proofs of relative consistency. The method of models	p. 138
Topics in combinatorial set theory	p. 143
Stationary sets	p. 145
[Delta]-systems	p. 159
Ramsey's Theorem and its generalizations. Partition calculus	p. 164
Inaccessible cardinals. Mahlo cardinals	p. 184
Measurable cardinals	p. 190
Real-valued measurable cardinals, saturated ideals	p. 203
Weakly compact and Ramsey cardinals	p. 216
Set mappings	p. 228
The square-bracket symbol. Strengthenings of the Ramsey counterexamples	p. 234
Properties of the power operation. Results on the singular cardinal problem	p. 243
Powers of singular cardinals. Shelah's Theorem	p. 259
Hints for solving problems of Part II	p. 272
Bibliography	p. 295
List of symbols	p. 297
Name index	p. 301
Subject index	p. 303
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Set Theory

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