Set Theory

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More About This Textbook

Overview

This is a classic introduction to set theory, suitable for students with no previous knowledge of the subject. Providing complete, up-to-date coverage, the book is based in large part on courses given over many years by Professor Hajnal. The first part introduces all the standard notions of the subject; the second part concentrates on combinatorial set theory. Exercises are included throughout and a new section of hints has been added to assist the reader.

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Editorial Reviews

From the Publisher
"The great number of exercises and problems (some of them with hints for solution) makes this book a valuable text for mathematics majors and also for an undergraduate course in mathematical logic." Monatshefte fur Mathematik
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Product Details

Table of Contents

Preface
Pt. I Introduction to set theory 1
1 Notation, conventions 5
2 Definition of equivalence. The concept of cardinality. The Axiom of Choice 11
3 Countable cardinal, continuum cardinal 15
4 Comparison of cardinals 21
5 Operations with sets and cardinals 28
6 Examples 36
7 Ordered sets. Order types. Ordinals 41
8 Properties of wellordered sets. Good sets. The ordinal operation 54
9 Transfinite induction and recursion. Some consequences of the Axiom of Choice, the Wellordering Theorem 66
10 Definition of the cardinality operation. Properties of cardinalities. The cofinality operation 77
11 Properties of the power operation 93
App An axiomatic development of set theory 107
A1 The Zermelo-Fraenkel axiom system of set theory 111
A2 Definition of concepts; extension of the language 114
A3 A sketch of the development. Metatheorems 117
A4 A sketch of the development. Definitions of simple operations and properties (continued) 122
A5 A sketch of the development. Basic theorems, the introduction of [omega] and R (continued) 124
A6 The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2-7 128
A7 The role of the Axiom of Regularity 130
A8 Proofs of relative consistency. The method of interpretation 133
A9 Proofs of relative consistency. The method of models 138
Pt. II Topics in combinatorial set theory 143
12 Stationary sets 145
13 [Delta]-systems 159
14 Ramsey's Theorem and its generalizations. Partition calculus 164
15 Inaccessible cardinals. Mahlo cardinals 184
16 Measurable cardinals 190
17 Real-valued measurable cardinals, saturated ideals 203
18 Weakly compact and Ramsey cardinals 216
19 Set mappings 228
20 The square-bracket symbol. Strengthenings of the Ramsey counterexamples 234
21 Properties of the power operation. Results on the singular cardinal problem 243
22 Powers of singular cardinals. Shelah's Theorem 259
Bibliography 295
List of symbols 297
Name index 301
Subject index 303
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