Axiomatic Set Theory

AXIOMATIC SET THEORY......Page 1
The University Series in Undergraduate Mathematics......Page 2
Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 6
Preface......Page 8
Table of Contents......Page 12
1.1 Set Theory and the Foundations of Mathematics......Page 14
1.2 Logic and Notation......Page 16
1.3 Axiom Schema of Abstraction and Russell\'s Paradox......Page 18
1.4 More Paradoxes......Page 21
1.5 Preview of Axioms......Page 25
2.1 Preliminaries: Formulas and Definitions......Page 27
2.2 Axioms of Extensionality and Separation......Page 32
2.3 Intersection, Union, and Difference of Sets......Page 37
2.4 Pairing Axiom and Ordered Pairs......Page 43
2.5 Definition by Abstraction......Page 46
2.6 Sum Axiom and Families of Sets......Page 50
2.7 Power Set Axiom......Page 59
2.8 Cartesian Product of Sets......Page 62
2.9 Axiom of Regularity......Page 66
2.10 Summary of Axioms......Page 69
3.1 Operations on Binary Relations......Page 70
3.2 Ordering Relations......Page 81
3.3 Equivalence Relations and Partitions......Page 93
3.4 Functions......Page 99
4.1 Equipollence......Page 104
4.2 Finite Sets......Page 111
4.3 Cardinal Numbers......Page 122
4.4 Finite Cardinals......Page 134
5.1 Definition and General Properties of Ordinals......Page 140
5.2 Finite Ordinals and Recursive Definitions......Page 148
5.3 Denumerable Sets......Page 163
6.1 Introduction......Page 172
6.2 Fractions......Page 174
6.3 Non-negative Rational Numbers......Page 179
6.4 Rational Numbers......Page 183
6.5 Cauchy Sequences of Rational Numbers......Page 187
6.6 Real Numbers......Page 194
6.7 Sets of the Power of the Continuum......Page 202
7.1 Transfinite Induction and Definition by Transfinite Recursion......Page 208
7.2 Elements of Ordinal Arithmetic......Page 218
7.3 Cardinal Numbers Again and Alephs......Page 237
7.4 Well-Ordered Sets......Page 243
7.5 Revised Summary of Axioms......Page 250
8.1 Some Applications of the Axiom of Choice......Page 252
8.2 Equivalents of the Axiom of Choice......Page 256
8.3 Axioms Which Imply the Axiom of Choice......Page 264
References......Page 266
Glossary of Symbols......Page 270
Author Index......Page 272
Subject Index......Page 274
Back Cover......Page 280