Notes on Set Theory

Jeffrey Harris, Mike Latimer9780672327490, 067232749X

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. At the same time, it is often viewed as a foundation of mathematics so that in the most prevalent, current mathematical practice “to make a notion precise” simply means “to define it in set theory.” This book tries to do justice to both aspects of the subject: it gives a solid introduction to “pure set theory” through transfinite recursion and the construction of the cumulative hierarchy of sets (including the basic results that have applications to computer science), but it also attempts to explain precisely how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author added solutions to the exercises, and rearranged and reworked the text in several places to improve the presentation.

The book is aimed at advanced undergraduate or beginning graduate mathematics students and at mathematically minded graduate students of computer science and philosophy.

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Table of contents :
Cover……Page 1
Undergraduate Texts in Mathematics……Page 3
Notes on Set Theory……Page 4
PREFACE……Page 7
CONTENTS……Page 11
CHAPTER 1 INTRODUCTION……Page 13
CHAPTER 2 EQUINUMEROSITY……Page 19
CHAPTER 3 PARADOXES AND AXIOMS……Page 31
CHAPTER 4 ARE SETS ALL THERE IS?……Page 44
CHAPTER 5 THE NATURAL NUMBERS……Page 61
CHAPTER 6 FIXED POINTS……Page 81
CHAPTER 7 WELL ORDERED SETS……Page 99
CHAPTER 8 CHOICES……Page 119
CHAPTER 9 CHOICE’S CONSEQUENCES……Page 131
CHAPTER 10 BAIRE SPACE……Page 145
CHAPTER 11 REPLACEMENT AND OTHER AXIOMS……Page 167
CHAPTER 12 ORDINAL NUMBERS……Page 185
APPENDIX A THE REAL NUMBERS……Page 208
APPENDIX B AXIOMS AND UNIVERSES……Page 233
SOLUTIONS TO THE EXERCISES IN CHAPTERS 1 – 12……Page 257
INDEX……Page 278
Undergraduate Texts in Mathematics……Page 284