内容简介

集合论是数学的一个基本分支，在数学中占据着独特的地位，其基本概念已渗透到数学的所有领域。本书从集合论中最基本的概念开始，循序渐进，深入浅出。主要内容有：公理及运算、关系与函数、自然数、实数的构造、基数与选择公理、秩序与序数、序数与序型等。本书附有大约300道习题。本书可作为数学、计算机及其他相关专业本科生教材。

作品目录

Chapter 1　INTRODUCTION　1

Baby Set Theory　1

Sets-An Informal View 7

Classes　10

Axiomatic method　10

Notation　13

Historical Notes　14

Chatper 2　AXIOMS AND OPERATIONS　17

Axioms　17

Arbitrary Unions and Intersections　23

Algebra of Sets　27

Epilogue　33

Review Exercises　33

Chapter 3　RELATIONS AND FUNCTIONS　35

Ordered Pairs　35

Relations　39

n-Ary Relations　41

Functions　42

Infinite Cartesian Products　54

Equivalence Relations　55

Ordering Relations　62

Review Exercises　64

Chapter 4　NATURAL NUMBERS　67

Inductive Sets　67

Peano's Postulates　70

Recursion on　73

Arithmetic　79

Ordering on　83

Review Exercises　88

Chapter 5　CONSTRUCTION OF THE REAL NUMBERS　90

Integers　90

Rational Numbers　101

Real Numbers　111

Summaries　121

Two　123

Chapter 6　CARDINAL NUMBERS AND THE AXIOM OF CHIOCE　128

Equinumerosity　128

Finite Sets　133

Cardinal Arithmetic　138

Ordering Cardinal Numbers　145

Axiom of Choice　151

Countable Sets　159

Arithmetic of Infinite Cardinals　162

Continuum Hypothesis　165

Chapter 7　ORDERINGS AND ORDINALS　167

Partial Orderings　167

Well Orderings　172

Replacement Axioms　179

Epsilon-Images　182

Isomorphisms　184

Ordinal Numbers　187

Debts Paid　195

Rank　200

Chapter 8　ORDINALS AND ORDER TYPES　209

Transfinite Recursion Again　209

Alephs　212

Ordinal Operations　215

Isomorphism Types　220

Arithmetic of Order Types　222

Ordinal Arithmetic　227

Chapter 9　SPECIAL TOPICS　241

Well-Founded Relations　241

Natural Models　249

Cofinality　257

Appendix　NOTATION, LOGIC, AND PROOFS　263

Selected References for Further Study　269

List of Axioms　271

Index　273