集合论基础PDF电子书下载

Chapter 1 INTRODUCTION 1

Baby Set Theory 1

Sets—An Informal View 7

Classes 10

Axiomatic Method 10

Notation 13

Historical Notes 14

Chapter 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 66

Inductive Sets 67

Peano’s Postulates 70

Recursion on ω 73

Arithmetic 79

Ordering on ω 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 CHOICE 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