內容介紹
集合論是數學的一個基本分支，在數學中占據著獨特的地位，其基本概念已滲透到數學的所有領域。本書從集合論中最基本的概念開始，循序漸進，深入淺出。主要內容有：公理及運算、關係與函式、自然數、實數的構造、基數與選擇公理、秩序與序數、序數與序型等。本書附有大約300道習題。本書可作為數學、計算機及其他相關專業本科生教材。作品目錄
Chapter 1 INTRODUCTION 1Baby 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
