集合论基础
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。
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作者简介
Herbert B.Enderton 哈佛大学博士，师从著名哲学家Hilary Putnam。曾任教于加州大学伯克利分校。现为加州大学洛杉矶分校数学系兼职教授，该校“逻辑学论坛”主席，曾担任《符号逻辑学会评论》杂志的主编。除本书外，他还著有A Mathematical Introduction to Logic和Linear Algebra。
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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
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读书文摘
Principle of Extensionality If two sets have exactly the same members, then they are equal.
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