章节目录
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
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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