Elements of Set Theory
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Contents
Preface
List of Symbols
Chapter 1 Introduction
Baby Set Theory
Sets—An Informal View
Classes
Axiomatic Method
Notation
Historical Notes
Chapter 2 Axioms and Operations
Axioms
Arbitrary Unions and Intersections
Algebra of Sets
Epilogue
Review Exercises
Chapter 3 Relations and Functions
Ordered Pairs
Relations
n-Ary Relations
Functions
Infinite Cartesian Products
Equivalence Relations
Ordering Relations
Review Exercises
Chapter 4 Natural Numbers
Inductive Sets
Peano's Postulates
Recursion on ?
Arithmetic
Ordering on ?
Review Exercises
Chapter 5 Construction of the Real Numbers
Integers
Rational Numbers
Real Numbers
Summaries
Two
Chapter 6 Cardinal Numbers and the Axiom of Choice
Equinumerosity
Finite Sets
Cardinal Arithmetic
Ordering Cardinal Numbers
Axiom of Choice
Countable Sets
Arithmetic of Infinite Cardinals
Continuum Hypothesis
Chapter 7 Orderings and Ordinals
Partial Orderings
Well Orderings
Replacement Axioms
Epsilon-Images
Isomorphisms
Ordinal Numbers
Debts Paid
Rank
Chapter 8 Ordinals and Order Types
Transfinite Recursion Again
Alephs
Ordinal Operations
Isomorphism Types
Arithmetic of Order Types
Ordinal Arithmetic
Chapter 9 Special Topics
Well-Founded Relations
Natural Models
Cofinality
Appendix Notation, Logic, and Proofs
Selected References for Further Study
List of Axioms
Index