Elements of Set Theory (Hardcover)

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 ω


Arithmetic


Ordering on ω


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