The Joy of Sets

1 Naive Set Theory. - 1. 1 What is a Set?. - 1. 2 Operations on Sets. - 1. 3 Notation for Sets. - 1. 4 Sets of Sets. - 1. 5 Relations. - 1. 6 Functions. - 1. 7 Well-Or der ings and Ordinals. - 1. 8 Problems. - 2 The ZermeloFraenkel Axioms. - 2. 1 The Language of Set Theory. - 2. 2 The Cumulative Hierarchy of Sets. - 2. 3 The ZermeloFraenkel Axioms. - 2. 4 Classes. - 2. 5 Set Theory as an Axiomatic Theory. - 2. 6 The Recursion Principle. - 2. 7 The Axiom of Choice. - 2. 8 Problems. - 3 Ordinal and Cardinal Numbers. - 3. 1 Ordinal Numbers. - 3. 2 Addition of Ordinals. - 3. 3 Multiplication of Ordinals. - 3. 4 Sequences of Ordinals. - 3. 5 Ordinal Exponentiation. - 3. 6 Cardinality Cardinal Numbers. - 3. 7 Arithmetic of Cardinal Numbers. - 3. 8 Regular and Singular Cardinals. - 3. 9 Cardinal Exponentiation. - 3. 10 Inaccessible Cardinals. - 3. 11 Problems. - 4 Topics in Pure Set Theory. - 4. 1 The Borel Hierarchy. - 4. 2 Closed Unbounded Sets. - 4. 3 Stationary Sets and Regressive Functions. - 4. 4 Trees. - 4. 5 Extensions of Lebesgue Measure. - 4. 6 A Result About the GCH. - 5 The Axiom of Constructibility. - 5. 1 Constructible Sets. - 5. 2 The Constructible Hierarchy. - 5. 3 The Axiom of Constructibility. - 5. 4 The Consistency of V = L. - 5. 5 Use of the Axiom of Constructibility. - 6 Independence Proofs in Set Theory. - 6. 1 Some Undecidable Statements. - 6. 2 The Idea of a Boolean-Valued Universe. - 6. 3 The Boolean-Valued Universe. - 6. 4 VB and V. - 6. 5 Boolean-Valued Sets and Independence Proofs. - 6. 6 The Nonprovability of the CH. - 7 Non-Well-Founded Set Theory. - 7. 1 Set-Membership Diagrams. - 7. 2 The Anti-Foundation Axiom. - 7. 3 The Solution Lemma. - 7. 4 Inductive Definitions Under AFA. - 7. 5 Graphs and Systems. - 7. 6 Proof of the Solution Lemma. - 7. 7 Co-Inductive Definitions. - 7. 8 A Model of ZF- +AFA. - Glossary of Symbols. Language: English