The Joy of Sets

This new edition has been completely rewritten; it includes a new chapter on non-well-founded set theory, a subject of considerable importance in computer science. Written in an easy-to-follow, intuitive style, the book is intended for upper-level undergraduate or beginning graduate students in mathematics, logic, philosophy, or computer science.

Zusammenfassung
This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "naive" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers.

Inhalt
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.

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