Book:Keith Devlin/The Joy of Sets: Fundamentals of Contemporary Set Theory/Second Edition

Subject Matter

 * Set Theory

Second edition, considerably rewritten and expanded, of the $1979$ work.

Contents

 * Preface


 * 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-Orderings and Ordinals
 * 1.8 Problems


 * 2 The Zermelo-Fraenkel Axioms
 * 2.1 The Language of Set Theory
 * 2.2 The Cumulative Hierarchy of Sets
 * 2.3 The Zermelo-Fraenkel 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 $V^{\mathcal B}$ 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 $\mathrm{ZF}^- +\mathrm{AFA}$


 * Bibliography
 * Glossary of Symbols
 * Index