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

## Keith Devlin: *The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd Edition)*

Published $1993$, **Springer**

- ISBN 0-387-94094-4.

### Subject Matter

Second edition, considerably rewritten and expanded, of the $1979$ work *Fundamentals of Contemporary Set Theory*.

### 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