# 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 $\text {1993}$, Springer

ISBN 0-387-94094-4

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
3.3 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

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