# Book:Keith J. Devlin/Fundamentals of Contemporary Set Theory

## Keith J. Devlin: Fundamentals of Contemporary Set Theory

Published $\text {1979}$, Springer Verlag

ISBN 978-0-387-90441-2.

### Contents

CHAPTER I. NAIVE SET THEORY
1. What is a set?
2. Operations on sets.
3. Notation for sets.
4. Sets of sets.
5. Relations.
6. Functions.
7. Well-orderings and ordinals.
CHAPTER II. THE ZERMELO-FRAENKEL AXIOMS
1. The language of set theory.
2. The cumulative hierarchy of sets.
3. Zermelo-Fraenkel set theory.
4. Axioms for set theory.
5. Summary of the Zermelo-Fraenkel axioms
6. Classes.
7. Set theory as an axiomatic theory.
8. The recursion principle.
9. The axiom of choice.
CHAPTER III. ORDINAL AND CARDINAL NUMBERS
1. Ordinal numbers.
3. Multiplication of ordinals.
4. Sequences of ordinals.
5. Ordinal exponentiation.
6. Cardinality. Cardinal numbers.
7. Arithmetic of cardinal numbers.
8. Cofinality. Singular and regular cardinals.
9. Cardinal exponentiation.
10. Inaccessible cardinals.
CHAPTER IV. SOME TOPICS IN PURE SET THEORY.
1. The Borel hierarchy.
2. Closed unbounded sets.
3. Stationary sets and regressive functions.
4. Trees.
5. Extensions of Lebesgue measure.
6. A result about the GCH.
CHAPTER V. THE AXIOM OF CONSTRUCTIBILITY.
1. Constructible sets.
2. The constructible hierarchy.
3. The axiom of constructibility.
4. The consistency of constructible set theory.
5. Use of the axiom of constructibility.
CHAPTER VI. INDEPENDENCE PROOFS IN SET THEORY.
1. Some examples of undecidable statements.
2. The idea of a boolean-valued universe.
3. The boolean-valued universe.
4. $V^{\Bbb B}$ and $V$.
5. Boolean-valued sets and independence proofs.
6. The non-provability of CH.
BIBLIOGRAPHY
GLOSSARY OF NOTATION
INDEX