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

Subject Matter

 * Set Theory

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.
 * 2. Addition of ordinals.
 * 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