Set Theory

Front Cover
Front Matter
Preface
Part I. Introduction to set theory
	Introduction
	1. Notation, conventions
	2. Definition of equivalence. The concept of cardinality. Axiom of Choice
	3. Countable cardinal, continuum cardinal
	4. Comparison of cardinals
	5. Operations with sets and cardinals
	6. Examples
	7. Ordered sets. Order types. Ordinals
	8. Properties of wellordered sets. Good sets. The ordinal operation
	9. Transfinite induction and recursion. Some consequences of the Axiom of Choice, the Wellordering Theorem
	10. Definition of the cardinality operation. Properties of cardinalities. The cofinality operation
	11. Properties of the power operation
	Hints for solving problems marked with * in Part I
Appendix. An axiomatic development of set theory
	Introduction
	AI. The Zermelo-Fraenkel axiom system of set theory
	A2. Definition of concepts; extension of the language
	A3. A sketch of the development. Metatheorems
	A4. A sketch of the development. Definitions of simple operations and properties (continued)
	A5. A sketch of the development. Basic theorems, the introduction of omega and R (continued)
	A6. The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2-7
	A7. The role of the Axiom of Regularity
	A8. Proofs of relative consistency. The method of interpretation
	A9. Proofs of relative consistency. The method of models
Part II. Topics in combinatorial set theory
	12. Stationary sets
	13. Delta-systems
	14. Ramsey\'s Theorem and its generalizations. Partition calculus
	15. Inaccessible cardinals. Mahlo cardinals
	16. Measurable cardinals
	17. Real-valued measurable cardinals, saturated ideals
	18. Weakly compact and Ramsey cardinals
	19. Set mappings
	20. The square-bracket symbol. Strengthenings of the Ramsey counterexamples
	21. Properties of the power operation. Results on the singular cardinal problem
	22. Powers of singular cardinals. Shelah\'s Theorem
	Hints for solving problems of Part II
Bibliography
List of symbols
Name index
Subject index
Back Cover