Set Theory

Description

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