Set Theory

This is a classic introduction to set theory, suitable for students with no previous knowledge of the subject. Providing complete, up-to-date coverage, the book is based in large part on courses given over many years by Professor Hajnal. The first part introduces all the standard notions of the subject; the second part concentrates on combinatorial set theory. Exercises are included throughout and a new section of hints has been added to assist the reader.

Part I. Introduction to Set Theory: 1. Notation, conventions; 2. Definition of equivalence. The concept of cardinality. The 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 well-ordered sets. Good sets. The ordinal operation; 9. Transfinite induction and recursion; 10. Definition of the cardinality operation. Properties of cardinalities. The confinality operation; 11. Properties of the power operation; Appendix. An axiomatic development of set theory; 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 ideas; 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; 22. Powers of singular cardinals. Shelah's theorem.