Set Theory

This is a classic introduction to set theory, from the basics through to the modern tools of combinatorial set theory.

Andras Hajnal (Author), Peter Hamburger (Author), Attila Mate (Translated by)

9780521596671, Cambridge University Press

Paperback, published 11 November 1999

328 pages
22.9 x 15.2 x 1.9 cm, 0.48 kg

'Give this book fifteen minutes attention - you will find that you must buy it! European Maths Society Journal

This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

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.

Subject Areas: Set theory [PBCH]