Set Theory

Set Theory

András Hajnal

Rutgers University

Peter Hamburger

Indiana University - Purdue University Fort Wayne

Translated from Hungarian to English by Attila Máté

Summary:

This is a classical 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, Δ-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.

Contents:

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 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 cardinals. The cofinality operation.
11. Properties of the power operation.

• Hints for solving * problems in Part I.

Appendix. An axiomatic development of set theory:

• The Zermelo-Frankel axiom system of set theory.
• Definition of concepts; extension of the language.
• A sketch of the development. Metatheorems.
• Definitions of simple operations and properties (continued).
• Basic theorems, the introduction of ω and R (continued).
• The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2-7.
• The role of the Axiom of Regularity.
• Proofs of relative consistency. The method of interpretation.
• The method of models.

Part II. Topics in combinatorial set theory:

1. Stationary sets.
2. D-systems.
3. Ramsey's theorem and its generalizations. Partition calculus.
4. Inaccessible cardinals. Mahlo cardinals.
5. Measurable cardinals.
6. Real-valued measurable cardinals, saturated ideals.
7. Weakly compact cardinals and Ramsey cardinals.
8. Set mappings.
9. The square-bracket symbol. Strengthenings of the Ramsey counterexamples.
10. Properties of the power operation. Results on the singular cardinal problem.
11. Powers of singular cardinals. Shelah's theorem.

• Hints for solving problems of Part II.

Errata:

• None currently known.

Published by Cambridge University Press in Hardback (0521593441) and Paperback (052159667X). London Mathematical Society Student Texts, 48, 1999, 349 pages.