Set Theory

by Andras Hajnal , Peter Hamburger , Translated by Attila Mate

• ISBN13: 9780521596671
• ISBN10: 052159667X
• Format: Paperback
• Copyright: 1999-11-28
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	vii
Part I. Introduction to set theory	1	(106)
Introduction	3	(2)
Notation, conventions	5	(6)
Definition of equivalence. The concept of cardinality. The Axiom of Choice	11	(4)
Countable cardinal, continuum cardinal	15	(6)
Comparison of cardinals	21	(7)
Operations with sets and cardinals	28	(8)
Examples	36	(5)
Ordered sets. Order types. Ordinals	41	(13)
Properties of wellordered sets. Good sets. The ordinal operation	54	(12)
Transfinite induction and recursion. Some consequences of the Axiom of Choice, the Wellordering Theorem	66	(11)
Definition of the cardinality operation. Properties of cardinalities. The cofinality operation	77	(16)
Properties of the power operation	93	(14)
Hints for solving problems marked with * in Part I	101	(6)
Appendix. An axiomatic development of set theory	107	(36)
Introduction	109	(2)
A1. The Zermelo-Fraenkel axiom system of set theory	111	(3)
A2. Definition of concepts; extension of the language	114	(3)
A3. A sketch of the development. Metatheorems	117	(5)
A4. A sketch of the development. Definitions of simple operations and properties (continued)	122	(2)
A5. A sketch of the development. Basic theorems, the introduction of ωw and R (continued)	124	(4)
A6. The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2--7	128	(2)
A7. The role of the Axiom of Regularity	130	(3)
A8. Proofs of relative consistency. The method of interpretation	133	(5)
A9. Proofs of relative consistency. The method of models	138	(5)
Part II. Topics in combinatorial set theory	143	(152)
Stationary sets	145	(14)
Δ-systems	159	(5)
Ramsey's Theorem and its generalizations. Partition calculus	164	(20)
Inaccessible cardinals. Mahlo cardinals	184	(6)
Measurable cardinals	190	(13)
Real-valued measurable cardinals, saturated ideals	203	(13)
Weakly compact and Ramsey cardinals	216	(12)
Set mappings	228	(6)
The square-bracket symbol. Strengthenings of the Ramsey counterexamples	234	(9)
Properties of the power operation. Results on the singular cardinal problem	243	(16)
Powers of singular cardinals. Shelah's Theorem	259	(36)
Hints for solving problems of Part II	272	(23)
Bibliography	295	(2)
List of symbols	297	(4)
Name index	301	(2)
Subject index	303