Set Theory Book

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