Introduction to Set Theory, Third Edition, Revised and Expanded

by Hrbacek; Karel

• ISBN13: 9780824779153
• ISBN10: 0824779150
• Edition: 3rd
• Format: Hardcover
• Copyright: 1999-06-22
• Publisher: CRC Press

Summary

Thoroughly revised, updated, expanded, and reorganized to serve as a primary text for mathematics courses, Introduction to Set Theory, Third Edition covers the basics: relations, functions, orderings, finite, countable, and uncountable sets, and cardinal and ordinal numbers. It also provides five additional self-contained chapters, consolidates the material on real numbers into a single updated chapter affording flexibility in course design, supplies end-of-section problems, with hints, of varying degrees of difficulty, includes new material on normal forms and Goodstein sequences, and adds important recent ideas including filters, ultrafilters, closed unbounded and stationary sets, and partitions.

Table of Contents

Preface to the Third Edition	iii
Preface to the Second Edition	v
Sets	1	(16)
Introduction to Sets	1	(2)
Properties	3	(4)
The Axioms	7	(5)
Elementary Operations on Sets	12	(5)
Relations, Functions, and Orderings	17	(22)
Ordered Pairs	17	(1)
Relations	18	(5)
Functions	23	(6)
Equivalences and Partitions	29	(4)
Orderings	33	(6)
Natural Numbers	39	(26)
Introduction to Natural Numbers	39	(3)
Properties of Natural Numbers	42	(4)
The Recursion Theorem	46	(6)
Arithmetic of Natural Numbers	52	(3)
Operations and Structures	55	(10)
Finite, Countable, and Uncountable Sets	65	(28)
Cardinality of Sets	65	(4)
Finite Sets	69	(5)
Countable Sets	74	(5)
Linear Orderings	79	(7)
Complete Linear Orderings	86	(4)
Uncountable Sets	90	(3)
Cardinal Numbers	93	(10)
Cardinal Arithmetic	93	(5)
The Cardinality of the Continuum	98	(5)
Ordinal Numbers	103	(26)
Well-Ordered Sets	103	(4)
Ordinal Numbers	107	(4)
The Axiom of Replacement	111	(3)
Transfinite Induction and Recursion	114	(5)
Ordinal Arithmetic	119	(5)
The Normal Form	124	(5)
Alephs	129	(8)
Initial Ordinals	129	(4)
Addition and Multiplication of Alephs	133	(4)
The Axiom of Choice	137	(18)
The Axiom of Choice and its Equivalents	137	(7)
The Use of the Axiom of Choice in Mathematics	144	(11)
Arithmetic of Cardinal Numbers	155	(16)
Infinite Sums and Products of Cardinal Numbers	155	(5)
Regular and Singular Cardinals	160	(4)
Exponentiation of Cardinals	164	(7)
Sets of Real Numbers	171	(30)
Integers and Rational Numbers	171	(4)
Real Numbers	175	(4)
Topology of the Real Line	179	(9)
Sets of Real Numbers	188	(6)
Borel Sets	194	(7)
Filters and Ultrafilters	201	(16)
Filters and Ideals	201	(4)
Ultrafilters	205	(3)
Closed Unbounded and Stationary Sets	208	(4)
Silver's Theorem	212	(5)
Combinatorial Set Theory	217	(24)
Ramsey's Theorems	217	(4)
Partition Calculus for Uncountable Cardinals	221	(4)
Trees	225	(5)
Suslin's Problem	230	(3)
Combinatorial Principles	233	(8)
Large Cardinals	241	(10)
The Measure Problem	241	(5)
Large Cardinals	246	(5)
The Axiom of Foundation	251	(16)
Well-Founded Relations	251	(5)
Well-Founded Sets	256	(4)
Non-Well-Founded Sets	260	(7)
The Axiomatic Set Theory	267	(18)
The Zermelo-Fraenkel Set Theory With Choice	267	(3)
Consistency and Independence	270	(7)
The Universe of Set Theory	277	(8)
Bibliography	285	(1)
Index	286