Thoroughly revised, updated, expanded, and reorganized to serve as a primary text for mathematics courses. DLC: Set theory.
Author(s): Wendy Willard
Series: Pure and Applied Mathematics
Edition: 3
Publisher: CRC Press, Marcel Dekker
Year: 1999
Language: English
Pages: 308
Preface to the Third Edition ......Page 10
Preface to the Second Edition ......Page 12
Contents ......Page 15
1.1 Introduction to Sets ......Page 18
1.2 Properties ......Page 20
1.3 The Axioms ......Page 24
1.4 Elementary Operations on Sets ......Page 29
2.1 Ordered Pairs ......Page 34
2.2 Relations ......Page 35
2.3 Functions ......Page 40
2.4 Equivalences and Partitions ......Page 46
2.5 Orderings ......Page 50
3.1 Introduction to Natural Numbers ......Page 56
3.2 Properties of Natural Numbers ......Page 59
3.3 The Recursion Theorem ......Page 63
3.4 Arithmetic of Natural Numbers ......Page 69
3.5 Operations and Structures ......Page 72
4.1 Cardinality of Sets ......Page 82
4.2 Finite Sets ......Page 86
4.3 Countable Sets ......Page 91
4.4 Linear Orderings ......Page 96
4.5 Complete Linear Orderings ......Page 103
4.6 Uncountable Sets ......Page 107
5.1 Cardinal Arithmetic ......Page 110
5.2 The Cardinality of the Continuum ......Page 115
6.1 Well-Ordered Sets ......Page 120
6.2 Ordinal Numbers ......Page 124
6.3 The Axiom of Replacement 1 ......Page 128
6.4 38nsfinite Induction and Recursion ......Page 131
6.5 Ordinal Arithmetic ......Page 136
6.6 The Normal Form ......Page 141
7.1 Initial Ordinals ......Page 146
7.2 Addition and Multiplication of Alephs ......Page 150
8.1 The Axiom of Choice end its Equivalents ......Page 154
8.2 The Use of the Axiom of Choice in Mathematics ......Page 161
9.1 Infinite Sums and Products of Cardinal Numbers ......Page 172
9.2 Regular and Singular Cardinals ......Page 177
9.3 Exponentiation of Cardinals ......Page 181
10.1 Integers and Rational Numbers ......Page 188
10.2 Real Numbers ......Page 192
10.3 Topology of the Real Lie ......Page 196
10.4 Sets of real Numbers ......Page 205
10.5 Bore1 Sets ......Page 211
11.1 Filters and Ideals ......Page 218
11.2 Ultrafilters ......Page 222
11.3 Closed Unbounded and Stationary Sets ......Page 225
11.4 Silver's Theorem ......Page 229
12.1 Ramsey's Theorem ......Page 234
12.2 Partition Calculus for Uncountable Cardinals ......Page 238
12.3 Trees ......Page 242
12.4 Suslin's Problem ......Page 247
12.5 Combinatorial Principles 2 ......Page 250
13.1 The Measure Problem ......Page 258
13.2 Large Cardinals ......Page 263
14.1 Well-Founded Relations ......Page 268
14.2 Well-Founded Sets ......Page 273
14.3 Non-Well-Founded Sets ......Page 277
15.l The Zermelo Fraenkel Set Theory With Choice ......Page 284
15.2 Consistency and Independence ......Page 287
15.3 The Universe of Set Theory ......Page 294
Bibliography ......Page 302
C ......Page 303
D ......Page 304
E ......Page 306
F ......Page 308
Z ......Page 0
