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Tables of Contents for Introduction to Set Theory
Chapter/Section Title
Page #
Page Count
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