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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 144(11)
Mathematics
Arithmetic of Cardinal Numbers 155(16)
Infinite Sums and Products of Cardinal 155(5)
Numbers
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