Introduction to Set Theory

Preface
1 Sets
1. Introduction to Sets
2. Properties
3. The Axioms
4. Elementary Operations on Sets
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Preface
1 Sets
1. Introduction to Sets
2. Properties
3. The Axioms
4. Elementary Operations on Sets
2 Relations, Functions and Orderings
1. Ordered Pairs
2. Relations
3. Functions
4. Equivalences and Partitions
5. Orderings
6. Operations and Structures
3 Natural Numbers
1. Introduction to Natural Numbers
2. Properties of Natural Numbers
3. The Recursion Theorem
4. Relations, Operations, and Structures Revisited
4 Finite and Countable Sets
1. Cardinality of Sets
2. Finite Sets
3. Countable Sets
4. Linear Orderings
5. Uncountable Sets
5 Real Numbers
1. Integers and Rational Numbers
2. Real Numbers
3. Topology of the Real Line
6 Cardinal Numbers
1. Cardinal Arithmetic
2. The Cardinality of the Continuum
3. Sets of Real Numbers
7 Ordinal Numbers
1. Well-Ordered Sets
2. Ordinal Numbers
3. The Axiom of Replacement
4. Transfinite Induction and Recursion
5. Ordinal Arithmetic
8 Alephs
1. Initial Ordinals
2. Addition and Multiplication of Alephs
9 The Axiom of Choice
1. The Axiom of Choice and Its Equivalents
2. The Use of the Axiom of Choice in Mathematics
3. Sets of Real Numbers Revisited
10 Arithmetic of Cardinal Numbers
1. Infinite Sums and Products of Cardinal Numbers
2. Regular and Singular Cardinals
3. Exponentiation of Cardinals
11 Uncountable Sets
1. Filters and Ideals
2. Closed Unbounded and Stationary Sets
3. The Measure Problem
4. Large Cardinals
12 The Axiomatic Set Theory
1. The Zermelo-Fraenkel Set Theory with Choice
2. Consistency and Independence
3. The Universe of Set Theory
Index
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