Contents
Preface xi
How to Read this Book xiii
Basic Notations xvii
Introduction 1
Part 1. Not Entirely Naive Set Theory 9
Chapter 1. Pairs, Relations, and Functions 11
Chapter 2. Partial Order Relations 17
Chapter 3. Cardinality 29
Chapter 4. Induction 43
4.1. Induction and recursion over the set of natural numbers 43
4.2. Induction and recursion over wellfounded sets 55
Part 2. An Axiomatic Foundation of Set Theory 69
Chapter 5. Formal Languages and Models 71
Chapter 6. Power and Limitations of the Axiomatic Method 81
6.1. Complete theories 81
6.2. The Incompleteness Phenomenon 92
6.3. Definability 101
Chapter 7. The Axioms 107
Chapter 8. Classes 121
Chapter 9. Versions of the Axiom of Choice 129
9.1. Statements Equivalent to the Axiom of Choice 129
9.2. Set Theory without the Axiom of Choice 143
9.3. The Axiom of Determinacy 148
9.4. The Banach-Tarski Paradox 151
Chapter 10. The Ordinals 155
10.1. The Class ON 155
10.2. Ordinal Arithmetic 163
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