Table of Contents |
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Preface | vii | |
I | Basics of set theory | 1 |
1 | Axiomatic set theory | 3 |
1.1 Why axiomatic set theory? | 3 | |
1.2 The language and the basic axioms | 6 | |
2 | Relations, functions and Cartesian product | 12 |
2.1 Relations and the axiom of choice | 12 | |
2.2 Functions and the replacement scheme axiom | 16 | |
2.3 Generalized union, intersection and Cartesian product | 19 | |
2.4 Partial and linear order relations | 21 | |
3 | Natural numbers, integers, and real numbers | 25 |
3.1 Natural numbers | 25 | |
3.2 Integers and rational numbers | 30 | |
3.3 Real numbers | 31 | |
II | Fundamental tools of set theory | 35 |
4 | Well orderings and transfinite induction | 37 |
4.1 Well-ordered sets and the axiom of foundation | 37 | |
4.2 Ordinal numbers | 44 | |
4.3 Definitions by transfinite induction | 49 | |
4.4 Zorn's Lemma in algebra, analysis and topology | 54 | |
5 | Cardinal numbers | 61 |
5.1 Cardinal numbers and the continuum hypothesis | 61 | |
5.2 Cardinal arithmetic | 68 | |
5.3 Cofinality | 74 | |
III | The Power of recursive definitions | 77 |
6 | Subsets of $$R^{n} | 79 |
6.1 Strange subsets of $$R^{n} and diagonalization argument | 79 | |
6.2 Closed sets and Borel sets | 89 | |
6.3 Lebesgue-measurable sets and sets with Baire property | 98 | |
7 | Strange real functions | 104 |
7.1 Measurable and nonmeasurable functions | 104 | |
7.2 Darboux functions | 106 | |
7.3 Additive functions and Hamel basis | 111 | |
7.4 Symmetrically discontinuous functions | 118 | |
IV | When induction is too short | 127 |
8 | Martin's axiom | 129 |
8.1 Rasiowa-Sikorski lemma | 129 | |
8.2 Martin's axiom | 139 | |
8.3 Suslin hypothesis and diamondsuit principle | 154 | |
9 | Forcing | 164 |
9.1 Elements of logic and other forcing preliminaries | 164 | |
9.2 Forcing method and a model for non-CH | 168 | |
9.3 Model for CH and Diamondsuit principle | 182 | |
9.4 Product lemma and Cohen model | 189 | |
9.5 Model for MA+non-CH | 196 | |
A | Axioms of set theory | 211 |
B | Comments on the forcing method | 215 |
C | Notation | 220 |
References | 225 | |
Index | 229 |
Last modified July 18, 2015.