# Set Theory for the Working Mathematician

### Cambridge University Press, 1997.

Hardback ISBN 0-521-59441-3; paperback ISBN 0-521-59465-0.

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 Rn 79
6.1 Strange subsets of Rn 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