Set Theory For The Working Mathematician

v To Monika, Agata, and Victoria and to the memory of my Mother, who taught me mathematics Contents Preface Part I ix Basics of set theory 1 Axiomatic set theory 1.1 Why axiomatic set theory? 1.2 The language and the basic axioms 1 3 3 6 2 Relations, functions, and Cartesian product 2.1 Relations and the axiom of choice 2.2 Functions and the replacement scheme axiom 2.3 Generalized union, intersection, and Cartesian product 2.4 Partial- and linear-order relations 12 12 16 19 21 3 Natural numbers, integers, and real numbers 3.1 Natural numbers 3.2 Integers and rational numbers 3.3 Real numbers 25 25 30 31 Part II 35 Fundamental tools of set theory 4 Well orderings and transfinite induction 4.1 Well-ordered sets and the axiom of foundation 4.2 Ordinal numbers 4.3 Definitions by transfinite induction 4.4 Zorn’s lemma in algebra, analysis, and topology 37 37 44 49 54 5 Cardinal numbers 5.1 Cardinal numbers and the continuum hypothesis 5.2 Cardinal arithmetic 5.3 Cofinality 61 61 68 74 vii viii Contents Part III The power of recursive definitions 6 Subsets of Rn 6.1 Strange subsets of Rn and the diagonalization argument 6.2 Closed sets and Borel sets 6.3 Lebesgue-measurable sets and sets with the Baire property 77 79 79 89 98 7 Strange real functions 7.1 Measurable and nonmeasurable functions 7.2 Darboux functions 7.3 Additive functions and Hamel bases 7.4 Symmetrically discontinuous functions 104 104 106 111 118 Part IV 127 When induction is too short 8 Martin’s axiom 8.1 Rasiowa–Sikorski lemma 8.2 Martin’s axiom 8.3 Suslin hypothesis and diamond principle 129 129 139 154 9 Forcing 9.1 Elements of logic and other forcing preliminaries 9.2 Forcing method and a model for ¬CH 9.3 Model for CH and ♦ 9.4 Product lemma and Cohen model 9.5 Model for MA+¬CH 164 164 168 182 189 196 A Axioms of set theory 211 B Comments on the forcing method 215 C Notation 220 References 225 Index 229 Preface The course presented in this text concentrates on the typical methods of modern set theory: transfinite induction, Zorn’s lemma, the continuum hypothesis, Martin’s axiom, the diamond principle ♦, and elements of forcing. The choice of the topics and the way in which they are presented is subordinate to one purpose – to get the tools that are most useful in applications, especially in abstract geometry, analysis, topology, and algebra. In particular, most of the methods presented in this course are accompanied by many applications in abstract geometry, real analysis, and, in a few cases, topology and algebra. Thus the text is dedicated to all readers that would like to apply set-theoret