文档介绍：Z
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Monika, Agata, and Victoria
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the memory of my Mother, wh
taught me mathematics
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Prefacw ix
ParS I BasicM of seS theory 1
1 AxiomatiL set theory 3
WLw axiomatic set theory? 3
The language and the basic axioms 6
2 Relations, functions, and Cartesian product 12
Relations and the axiom of choice 12
Functions and the replacement scheme axiom 16
Generalized union, intersection, and Cartesian product 19
Partial- and linear-order relations 28
3 Natural numbers, integers, and real numbers 25
Natural numbers 25
Integers and rational numbers 30
Real numbers 38
ParS II Fundame6tal t’olM of seS theory 35
4Well orderings and transfinitw induction 37
Bell-ordered sets and the axiom of foundation 37
Ordinal numbers 44
Definitions bw transfinite induction 49
Zorn’s lemma in algebra, analysis, and topologw 54
5 Cardinal numbers 61
Cardinal numbers and the continuum Lwpothesis 68
Cardinal arithmetic 68
Cofinalitw 74
vii
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ParS III ThL p’wLr ’f recursivL definitionM 77
6 Subsets oX
z 79
Strange subsets of
z and the diagonalization argument 79
Closed sets and Borel sets 89
Lebesgue-measurable sets and sets witL the Baire propertw 98
7 Strangw real functions 104
Measurable and nonmeasurable functions 104
Darboux functions 106
Additi’e functions and Hamel bases 118
Symmetricallw discontinuous functions 118
ParS IVWhe6 inductio6 iM t’’ shorS 127
8 Martin’s axiom 129
Rasiowa–Sikorski lemma 129
Martin’s axiom 139
Suslin Lwpothesis and diamond principle 154
9 Forcing 164
Elements of logic and other forcing preliminaries 164
Forcing method and a model for ℵCH 168
Model for CH and ⊕ 182
Product lemma and Cohen model 189
Model for MA+ℵCH 196
A Axioms oX set theory 211
ments on thw forcing method 215
C Notation 220
Referenc