Set Theory for the Working MathematicianCambridge University Press, 1997 M08 28 - 236 páginas This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields. |
Contenido
Axiomatic set theory | 3 |
12 The language and the basic axioms | 6 |
Relations functions and Cartesian product | 12 |
22 Functions and the replacement scheme axiom | 16 |
23 Generalized union intersection and Cartesian product | 19 |
24 Partial and linearorder relations | 21 |
Natural numbers integers and real numbers | 25 |
32 Integers and rational numbers | 30 |
63 Lebesguemeasurable sets and sets with the Baire property | 98 |
Strange real functions | 104 |
72 Darboux functions | 106 |
73 Additive functions and Hamel bases | 111 |
74 Symmetrically discontinuous functions | 118 |
When induction is too short | 127 |
Martins axiom | 129 |
82 Martins axiom | 139 |
33 Real numbers | 31 |
Fundamental tools of set theory | 35 |
Well orderings and transfinite induction | 37 |
42 Ordinal numbers | 44 |
43 Definitions by transfinite induction | 49 |
44 Zorns lemma in algebra analysis and topology | 54 |
Cardinal numbers | 61 |
52 Cardinal arithmetic | 68 |
53 Cofinality | 74 |
The power of recursive definitions | 77 |
Subsets of ℝⁿ | 79 |
62 Closed sets and Borel sets | 89 |
83 Suslin hypothesis and diamond principle | 154 |
Forcing | 164 |
92 Forcing method and a model for CH | 168 |
93 Model for CH and | 182 |
94 Product lemma and Cohen model | 189 |
95 Model for MA+CH | 196 |
A Axioms of set theory | 211 |
B Comments on the forcing method | 215 |
C Notation | 220 |
| 225 | |
| 229 | |
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Términos y frases comunes
antichain axiom of choice bijection ccc forcing cf(a choose closed compatible condition contains continuum hypothesis Corollary countable transitive model Darboux function define definition denoted dense subset disjoint dom(p equivalence relation exercise exists a set extends family F filter F finish the proof finite formula Func(X function f G₁ Hamel basis Hence implies intersects interval Lemma Let F Let G limit ordinal linearly ordered set M-generic filter M[Go Martin's axiom model of ZFC natural numbers Notice obtain a contradiction one-to-one order isomorphism order type ordinal number P-name P₁ pairwise-disjoint partially ordered set particular Proof Let proof of Theorem proper initial segment Proposition prove Rasiowa-Sikorski lemma real numbers recursion satisfies sequence set theory smallest element strictly increasing strongly Darboux Suslin line transfinite induction Un<w uncountable union valg w₁ w₂ well-ordered set ZFC axioms