Notes on Set TheorySpringer Science & Business Media, 2013 M04 17 - 273 páginas What this book is about. The theory of sets is a vibrant, exciting math ematical theory, with its own basic notions, fundamental results and deep open problems, and with significant applications to other mathematical theories. At the same time, axiomatic set theory is often viewed as a foun dation ofmathematics: it is alleged that all mathematical objects are sets, and their properties can be derived from the relatively few and elegant axioms about sets. Nothing so simple-minded can be quite true, but there is little doubt that in standard, current mathematical practice, "making a notion precise" is essentially synonymous with "defining it in set theory. " Set theory is the official language of mathematics, just as mathematics is the official language of science. Like most authors of elementary, introductory books about sets, I have tried to do justice to both aspects of the subject. From straight set theory, these Notes cover the basic facts about "ab stract sets," including the Axiom of Choice, transfinite recursion, and car dinal and ordinal numbers. Somewhat less common is the inclusion of a chapter on "pointsets" which focuses on results of interest to analysts and introduces the reader to the Continuum Problem, central to set theory from the very beginning. |
Contenido
2 | |
5 | |
Paradoxes and axioms | 19 |
Are sets all there is? | 33 |
Fixed points | 78 |
Transfinite induction | 98 |
Wellfoundedness of | 105 |
Choices | 128 |
Choices consequences 131 | 130 |
Baire space | 147 |
Replacement and other axioms | 169 |
Ordinal numbers | 189 |
A The real numbers | 209 |
B Axioms and universes | 239 |
266 | |
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atoms Axiom of Choice Axiom of Replacement axiomatic Baire space basic bijection binary bisimulation Borel Cantor cardinal numbers Cauchy Chapter closure compute construct Continuum Hypothesis contradiction countable set define definite condition definite operation domain edge relation equinumerosity equinumerous Exercise exists exactly finite function f graph G grounded graph grounded set hence ill founded implies inductive poset infinite initial segment injection isomorphism Least Fixed Point least upper bound Lemma M₁ mathematical monotone natural numbers Neumann cardinals node non-empty objects ordered field ordered set ordinals pair partial function pointed graphs pointset poset powerset Problem Proof proposition prove pure rationals real numbers relativization Rieger universe satisfies the identity sequence set theory subset Suppose surjection TC(A topological space transitive set uncountable unique verify wellordering Z-F universe Zermelo universe ZFDC