Notes on Set TheorySpringer Science & Business Media, 1994 M02 18 - 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
Introduction | 1 |
Paradoxes and axioms | 19 |
Are sets all there | 33 |
7 | 47 |
The natural numbers | 53 |
28 | 66 |
Fixed points | 78 |
33 | 85 |
Choices | 117 |
Baire space | 147 |
Replacement and other axioms | 169 |
Ordinal numbers | 189 |
43 | 205 |
A The real numbers | 209 |
B Axioms and universes | 239 |
267 | |
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atoms Axiom of Choice Axiom of Dependent Axiom of Replacement axiomatic Baire space basic bijection binary bisimulation Borel Cantor cardinal number chain Chapter compute construct Continuum Hypothesis contradiction countable sets countably continuous define definite condition definite operation Dependent Choices domain equinumerosities equinumerous equivalence Exercise exists exactly Fixed Point Theorem full Axiom function f graph G grounded graph grounded set hence implies inductive poset infinite initial segment injection Least Fixed Point least upper bound Lemma mathematical monotone mapping natural numbers node non-empty ordered field ordered set ordinals pair partial function partial ordering pointset powerset Problem Proof properties proposition prove real numbers Recursion Theorem relation satisfies the identity seg(x sequence set theory subset Suppose surjection TC(A topological space transfinite transitive set trivial unique verify wellordering Zermelo universe