Notes on Set Theory
Springer Science & Business Media, 1994 - 272 páginas
"The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. At the same time, it is often viewed as a foundation of mathematics so that in the most prevalent, current mathematical practice "to make a notion precise" simply means "to define it in set theory." This book tries to do justice to both aspects of the subject: it gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets (including the basic results that have applications to computer science), but it also attempts to explain precisely how mathematical objects can be faithfully modeled within the universe of sets." "Topics covered include the naive theory of equinumerosity; paradoxes and axioms; modeling mathematical notions by sets; cardinal numbers; natural numbers; fixed points (continuous least-fixed-point theorem); well-ordered sets (transfinite induction and recursion, Hartogs' theorem, comparability of well-ordered sets, least-fixed-point theorem); the Axiom of Choice and its consequences; Baire space (Cantor-Bendixson theorem, analytic pointsets, perfect set theorem); Replacement and other axioms; ordinal numbers. There is an Appendix on the real numbers and another on natural models, including the antifounded universe." "The book is aimed at advanced undergraduate or beginning graduate mathematics students and at mathematically minded graduate students of computer science and philosophy."--BOOK JACKET.
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Paradoxes and axioms
Are sets all there is?
The natural numbers
Well ordered sets
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addition apply argument assume atoms Axiom of Choice basic binary Cantor cardinal cardinal numbers Chapter closed complete compute condition consider construct contains continuous contradiction countable define definite condition definite operation definition determined easily element equivalence exactly example Exercise exists fact Figure finite Fixed Point function gives graph grounded hence holds hypothesis immediately implies inductive infinite initial segment injection least Lemma limit mapping mathematical means monotone natural numbers non-empty notion objects obvious operation ordered field ordered set pair partial function pointset poset precisely Principle Problem Proof properties proposition prove pure rationals recursion relation Replacement satisfies sequence set theory similar simple space structured subset Suppose surjection Theorem topological transitive trivial true union unique universe upper bound verify wellordering Zermelo
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Introduction to Set Theory, Third Edition, Revised and Expanded
Karel Hrbacek,Thomas Jech
Vista previa limitada - 1999