## Notes on Set Theory"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."--Jacket. |

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atoms Axiom of Choice Axiom of Replacement axiomatic Baire space basic bijection bisimulation Borel Cantor cardinal numbers Cauchy Chapter compute construct Continuum Hypothesis contradiction countable set define definite condition definite operation element equinumerosity equinumerous equivalence relation Exercise exists exactly Fixed Point Theorem function f graph G grounded graph grounded set hence hypothesis implies inductive poset infinite initial segment injection intuitive isomorphism Least Fixed Point least upper bound Lemma mathematical natural numbers Neumann cardinals node non-empty notation notion objects ordered field ordered set ordinals pair partial function partial ordering pointed graphs pointset powerset Problem Proof properties proposition prove rationals real numbers Recursion Theorem relativization Rieger universe satisfies the identity sequence set theory structured set subset Suppose surjection topological space transitive transitive set trivial unique verify wellordering Zermelo universe