Transcription of Set Theory - UCLA Mathematics
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Set TheoryAndrew MarksJuly 22, 2020 These notes cover introductory set Theory . Starred sections below are op-tional. They discuss interesting Mathematics connected to concepts covered inthe course. A huge thanks to Spencer Unger for enlightening conversations, andthe students in the class who asked excellent questions, and corrected countlesstypos in the midst of a global Independence in modern set Theory * ..62 The axioms Classes and von Neumann-Bernays-G odel set Theory * .. 123 Wellorderings144 Ordinals165 Transfinite induction and Goodstein s theorem * .. 236 The cumulative hierarchy257 The Mostowski collapse288 The axiom of Fragments of the axiom of choice* .. 319 Cardinality Cardinality in models of the axiom of determinacy* .. Resurrecting Tarski s Theory of cardinal algebras*.
Set theory began with Cantor’s proof in 1874 that the natural numbers do not have the same cardinality as the real numbers. Cantor’s original motivation was to give a new proof of Liouville’s theorem that there are non-algebraic real numbers1. However, Cantor soon began researching set theory for its own sake.
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