Transcription of Set Theory - UCLA Mathematics
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Set Theory - UCLA Mathematics
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* .. 3610 Cofinality3811 Cardinal arithmetic The singular cardinals hypothesis*.
We use ZFC to denote ZF+ the axiom of choice. The rst part of this class will be discussing these axioms of ZFC and axiomatic set theory. Figure 2: A picture of the set theoretic universe, known as V. At step , we construct all sets of \rank" . V denotes all sets of rank less than . Note that we will never de ne what a set is in these notes.
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