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*.
3Other alternatives to ZFC have been also explored such as Russell’s type theory, or Quine’s new foundations. They are rarely considered in modern set theory. 4Indeed, G odel’s incompleteness theorem says that it’s hopeless to try and axiomatize all true sentences about the natural numbers. It is similarly hopeless to try and axiomatize all
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