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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*.

The smallest size of in nite set is N(see Exercise 1.2). Finally, say a set Xis countable if jXj jNj. Exercise 1.2. If Xis a set, either Xhas the same cardinality as a nite set, or jNj jXj. Exercise 1.3 (A countable union of countable sets is countable.). If X i is a countable set for every i2N, then S i X i is countable. Exercise 1.4.

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