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Graduate Courses 21-602
Introduction to Set Theory
Fall: 12 units
The axioms of ZFC, ordinal arithmetic, cardinal arithmetic including Kőnig's lemma, class length induction and recursion, the rank hierarchy, the Mostowski collapse theorem, the H(λ) hierarchy, the Δ1 absoluteness theorem, the absoluteness of wellfoundedness, the reflection theorem for hierarchies of sets, ordinal definability, the model HOD, relative consistency, Gödel's theorem that HOD is a model of ZFC, constructibility, Gödel's theorem that L is a model of ZFC + GCH, the Borel and Projective hierarchies and their effective versions, Suslin representations for Σ11, Π11 and Σ12, sets of reals, Shoenfield's absoluteness theorem, the complexity of the set of constructible reals, the combinatorics of club and stationary sets (including the diagonal intersection, the normality of the club filter and Fodor's lemma), Solovay's splitting theorem, model theoretic techniques commonly applied in set theory (e.g., elementary substructures, chains of models and ultrapowers), club and stationary subsets of [X]ω (including a generalization of Fodor's lemma and and connections with elementary substructures), Jensen's diamond principles and his proofs that they hold in L, Gregory's theorem, constructions of various kinds of uncountable trees (including Aronszajn, special, Suslin, Kurepa), Jensen's square principles and elementary applications, the basic theory of large cardinals (including inaccesssible, Mahlo, weakly compact and measurable cardinals), Scott's theorem that there are no measurable cardinals in L, Kunen's theorem that the only elementary embedding from V to V is the identity.
Prerequisites for 21-602The minimum background for 21-602 is the equivalent of undergraduate set theory (e.g., 21-329) and the fundamentals of logic (e.g., 21-600). Students should arrive with a working knowledge of basic ordinal and cardinal arithmetic, Gödel's completeness theorem and the downward Loewenheim-Skolem theorem. An understanding of the statement of Gödel's incompleteness theorem is also assumed. (This theorem is mentioned in 21-600 but proved in 21-700.)