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 Δ_{1} 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
Σ^{1}_{1},
Π^{1}_{1} and
Σ^{1}_{2},
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 21-602

The 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.)