I. Set Theory
-Forcing Basics: Statement of Forcing Theorem. Force CH. Force not CH.
-Product Forcing: Product Lemma. Lévy Collapse. Easton support forcing a partial
-Force a Suslin Tree. Force Diamond.
-Iterated Forcing: Iterated Forcing Lemma. Force MA (and not CH).
-Force a Kurepa tree. Using an inaccessible cardinal, force no Kurepa trees.
2. Large Cardinals (Measurable, mostly)
-Measurable Cardinals: They are inaccessible. The least cardinal with a
non-trivial sigma-additive two-valued measure is measurable.
-Ultrapowers: Fundamental theorem of ultrapowers. Properties of ultrapower
embeddings of V. Scott's Theorem. There is a measurable cardinal iff there is a
non-trivial e.e. of V. Measurable implies Mahlo. Kunen's theorem.
-Normal Measures: Characterise normal measures. Every measurable
cardinal has one. Sets with normal measure 1 are stationary.
-Weakly Compact Cardinals: Ramsey's Theorem. Measurable implies weakly
compact implies inaccessible. Characterise weakly compact. Measurable implies
-Lévy-Solovay theorem for measurables
3. Infinitary Combinatorics
-Suslin's Problem: There is an Aronszajn Tree. There is a Suslin tree
iff there is a Suslin line. Diamond implies there is a Suslin tree. MA
implies there is not.
-Delta Lemma (x2)
-Theorems of MA: c is regular. ccc is preserved by arbitrary
products. ccc is the same as strong ccc. Every Aronszajn tree is regular.
II. Classical Groups
-Chapters 1 - 5, 7 of "The Geometry of Classical Groups" by Donald Taylor