I. Set Theory

1. Forcing


-Forcing Basics: Statement of Forcing Theorem. Force CH. Force not CH.


-Product Forcing: Product Lemma. Lévy Collapse. Easton support forcing a partial

continuum function.


-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

Ramsey.


-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

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