Dissertation Subject Oral Syllabus

(a) Definition of a ∆-system, statement and proof of the ∆-system lemma. (b) Statement of MA (Martin’s Axiom) and MAκ, proof that MAω holds and MAc fails. (c) Definition of a club and a stationary subset of κ, of the club filter and proof that the club filter on a regular cardinal κ is κ-complete. (d) Statement and proof of the Pressing Down Lemma. i. An application: Statement and proof of Thomas’s theorem on automorphism towers of centreless groups. (e) Statement of ♦ (the Diamond Axiom) and ♣ (the Club Axiom). (f) Trees i. Statement and proof of König’s Lemma. ii. Definition of a κ-tree, a κ-Aronszajn tree and a κ-Kurepa tree. iii. Proof of existence of an ω1-Aronszajn tree. iv. Definition of a κ-Suslin tree, proof that κ-Suslin implies κ-Aronszajn. v. Definition of a Suslin line and of Suslin’s hypothesis SH, proof that the existence of a Suslin tree is equivalent to the existence of a Suslin Line. vi. MAω1 implies that there is no ω1-Suslin tree. vii. ♦ implies existence of a Suslin Tree. (g) Definition of Def(X) (Definable subsets of a set X), the constructible hierarchy, definition of L, the Axiom of Constructibility V = L. (h) Statement only: L V = L, V = L implies that there is a definable well-order of the universe; the Condensation Lemma. (i) L AC + GCH, L ♦.