Axiomatic Set Theory

Lecture 1, 07.03.: We made a review of the material covered in Chapter I of [3], up to Theorem I.9.11 (Transfinite Recursion on Well-founded Relations). Lecture 2, 14.03.: We discussed the notion of a rank, as well as the Mostowski collapsing function material corresponding to Section 9 of [3]. Lecture 3, 04.04.: We discussed hereditarily transitive sets, the DownwardLöwenheim-Skolem-Tarksi Theorem, the notions of definability and absoluteness (see sections I.15 and I.16 of [3]). Lecture 4, 11.04.: We covered [3, Section II.2]. Lecture 5, 18.04.: We continued our discussion of absoluteness and covered section II.4 of [3]. Lecture 6, 25.04. We covered the reflection theorems from section II.4 of [3] and started our discussion of the Constructible Universe. Lecture 7, 02.05. We covered section II.6 of [3] up to and including Lemma II.6.28. Lecture 8, 09.05. We proved the Delta System Lemma, introduced Solovay’s almost disjoint set partial order and proved that under MA 2ω is regular. Lecture 9, 23.05. We showed that MA(κ) implies each of the following: add(M) > κ and add(N ) > κ. In addition, we proved that MA(ω1) implies that any product of ccc spaces is ccc and introduced the notion of a Suslin line.