Fields institute, Room 210
Speaker: Teruyuki Yorioka (Shizuoka)
Title: An application of Aspero-Mota’s iteration to Todorcevic’s OCA
Abstract: It is an open question whether it is consistent that Todorcevic’s OCA holds together with the continuum larger than $\aleph_2$.
As many people knows, it is sometimes trouble to force set theoretic statements together with the continuum larger than $\aleph_2$.
Aspero and Mota gave a new method of a forcing iteration which forces that
the size of the continuum is larger than $\aleph_2$, and they have proved that, for example, it is consistent that the axiom $\mho$ holds together with the continuum larger than $\aleph_2$.
In the middle of 1990’s, Ilijas Farah proved that it is consistent that Todorcevic’s OCA for separable metric spaces of size $\aleph_1$ holds together with the continuum larger than $\aleph_2$.
It is given a different proof of this result using Aspero-Mota’s iteration.
Fields institute, Room 230
Speaker: Dima Sinapova (UIC)
Title: A bad scale and not SCH at $\aleph_\omega$
Abstract: Starting from a supercompact, we construct a model in which SCH fails at $\aleph_\omega$ and there is a bad scale at $\aleph_\omega$. The existence of a bad scale implies the failure of weak square.
The construction uses two Prikry type forcings defined in different ground models and a suitably defined projection between them. This is joint work with Spencer Unger.