Set theory seminar -Forcing axioms and inner models IV

October 3, 2008

We proved Baumgartner’s result that under {\sf BPFA}, every tree of height and size \omega_1 is sealed in the sense that no outer model can add a new uncountable branch. From this we concluded Todorcevic’s result that under {\sf BPFA} any forcing adding a subset of \omega_1 either adds a real or else it collapses \omega_2. We also drew some conclusions about inner models of {\sf GCH}.