## Set theory seminar -Forcing axioms and inner models IV

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 ${\sf GCH}$.