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 ${sf GCH}$.

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