We isolate a new forcing axiom, ${\rm wPFA}$, which is strictly between ${\rm BPFA}$ and ${\rm PFA}$. ${\rm wPFA}$ is equiconsistent with a remarkable cardinal, it implies the failure of $\square_{\omega_1}$, but it is compatible with $\square_\kappa$ for all $\kappa \geq \omega_2$. This is part of joint work with J. Bagaria and V. Gitman.

Goldstern and Shelah (1995) introduced the class of bounded forcing axioms, that is forcing axioms for families of antichains of bounded size. For example, the bounded proper forcing axiom ${\rm BPFA}$ asserts that for any proper forcing notion $\mathbb{P}$ and any collection $D$ of at most $\aleph_1$ many maximal antichains in $\mathbb{P}$, each of size at most $\aleph_1$, there is a filter on $\mathbb{P}$ meeting each antichain in $D$. The speaker will present a theorem of Joan Bagaria (2000) that characterizes bounded forcing axioms in terms of generic absoluteness: for instance, Bagaria’s result shows that ${\rm BPFA}$ is equivalent to the assertion that if a $\Sigma_1$ sentence of the language of set theory with parameters of hereditary size at most $\aleph_1$ is true in some proper forcing extension, then it is already true in the ground model.