## Logic Seminar

It is a well established fact that we can get the consistency of axioms like $MM$ or $PFA$ from a supercompact cardinal. However if we restrict ourselves to looking at just one forcing, it is not necessarily clear that the forcing axiom requires or entails any large cardinal strength. So after Jensen showed that the forcing axiom for Namba forcing is consistent with $CH$, it was a question of Moore as to whether or not $CH$ actually implied the axiom. Zapletal has answered this question in the negative, while at the same time showing some of the first effects of Namba forcing on the combinatorics of $\aleph_1$ and providing a potential method for further investigating any consistency strength of the forcing axiom for Namba forcing. I will give a brief overview of forcing axioms and bounded forcing axioms, and in particular discuss where Namba forcing fits in to the established picture. Then I will present Zapletal's paper ``Bounded Namba forcing axiom may fail", and discuss potential ways of reworking the construction in order to prove more about the forcing axiom.