August 31
Gunter Fuchs, CUNY
What is the bounded forcing axiom for countably closed forcing?

I will give an overview over the theory of bounded forcing axioms, and explore some that are compatible with the continuum hypothesis. One of these is an axiom I propose to call the bounded forcing axiom for countably closed forcing (which at first may sound like an absurd concept, since already $MA_{\omega_1}$ for countably closed forcing is a theorem of ZFC). I will then focus on the bounded forcing axiom for subcomplete forcing (BSCFA), and show a result that was obtained in joined work with Kaethe Minden, that in the presence of CH, this axiom is equivalent to the absoluteness of the set of branches of any tree of height and width $\omega_1$. Time permitting, I may also mention some 'predicative' strengthenings of bounded forcing axioms whose versions for subcomplete forcing allow us to conclude the existence of a definable well-order of the power set of $\omega_1$, assuming the mantle is a ground.