Bounded Martin’s Maximum with an Asterisk

Abstract

We isolate natural strengthenings of Bounded Martin’s Maximum which we call ${\mathsf{BMM}}^{\ast }$ and $A-{\mathsf{BMM}}^{\ast ,++}$ (where $A$ is a universally Baire set of reals), and we investigate their consequences. We also show that if $A-{\mathsf{BMM}}^{\ast ,++}$ holds true for every set of reals $A$ in $L\left(\mathbb{R}\right)$, then Woodin’s axiom $(\ast )$ holds true. We conjecture that ${\mathsf{MM}}^{++}$ implies $A-{\mathsf{BMM}}^{\ast ,++}$ for every $A$ which is universally Baire.