Martin's Maximum${}_{\mathfrak{c}}^{\ast ,++}$ in ${\mathbb{P}}_{\max }$ extensions of strong models of determinacy

Abstract

We study a strengthening of ${\mathrm{MM}}^{++}$ which is called ${\mathrm{MM}}^{\ast ,++}$ and which was introduced by Asperó and Schindler. We force its bounded version ${\mathrm{MM}}_{\mathfrak{c}}^{\ast ,++}$, which is stronger than both ${\mathrm{MM}}^{++}(\mathfrak{c})$ as well as ${\mathrm{BMM}}^{++}$, by ${\mathbb{P}}_{\max }$ forcing over a determinacy model $L^{F_{\mathrm{uB}}}({\mathbb R}^*,\mbox{Hom}^{\ast})$. The construction of the ground model $L^{F_{\mathrm{uB}}}({\mathbb R}^{\ast},\mbox{Hom}^{\ast})$ builds upon Gappo and Sargsyan, and the derived model construction of Larson, Sargsyan, and Wilson.