Martin's Maximum${}^{\ast, ++}_{\mathfrak{c}}$ in $\mathbb{P}_{\max}$
extensions of strong models of determinacy
Martin's Maximum${}^{\ast, ++}_{\mathfrak{c}}$ in $\mathbb{P}_{\max}$
extensions of strong models of determinacy
We study a strengthening of $\mathrm{MM}^{++}$ which is called $\mathrm{MM}^{\ast, ++}$ and which was introduced by Asper\'o and Schindler. We force its bounded version $\mathrm{MM}^{\ast, ++}_{\mathfrak{c}}$, 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 …