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

TL;DR

This paper explores a strengthened form of Martin's Maximum, called MM^{**,++}, and demonstrates how to force its bounded version over a determinacy model using advanced set-theoretic techniques.

Contribution

It introduces a new strengthening of Martin's Maximum and constructs its bounded version via ext{P}_{ ext{max}} forcing over a determinacy model, extending previous work by Gappo, Sargsyan, Larson, and Wilson.

Findings

Successfully forces MM^{**,++}_c over a determinacy model.

Shows the strengthened forcing is stronger than existing models.

Builds on and extends previous models by Gappo, Sargsyan, Larson, and Wilson.

Abstract

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 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.