# Statement **Martin's maximum $\mathrm{MM}$**: Suppose $(\mathbb{P},\leq)$ is a forcing notion that preserves stationary subsets of $\omega_1$, and that $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. >[!note] Remark > If $\mathbb{P}$ is a forcing notion which doesn't preserve stationary sets, then there is a collection of $\aleph_{1}$ dense sets s.t. no filter meets them all. > So this axiom cannot be strengthened by extending to a larger class of forcing notions, hence the "maximum". **Martin's Maximum ${ }^{+\beta}$ ($\mathrm{MM}^{+\beta}$ )**: Suppose $(\mathbb{P},\leq)$ is a forcing notion which preserves stationary subsets of $\omega_1,\left\langle D_\alpha: \alpha<\omega_1\right\rangle$ is a sequence of dense subsets of $P$ and $\left\langle\tau_\alpha: \alpha<\beta\right\rangle$ is a sequence of $P$-names for stationary subsets of $\omega_1$, there is a filter $G \subseteq P$ such that $G \cap D_\alpha$ is nonempty for each $\alpha<\omega_1$ and $\left\{\gamma<\omega_1 \mid \exists p \in G\ p \Vdash \check{\gamma} \in \tau_\alpha\right\}$ is stationary for each $\alpha<\beta$. $\mathrm{MM}^{+}$ = $\mathrm{MM}^{+1}$ $\mathrm{MM}^{++}$ = $\mathrm{MM}^{+\omega_{1}}$ **** # Implications of MM - [[Proper forcing axiom]] and [[Martin's Axiom]] - The nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated - For all $\kappa\geq\aleph_2$, if $\kappa$ is regular then $\kappa^{\aleph_0}=\kappa$. # Consistency of MM - Consistency follows from the existence of a [[Supercompact]] cardinal.