I've been told that if $V$ has CH and large cardinals, then there is an inner model in which MM holds. The sketch is as follows: Any $\Sigma^2_1$ sentence that can be forced is already true in such $V$. Therefore there exist sets of reals $X,Y$ satisfying the following:
- $X$ codes a transitive model of $\mathrm{ZFC+MM}$ containing all of $\mathbb{R}$,
- $Y$ is the sharp of $(X,\mathbb{R})$.
Then one can proceed to extract an inner model containing all of $\mathbb{R}$ in which MM holds.
Can someone supply more details to the argument? More specifically:
- Where can I find the theorem asserting the generic absoluteness in the beginning?
- How do you find the transitive model of $\mathrm{ZFC+MM}$ containing all of $\mathbb{R}$, and how do you code it with a set of real?
- How do you obtain the inner model from $X,Y$? (I don't know much about sharps.)
