I am curious about the existence of inner models of $\mathrm{ZFC}$ in conjunction with forcing axioms, under assumptions inconsistent with such theories. For example:


* can we prove under any extension of $\mathrm{ZF} + \mathrm{AD}$, that there is an inner model of $\mathrm{ZFC} + \mathrm{MA}$?
* can we prove under any extension of $``V$ is $L$-like$"$, that there is an inner model of $\mathrm{ZFC} + \mathrm{PFA}$?



Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions
All you have to have is an inner model where some set that it thinks is large, is actually countable. To answer the first question, ZF+AD implies that $0^\sharp$ exists, so there exists a generic for a poset forcing MA over $L$. For the second question, the best known upper bound for PFA is a supercompact, and we don’t have any $L$-like inner model theory at that level.


