# Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions

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 $$\mathrm{ZFC} + \mathrm{PFA}$$?
• What exactly is $L$-like? Fine structural? $L[x]$? Apr 1, 2021 at 10:38
• I think fine-structural is a good description. Apr 1, 2021 at 11:02
• Well, if Woodin's Ultimate-$L$ project will bear fruits, then the answer to your second question is positive. Apr 1, 2021 at 11:03

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.

• Am I right to say that, if I replace $\mathrm{MA}$ with $\mathrm{PFA}$ in the first question, the answer is also unknown? Apr 1, 2021 at 9:42
• @Zoorado Yes, that sounds right. Apr 1, 2021 at 9:46
• @Zoorado: PFA implies $\sf AD$ in $L(\Bbb R)$, and I seem to recall that it implies even more than just that. If that is indeed the case, then Con(ZFC+PFA) is strictly greater than Con(ZF+AD). Apr 1, 2021 at 10:36
• Yes, you are right. I was mainly curious if there were proper extensions of ZF+AD which can give rise to inner models of ZFC+PFA. Apr 1, 2021 at 11:01
• @Zoorado This is a current conjecture as far as I understand. Namely that some strong hypothesis should imply that the HOD of a canonical inner model of AD (larger than $L(\mathbb R)$) should contain large enough cardinals to force PFA. Apr 1, 2021 at 16:29

I want to comment that both AD and "$$V$$ is $$L$$-like" are consistent with the existence of inner models of very strong theories.

For AD, this is actually quite simple. Suppose there is a supercompact cardinal with a measurable $$\kappa$$ above it. Let $$P$$ be a countable transitive model of $$\text{ZFC}^-$$ that admits an elementary embedding $$\pi : P\to H((2^\kappa)^+)$$. Then $$P$$ belongs to $$L(\mathbb R)$$ (which in this context is a model of AD), and $$L(\mathbb R)$$ can iterate away a measure on $$\pi^{-1}(\kappa)$$ to obtain an inner model of ZFC + there is a supercompact cardinal. The iteration is wellfounded by a "realizability argument" because $$P$$ embeds into $$H((2^\kappa)^+)$$.

The "$$V$$ is $$L$$-like" situation is slightly more subtle. Under the same large cardinal hypotheses, $$M_1^\#$$ (the minimal active $$(\omega,\omega_1+1)$$-iterable $$\omega$$-sound mouse with a Woodin cardinal) exists and is fully iterable. Suppose $$N$$ is an inner model in which $$\delta^{+M_1}$$ is countable where $$\delta$$ is the Woodin cardinal of $$M_1$$. Then I claim $$N$$ has an inner model with a supercompact cardinal. First, one can do a genericity iteration to make $$V_{\kappa+3}$$ set-generic over an iterate of $$M_1$$ for the extender algebra. As a consequence, in $$M_1$$, there is a condition in the extender algebra at $$\delta$$ that forces that there is a measurable cardinal $$\bar \kappa < \delta$$ such that $$V_{\bar \kappa}$$ has a supercompact cardinal. Since $$\delta^{+M_1}$$ is countable in $$N$$, $$N$$ can build a forcing extension $$P$$ of $$M_1$$ below this condition. Iterating away a measure on $$\bar \kappa$$ in $$P$$ yields an inner model of $$P$$ with a supercompact, and this inner model is contained in $$N$$. Similarly, $$N$$ will have an inner model of PFA (or MM). This shows, for example, that under large cardinal hypotheses, $$M_2$$ has an inner model of MM. (Actually $$M_1$$ does not have such an inner model.)

Therefore, we can in fact produce extensions of the theories ZF + AD and ZFC + "$$V$$ is $$L$$-like" that are consistent relative to large cardinals and that prove the existence of inner models of very strong theories, although these extensions are totally ad hoc: e.g., ZF + AD + "there is an inner model of ZFC with a cardinal that is supercompact to a measurable." This is closely analogous to the fact that assuming large cardinals, $$L$$ has a countable transitive model containing a supercompact cardinal.