MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this community
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:
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.
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.
