## Set theory seminar -Forcing axioms and inner models VI

October 17, 2008

I presented a sketch of a nice proof due to Todorcevic that ${\sf PFA}$ implies the P-ideal dichotomy ${\sf PID}$. I then introduced Viale’s covering property ${\sf CP}$ and showed that it follows from ${\sf PID}$. Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:

Theorem (Viale). Assume $M\subseteq V$ is an inner model.

1. If ${\sf PFA}$ holds in $V$ and $M$ computes cardinals correctly, then it also computes correctly ordinals of cofinality $\omega$.
2. If ${\sf MM}$ holds in $V$, $\kappa$ is a strong limit cardinal, $(\kappa^+)^M=\kappa^+$, and in $M$ we have that $\kappa$ is regular, then in $V$, the cofinality of $\kappa$ cannot be $\omega_1$.

It follows from this result and the last theorem from last time that if $V$ is a model of ${\sf MM}$ and a forcing extension of an inner model $M$ by a cardinal preserving forcing, then ${\sf ORD}^{\omega_1}\subset M$.

In fact, the argument from last time shows that we can weaken the assumption that $V$ is a forcing extension to the assumption that for all $\kappa$ there is a regular cardinal $\lambda\ge\kappa$ such that  in $M$ we have a partition $S^\lambda_\omega=\sqcup_{\alpha<\kappa}S_\alpha$ where each $S_\alpha$ is stationary in $V$.

It is possible that this assumption actually follows from ${\sf MM}$ in $V$. However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable $V$ cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of $V$ that preserves cardinals, does not add reals, and (for some cardinal $\kappa$) the set of points of countable $V$-cofinality in $\lambda$ is nonstationary for every regular $\lambda\ge\kappa^+$. Obviously, this situation is incompatible with ${\sf PFA}$ in $V$, by Viale’s result.