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

I concluded my series of talks by showing the following theorem of Viale:

Theorem (Viale). Assume ${sf CP}(kappa^+)$ and let $Msubseteq V$ be an inner model where $kappa$ is regular and such that $(kappa^+)^M=kappa^+$. Then ${rm cf}(kappa)neomega$.

This allows us to conclude, via the results shown last time, that if ${sf PFA}$ holds in $V$ and $M$ computes cardinals correctly, then it also computes correctly ordinals of cofinality $omega$.

An elaboration of this argument is expected to show that, at least  if we strengthen the assumption of ${sf PFA}$ to ${sf MM}$, then $M$ computes correctly ordinals of cofinality $omega_1$

Under an additional assumption, Viale has shown this:  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 $omega_1$. The new assumption on $kappa$ allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and $I_{, Fund. Math. 148 (1995), 165-198, in place of the structure imposed by ${sf CP}(kappa^+)$. It is still open if the corresponding covering statement ${sf CP}(kappa^+,omega_1)$ follows from ${sf MM}$, which would eliminate the need for this the strong limit requirement.

• Go to the intermezzo for a discussion of consistency strengths.