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 $M\subseteq V$ be an inner model where $\kappa$ is regular and such that $(\kappa^+)^M=\kappa^+.$ Then ${\rm cf}(\kappa)\ne\omega$.

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.
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