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_{<f} [\lambda], 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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