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

**Theorem** (Viale). Assume and let be an inner model where is regular and such that . Then .

This allows us to conclude, via the results shown last time, that if holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .

An elaboration of this argument is expected to show that, at least if we strengthen the assumption of to , then computes correctly ordinals of cofinality .

Under an additional assumption, Viale has shown this: If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be . The new assumption on allows us to use a result of Dzamonja and Shelah, *On squares, outside guessing of clubs and *, Fund. Math. **148** (1995), 165-198, in place of the structure imposed by . It is still open if the corresponding covering statement follows from , which would eliminate the need for this the strong limit requirement.

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