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

Theorem (Viale). Assume is an inner model.

If holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .

If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be .

It follows from this result and the last theorem from last time that if is a model of and a forcing extension of an inner model by a cardinal preserving forcing, then .

In fact, the argument from last time shows that we can weaken the assumption that is a forcing extension to the assumption that for all there is a regular cardinal such that in we have a partition where each is stationary in .

It is possible that this assumption actually follows from in . However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable 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 that preserves cardinals, does not add reals, and (for some cardinal ) the set of points of countable -cofinality in is nonstationary for every regular . Obviously, this situation is incompatible with in , by Viale’s result.

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5 Responses to Set theory seminar -Forcing axioms and inner models VI

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