March 12, 2009
4. Strongly compact cardinals and
Definition 1 A cardinal is strongly compact iff it is uncountable, and any -complete filter (over any set ) can be extended to a -complete ultrafilter over
The notion of strong compactness has its origin in infinitary logic, and was formulated by Tarski as a natural generalization of the compactness of first order logic. Many distinct characterizations have been found.
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October 24, 2008
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.
October 17, 2008
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.