## Updates

November 14, 2011 at 17:09 | Posted in Uncategorized | Leave a commentThe quarter is ending here, so I should have a little more time for maintaining this blog. I mentioned before that I figured out how to fix the proof of Claim 3.5, which then gives a fairly transparent proof of the cov vs. pp theorem. I need to clarify this a little, because what I’ve got doesn’t recover the full result originally claimed by Shelah. Here’s what I can show:

Theorem 1Suppose is singular, and with and regular. Then the following two statements are equivalent for a regular cardinal :

- .
- .

This is enough to deduce Shelah’s cov vs. pp theorem (Theorem 5.4 of Chapter II in The Book), which states:

Theorem 2If is a regular uncountable cardinal, and , then

Now the issue left unresolved concerns the attainment of suprema in one particular case: if is singular and is a weakly inaccessible cardinal , must there exist a cofinal subset of of cardinal less than and a -complete ideal on containing the bounded subsets of such that

Said another way, if is regular, then must the supremum in the definition of be attained?

PS: I’m still very sleep-deprived…it’s all part of being the father of young children.

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