## Conclusion 3.2A (part 2)

June 16, 2011 at 14:10 | Posted in Uncategorized | Leave a commentRecall from the last post that I’ve committed myself to providing a proof of the following:

Theorem 1 (Conclusion 3.2A (i) page 378, reformulated)Suppose is a singular cardinal of cofinality with . Then for some ,

We’ve already discussed what the notation means, so I’m going to plunge right in. I’ll probably split this up into a few posts; the result is true, but the proof given in The Book has the unfortunate property that at a key moment, the reader is given an incorrect reference as explanation, and it might take me a post or two to fill in the necessary details.

*Proof:* We’ll start with the easy stuff and prove first the inequality

whenever is such that is progressive.

Let be given (such exist because — see posts of June 15), and define

By Claim 3.2 (June 15, 2011), we know

So let us assume that is a progressive set of regular cardinals cofinal in , is an ideal on containing the bounded subsets of , and

To establish our inequality, we must show

This is actually quite easy. Let us define an ideal on by

Set , and note

- (as ), and
- (as contains the set ).

Thus, to finish this part of the proof we need only check

but this follows easily from (5).

I’ll end this post here, and pick up with establishing the other inequality (and the “” part) tomorrow.

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