## The real theorem

June 24, 2011 at 12:56 | Posted in Uncategorized | Leave a commentThis post is a bit of a detour, but I wanted to point out that Conclusion 3.2A is really just Claim 3.2 grafted onto a nice characterization of . Shelah implicitly uses this characterization in his proof of Conclusion 3.2A, but I can’t find a formulation of it anywhere in The Book.

Theorem 1Let be a singular cardinal. Then the following three cardinals are equal:

- .

Notice that in *2* and *3*, the suprema involved can only decrease as increases, and this tells us a couple of things. First, the cardinals in *2* and *3* are easily seen to be equal, as without loss of generality , hence the collection of relevant sets is exactly the same in each case, and we finish using the equality

for progressive .

Second, we know that as approaches , the suprema involve eventually stabilize and hence the minimum value is actually attained by some . What the theorem shows is that this stable value is exactly . Notice as well that although is computed using sets cofinal in , we do not assume this for the involved in *2* and *3*.

We turn now to the proof:

*Proof:* We have already noted that *2* and *3* are equal for easy reasons. The fact that *1* is less than or equal to *2* is also trivial from the definition of :

Suppose . Then there is a progressive , a cardinal , and an ultrafilter on such that

- is disjoint to the bounded ideal on ,
- , and
- .

Given , we know as , hence

Thus,

for each and the desired inequality follows immediately.

What about the other inequality? This is exactly Lemma 1 from our June 17, 2011 post, which was written with this application in mind.

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