## First steps in the hard direction

May 10, 2011 at 16:01 | Posted in Uncategorized | Leave a commentNow what about the “hard direction”? Given cardinals , , and such that is regular and , our aim is to prove

and moreover, if is regular, then there exist and such that

- is a progressive set of regular cardinals with and ,
- is a -complete ideal on extending the bounded ideal,
- the structure has a true cofinality, and
- .

Notice that by the preceding post, condition 4 is equivalent to the statement

and so the equality

holds (at least in the case where ) in a strong sense. Note that the underlying issue here is that the pp number is defined as the supremum of a certain set of regular cardinals; we will show that if this supremum itself is a regular cardinal (and hence weakly inaccessible), then it actually occurs as the true cofinality of for some relevant and .

We’ll do this by proving the following:

Theorem 1Suppose is a regular cardinal such that whenever and satisfy conditions 1, 2, and 3 above. Then as well.

The usual caveats apply here: it is closer to the truth to say that I will *attempt* to prove the above theorem. We’ll see what happens…

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