## Meanderings

November 15, 2011 at 15:16 | Posted in Uncategorized | Leave a commentI wanted to use this post as an opportunity to write down some things that I think I know regarding the unresolved questions raised in my last post.

To set the stage, suppose with and regular. Let us define the set to be of all cardinals such that

where

- is a progressive set of regular cardinals cofinal in ,
- ,
- is a -complete ideal on containing the bounded subsets of , and
- exists,

so that

We are interested in the following:

Suppose is a regular cardinal . Is ?

If the answer to the above question is “NO” for some , then I *THINK* I know the following:

- There is a cardinal such that and .
- There is a cofinal and progressive with unbounded in .
- There is cofinal and progressive of cardinality for which modulo the ideal has true cofinality at least .

What I want to do over our Christmas break is to work out the details of the above and see if I *REALLY* know this. The idea is to keep mining pcf theory for more and more structure, with the hope of getting a contradiction eventually. I’ll post my work on the blog as it progresses.

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