## A technical result (continued)

April 25, 2011 at 17:01 | Posted in Uncategorized | Leave a commentThe following proposition can be viewed as a continuation of the technical result from the last post. What I’m trying to do is to look at from different points of view so that we get a better picture of what’s going on.

[Warning: This post may be revised as I haven’t had enough time to proofread it properly]

Proposition 1Let be cardinals, with and regular. Then the following two statements are equivalent for a cardinal :

- there is a progressive set of regular cardinals cofinal in such that

- ,
- , and
- is cofinal in .

*Proof:* Assume first that , and let and witness this fact. These objections and put into , so we need only verify that the generator is unbounded in . This follows easily, though, as contains all the bounded subsets of .

Now suppose that is a progressive set of regular cardinal cofinal in and of cardinality less than , and that is in with unbounded in . Let be the -complete ideal generated by together with the bounded subsets of .

Our assumptions imply that (since is in we know cannot be covered by fewer than sets from ). Basic pcf-theory tells us

and it follows

as well.

But now we see that and witness that is a member of , as was claimed.

The following corollary is an application of the above result; it is the main point of this post, and will be important in the course of later discussions.

Corollary 2Let be cardinals, with and regular. If is a progressive set of regular cardinals cofinal in with , then there is a such that

*Proof:* Let be a set of cardinals as in the corollary, and define

Let be the -complete ideal on generated by together with the bounded subsets of . We claim that is not a proper ideal, i.e.,

Suppose this fails. Let be least such that . Then

and since is -complete, it follows that

Note also that

and hence

Finally, extends the bounded ideal on and so we know that must be unbounded in . Given our proposition, this would imply

yielding a contradiction.

Thus, is not a proper ideal. Decoding what this means, we find that there exists a such that can be covered by a union of fewer than sets from , and this implies

But is a collection of cardinals, and given the definition of , the conclusion of the corollary is immediate.

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