## Ignore that last post…

September 21, 2011 at 14:25 | Posted in Uncategorized | Leave a commentActually, I think I lied to you in the last post, at least as far as our strategy is concerned. I’ll outline another route to the result that is closer to what Shelah does; it’s actually shorter than what I had in mind originally.

Recalling our situation, is a singular cardinal, and is as usual. We have also assumed . Let us assume that is a family of functions as required in the definition of , i.e., a collection of functions that are “-cofinal in any subproduct of size constructed from ” (see previous posts for a more precise definition, but things should be clear from the construction).

Given a finite subset of , we define

and define

Note that each is a subset of of cardinality , so

Also, it should be clear that

so we will be finished if we can prove that any can be covered by a union of fewer than sets drawn from .

How will we do this? Given , we will produce a family satisfying the following two properties:

and

This suffices, because

and since , we will have shown that can be covered by a union of fewer than sets from .

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