## Conclusion 3.2A (part 4)

June 17, 2011 at 13:54 | Posted in Uncategorized | Leave a commentSo let us now return to the proof of Conclusion 3.2A. We’ll keep notation the same; what remains to be shown is that if is singular of cardinality and , then there is a for which

We have assumed , so for all sufficiently large we know is progressive. Given such a , we can apply Claim 3.2 (June 15, 2011) to conclude

So we fix can fix large enough so that

- is progressive,
- , and
- ,

where for the last statement we use the result of the previous posting.

We will get what we need by comparing the last two bullet points:

Suppose . This means that we there is an ultrafilter and a set of cardinality such that

- , and
- .

Clearly we may assume that has cardinality exactly , and the second bullet implies

as well, where we view as an ultrafilter on . Chaining things together, we have

So for our choice of , we have

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