## Continuation on pp

February 8, 2011 at 21:02 | Posted in Uncategorized | Leave a commentThe main result of this post is natural, but unsurprising. Oddly enough, though, I can’t find this exact version published anywhere. I mentioned something in this direction in my Handbook of Set Theory argument; I wish I had thought about it a bit more back then, because the proof is certainly easy enough. Anyway, the result is the following:

Theorem 1The following statements are equivalent for a singular cardinal :

- If is a cofinal progressive set of regular cardinals, then carries a scale of length modulo the ideal of bounded subsets of .

*Proof:*

We prove first that (2) implies (1), as this implication is almost immediate. Suppose is a progressive set of regular cardinals cofinal in and of cardinality . Our assumptions imply that there is a sequence of functions from such that

- , and
- if , then there is an such that .

If is any ultrafilter on disjoint to , then the two above statements continue to hold modulo , and hence has cofinality at most . But clearly this cofinality must be at least as extends the co-bounded filter on . The statement now follows from the definitions involved.

For the other direction, suppose , let satisfy the assumptions of statement (2), and let denote the ideal . It is clear that

(see Claim 1.3(4) on page 4 of Cardinal Arithmetic), that is, given a collection of functions in of cardinality at most , there is a single function in above them all modulo the ideal .

By Conclusion 3.2 on page 62 of Cardinal Arithmetic, is equal to the supremum of all cardinals of the form , where is an ultrafilter on disjoint to . This latter cardinal is at most , and so it follows that

Given (1) and (2), it is straightforward to construct the desired scale.

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