## What is pp?

February 5, 2011 at 15:12 | Posted in Uncategorized | Leave a commentWe may as well make a short post dealing with the definition of for singular; this will also let me test the latex2wp setup at home.

Definition 1If is a singular cardinal, then is the supremum of all cardinals of the formwhere

- is a set of regular cardinals cofinal in ,
- , and
- is an ultrafilter on extending the co-bounded filter.

This is not the “official” definition given as Definition 1.1 on page 41 of Cardinal Arithmetic, but it is equivalent. The official definition is designed with a nod towards generalization and refinement.

Note that the above definition is not the same as looking at

The reason is that in the definition of we do not require the ultrafilters under consideration to extend the co-bounded filter on , and this can make a difference if : you can potentially get a very large in by virtue of an ultrafilter concentrating on an initial segment of . In this situation, though, the fact that is in isn’t really anything to do with , and so the definition of is designed to ignore this phenomenon.

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