What is pp?

February 5, 2011 at 15:12 | Posted in Uncategorized | Leave a comment

We may as well make a short post dealing with the definition of {\rm{pp}(\mu)} for {\mu} singular; this will also let me test the latex2wp setup at home.

Definition 1 If {\mu} is a singular cardinal, then {{\rm pp}(\mu)} is the supremum of all cardinals of the form

\displaystyle  {\rm cf}\left(\prod\mathfrak{a}, <_D\right) \ \ \ \ \ (1)

where

  • {\mathfrak{a}} is a set of regular cardinals cofinal in {\mu},
  • {|\mathfrak{a}|={\rm cf}(\mu)}, and
  • {D} is an ultrafilter on {\mathfrak{a}} 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

\displaystyle  \sup\{\theta:\theta\in{\rm pcf}\mathfrak{a}\text{ for some }\mathfrak{a}\text{ as above}\}. \ \ \ \ \ (2)

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

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