What is pp?

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.