## A technical result

This post is concerned with a cosmetic reformulation ${{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)}$; I may add the proof later on when I get some time (mainly to make sure it’s all formulated exactly right), but the argument is a straightforward one.

Anyway, let us suppose ${\sigma}$ and ${\theta}$ are regular cardinals, and ${\sigma\leq{\rm cf}(\mu)<\theta<\mu}$.

We define the ${{\rm PP}^0_{\Gamma(\theta,\sigma)}(\mu)}$ to consist of all cardinals ${\kappa}$ for which there is a cardinal ${\tau<\theta}$, a ${\sigma}$-complete ideal ${I}$ on ${\tau}$, and a sequence ${\langle \mu_i:i<\tau\rangle}$ of regular cardinals below ${\mu}$ such that

• ${\{i<\tau:\mu_i<\epsilon\}\in I}$ for each ${\epsilon<\mu}$, and
• ${\kappa={\rm tcf}\left(\prod_{i<\tau}\mu_i, <_I\right)}$.

Notice that the above definition is tailored to guarantee

$\displaystyle {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)=\sup{\rm PP}^0_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (1)$

On the other hand, the set ${{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}$ consists of all cardinals ${\kappa}$ such that

$\displaystyle \kappa={\rm tcf}\left(\prod\mathfrak{a}, <_I\right), \ \ \ \ \ (2)$

where

• ${\mathfrak{a}}$ is a progressive set of regular cardinals cofinal in ${\mu}$,
• ${|\mathfrak{a}|<\theta}$,
• ${I}$ is a ${\sigma}$-complete ideal on ${\mathfrak{a}}$ containing the bounded subsets of ${\mathfrak{a}}$, and
• ${{\rm tcf}\left(\prod\mathfrak{a}, <_I\right)}$ exists.

Proposition 1 With notation as above, we have

$\displaystyle {\rm PP}^0_{\Gamma(\theta,\sigma)}(\mu)={\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (3)$

and so

$\displaystyle {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)=\sup{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (4)$

We will tend to use the above result as a definition in the future.