## Meanderings

I wanted to use this post as an opportunity to write down some things that I think I know regarding the unresolved questions raised in my last post.

To set the stage, suppose ${\aleph_0<\sigma\leq{\rm cf}(\mu)<\theta<\mu}$ with ${\sigma}$ and ${\theta}$ regular. Let us define the set ${{\rm PP}_{\Gamma(\theta,\sigma)}(\mu)}$ to be of all cardinals ${\kappa}$ such that

$\displaystyle \kappa={\rm tcf}\left(\prod\mathfrak{a}, <_I\right), \ \ \ \ \ (1)$

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,

so that

$\displaystyle {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)=\sup{\rm PP}_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (2)$

We are interested in the following:

Suppose ${{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)}$ is a regular cardinal ${\kappa}$. Is ${\kappa\in{\rm PP}_{\Gamma(\theta,\sigma)}(\mu)}$?

If the answer to the above question is “NO” for some ${\mu}$, then I THINK I know the following:

• There is a cardinal ${\lambda}$ such that ${\sigma\leq{\rm cf}(\lambda)<\theta\leq\lambda<\mu}$ and ${\kappa\in{\rm PP}_{\Gamma(\theta,\sigma)}(\lambda)}$.
• There is a cofinal and progressive ${\mathfrak{a}\subseteq\mu}$ with ${{\rm pcf}_{\Gamma(\theta,\sigma)}(\mathfrak{a})}$ unbounded in ${\kappa}$.
• There is cofinal and progressive ${\mathfrak{b}\subseteq\mu\cap{\sf Reg}}$ of cardinality ${\theta}$ for which ${\prod\mathfrak{b}}$ modulo the ideal ${[\mathfrak{b}]^{<\theta}}$ has true cofinality at least ${\kappa}$.

What I want to do over our Christmas break is to work out the details of the above and see if I REALLY know this. The idea is to keep mining pcf theory for more and more structure, with the hope of getting a contradiction eventually. I’ll post my work on the blog as it progresses.