Meanderings

November 15, 2011 at 15:16 | Posted in Uncategorized | Leave a comment

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.

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