A technical result

April 25, 2011 at 15:36 | Posted in Uncategorized | Leave a comment

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.

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