A technical result (continued)

April 25, 2011 at 17:01 | Posted in Uncategorized | Leave a comment

The following proposition can be viewed as a continuation of the technical result from the last post. What I’m trying to do is to look at {{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)} from different points of view so that we get a better picture of what’s going on.

[Warning: This post may be revised as I haven’t had enough time to proofread it properly]

Proposition 1 Let {\sigma\leq{\rm cf}(\mu)<\theta<\mu} be cardinals, with {\sigma} and {\theta} regular. Then the following two statements are equivalent for a cardinal {\kappa}:

  1. {\kappa\in{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}
  2. there is a progressive set of regular cardinals {\mathfrak{a}} cofinal in {\mu} such that
    • {|\mathfrak{a}|<\theta},
    • {\kappa\in{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})}, and
    • {\mathfrak{b}_{\kappa}[\mathfrak{a}]} is cofinal in {\mathfrak{a}}.

Proof: Assume first that {\kappa\in{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}, and let {\mathfrak{a}} and {I} witness this fact. These objections {\mathfrak{a}} and {I} put {\kappa} into {{\rm pcf}_{\sigma\text{-complete}}(\mu)}, so we need only verify that the generator {\mathfrak{b}_{\kappa}[\mathfrak{a}]} is unbounded in {\mathfrak{a}}. This follows easily, though, as {I} contains all the bounded subsets of {\mathfrak{a}}.

Now suppose that {\mathfrak{a}} is a progressive set of regular cardinal cofinal in {\mu} and of cardinality less than {\theta}, and that {\kappa} is in {{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})} with {\mathfrak{b}_{\kappa}[\mathfrak{a}]} unbounded in {\mathfrak{a}}. Let {J} be the {\sigma}-complete ideal generated by {J_{<\kappa}[\mathfrak{a}]} together with the bounded subsets of {\mathfrak{a}}.

Our assumptions imply that {\mathfrak{b}_\kappa[\mathfrak{a}]\notin J} (since {\kappa} is in {{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})} we know {\mathfrak{b}_{\kappa}[\mathfrak{a}]} cannot be covered by fewer than {\sigma} sets from {J_{<\kappa}[\mathfrak{a}]}). Basic pcf-theory tells us

\displaystyle  {\rm tcf}\left(\prod\mathfrak{b}_\kappa[\mathfrak{a}], <_{J_{<\kappa}[\mathfrak{a}]}\right)=\kappa, \ \ \ \ \ (1)

and it follows

\displaystyle  \kappa={\rm tcf}\left(\prod\mathfrak{b}_{\kappa}[\mathfrak{a}], <_J\right) \ \ \ \ \ (2)

as well.

But now we see that {\mathfrak{b}_\kappa[\mathfrak{a}]} and {J} witness that {\kappa} is a member of {{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}, as was claimed. \Box

The following corollary is an application of the above result; it is the main point of this post, and will be important in the course of later discussions.

Corollary 2 Let {\sigma\leq{\rm cf}(\mu)<\theta<\mu} be cardinals, with {\sigma} and {\theta} regular. If {\mathfrak{a}} is a progressive set of regular cardinals cofinal in {\mu} with {|\mathfrak{a}|<\theta}, then there is a {\xi<\mu} such that

\displaystyle  \sup{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}\setminus\xi)\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (3)

Proof: Let {\mathfrak{a}} be a set of cardinals as in the corollary, and define

\displaystyle  \tau=({\rm pp}_{\Gamma(\theta,\sigma)}(\mu))^+. \ \ \ \ \ (4)

Let {J} be the {\sigma}-complete ideal on {\mathfrak{a}} generated by {J_{<\tau}[\mathfrak{a}]} together with the bounded subsets of {\mathfrak{a}}. We claim that {J} is not a proper ideal, i.e.,

\displaystyle  \mathfrak{a}\in J. \ \ \ \ \ (5)

Suppose this fails. Let {\kappa\in{\rm pcf}(\mathfrak{a})} be least such that {\mathfrak{b}_\kappa[\mathfrak{a}]\notin J}. Then

\displaystyle  J_{<\kappa}[\mathfrak{a}]\subseteq J, \ \ \ \ \ (6)

and since {J} is {\sigma}-complete, it follows that

\displaystyle  \kappa\in{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}). \ \ \ \ \ (7)

Note also that

\displaystyle  J_{<\tau}[\mathfrak{a}]\subseteq J\Longrightarrow \tau\leq \kappa, \ \ \ \ \ (8)

and hence

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)<\kappa. \ \ \ \ \ (9)

Finally, {J} extends the bounded ideal on {\mathfrak{a}} and so we know that {\mathfrak{b}_{\kappa}[\mathfrak{a}]} must be unbounded in {\mathfrak{a}}. Given our proposition, this would imply

\displaystyle  \kappa\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (10)

yielding a contradiction.

Thus, {J} is not a proper ideal. Decoding what this means, we find that there exists a {\xi<\mu} such that {\mathfrak{a}\setminus\xi} can be covered by a union of fewer than {\sigma} sets from {J_{<\tau}[\mathfrak{a}]}, and this implies

\displaystyle  {\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}\setminus\xi)\subseteq\tau. \ \ \ \ \ (11)

But {{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})} is a collection of cardinals, and given the definition of {\tau}, the conclusion of the corollary is immediate. \Box

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