Conclusion 3.2A (part 2)

June 16, 2011 at 14:10 | Posted in Uncategorized | Leave a comment

Recall from the last post that I’ve committed myself to providing a proof of the following:

Theorem 1 (Conclusion 3.2A (i) page 378, reformulated) Suppose {\mu} is a singular cardinal of cofinality {\kappa} with {\mu<\aleph_\mu}. Then for some {\tau<\mu},

\displaystyle   {\rm pp}(\mu)=^+{\rm cf}_{\kappa}\left(\prod(\tau,\mu)\cap{\sf Reg}\right). \ \ \ \ \ (1)

We’ve already discussed what the notation means, so I’m going to plunge right in. I’ll probably split this up into a few posts; the result is true, but the proof given in The Book has the unfortunate property that at a key moment, the reader is given an incorrect reference as explanation, and it might take me a post or two to fill in the necessary details.

Proof: We’ll start with the easy stuff and prove first the inequality

\displaystyle  {\rm pp}(\mu)\leq {\rm cf}_{\kappa}\left(\prod(\tau,\mu)\cap{\sf Reg}\right) \ \ \ \ \ (2)

whenever {\tau<\mu} is such that {(\tau,\mu)\cap{\sf Reg}} is progressive.

Let {\tau} be given (such {\tau} exist because {\mu<\aleph_\mu} — see posts of June 15), and define

\displaystyle  \mathfrak{a}:=(\tau,\mu)\cap{\sf Reg}. \ \ \ \ \ (3)

By Claim 3.2 (June 15, 2011), we know

\displaystyle  {\rm cf}_{\kappa}\left(\prod(\tau,\mu)\cap{\sf Reg}\right)={\rm cf}^{\aleph_0}_{<\kappa^+}(\prod\mathfrak{a})=\sup{\rm pcf}_{\Gamma(\kappa^+,\aleph_0)}(\mathfrak{a}). \ \ \ \ \ (4)

So let us assume that {\mathfrak{b}} is a progressive set of regular cardinals cofinal in {\mu}, {J} is an ideal on {\mathfrak{b}} containing the bounded subsets of {\mathfrak{b}}, and

\displaystyle   \nu = {\rm tcf}\left(\prod\mathfrak{b}, <_J\right). \ \ \ \ \ (5)

To establish our inequality, we must show

\displaystyle  \nu\in{\rm pcf}_{\Gamma(\kappa^+,\aleph_0)}(\mathfrak{a}). \ \ \ \ \ (6)

This is actually quite easy. Let us define an ideal {I} on {\mathfrak{a}} by

\displaystyle  A\in I\Longleftrightarrow A\cap \mathfrak{b}\in J. \ \ \ \ \ (7)

Set {\mathfrak{c}:=\mathfrak{b}\cap (\tau,\mu)}, and note

  • {|\mathfrak{c}|\leq\kappa<\kappa^+} (as {\mathfrak{c}\subseteq\mathfrak{b}}), and
  • {\mathfrak{a}\setminus\mathfrak{c}\in I} (as {J} contains the set {\mathfrak{b}\cap\tau^+}).

Thus, to finish this part of the proof we need only check

\displaystyle  \nu = {\rm tcf}\left(\prod\mathfrak{a}, <_I\right), \ \ \ \ \ (8)

but this follows easily from (5).

I’ll end this post here, and pick up with establishing the other inequality (and the “{+}” part) tomorrow.

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