## Conclusion 3.2A (part 2)

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.

$\Box$