The real theorem

This post is a bit of a detour, but I wanted to point out that Conclusion 3.2A is really just Claim 3.2 grafted onto a nice characterization of ${{\rm pp}(\mu)}$. Shelah implicitly uses this characterization in his proof of Conclusion 3.2A, but I can’t find a formulation of it anywhere in The Book.

Theorem 1 Let ${\mu}$ be a singular cardinal. Then the following three cardinals are equal:

1. ${{\rm pp}(\mu)}$
2. ${\min_{\tau<\mu}\sup\{{\rm cf}(\prod\mathfrak{b}, <):\mathfrak{b}\subseteq (\tau,\mu)\cap{\sf Reg}\wedge|\mathfrak{b}|\leq{\rm cf}(\mu)\}}$
3. ${\min_{\tau<\mu}\sup\{\max{\rm pcf}(\mathfrak{b}):\mathfrak{b}\subseteq(\tau,\mu)\cap{\sf Reg}\wedge |\mathfrak{b}|\leq{\rm cf}(\mu)<\min(\mathfrak{b})\}}$.

Notice that in 2 and 3, the suprema involved can only decrease as ${\tau}$ increases, and this tells us a couple of things. First, the cardinals in 2 and 3 are easily seen to be equal, as without loss of generality ${{\rm cf}(\mu)<\tau}$, hence the collection of relevant sets ${\mathfrak{b}}$ is exactly the same in each case, and we finish using the equality

$\displaystyle {\rm cf}\left(\prod\mathfrak{b}, <\right)=\max{\rm pcf}(\mathfrak{b}) \ \ \ \ \ (1)$

for progressive ${\mathfrak{b}}$.

Second, we know that as ${\tau}$ approaches ${\mu}$, the suprema involve eventually stabilize and hence the minimum value is actually attained by some ${\tau<\mu}$. What the theorem shows is that this stable value is exactly ${{\rm pp}(\mu)}$. Notice as well that although ${{\rm pp}(\mu)}$ is computed using sets cofinal in ${\mu}$, we do not assume this for the ${\mathfrak{b}}$ involved in 2 and 3.

We turn now to the proof:

Proof: We have already noted that 2 and 3 are equal for easy reasons. The fact that 1 is less than or equal to 2 is also trivial from the definition of ${{\rm pp}(\mu)}$:

Suppose ${\eta<{\rm pp}(\mu)}$. Then there is a progressive ${\mathfrak{b}\subseteq \mu\cap{\sf Reg}}$, a cardinal ${\kappa}$, and an ultrafilter ${D}$ on ${\mathfrak{b}}$ such that

• ${|\mathfrak{b}|={\rm cf}(\kappa)}$
• ${D}$ is disjoint to the bounded ideal on ${\mathfrak{b}}$,
• ${\kappa={\rm cf}(\prod\mathfrak{b}/D)}$, and
• ${\eta<\kappa}$.

Given ${\tau<\mu}$, we know ${\kappa\in{\rm pcf}(\mathfrak{b}\cap(\tau,\mu))}$ as ${\mathfrak{b}\cap (\tau,\mu)\in D}$, hence

$\displaystyle \tau<\kappa\leq\max{\rm pcf}(\mathfrak{b})\cap(\tau,\mu). \ \ \ \ \ (2)$

Thus,

$\displaystyle \eta<\kappa\leq\sup\{\max{\rm pcf}(\mathfrak{b}):\mathfrak{b}\subseteq(\tau,\mu)\cap{\sf Reg}\wedge |\mathfrak{b}|\leq{\rm cf}(\mu)<\min(\mathfrak{b})\} \ \ \ \ \ (3)$

for each ${\tau<\mu}$ and the desired inequality follows immediately.

What about the other inequality? This is exactly Lemma 1 from our June 17, 2011 post, which was written with this application in mind.

$\Box$