The real theorem

June 24, 2011 at 12:56 | Posted in Uncategorized | Leave a comment

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.

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