Bounds on $\max \mathrm{pcf}(A)$ if $\Pi A$ is big

Question

For concreteness, let $A = \{\aleph_n : n < \omega\}$. We know $\max \mathrm{pcf}(A) \in [\aleph_{\omega+1},\Pi A]$. My question is, if $\Pi A$ is big (say, $\aleph_{\omega_1+1}$), then which cardinals in that interval can $\max \mathrm{pcf}(A)$ really be?

My second question, which motivates the first, is: Let $E = \{\aleph_{2n} : n < \omega\}$ and $O = \{\aleph_{2n+1} : n < \omega \}$; then is it possible for $\max \mathrm{pcf}(A) = \max \mathrm{pcf}(E) \gg \max \mathrm{pcf}(O)$ (where $\gg$ means "much larger than" in any way you care to make precise)?

If the answer to the first question is that $\max \mathrm{pcf}(A)$ can never really be that big, then the second question doesn't matter. But if it can be big, and the answer to the second question is that the even alephs and the odd alephs can have very different $\max\mathrm{pcf}$'s, I would find that somewhat disturbing.

Answer 1

Shelah proved that it is consistent that GCH holds below $\aleph_\omega$, while $2^{\aleph_\omega}=\aleph_{\omega+\alpha+1}$ for any countable ordinal $\alpha$ you care to choose. (See Theorem 36.5 of Jech's book, for example).

In such a model, ${\rm max pcf}(A)=\aleph_{\omega+\alpha+1}$ as well. Now if you add $\aleph_{\omega_1+1}$ Cohen reals (which has no effect on the pcf structure) you end up with a model where

• $|\prod A| = \aleph_{\omega_1+1}$, and
• ${\rm max pcf}(A)=\aleph_{\omega+\alpha+1}$.

So ${\rm max pcf} (A)$ could potentially be any successor cardinal below $\aleph_{\omega_1}$.

(Of course, it's still unknown if $\aleph_{\omega_1}\leq{\rm max pcf }(A)$ is possible, so this is the best answer we can hope for given our current knowledge.)

I don't know the answer to your "evens and odds" question, but certainly you can split $A$ up into two disjoint pieces whose "gap" is as large as possible:

Let $\tau$ denote ${\rm max pcf}(A)$, and suppose $\aleph_{\omega+1}<\tau$.

We know there exists an unbounded $B\subseteq A$ such that $\prod B$ contains a scale (mod finite) of length $\aleph_{\omega+1}$. This implies ${\rm max pcf}(B)=\aleph_{\omega+1}$.

The set $A\setminus B$ cannot be in the ideal $J_{<\tau}[A]$ (otherwise, we contradict ${\rm max pcf}(A)=\tau$), and so we must conclude ${\rm max pcf}(A\setminus B)=\tau$ as well.