Consequences of no inaccessible accumulation point

November 28, 2011 at 14:05 | Posted in Uncategorized | Leave a comment

This post is just an organizational one without any proofs. I wanted to list a few consequences of the following conjecture of Shelah:

• If ${\mathfrak{a}}$ is a progressive set of regular cardinals, then ${{\rm pcf}(\mathfrak{a})}$ does not have a weakly inaccessible point of accumulation.

Equivalently, if ${\mathfrak{a}}$ is a progressive set of regular cardinals, then ${{\rm pcf}(\mathfrak{a})\cap\kappa}$ is bounded in ${\kappa}$ for every weakly inaccessible ${\kappa}$.

In his paper [Sh:666], Shelah argues that this conjecture is “a significant dividing line between chaos and order”.

Why?

One answer is outlined in [Sh:666]: if the conjecture is true, then for any progressive set of regular cardinals ${\mathfrak{a}}$ we have

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

while if the conjecture fails one can force a counterexample to the above statement.

The above is really just an outcropping of deeper results from the third section of Chapter VIII of The Book, where Shelah proves that a subset ${\mathfrak{b}}$ of ${{\rm pcf}(\mathfrak{a})}$ which does not have a weakly inaccessible accumulation point still has a nice pcf structure, even though it may be the case that ${\mathfrak{b}}$ is not progressive. In particular, we have ${{\rm pcf}(\mathfrak{b})\subseteq{\rm pcf}(\mathfrak{a})}$ for such a ${\mathfrak{b}}$. As a corollary, we see that if Shelah’s conjecture is true, then

$\displaystyle {\rm pcf}({\rm pcf}(\mathfrak{a}))={\rm pcf}(\mathfrak{a}) \ \ \ \ \ (2)$

for any progressive set of regular cardinals ${\mathfrak{a}}$.

But Shelah’s Conjecture also has consequences for cardinal arithmetic as well: this is the content of the fourth section of [Sh:430]. I invite the adventurous reader to take a look at that particular piece of Shelah’s oeuvre, because at this point I have no idea what the theorems say. Well, that’s not quite true, as I have a vague idea of what they say, but they are couched in the language of nice filters originating in Chapter V of The Book, and that’s a language I’ve not yet tried to learn. In [Sh:666], he says that if the conjecture holds and ${\aleph_\delta}$ is the ${\omega_1}$th fixed point (strong limit), then ${{\rm pp}(\aleph_\delta)}$ is less than then ${\omega_4}$th fixed point. (But I don’t actually see this in [Sh:430] so it’s possible that something was retracted.).