An Interlude and Some Motivation (Sh506)

I’m interested in the following open question:

Is there a singular cardinal $\mu$ such that $\mu^+$ is a Jónsson cardinal?

I think all of the projects that I’m working on are somehow related to the above question. In particular, that’s been my motivation for working through Sh506. So, I figured I would take some space to make that motivation precise. We’re going to start with a couple of definitions:

Definition: Suppose that $I$ is an ideal over some cardinal $\kappa$. We say that $I$ is $\theta$-indecomposable for a regular cardinal $\theta$ if, for every $I$-positive set $A$ and every function $f:A\to \theta$, there exists a $\rho<\theta$ such that $f^{-1}[\{\rho\}]$ is $I$-positive.

Definition: We say that $I$ is $\theta$-weakly saturated if any collection $\{A_i : i<\theta\}$ of $\theta$-many $I$-positive sets cannot be disjoint.

Both of these properties say something about our ability to split up sets with respect to my ideal. Indecomposability says that partitioning a positive set into many pieces yields a positive set in one of the cells. On the other hand, weak saturation tells me that I cannot partition $\kappa$ into many medium-sized pieces. The conjunction of these two statements turns out to be somewhat powerful. The following result appears as Theorem 2 of This paper by Todd Eisworth :

Theorem(Eisworth): Suppose that $\kappa$ is a cardinal with $I$ an ideal over $\kappa$ with the property that there is some regular $\theta<\kappa$ such that $I$ is $\theta$-indecomposable and $\theta$-weakly saturated. Then if $\lambda$ is the completeness of $I$, there is a stationary set $S\subseteq\kappa$ such that every sequence $\langle T_i : i<\xi \rangle$ of stationary subsets of $S$ with $\xi<\lambda$ reflects simultaneously.

We then have the following result due to Shelah:

Theorem(Shelah): If $\mu$ is a singular cardinal with $\mu^+$ Jónsson, then there is a regular cardinal $\theta<\mu$, and an ideal $I$ which is $cf(\mu)$-complete, $\theta$-indecomposable, and $\theta$-weakly saturated.

Now, it’s known that the existence of such an ideal is consistent at $\mu^+$ for $\mu$ a singular cardinal of countable cofinality. This note works through the proof of this. It was also shown in the paper by Eisworth that, if $I$ is an ideal which is $\theta$-weakly saturated and $\theta$-indecomposable, then any collection of $\theta$-many $I$-positive sets has an unbounded subcollection with non-empty intersection. Say that the ideal $I$ satisfies $(*)_\theta$ if this holds. Suppose now that $\mu$ is a singular cardinal and $\mu^+$ carries an ideal satisfying $(*)_\theta$. Define a map

$\pi: \mu^+\to \mu$ given by $\pi(\alpha)=cf(\alpha)$.

Let $J$ be the Rudin-Keisler projection of $I$ to $\mu$ by way of $\pi$. Then we see that $J$ also satisfies $(*)_\theta$ (also, $J$ is $\theta$-weakly saturated). If $\mu^+$ is Jónsson, a theorem of Shelah’s tells us that $\mu$ must be the limit of inaccessible cardinals, and hence an $\aleph$-fixed point. So, we can try to do pcf theory at $\mu$ with this ideal $J$ that we have. Now, I also mentioned in the previous post that I suspect that the trichotomy theorem holds under our original assumptions. I was unable to make it go through with just weak saturation, but it does hold if we assume the ideal we are working over satisfies $(*)_\theta$. So in the next blog entry, I plan on showing this (and then some). So now that we have trichotomy and $\lambda$-directedness of $\prod A/J_{<\lambda}$, many of the big pcf theorems will go through pretty easily. In particular, we will be able to show that the pcf theorem holds in this case.

I do want to point out though, that Shelah has a proof of the pcf theorem in Sh506 without assuming $(*)_\theta$ which makes our approach suboptimal. However, the proof we will give is cleaner, and is basically the same as the ones from the Abraham-Magidor chapter in the Handbook of Set Theory. Further, my motivation is wanting to do pcf theory at singular cardinals $\mu$ such that $\mu^+$ is Jónsson, so it all works out.

Anyway, my plan future entries is to work through:

The existence of universal cofinal sequences (assuming just weak saturation);

The Trichotomy Theorem (assuming $(*)_\theta$);

The pcf theorem (assuming $(*)_\theta$).