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).

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