I’m interested in the following open question:
Is there a singular cardinal such that 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 is an ideal over some cardinal . We say that is -indecomposable for a regular cardinal if, for every -positive set and every function , there exists a such that is -positive.
Definition: We say that is -weakly saturated if any collection of -many -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 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 is a cardinal with an ideal over with the property that there is some regular such that is -indecomposable and -weakly saturated. Then if is the completeness of , there is a stationary set such that every sequence of stationary subsets of with reflects simultaneously.
We then have the following result due to Shelah:
Theorem(Shelah): If is a singular cardinal with Jónsson, then there is a regular cardinal , and an ideal which is -complete, -indecomposable, and -weakly saturated.
Now, it’s known that the existence of such an ideal is consistent at for 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 is an ideal which is -weakly saturated and -indecomposable, then any collection of -many -positive sets has an unbounded subcollection with non-empty intersection. Say that the ideal satisfies if this holds. Suppose now that is a singular cardinal and carries an ideal satisfying . Define a map
given by .
Let be the Rudin-Keisler projection of to by way of . Then we see that also satisfies (also, is -weakly saturated). If is Jónsson, a theorem of Shelah’s tells us that must be the limit of inaccessible cardinals, and hence an -fixed point. So, we can try to do pcf theory at with this ideal 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 . So in the next blog entry, I plan on showing this (and then some). So now that we have trichotomy and -directedness of , 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 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 such that 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 );
The pcf theorem (assuming ).