Some Projects

I felt like I’ve been neglecting this thing for a while, but I do have a good reason! I managed to take and pass the algebra comprehensive exam here, which means I only have to pass one more comprehensive, take one more course next semester, and finish up my dissertation proposal in order to advance to candidacy. So, I feel pretty okay right about now. Since I’ve started delving into set theory stuff again, I do plan on updating this blog as a way to work through a few projects that I want to work on this semester.

Club Guessing and Jónsson Cardinals: Let $\mu$ be a singular cardinal, and let $S\subseteq \mu^+$ be stationary. We say that a sequence of the form $\bar{C}= \langle C_\delta : \delta\in S\rangle$ an $S$-club system if $C_\delta\subseteq\delta$ is club in $\delta$ for each $\delta\in S$. Now, let $\bar{I}=\langle I_\delta : \delta\in S\rangle$ be a sequence of proper ideals on $\delta$. We can use these ideal to measure how close the system $\bar{C}$ is to being some sort of guessing sequence. In Cardinal Arithmetic, Shelah defines the ideal $Id_p(\bar{C},\bar{I})$ by saying that $A\in Id_p(\bar{C},\bar{I})$ if and only if there is some club $E\subseteq \mu^+$ such that $A$ runs away from $C_\delta$ (with respect to $I_\delta$) stationary often. That is, for every $\delta\in S\cap E$, the set $A\cap E\cap C_\delta$ is $I_\delta$-positive. It turns out that, if $\mu^+$ is Jónsson, then the ideal $Id_p(\bar{C}, \bar{I})$ must have some nice properties. So, I plan on using this place to work through chapter III of Cardinal Arithmetic where much of this material is located.

Stationary Reflection and the failure of SCH: Let $\kappa$ be measurable, and let $\mathbb{P}$ be vanilla Prikry forcing over a uniform, normal measure over $\kappa$. In the extension, $\kappa$ becomes singular of countable cofinality, and the easiest way to exhibit a failure of SCH is to blow up $2^\kappa$ to $\kappa^{++}$ and follow it up with Prikry forcing. But, since $\mathbb{P}$ is $kappa^+$-c.c., it follows that $S=(S^{\kappa^+}_\kappa)^V$ is still stationary in the extension by Prikry forcing. Noting that $S\subseteq S^{\kappa^+}_\omega$ in the extension, by upward absoluteness of non-reflection, it follows that $S^{\kappa^+}_\omega$ has a non-reflecting stationary subset. In fact, this argument shows that any sort of Prikry forcing on a single cardinal will have the unfortunate consequence of producing a non-reflecting stationary subset at a successor of the singular that we forced over. So, the natural question is whether or not we can have a failure of SCH and at some cardinal $\kappa$ and have full reflection at $\kappa^+$.

In his thesis, Assaf Sharon gives an affirmative answer to this. My understanding of the proof is the following: The basic idea is to first do an iterated Laver preparation and produce an $\omega$-sequence of Laver-indestructible supercompact cardinals. Then, blow up the power of the limit by using Gitik’s long extender forcing. Since the tail forcings are highly closed, we should be able to use something like generic supercompactness to show that $\kappa^+$ is close to fully reflecting. Then, in order to take care of the remaining bad sets, Sharon defines a sort of Prikry club shooting, and iterates it. The resulting model is one in which $\kappa$ is strong limit and singular of countable cofinality,  $2^\kappa\geq\kappa^{++}$ and $\kappa^+$ fully reflects.

Some motivation for using the above method: First of all, the long extender forcing is not $\kappa^+$-c.c. (though $\kappa^+$ is preserved in the extension), so that’s the best hope for using a Prikry-type forcing and ending up with full reflection at a successor of a singular at which SCH fails in the extension. In a recent paper, Sinapova and Unger have shown that, if one forces with a modified version of the extender based forcing over $\kappa$, and if the “bottom” cardinal in the sequence is indestructibly supercompact, then the closure of the tail forcing can be used to project a bad scale up to $\kappa^+$. Furthermore, in Magidor’s paper, in which he forced a model of $refl(\aleph_{\omega+1})$, he used the notion of generic supercompactness to accomplish this (along with some results due to Shelah about stationary sets in forcing extensions). He mentions, however, that the original paper utilized club shooting to get rid of the bad sets, and that this ended up being unnecessary thanks to a comment from Shelah. Finally, the big motivator of this is the result due to Solovay that, if $\kappa$ is the limit of a sequence of supercompacts, then full reflection holds at $\kappa^+$. Putting all of this together, we see that this method should preserve enough “supercompactness” in some useful sense in the extension, that it’s possible to end up with a model of full reflection at a successor of a singular where SCH fails. At least, that’s the intuition.

Anyway, those are the two big things that I’m going to be using this blog to go through in the near future. The primary motivation here is the problem of whether or not it’s possible for a successor of a singular cardinal to be Jónsson. It turns out that, in order to have one floating around, a number of extremely stringent conditions must be satisfied, including a failure of SCH at said singular and having full reflection at said successor (in addition to instances of simultaneous reflection). In other words, if such a thing is consistent, it will have relatively high consistency strength.

As a warning, if I said anything horribly wrong, this was written completely off the top of my head and right before heading to bed (I teach at 7:30am).