Justin Moore started off day 2 with a tutorial on "Iterated Forcing and the Continuum Hypothesis." The goal is try and develop forcings which preserve CH while obtaining partial consequences of combinatorial principles which in their full strength contradict CH. As most of the forcings required to instantiate these principles require iterated forcing, the bulk of this talk focused on possible conditions on a poset which guarantee that even after an iteration of some length, new reals have not been added. This is is more subtle than it may initially seem, as the act of iteration can add reals even when none of the forcings being iterated do so on their own. The first thing one of these posets must do is preserve stationary subsets of aleph_1. There is a reasonably straightforward example which adds reals after being iterated, and the precise reason it does so is because it fails to preserve stationary sets. This leads to the notion of proper forcing. However, this is not enough. A more subtle example shows that an iteration of proper posets which individually do not add reals can still add in the end. One answer is to look at something called completely proper posets. This is a technical definition, but the upshot is that for a completely proper poset, certain information about its generic don't depend on the model being forced over. A theorem of Shelah shows that countable support iterations of completely proper posets with one of two extra assumptions are still completely proper, and thus do not add reals. Interestingly enough, these extra assumptions really are separate and not obviously subsumed by some more general assumption. Again, these advanced forcing talks really are beyond my skill set with forcing, but I am enjoying soaking up the culture and philosophy around them.
In first part of the afternoon, Manachem Magidor finished up his tutorial on "Compactness and Incompactness Principles for Chromatic Numbers and Other Cardinal Sins." This talk started off with examples of both compactness and incompactness theorems for bounds on chromatic number under different settings. If stationary reflection fails, then bounds for chromatic numbers are not compact. On the other hand, in the presence of a countably complete ultrafilter, having chromatic number bounded by aleph_0 is a strongly compact property. After this, Magidor talked some of his joint with with Bagaria in which they relate compactness properties to other abstract compactness principles. While compactness principles are obtainable for large cardinals, getting them on small cardinals, say aleph_n for some n, is another matter, and involves large cardinals of noticeably higher consistency strength. I do want to check out how some of these chromatic number play out under AD.
Omar Ben Neria went next, He spoke on "The Distance Between HOD and V." The question is, up to consistency, how small can HOD be? There are a couple ways to measure this, one way for HOD to be large is for it satisfy a kind of coloring lemma, and a way for HOD to be small is for cardinals in V to be large cardinals in HOD. Woodin has shown from an extendible cardinal that a version of Jensen's dichotomy holds between V and HOD. Ben Neria and Unger were able to strengthen a result of Cummings, Friedman, and Golshani which shows that it is consistent for covering to fail for HOD. In fact, it is consistent that every uncountable regular cardinal in V is huge in HOD. One important piece of technology for the proof is weakly homogeneous forcings. These ensure that the HOD of the forcing extension do not escape the ground model. Another important technique is to use iterated forcing with non-stationary support. I don't understand the intense modified iterated forcing going on here, but the result is nice. I asked about getting all uncountable cardinals of V to be Jonsson cardinals in HOD, and it seems that the their construction does achieve this, although at the cost of at least a measurable cardinal at the moment. I'm not sure what an optimal consistency strength would be. Ben was kind of enough to point to me some information which may help me to attack the consistency problem of all uncountable cardinalsbelow Theta being Jonsson in L(R).
To finish out the day, Nam Trang gave talk on the "Compactness of Omega_1." Under the axiom of determinacy, omega_1 has some supercompactness type properties. For some fixed set X, Trang was interested in finding minimal models satisfying the claim "omega_1 is X-supercompact." Even under ZF, this implies some failures of square. More than just implying that omega_1 is strongly compact with respect to the reals, it turns out that AD and thus property are equiconsistent. The statement that omega_1 is supercompact with respect to the reals is equiconsistent to an even stronger form of determinacy. Moving up with the power set operation on R, the compactness properties become equiconsistent with stronger forms of determinacy still. Trang, in joint work with Rodriguez was able to show that there is a unique model of certain form which thinks that omega_1 is R-supercompact. I always enjoy hearing about these higher forms of determinacy and their corresponding models, and these equiconsistency results add yet another connection between the theory of large cardinals and the theory of determinacy.
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