5/13/2016

Day 2 of the CUNY Graduate Students in Set Theory Conference

Kameryn Williams started the day out with a talk on minimal models of second order theories. He discussed some second order alternates of ZFC: GBC and KM. While GBC, the weaker of the two extensions has least models, KM does not. This involved an intriguing exploitation of inaccessible cardinals and technique that allows one to build ill-founded models over a fixed well-founded part. There are number of good open questions surrounding this result in the form of slight weakenings on "least" or the consideration of specific nice families of models.

After this, James Walsh talked about extracting ordinal notations from theories. There are countable ordinals that can be associated with different theories of arithmetic, but in the past this has always been ad hoc and relied on seemingly inessential syntactic features. The work Walsh presented is exploring a way to more canonically and uniformly find these ordinals. You have to love math, for perfectly acceptable reasons that Walsh explained to me later, there are objects in this theory called worms. One theorem was proved by "riding the worm." Amazing. In all seriousness though, this work is pretty cool; I always found epsilon_0 interesting and this work is a mega extension of that. Moreover, once he has obtained an ordinal notation system in the manner he described, he has a computable way of generating the cantor normal form of the ordinal.

After the coffee break, Miha Habic gave a talk on grounded Martin's axiom. Even though this was a forcing talk about a fairly technical (though natural) generalization of MA, Habic made a thoroughly understandable 20 minute talk. There was brief mention of a hierarchy of Martin type axioms. It looks like it may eventually be as messy as the choice hierarchy. grMA doesn't do quite as much as MA, but it can still do a lot, and it seems to be quite a bit less restrictive than MA. The talk ended on an intriguing note about the grounded PFA and the possibility of it having weaker consistency strength than PFA.

Finishing out the morning, we had a philosophy student from Italy. Maria Foglioni gave a defense of logic's basic rules of inference. While Foglioni was a fine speaker, I have a hard time getting excited about this material. Basically, some people quasi-formalized (in that vague way only possible in philosophy) the idea that you can't convince someone that logic makes sense. The only way to do that would depend on them accepting logic in the first place. On the other hand, this guy McGee wrote up what he thought was an empirical counter example to modus ponens. Except the second line of the MP isn't strictly speaking true, only probable, and on close inspection the third line doesn't actually line up with the consequent in the first. Great. Again, Foglioni spoke well on the material, but It was still just odd to me.

We broke for lunch. More Asian food. Mathematicians always want Asian food. (Unless you are with Greek mathematicians in which case they want Greek food)

After lunch, Shehzad Ahmed talked about Jonsson cardinals and PCF theory. PCF theory is very hard to get into a 20 minute talk. So much notation, so many ideals. Still, Ahmed did an admirable job  motivating what was happening and why we should be interested in the old question of whether or not successors of singular cardinals can be Jonsson. It is strange to see a talk on Jonsson cardinals and have basically no familiarity with the techniques discussed. Choice based combinatorics and determinacy based combinatorics are very different animals. We did find some common ground though, and I think some of the constructions from the choice context will have relevance on the consistency strength questions I have about Jonsson cardinals without choice.

Francis Adams spoke next about definable graphs and dominating reals. Essentially, he wants to know when a graph can be countably colored, allowing for this to happen in a generic extension. He obtains a nice sufficient condition that is essentially combinatoric. The interactions between this question and the question of definable colorings are not straightforward however, if they exist at all. So this may represent an effectively new kind of question about graphs.

After this we broke for an "open problem" session. We all submitted our favorite problem and then people voted and we took the top three or four. I continued to evangelize for recursive tic-tac-toe and that got picked. So we spent a good bit of time playing with recursive tic-tac-toe, which was great. Some joker submitted the PFA conjecture, and some other jokers voted for it. Wisely, no one attempted it.

Hossein Ramandi gave a chalk talk on the minimality of non sigma-scattered linear orders. This is a large class of linear orders which extends the ordinals. Surprisingly, the talk veered immediately into supercompact cardinals and some intense large cardinal forcing constructions. Its always surprising which quesions are connected to each other. Ramansi ended up focusing on a particular kind of linear order and is able to get some results if these are of size at most omega_1.

Ian Smythe closed out the conference with "Towards a Local Gowers Dichotomy." Yay Ramsey theory. Having finally learned why happy families are called happy, I'm not sure if I'm amazed or dissapointed. Happy is not the opposite of m.a.d. These local versions of the Ramsey property that Smythe discusses are interesting, and I think they are true for the combinatorial results I have been proving about R (as a space. The older results are on the integers and are coded into reals). Smythe introduced a theorem of Rosendal which is able to prove a Ramsey type theorem in a functional analysis context. This is pretty exciting as there is not even a straight forward pigeon-hole principle in this context. Smythe got to briefly his local version of this problem, but ran out of time. I wish he had been able to talk more about his part, it sounds very interesting.