research update

The plan for last semester was to finish a chapter in choiceless higher Chang conjectures. There was a change of plans when (luckily) Arthur Apter asked me to give a talk explaining the construction in Gitik's "All uncountable cardinals can be singular". Trying to simplify the construction I was busy with it all semester. I managed to simplify it only a little bit, in that requirement (4) in the definition of P₂ wasn't really necessary and the proof of ZFC-powerset in the generic extension is replaced by a proof that the forcing is pretame. Moreover now the construction is more susceptible to modifications. This together with some applications of this construction are going to be part of my upcoming PhD thesis. Recent estimates for my graduation are April to May 2009.

Right now I'm finally finishing up the chapter on the higher Chang conjectures. I will give a talk about that in the Colloquium Logicum 2008, next week in Darmstadt. Also, it turns out that Gitik's "everything singular"-method had something to say about these Chang conjectures too. What a great construction!

research update

Last semester I've learned a bit more of fine structure theory from Peter Koepke's lecture and the next I'll be employed for the graduate seminar which will focus on fine structure theory. Still though I am reading more about Radin forcing whilst trying to finish what I hope to be an entire chapter in my PhD thesis. It's going to be an analysis of the consistency strengths of as much higher choiceless Chang's conjectures I possibly can (and it seems there will be at least a narrow gap between the upper and lower bound for most of them). Last summer I have read and almost entirely understood Gitik's paper "All uncountable cardinals can be singular", a very interesting and involved construction indeed.

In general, I do set theory without choice, and in particular infinitary combinatorics and large cardinals without choice. As Mitchell Spector says in his "Ultrapowers without the axiom of choice", most set theorists strongly prefer working with choice when dealing with large cardinals because of the importance of the ultrapower construction and the fact that the fundamental theorem for these constructions fails without DC. Many of the symmetric models I work with satisfy DC and more. For me the ones that don't are a welcome challenge.