the first paper

PDL for Ordered Trees

Loredana Afanasiev, Patrick Blackburn, Ioanna Dimitriou, Bertrand Gaiffe, Evan Goris, Maarten Marx, Maarten de Rijke.
Journal of Applied Non-Classical Logics 15(2): 115-135 (2005)

This is the first paper I participated in as a master's student in the ILLC.

official statement of my PhD project

Many combinatorial principles attain their set theoretic strength only in the presence of the axiom of choice. Without it, it is possible that small cardinals like ω₁ can have large cardinal properties like being measurable or satisfying strong partition properties. The intended research area is to determine the consistency strengths of various infinitary combinatorial properties with respect to the Zermelo-Fraenkel axioms ZF, i.e., without the axiom of choice. This is a wide field, since the classical questions about generalised Chang's conjectures, ℵ_ω being Jonsson or Rowbottom, mutual stationarity, pcf-theory and others can be examined from this perspective.

Peter Koepke is working in this area together with Arthur Apter, CUNY. The PhD project will be embedded into this collaboration. The work will combine forcing techniques and inner model arguments. We indicate this in the case of a simple example: To prove the conjecture that Chang's conjecture (ω₃, ω₂) --> (ω₂, ω₁) with ZF is equiconsistent with the existence of an ω₂-Erdos cardinal, one uses forcing techniques of Apter and inner model techniques of Koepke. An ω₂-Erdos cardinal is collapsed by a symmetric subcollapse of a Levy collapse to ω₃. Conversely, one applies the Chang property, moves to a submodel with the axiom of choice which contains the Chang substructure, and applies known core model techniques. The PhD project will involve studying several such properties of increasing degrees of complexity.

New space!

Welcome to my new personal space. First, I'll transfer the old files as posts from my previous webpage. Then I'll give you a proper welcome.

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.

AD or AC?

the diagram

A fancy diagram of the consequences of real choice. This was part of a project during my masters' studies at the ILLC concerning the relationship between choice and determinacy. I gave a talk on that project at the Logic Colloquium 2005 in Athens. Later on it may be modified to include RGB colouring according to compatibility with determinacy. Red will mark an incompatible with AD fragment, green a fragment that is also fragment of AD and blue will mark consistency and/or independence from AD. Any comments will be very welcome. References will be given after request.

second order arithmetic talk

GLLC 14½ talk: Topological regularities in second order arithmetic

I am very happy that I was invited at the “Games in Logic, Language and Computation 14½" meeting at the ILLC in Amsterdam (where I finished my masters) to give this talk. The talk is based on work by Peter Koepke and Michael Möllerfeld. It shows that ZFC is equiconsistent with full second order arithmetic (SOA) plus all sets of reals are Lebesgue measurable, have the Baire property and the perfect set property. I helped finish off the forcing side (which admittedly is a bit disappointingly easy). These are the slides.