Introduction to symmetric models and the approximation lemma.

Last June, 2009, I gave a lecture on how to construct symmetric models at the Universiteit van Amsterdam. One student attended, Rogier Jacobs, and him I have to thank for attending and giving his input on the first version of the lecture notes. Here you can find the 'final' version of these notes:

Tutorial on symmetric models and the approximation lemma.

The notes assume basic knowledge of set theory and forcing, as given in Kunens "Set Theory", up to Chapter VII. In them it's shown how to construct symmetric models, and three basic examples are given. These are:

• the Feferman-Lévy model, where ω₁ is singular and the reals are a countable union of countable sets,
• a generalisation of the Jech model, where a successor cardinal is measurable, and
• a similar model to the Jech model, where a successor cardinal is weakly compact.

Soon there will be a follow-up note, with a small fragment of Gitik's construction in "All uncountable cardinals can be singular". In this small version of it, only one strongly compact cardinal is used (as opposed to the class of strongly compacts in Gitik's model), and the result is a symmetric model in which ω₁ is singular. Clearly this isn't an interesting result per se, but it will be presented as an introduction to the full Gitik construction. So there will be more notes appearing here, with the final goal of presenting Gitik's model where all uncountable cardinals are singular.

Comments on the pdf, and solved exercises for correction are very welcome!

Combinatorial principles in ZFC

The mathematical institute of Bonn has moved to a new, better building and today is the first day that our offices are usable. To celebrate, here's a chart of some direct implications between combinatorial principals in ZFC. This will be updated to include more principles.

welcome!

Welcome to my new personal space. This is intended as a long term solution to my professional internet space needs, independent from the university with which I am affiliated at any time. Don't get me wrong, I am a team player and currently am devoted to the mathematical logic group of Bonn:



But official spaces tend to be a little formal and I don't prefer formal. This website is in fact a blog. I chose this format because it's the easiest to update and it looks very good with minimal effort. On the right hand side you can use the "labels" as my menu. I plan to post news on my research projects and plans every semester or so, in hopes of attracting future collaborators. Please, feel free to email me or leave comments on my posts.

Thanks for reading,
Ioanna.

curriculum vitae

I was born on November 27, 1978 in Thessaloniki, Greece, where I grew up. I graduated in pure mathematics from the mathematics department of the Aristotle University of Thessaloniki on July 22, 2003. On September 1, 2003 I started my master's in logic (MoL) at the ILLC in the Universiteit van Amsterdam. I wrote my master's thesis under the supervision of Dr. Benedikt Löwe on the topic of set theory, in particular forcing and the negation of the axiom of choice. I graduated my masters on January 30, 2006, after having started my PhD project at January 2, 2006 at the mathematical institute of the Universität Bonn. My advisor in Bonn is Prof. Peter Koepke and my PhD project is about infinitary combinatorics without the axiom of choice. Most of my research involves still forcing and the negation of the axiom of choice. Being a student of Peter Koepke I also have learned some things about core model theory.

I am still in Bonn at the moment and I am an active member of the mathematical logic group here. I take care of the website and have helped organise all conferences and workshops that our group was involved in, as long as I'm here. I also prepared extensive workshop reports for three logic meetings that took place here in Bonn during my stay. I have taught a couple of tutorials on basic logic and set theory but mostly I have been busy with the graduate logic seminar, where selected topics in set theory are presented. Initially I was funded by these teaching duties but now I am funded by a DFG-NWO collaboration grant between Benedikt Löwe in Amsterdam and Peter Koepke in Bonn, entitled "Infinitary combinatorics without the axiom of choice". This semester, together with Arthur Apter, Peter Koepke, and Benedikt Loewe, we are organising the first workshop of this collaboration project.

I have co-authored a couple of papers published in logic journals, I refereed one paper in a mathematical logic journal, and I have a few projects on the way (see label "publications"). I am currently writing up my PhD thesis, that involves a detailed study of several patterns of singular cardinals under the negation of the axiom of choice, and some results on the surprisingly low consistency strength of higher Chang's conjectures without the axiom of choice.

I am a member of the ASL (Association for Symbolic Logic), BIGS (Bonn International Graduate School), and of the DVMLG (Deutsche Vereinigung für Mathematische Logik und für Grundlagen der Exakten Wissenschaften).

Finally, my native tongue is Greek, and my mother's native is Chilean Spanish (I was bilingual as a young child). I am very fluent in English, and I have some intermediate to beginner skills in Dutch and German.

books I want

This is a list with books I really want to own but still I don't. My birthday is at the 27th of November :)

Handbook of Mathematical Logic, edited by Jon Barwise,

Set theory by Thomas Jech, 3rd edition,

Set theory by Thomas Jech, 1st edition,

Set theory, an introduction to independence proofs by Kenneth Kunen,

Consequences of the axiom of choice by Paul Howard and Jean E. Rubin,

The axiom of choice by Thomas Jech, Thanks to Ben, Daisuke, Merlin, and Thilo!!

Constructibility by Keith Devlin, the '84 book!

master's thesis

"Strong limits and inaccessibility with non-wellorderable powersets''

My master's thesis at the ILLC. The thesis discusses four notions of inaccessibility that are equivalent in ZFC. It is shown that three of them are pairwise not equivalent in ZF alone. The separation of the fourth notion comes from Blass and it's discussed in the resulting paper. The main technique used is the technique of symmetric models. Its publication number in the ILLC publication series is MoL-2006-3.

second order arithmetic

Topological regularities in second order arithmetic.

Ioanna Dimitriou, Peter Koepke, Michael Möllerfeld
In preparation.

This is a project of Peter Koepke with Michael Möllerfeld which I helped round up. See also these slides.

results from my master's

Inaccessible cardinals without the axiom of choice.

Andreas Blass, Ioanna M. Dimitriou, Benedikt Löwe
Fundamenta Mathematicae , vol.194, pp. 179-189

This is the paper with the main results from my master's thesis plus the answer to my open question, given by Blass.

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.