Non-wellfounded Sets
Second Semester 2004-2005, period c, June 2005
Instructors: Benedikt Löwe ( bloewe@science.uva.nl ), Maricarmen
Martinez (mmartine@science.uva.nl),
and Fabrice Nauze ( fnauze@science.uva.nl).
The topic: The aim of this project is to
familiarize the student with the theory of non-wellfounded sets. In this
version of set theory, the universe includes all of the usual well-founded sets
allowed by Zermelo Fraenkel theory, and many more. For example, the theory
guarantees that there is a set x satisfying the
constraint x = { x}. The fact that the theory allows for such
sets makes it very suitable for modeling several kinds of circular phenomena of
interest in computer science (i.e. data structures), linguistics (i.e.
self-reference), philosophy (i.e. paradoxes), and other fields. While we will
choose one application to look at in some detail, in this project we will
mainly focus on some mathematical aspects of the topic. Some of the relevant
questions are: What does the universe of non-wellfounded sets look like? How is
it constructed? Suppose that you have two descriptions of sets given in terms
of equations (constraints) like the one given above. When do the two
descriptions define the same set?
Background: The prerequisite for this project is a good knowledge of set theory equivalent to the Axiomatic Set theory course.
Main reference: The project will be based on part III of Barwise & Moss' book Vicious Circles.
Evaluation: Each
student will have to present some part of the textbook and submit at the end of
the project a written version of his or her presentation. This write-up, which
is due on
Schedule and presentations: The assignment of
presenters to presentations in this schedule is preliminary. In principle,
students can arrange to swap topics, but the instructors should be notified
about such changes by the end of the first meeting. You can expect the covered
material to increase in mathematical sophistication each week. Each meeting
should take up to 90 minutes. Thus, you should plan for your presentations to
be about 45 minutes long, so to allow for questions and discussion.
|
Meeting date and place |
Topic |
Presenter |
Relevant materials, references, etc. |
|
Tuesday, June 7 |
Introduction and motivation: two applications of non-wellfounded sets. |
Fabrice and Maricarmen |
· Slides about the muddy children. · Slides about Hypergame. ·
Gerbrandy and Groeneveld’s paper about
information update: click here,
search for |
|
Tuesday, June 14 |
Chapter 6: The solution lemma |
Julio |
|
|
Tuesday, June 21 |
Chapter 7: Bisimulation |
Caroline |
|
|
Tuesday, June 23 |
Chapter 8: Substitution |
Joost |
A correct version of the proof of Theorem 8.1 can be found
at |
|
Wednesday, June 29 |
Chapter 9: Building a model of ZFA |
Yurii |
|