Seminar on Mathematical Logic


Year: 2010
ECTS: 6
Lecturers: Dr. George Barmpalias and Dr. Benedikt Löwe

Meetings: The dates for the meetings for this course will be a proper subset of those mentioned in the schedule. This is because of some conflicts with other courses. These will be: 2Feb, 23Feb, 2Mar, 9Mar, 23Mar, 6Apr, 13 Apr, 20Apr, 27Apr, 4May, 11May, 18May, 25May.

There will be NO meetings on: 9Feb, 16Feb, 16Mar, 30Mar.

Description: The students attending this course will be given a collection of selected results from the areas of Descriptive Set Theory, Algorithmic Randomness and Computability theory. Each student is assigned a topic/result which (s)he is going to master and eventually present it before all students in the class and the two lecturers. Often the preparation and study of a result requires substantial work, especially on covering the background theory on which the result lies upon.

The topics and students responsible for them in the class of 2010, along with the dates of their talks are as follows:
  • (23 Feb 2010) Σ03 and Σ04 determinacy (references 4,7).
    Speaker: Martijn Baartse

  • (9 Mar 2010) Cancelled

  • (23 Mar 2010) Π11 Uniformization (Kondo) (reference 1)
    Speaker: Gabriela Rino

  • (6 Apr 2010) A new proof of Π11 uniformization (Blackwell) (reference 9).
    Speaker: Helene Tourigny

  • (13 Apr 2010) First Periodicity Theorem (references 1,8)
    Speaker: Zhenhao Li

  • (20 Apr 2010) Second Periodicity Theorem (references 1,8)
    Speaker: Purbita Jana

  • (27 Apr 2010) Mauldin's Theorem: the set of nowhere differentiable continuous functions form a Π11-complete set (reference 1).
    Speaker: Adam Lesnikowski

  • (4 May 2010). Turing incomparability in Scott sets (reference ).
    Speaker: Nicola Di Giorgio

  • 11 May 2010. Extracting information is hard (reference 2)
    Speaker: Tom Sterkenburg.

  • (18 May 2010) Unprovability of Σ04 determinacy in second order arithmetic (references 5, 10)
    Speakers: G. Barmpalias and B. Loewe
Assesment: Students will be assessed on the basis of their preparation of their presentation and their performance during their talk.


Prerequisites: The project is particularly suitable for those who have some mathematical maturity and can study with some degree of independence. Some interest in definability theory (and in particular Descriptive Set Theory, Algorithmic Randomness and Computability theory) is highly desirable, as well as feeling comfortable with self-study.

Bibliography:

  1. Classical Descriptive Set theory, by A. Kechris (Graduate texts in Mathematics)

  2. Extracting information is hard, by J. Miller, to appear in Adv. Math. pdf

  3. Turing incomparability in Scott sets by A. Kucera and T. Slaman, Proc. Amer. Math. Soc., 135:3723--3731, 2007. Preprint

  4. Proving determinacy, Draft of the unpublished book of D. Martin.

  5. Higher set theory and mathematical practice by Harvey M. Friedman, Ann. Math. Logic 2 1970/1971 no. 3, 325--357.

  6. Infinite games of perfect information by Morton Davis, Advances in game theory (1964) 85--101 Princeton Univ. Press, Princeton, N.J.

  7. ZF \vdash Σ04 determinateness by J. B. Paris, J. Symbolic Logic 37 (1972), 661--667.

  8. Descriptive set theory, by Y. Moschovakis (Studies in logic and the FOM)

  9. Infinite games and analytic sets, by David Blackwell, Proc. Nat. Acad. Sci. U.S.A. 58 1967 1836--1837

  10. The limits of determinacy in second order arithmetic, by A. Montalban and R. Shore, Preprint