Admissible computability and set theory (TU Wien)

Blocked course, Summer 2024 (around 35 hours of lectures to take place last two weeks of May).

Graduate course. The goal is to study some basic admissible set theory and applications to game theory, descriptive set theory, and other areas. We may cover some recent joint results with various people.

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Session 1: Preliminary meeting. Friday 05/17, 2pm–3pm. Freihaus large seminar room, 5th floor, green part. Meeting to discuss and decide the course contents and schedule. 

Session 2: Saturday 05/18, 10am–4pm. Green seminar room, Freihaus. 

Contents: History; generalized computability; Kripke-Platek set theory; Sigma-collection, Sigma-replacement, Delta-separation, Sigma-recursion; Ville's lemma; the next admissible set; the Barwise-Gandy-Moschovakis theorem.

References:

Barwise (1975), Admissible sets and structures.

Barwise, Gandy, and Moschovakis (1975). The next admissible set. In: Journal of Symbolic Logic.

Session 3: Tuesday 05/21, 2pm–8pm. Green seminar room, Freihaus. 

Contents: Skolem hulls; the Kleene-Suslin Theorem; the Church-Kleene ordinal.; the Spector-Gandy theorem; the strong Spector Gandy theorem; monotone Pi11 inductive definitions.

References:

Barwise (1975), Admissible sets and structures.

Moschovakis (2009), Descriptive Set Theory.

Sacks (1990). Higher Recursion Theorem.

Session 4: Thursday 05/23, 2pm–4pm; 5pm–8pm. Dissertation room (8th floor). Contents: the Gale-Stewart theorem; complexity of winning strategies for open and closed games; Svenonius' theorem; Solovay's proof of F_sigma determinacy.

References:
Barwise (1975), Admissible sets and structures.

Moschovakis (2009), Descriptive Set Theory.

Tanaka (1990), Weak axioms of determinacy and subsystems of analysis, I. In: Annals of Pure and Applied Logic.

Exercise session 1: Saturday 05/25, 10am–4pm. Green seminar room, Freihaus. 

Session 5: Monday 05/27, 1pm–3pm. 5th floor discussion room. Contents: monotone inductive definitions; recursively inaccessible ordinals; Mostowski's Axiom Beta; reflecting ordinals.

Barwise (1975), Admissible sets and structures.

Aczel and Richter's 1974 paper.

Session 6: Tuesday 05/28, 1pm–4pm; 6pm–8pm. Room: 1–4pm in the Goldenes Lamm lecture room; 6–8pm dissertation room (8th floor). Contents: the Aczel-Richter theorem; Gostanian's theorem; Aanderaa's theorem; Solovay's theorem; complexity of strategies for F_sigma games.

Aczel and Richter's 1974 paper.

Gostanian, the next admissible ordinal, AML.

Aguilera, the order of reflection. JSL.

Aguilera and Lubarsky, on winning strategies for F_sigma games. Preprint.

Exercise session 2: FRIDAY 05/31, 2pm–$\infty$. (Note change of date and time.) Zeichensaal 3, Freihaus. Speakers: Kouptchinsky (\Sigma_m separation), Stepanov (recursively Mahlo ordinals).