(Department of Mathematics, University of Bonn)

I will cover different aspects of the set theory of infinite games. Starting with the traditional approach of Gale and Stewart, I discuss the basics of the theory of determinacy. Then I will define more general games (imperfect information games, games with more players, etc.) and develop a theory of determinacy for these games.

**First Lecture**(*May 6th, 2004*). Properties of games: perfect information, perfect recall, number of players. Baire Space and its topology. Motivation: Harrington's Covering Game (without proof).**Second Lecture**(*May 13th, 2004*). Pointclasses. Closure properties of pointclasses. The Borel hierarchy. The projective hierarchy. The theorem of Gale and Stewart.- (
*May 20th, 2004*).*No lecture: Ascension Day.* **Third Lecture**(*May 27th, 2004*). The Hausdorff Difference Hierarchy. Combinatorial labellings.**Sigma**^{0}_{2}sets don't admit combinatorial labellings. Combinatorial labellings for differences of open sets.- (
*June 3rd, 2004*).*No lecture: Pentecostal Break.* **Fourth Lecture**(*June 8th, 2004*). [Note: Tuesday!] Strategic Trees.**AD**and**AC**: construction of a nondetermined set. Proof Sketch of Davis' theorem on the perfect set property.**AD**implies**AC**_{N}(**R**). Inconsistency of**AD**_{P(R)}.- (
*June 10th, 2004*).*No lecture: Corpus Christi.* **Fifth Lecture**(*June 11th, 2004*). [Note: Friday, 14-16!] Coding countable ordinals by reals. Inconsistency of**AD**_{omega1}. Regularity of*omega*_{1}under**AD**. Completing Davis' proof on the perfect set property. Turing degrees. Recursive ordinals. Ordinals recursive relative to an oracle.**AD**implies that all ultrafilters are*sigma*-complete (proof sketch). The Martin measure.**AD**implies that the Martin measure is an ultrafilter.**Sixth Lecture**(*June 17th, 2004*).**AD**implies that*aleph*_{1}is a measurable cardinal. Some metamathematical consequences: sharps, inner models with measurable cardinals. Coding of countable ordinals as real numbers. The set**WO**. Prewellorders and the prewellordering property. General Boundedness Lemma for prewellordered pointclasses.**Seventh Lecture**(*June 24th, 2004*). Solovay's Lemma (Every subset of*aleph*_{1}is coded by a real).**AD**implies that*aleph*_{2}is a measurable cardinal. Finitary descriptions of continuous functions.**Eighth Lecture**(*July 1st, 2004*). Continuous functions and Lipschitz functions. The Lipschitz game. The Wadge game. Wadge's Lemma. The Wadge jump.**Ninth Lecture**(*July 8th, 2004*). Differences between the Wadge and the Lipschitz hierarchy.**SLO**and the perfect set property. Lebesgue measure on Baire space. Flip sets and Lebesgue measure. The Martin-Monk Theorem.**Tenth Lecture**(*July 15th, 2004*).*Theta*. The length of the Wadge and the Lipschitz hierarchy. Basic properties of the Lipschitz hierarchy: successors are selfdual, countable limits are selfdual, selfdual degrees are countable limits. The Wadge and the Lipschitz hierarchy. The Wadge hierarchy on Cantor space.**Eleventh Lecture**(*July 22nd, 2004*). Imperfect information games. Blackwell determinacy.**AD**implies Blackwell determinacy. Some consequences of Blackwell determinacy (without proof). The Hierarchy of Norms. Wellordering proof under**AD**. The Blackwell Hierarchy of Norms.- (
*July 29th, 2004*).*No lecture: Logic Colloquium 2004.*

Last changed: July 22nd, 2004