Infinite Games

Sommersemester 2004

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. Sigma⁰₂ 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 omega₁ 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 aleph₁ 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 aleph₁ is coded by a real). AD implies that aleph₂ 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.