Mathematics, University of
Instructor: Dr Benedikt Löwe
Time: Thursday 10-12
Place: SR A
Course language: English
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 ACN(R).
Inconsistency of ADP(R).
- (June 10th, 2004). No lecture: Corpus Christi.
- Fifth Lecture (June 11th, 2004). [Note:
Friday, 14-16!] Coding countable ordinals by reals.
Inconsistency of ADomega1. 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
- 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
Last changed: July 22nd, 2004