Uniformity aspects of determinacy
Logic and Set Theory Seminar
8th January 2019, 3:00 pm – 4:30 pm
Howard House, 4th Floor Seminar Room
We consider uniformity aspects of determinacy for some low-level point-classes. The formal framework for this is Weihrauch reducibility, which will be introduced. We distinguish two cases: For games on Cantor space with winning sets from the Hausdorff difference hierarchy, we find that there is a player such that the knowledge that she will win does not help the task of a constructing a winning strategy. This does not hold for open winning sets on Baire space -- here knowing who wins the game makes it easier to construct a winning strategy. Open determinacy on Baire space shares all known properties with the perfect tree theorem (a closed subset of Baire space is either countable or contains a perfect subset), but it is an open question whether they are actually equivalent.
The results presented are from joint work with Takayuki Kihara and Alberto Marcone (https://arxiv.org/abs/1812.01549) and with Stephane Le Roux (https://arxiv.org/abs/1407.5587).