Cool Logic

Hugo Nobrega (ILLC)

Games in Descriptive Set Theory, or: it's all fun and games until someone loses the axiom of choice

May 22nd at 18:00, in F1.15 ILLC seminar room

Descriptive set theory (DST) is the study of the definable sets of real numbers and similar topological spaces. One of DST's main driving questions is: What can we say about a set if all that we know is that it is definable with a certain complexity? For example, we know that if a set is the projection of a closed subset of the real plane then it cannot be a counterexample to the Continuum Hypothesis. On the other hand, the usual axioms of set theory don't determine whether the same can be said of all *complements* of such sets! One especially interesting space studied in DST is the Baire space, composed of the infinite sequences of natural numbers. The topology of this space has a certain computational-combinatorial flavor which makes many arguments more intuitive than in other spaces. Another nice aspect of this space is that it lends itself quite naturally to analysis by infinite games, which I hope to convince you of in this talk. I will start with a brief description of some games which have far-reaching consequences for set theory and the foundation of mathematics. The main focus of the talk will be the games which characterize interesting classes of functions in Baire space, where I will describe results by Wadge, Duparc, Andretta, Semmes, and (time permitting) yours truly. I will assume no prior knowledge other than some basic mathematics, such as the definition of a topology.