Regularity Properties and Determinacy MSc

One of the most intriguing developments of modern set theory is the investigation of two-player infinite games of perfect information. Of course, it is clear that applied game theory, as any other branch of mathematics, can be modeled in set theory. But we are talking about the converse: the use of infinite games as a tool to study fundamental set theoretic questions. When such infinite games are played using integers as moves, a surprisingly rich theory appears, with connections and consequences in all fields of pure set theory, particularly the study of the continuum (the real numbers) and Descriptive Set Theory (the study of " definable " sets of reals). The concept of determinacy of games—a game is determined if one of the players has a winning strategy—plays a key role in this field. In the 1960s, the Polish mathematicians Jan Mycielski and Hugo Steinhaus [MySt62, My64, My66] proposed the famous Axiom of Determinacy (AD), which implies that all sets of reals are Lebesgue measurable, have the Baire property, the Perfect Set Property, and in general all the " regularity properties ". This contradicts the Axiom of Choice (AC) which allows us to construct irregular sets by using an enumeration of the continuum. A lot of work on determinacy is therefore done in ZF, i.e., Zermelo-Fraenkel set theory without the Axiom of Choice. In such a mathematical universe with AC replaced by AD, the pathological, non-constructive sets that form counterexamples to the regularity properties are altogether banished. But how should we understand determinacy in the context of ZFC, i.e., standard Zermelo-Fraenkel set theory with Choice? The easiest way is to look at determinacy as another kind of regularity property, D, where a set of reals A is determined if its corresponding game is determined. Since in the AD context infinite games are used to prove regularities, one would expect determinacy to be a kind of " mother regularity property " , one which subsumes and implies all the others. This is indeed true, but only in the " classwise " sense: assuming for some large collection Γ of sets that each of them is determined, we may conclude that each set in Γ has the regularity properties. Does determinacy actually have " pointwise " consequences, i.e., if we know of a set A that it is determined, does that imply that A is regular? In general, the answer is no. …