One of my ambitions in life is to understand projective determinacy. Fortunately, Tim Gowers has written a series of posts to explain Martin’s proof that Borel sets are determined.

The main source of interest in determinacy is that results suggest that it is the strongest regularity property that a set can have, in that it it tends to imply other nice properties such as Lebesgue measurability. Here is a short proof by Martin that determinacy implies Lebesgue measurability. Justin Palumbo has a nice set of lecture notes that relate determinacy to other regularity properties.

(One nuance is that determinacy for a single set usually doesn’t imply strong regularity properties — the proofs typically require several auxiliary games for a single set. The Martin and Palumbo links use the setting of the axiom of determinacy, which is the axiom that all sets are determined. This is actually false in ZFC: it contradicts the axiom of choice. There are analogous results that hold in ZFC where you keep track of which sets you need to have determined.)