Asaf Karagila (University of Leeds) – Large cardinals, small violations of choice: how large cardinals affect the universe

Woodin's HOD Dichotomy states that in the presence of large cardinals, the universe of sets is either very close to being HOD, or it is very very far from that. In this recent work with Hope Duncan and Gabe Goldberg we show that in the presence of a supercompact cardinal, a similar dichotomy theorem can be proved: either the axiom of choice fails very very badly, or else in many "reasonable inner models", the axiom of choice can be added generically