This is a talk for the group in logic and philosophy of language at the Munich Center for Mathematical Philosophy

24 June 2021, 4:15 pm Munich time (3:15 pm BST) on Zoom at 925 6562 2309 (contact Ursula Danninger office.leitgeb@lrz.uni-muenchen.de for password.)

**Abstract.** Zermelo famously proved that second-order ZFC is quasi-categorical—the models of this theory are precisely the rank-initial segments of the set-theoretic universe cut off at an inaccessible cardinal. Which are the fully categorical extensions of this theory? This question gives rise to the notion of categorical large cardinals, and opens the door to several puzzling philosophical issues, such as the conflict between categoricity as a fundamental value in mathematics and reflection principles in set theory. (This is joint work with Robin Solberg, Oxford.)