Can the Cumulative Hierarchy Be Categorically Characterized?

Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee’s result of the significance he and the realist want to attribute to it. I consider two strategies to deal with the problem — one of which is outlined by McGee himself (2000) — and argue that both of them fail. I end with some remarks on the prospects for mathematical realism in the light of my discussion.