This will be an online talk for the CUNY Set Theory Seminar, Friday 26 June 2020, 2 pm EST = 7 pm UK time. Contact Victoria Gitman for Zoom access. 

Abstract: Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory ZFC2 in his 1930 quasi-categoricity result asserting that the models of ZFC2 are precisely those isomorphic to a rank-initial segment 𝑉𝜅 of the cumulative set-theoretic universe 𝑉 cut off at an inaccessible cardinal 𝜅. I shall discuss the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the 𝑉𝜅 universes are categorically characterized by their sentences or theories. For example, if 𝜅 is the smallest inaccessible cardinal, then up to isomorphism 𝑉𝜅 is the unique model of ZFC2 plus the sentence “there are no inaccessible cardinals.” This cardinal 𝜅 is therefore an instance of what we call a first-order sententially categorical cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of ZFC2 by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford).