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 $\text{ZFC}_2$ in his 1930 quasi-categoricity result asserting that the models of $\text{ZFC}_2$ are precisely those isomorphic to a rank-initial segment $V_\kappa$ of the cumulative set-theoretic universe $V$ cut off at an inaccessible cardinal $\kappa$. 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 $V_\kappa$ universes are categorically characterized by their sentences or theories. For example, if $\kappa$ is the smallest inaccessible cardinal, then up to isomorphism $V_\kappa$ is the unique model of $\text{ZFC}_2$ plus the sentence “there are no inaccessible cardinals.” This cardinal $\kappa$ 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 $\text{ZFC}_2$ 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).

Some relevant discussions on the topic:

– https://mathoverflow.net/q/206172/7206

– https://mathoverflow.net/q/135995/7206

– https://math.stackexchange.com/q/317729/622

Thanks for those links, Asaf—I had totally forgotten about my posts there years ago! Meanwhile, the current paper (joint with Robin Solberg) aims to go somewhat beyond those observations. For example, we investigate the existence of gaps in the categorical cardinals, to separate the notions of sentententially categorical from theory categorical, in both first and second-order, and to look into the interaction of categoricity and forcing.

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