Categorical large cardinals and the tension between categoricity and settheoretic reflection
Abstract
Inspired by Zermelo's quasicategoricity result characterizing the models of secondorder ZermeloFraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a firstorder sentence, a firstorder theory, a secondorder sentence or a secondorder theory. The heights of these models, we define, are the categorical large cardinals. We subsequently consider various philosophical aspects of categoricity for structuralism and realism, including the tension between categoricity and settheoretic reflection, and we present (and criticize) a categorical characterization of the settheoretic universe $\langle V,\in\rangle$ in secondorder logic.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.07164
 Bibcode:
 2020arXiv200907164H
 Keywords:

 Mathematics  Logic
 EPrint:
 26 pages. Commentary about this article can be made on the first author's web page at http://jdh.hamkins.org/categoricallargecardinals