Categorical large cardinals and the tension between categoricity and set-theoretic reflection

TL;DR

This paper explores the conditions under which models of second-order set theory are fully categorical, analyzing large cardinals and examining the philosophical implications of categoricity versus reflection in set-theoretic foundations.

Contribution

It provides a mathematical analysis of categorical large cardinals and discusses the philosophical tension between categoricity and reflection in set theory.

Findings

Identification of conditions for full categoricity of models

Analysis of the tension between categoricity and set-theoretic reflection

Arguments favoring noncategoricity in foundational perspectives

Abstract

Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. Thus we mount an analysis of the categorical large cardinals. This mathematical analysis leads naturally to philosophical issues concerning structuralism and realism, including especially the tension between categoricity and reflection. Ultimately we identify grounds for the preference of noncategoricity in one's foundations.