On the Virtue of Categoricity

Abstract

A formal theory is categorical if any two of its models are isomorphic, that is, there is a structure-preserving bijection between them. It is natural to think that whether or not categoricity is a virtuous feature of a theory depends on whether we have some antecedent belief that the theory in question describes a unique structure. In this paper, I put forth a purpose-relative framework for assessing the virtuousness of a metamathematical property. According to this framework, whether categoricity (or any other property) is virtuous depends more deeply on what we take the purpose or function of formal theories to be. I then develop one important purpose we would like theories to serve which I call the instrumentalist conception of formal theories. I argue that given this instrumentalist conception, categoricity is best understood as having somewhat limited virtue, and that, in contrast, uncountable categoricity is highly virtuous.