The Plural Iterative Conception of Set

Abstract

Georg Cantor informally distinguished between “consistent” and “inconsistent” multiplicities as those many things that, respectively, can and cannot be thought of as one, i.e., as a set. To clarify this distinction, the recent debate filtered the logic of plurals through two main approaches to the process of set-formation: limitation of size (Burgess) or set-theoretic potentialism (Linnebo). In this paper I propose a third route through the development of a plural iterative conception of set. Inspired by Tim Button’s Level Theory, I define and axiomatize the notion of a plural level, which explains Cantor’s multiplicities either as level-bound (consistent) or level unbound (inconsistent) pluralities. While this framework is clearly in contrast with the limitation of size view, it also revives a plausible actualist framework prematurely dismissed by the advocates of potentialism.