Classes, and their Potential Parts
Abstract: We explore the prospects of a mereological account of proper classes within a potentialist framework. Actualism conceives of the set-theoretic hierarchy as a completed or actual multiplicity of sets and urelements, which form the domain of a binary relation governed by the Zermelo-Fraenkel axioms. The moral of Russell’s paradox, for the actualist, is that not all multiplicities form a set, e.g., there is no set of all and only non-self-membered sets, but there is no problem in principle to paraphrase apparent singular reference to proper classes in terms of plural reference to sets. On the other hand, Potentialism regards the existence of a set as potential relative to that of its members. So, even if there is no set of all and only actual non-self-membered sets, there can be one, which is not yet actual. There is no vantage point from which to conceive of the set-theoretic hierarchy as a completed or actual multiplicity of sets and urelements; no matter what sets may be actual, there can be further sets that are not yet actual. In the absence of a completed multiplicity of sets, there is no room for a plural interpretation of proper classes. We look at a mereological interpretation of proper classes as an alternative. We attempt to make sense of classes, whether proper or improper, as sums of singletons in a potentialist framework. We will, in particular, motivate a view of proper classes as nonrigid sums, which, unlike sets, change their parts as the set-theoretic hierarchy progressively unfolds. While conceived as urelements, proper classes lack elements, they nevertheless have singletons as atomic proper parts, which allows one to simulate membership. What makes proper classes distinctive on the present framework is the fact that there is no vantage point from which its parts have been completely given. Whatever their actual parts may be, they can have further parts that are not yet actual.
The talk will be hybrid (in-person and on Zoom).
Contact Laura.Crosilla [you know what goes here] gmail.com for a Zoom-invite.