This will be a talk at the OLOFOS seminar “Formal analysis of modality and pragmatics in science and mathematics”. You can get the slides here.
Abstract: A popular view in the philosophy of set theory is that of *potentialism*: the position that the set-theoretic universe unfolds as more sets come into existence or become accessible to us. A difficult question for the potentialist is to explain how *classes* (understood as intensional entities) behave on this framework, and in particular what logic governs them. In this talk we’ll see how category-theoretic resources can be brought to bear on this issue. I’ll first give a brief introduction to topos theory, and then I’ll explain how (drawing on work of Lawvere) we can think of intensional classes for the potentialist as given by a functor category. I’ll suggest some tentative directions for research here, including the possibility that this representation indicates that the logic of intentional classes should be intuitionistic rather than classical, and that the strength of the intuitionistic logic is dependent upon the partial order on the worlds.