Bergen Logic Seminar: Universes without Objects: An Arealist’s Guide to the Set-Theoretic Multiverse

Universes without Objects: An Arealist’s Guide to the Set-Theoretic Multiverse

Abstract

Since Cohen’s discovery of the forcing method in 1963, set-theoretic practice has been characterized by the independence of numerous natural mathematicalquestions from the axioms ZFC with large cardinals. Much of the work in the philosophy of set theory has centered on the question of how, and indeed whether, to determine the truth or falsity of stronger axioms which could settle these independent questions. In this talk, I will survey one recent approach which suggests that thesearch for a uniquely well-justified collection of stronger axioms is entirely wrong-headed: the set-theoretic multiverse. I will present two prominent approaches to the multiverse, noting important differences between them. I will then suggest that the overtly metaphysical nature of the current debate about the multiverse has lead to a quagmire, with little hope of resolving the relevant philosophical issues. I instead propose an alternate, arealist reframing of the debate which offers the promise of making progress on these issues. I conclude by examining how this understanding of the multiverse debate should alter our understanding of the virtues of multiversism. Note that this talk will assume no particular mathematical background knowledge.