On Multiversism

This paper isolates a surprising range of multiverse positions, revealing their sometimes-dubious metaphysical underpinnings and demonstrating that the distinction between multiversism and universism is often muddier than it might appear. Setting metaphysics aside, the operative questions cluster around Axiomatic Universism versus Multiversism—should our fundamental theory be couched in terms of sets or in terms of sets and classes?—and Heuristic Universism versus Multiversism—should set-theoretic practice be guided by an intuitive picture of an iterative hierarchy or of a collection of such hierarchies? It concludes that the mathematical benefits of a switch from (extensions of) ZFC to (extensions of) a multiverse theory like Steel’s MV have yet to be demonstrated and that a truly metaphysics-free perspective should happily welcome the heuristic benefits of both intuitive pictures.