OF SHEEP AND WOLVES: EQUIVALENCE AND DISAGREEMENT IN SET THEORY

Abstract This article offers a philosophical overview and investigation of the problem of incompleteness in set theory and what this entails for the ensuing debates about proposed extensions of $ZFC$ . The incompleteness of $ZFC$ is well-known and leaves us with a rich array of competing extensions. What should we make of disagreements between them? We start by considering second-order logic and its categoricity theorems and how they might be used to compare different set theories. We then aim to use interpretability as a way of understanding that some of these debates are insubstantial. This culminates in some discussion of the relationship between interpretability and the generic multiverse. The second half of the article then takes up a more modest goal: we search for common ground and settle for partial agreement between set theories in much the same way that physicists are often content with empirical agreement. We then aim to describe a natural bound on the amount of agreement that we can expect to obtain between reasonable extensions of $ZFC$ .