Against the iterative conception of set

Abstract

According to the iterative conception, each set is formed out of sets that are, in some sense, prior to it. Because priority plays an essential role in explanations of why contradiction-inducing sets, such as the universal set, do not exist, the success of these explanations depends on our ability to make sense of the relevant priority relation. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty set and raises serious epistemological concerns; but the leading realist interpretations–ontological and modal interpretations of priority—are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded.