Remarks on Ontological Dependence in Set Theory

In a recent paper, John Wigglesworth explicates the notion of a set's being grounded in or ontologically depending on its members by the modal statement that in any world (possible or impossible), that a set exists in that world entails that its members exist as well. After suggesting that variable-domain S5 captures an appropriate account of metaphysical necessity, Wigglesworth purports to prove that in any set theory satisfying the axiom Extensionality this condition holds, that is, that sets ontologically depend on their members with respect to extraordinarily weak notions of set. This paper diagnoses a number of problems concerning Wigglesworth's formal argument. For one, we will show that Wigglesworth's argument is invalid as it requires an appeal to hidden, extralogical theses concerning rigid designation and the persistence of sets across possible worlds. Having demonstrated the indispensability of these principles to Wigglesworth's argument, we will then show that even granted the enthymematic premises, the argument only proves the ontological dependence of singletons on their members and does not extend to sets in general. Finally, we will consider strengthenings of Wigglesworth's reasoning and suggest that even the weakest generalization will bear undesirable consequences.