Title: Identity and Extensionality in Boffa Set Theory

Abstract: Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger (2000) contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. Here we identify important similarities between Rieger’s argument, on one hand, and how non-rigid structures are conceived within mathematical structuralism, on the other. We argue that this opens the way for an objection against Rieger, whilst affording the theoretical resources for the start of a defence of Boffa set theory as a faithful description of set-theoretic reality.