Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger 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. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst affording the theoretical resources for a defence of Boffa set theory as a faithful description of set-theoretic reality.

中文翻译：

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Boffa 集合论中的同一性和外延性

Boffa 无充分根据的集合论允许几个不同的集合等于它们各自的单例，即所谓的“蒯因原子”。里格认为，这一理论不能忠实地描述集合论现实。他认为，即使在承认存在不成立的集合之后，“集合的外延性质”也排除了数字上不同的蒯因原子。在本文中，我们揭示了里格的论点与数学结构主义中如何构思非刚性结构之间的重要相似之处。这为反对里格开辟了道路，同时为捍卫博法集合论作为对集合论现实的忠实描述提供了理论资源。