Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism.

中文翻译：

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打破平局：重新审视贝纳塞拉夫的认同论点

大多数哲学家采用贝纳塞拉夫在“数字不可能是什么”中的论点成功地反驳了数字是集合的还原论观点。这种哲学共识与数学实践相矛盾，还原论在其中继续蓬勃发展。在这篇笔记中，我们通过对文献中几乎一致接受的中心前提提出质疑，对贝纳塞拉夫的论点提出了新的挑战。也就是说，我们认为——与正统相反——有形而上学相关的理由更喜欢冯诺依曼序数而不是其他集合论的算术简化。在这样做的过程中，我们提供了集合论事实，我们认为这些事实对于对还原论的知情评估至关重要。