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Title: Wittgenstein's Attack on the Orthodox Definition of Equinumerosity Reconsidered
Author(s): BÜTTNER, Kai Michael
Journal: Logique et Analyse
Volume: 259    Date: 2022   
Pages: 189-214
DOI: 10.2143/LEA.259.0.3291802

Abstract :
In set theory, equinumerosity of two sets is defined as consisting in the existence of a one-one correlation between their elements. One of the very few philosophers who reject this definition is Wittgenstein. The article critically examines his rarely discussed objections. Wittgenstein claims that equinumerosity of two sets does not entail the existence of a one-one correlation between their elements unless relations of a rather peculiar sort are admitted. But if these 'extensional relations', as he calls them, are admitted, the set theoretic definition of equinumerosity would become circular. This paper argues that Wittgenstein’s circularity charge ultimately fails, but that his arguments against conceiving extensional relations as genuine relations are more convincing. The upshot of these discussions will be that equinumerosity is better defined as the possibility, not the actuality, of a corresponding one-one correlation.

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