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The Axiom of Choice is false intuitionistically (in most contexts)

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    0572042 - FLÚ 2024 RIV GB eng J - Journal Article
    McCarty, Ch. - Shapiro, S. - Klev, Ansten
    The Axiom of Choice is false intuitionistically (in most contexts).
    Bulletin of Symbolic Logic. Roč. 29, č. 1 (2023), s. 71-96. ISSN 1079-8986. E-ISSN 1943-5894
    Grant - others:AV ČR(CZ) LQ300092101
    Program: Prémie Lumina quaeruntur
    Institutional support: RVO:67985955
    Keywords : choice principles * intuitionism * constructivism * extensionality
    OECD category: Philosophy, History and Philosophy of science and technology
    Impact factor: 0.6, year: 2022
    Method of publishing: Open access
    https://doi.org/10.1017/bsl.2022.22

    There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.
    Permanent Link: https://hdl.handle.net/11104/0343185

     
     
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