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

0572042 - FLÚ 2024 RIV GB eng J - Článek v odborném periodiku
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 ostatní: AV ČR(CZ) LQ300092101
Program: Prémie Lumina quaeruntur
Institucionální podpora: RVO:67985955
Klíčová slova: choice principles * intuitionism * constructivism * extensionality
Obor OECD: Philosophy, History and Philosophy of science and technology
Impakt faktor: 0.6, rok: 2022
Způsob publikování: 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.
Trvalý link: https://hdl.handle.net/11104/0343185