THE AXIOM OF CHOICE IS FALSE INTUITIONISTICALLY (IN MOST CONTEXTS)

C. McCarty, S. Shapiro, A. Klev
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Abstract

Abstract 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.
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选择公理在直觉上是错误的(在大多数情况下)
似乎有一种观点认为,直觉主义者不仅认为选择公理(AC)是正确的,而且认为这是他们基本假设的结果。不管是否普遍,这种观点在很大程度上是错误的。本文简要而全面地概述了交流在各种直觉主义和建构主义体系中的地位。调查清楚地表明,选择公理在大多数情况下不能成为定理,在一些重要的情况下甚至是完全错误的。在所调查的系统中,只有内涵类型理论使AC成为定理,但该理论中AC的范围不包括,例如,实分析。只需要少量的延伸性,直觉主义者就可以提供AC失效的明显证据。
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POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
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