In defense of Countabilism

Philosophical Studies 179 (7):2199-2236 (2022)
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Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not.

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Author Profiles

David Builes
Princeton University
Jessica M. Wilson
University of Toronto at Scarborough

Citations of this work

A Humean Non-Humeanism.David Builes - 2023 - Philosophical Studies 180 (3):1031-1048.
Modal Idealism.David Builes - forthcoming - Oxford Studies in Philosophy of Mind.
Cantor, Choice, and Paradox.Nicholas DiBella - forthcoming - The Philosophical Review.
Arithmetic is Necessary.Zachary Goodsell - forthcoming - Journal of Philosophical Logic.
Mathematical Modality: An Investigation in Higher-order Logic.Andrew Bacon - 2024 - Journal of Philosophical Logic 53 (1):131-179.

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On the Plurality of Worlds.David K. Lewis - 1986 - Malden, Mass.: Wiley-Blackwell.
Does conceivability entail possibility.David J. Chalmers - 2002 - In Tamar Gendler & John Hawthorne (eds.), Conceivability and Possibility. New York: Oxford University Press. pp. 145--200.
A treatise of human nature.David Hume & A. D. Lindsay - 1969 - Harmondsworth,: Penguin Books. Edited by Ernest Campbell Mossner.

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