In defense of Countabilism
Philosophical Studies 179 (7):2199-2236 (2022)
Abstract
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.Author Profiles
DOI
10.1007/s11098-021-01760-8
My notes
Similar books and articles
Making Sense of the Aristotelian Notion of Infinity.Hwan Sunwoo - 2018 - Proceedings of the XXIII World Congress of Philosophy 55:53-71.
Extending the Non-extendible: Shades of Infinity in Large Cardinals and Forcing Theories.Stathis Livadas - 2018 - Axiomathes 28 (5):565-586.
Actual versus Potential Infinity (BPhil manuscript.).Anne Newstead - 1997 - Dissertation, University of Oxford
The influence of Spinoza’s concept of infinity on Cantor’s set theory.Paolo Bussotti & Christian Tapp - 2009 - Studies in History and Philosophy of Science Part A 40 (1):25-35.
Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale.Ivette Fred Rivera & Jessica Leech (eds.) - 2018 - Oxford, England: Oxford University Press.
Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory.Valérie Lynn Therrien - 2012 - Ithaque 10:43-65.
Part IV. Perspectives on infinity from physics and cosmology : 7. Some considerations on infinity in physics / Carlo Rovelli ; 8. Cosmological intimations of infinity / Anthony Aguirre ; 9. Infinity and the nostalgia of the stars/ Marco Bersanelli ; 10. Infinities in cosmology. [REVIEW]Michael Heller - 2011 - In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press.
A conflict between van Inwagen’s metaontology and his ontology.David Fisher - 2019 - Synthese 198 (1):707-722.
Necessary Beings: An Essay on Ontology, Modality, & the Relations Between Them. By Bob Hale. Oxford University Press, 2013, pp. 320, £40. ISBN: 978-0-19-966957-8. [REVIEW]Katherine Hawley - 2015 - Philosophy 90 (4):706-710.
Problems for Modal Reductionism: Concrete Possible Worlds as a Test Case.Jonathan Nassim - 2015 - Dissertation, Birkbeck College
Concrete possible worlds.Phillip Bricker - 2008 - In Theodore Sider, John Hawthorne & Dean W. Zimmerman (eds.), Contemporary Debates in Metaphysics. Blackwell. pp. 111--134.
Analytics
Added to PP
2021-11-18
Downloads
700 (#12,709)
6 months
83 (#12,271)
2021-11-18
Downloads
700 (#12,709)
6 months
83 (#12,271)
Historical graph of downloads
Author Profiles
Citations of this work
References found in this work
The Conscious Mind: In Search of a Fundamental Theory.David J. Chalmers - 1996 - Oxford University Press.
Four Dimensionalism: An Ontology of Persistence and Time.Theodore Sider - 2001 - Oxford University Press.