Gödel's Argument for Cantorian Cardinality

Noûs 53 (2):375-393 (2017)
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Abstract

On the first page of “What is Cantor's Continuum Problem?”, Gödel argues that Cantor's theory of cardinality, where a bijection implies equal number, is in some sense uniquely determined. The argument, involving a thought experiment with sets of physical objects, is initially persuasive, but recent authors have developed alternative theories of cardinality that are consistent with the standard set theory ZFC and have appealing algebraic features that Cantor's powers lack, as well as some promise for applications. Here we diagnose Gödel's argument, showing that it fails in two important ways: Its premises are not sufficiently compelling to discredit countervailing intuitions and pragmatic considerations, nor pluralism, and its final inference, from the superiority of Cantor's theory as applied to sets of changeable physical objects to the unique acceptability of that theory for all sets, is irredeemably invalid.

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Matthew Parker
London School of Economics

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References found in this work

The foundations of arithmetic.Gottlob Frege - 1884/1950 - Evanston, Ill.,: Northwestern University Press.
Mathematical truth.Paul Benacerraf - 1973 - Journal of Philosophy 70 (19):661-679.
What is Cantor's Continuum Problem?Kurt Gödel - 1947 - The American Mathematical Monthly 54 (9):515--525.
Philosophy of mathematics: selected readings.Paul Benacerraf & Hilary Putnam (eds.) - 1983 - New York: Cambridge University Press.
Infinitesimal Probabilities.Vieri Benci, Leon Horsten & Sylvia Wenmackers - 2016 - British Journal for the Philosophy of Science 69 (2):509-552.

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