Size and Function

Erkenntnis 83 (4):853-873 (2018)
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Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity.

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Bruno Whittle
University of Wisconsin, Madison

Citations of this work

In defense of Countabilism.David Builes & Jessica M. Wilson - 2022 - Philosophical Studies 179 (7):2199-2236.
Cantor, Choice, and Paradox.Nicholas DiBella - forthcoming - The Philosophical Review.
Might All Infinities Be the Same Size?Alexander R. Pruss - 2020 - Australasian Journal of Philosophy 98 (3):604-617.

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References found in this work

What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
The Construction of Logical Space.Agustín Rayo - 2013 - Oxford, England: Oxford University Press.
Frege's conception of numbers as objects.Crispin Wright - 1983 - [Aberdeen]: Aberdeen University Press.
Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.

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