Measuring the size of infinite collections of natural numbers: Was Cantor's theory of infinite number inevitable?

Review of Symbolic Logic 2 (4):612-646 (2009)
Abstract
Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the ‘size’ of A should be less than the ‘size’ of B (part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher).
Keywords Cantor  Transfinite arithmetic  Infinite cardinals
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DOI 10.1017/S1755020309990128
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References found in this work BETA
Paradoxien des Unendlichen.Bernard Bolzano - 2012 - Felix Meiner Verlag.
The Nature of Mathematical Knowledge.Philip Kitcher - 1983 - Oxford University Press.
Principles of Mathematics.Bertrand Russell - 1903 - Cambridge University Press.
Wissenschaftslehre. [REVIEW]Bernard Bolzano - 2001 - Revue de Métaphysique et de Morale 2 (18):134-136.
Collected Works.Kurt Gödel - 1986 - Oxford University Press.

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Citations of this work BETA
Fair Infinite Lotteries.Sylvia Wenmackers & Leon Horsten - 2013 - Synthese 190 (1):37-61.
Set Size and the Part-Whole Principle.Matthew W. Parker - 2013 - Review of Symbolic Logic (4):1-24.

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