Why Numbers Are Sets

Synthese 133 (3):343-361 (2002)
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Abstract

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.

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Eric Steinhart
William Paterson University of New Jersey

Citations of this work

Conceptual engineering for mathematical concepts.Fenner Stanley Tanswell - 2018 - Inquiry: An Interdisciplinary Journal of Philosophy 61 (8):881-913.
On Number-Set Identity: A Study.Sean C. Ebels-Duggan - 2022 - Philosophia Mathematica 30 (2):223-244.
Multiple reductions revisited.Justin Clarke-Doane - 2008 - Philosophia Mathematica 16 (2):244-255.

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