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Why Numbers Are Sets

Eric Steinhart
Synthese
Vol. 133, No. 3 (Dec., 2002), pp. 343-361
Published by: Springer
Stable URL: http://www.jstor.org/stable/20117311
Page Count: 19
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Why Numbers Are Sets
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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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