An Aristotelian notion of size

The naive idea of size for collections seems to obey both Aristotle's Principle: the whole is greater than its parts and Cantor's Principle: 1-to-1 correspondences preserve size. Notoriously, Aristotle's and Cantor's principles are incompatible for infinite collections. Cantor's theory of cardinalities weakens the former principle to the part is not greater than the whole, but the outcoming cardinal arithmetic is very unusual. It does not allow for inverse operations, and so there is no direct way of introducing infinitesimal numbers. (Sizes are added by means of disjoint unions and multiplied by means of disjoint unions of equinumerous collections.) Here we maintain Aristotle's principle, instead halving Cantor's principle to equinumerous collections are in 1-1 correspondence. In this way we obtain a very nice arithmetic: in fact, our numerosities may be taken to be nonstandard integers. These numerosities appear naturally suited to sets of ordinals, but they depend, for generic sets, on a labelling of the universe by ordinals. The problem of finding a canonical way of attaching numerosities to all sets seems to be worth further investigation. (c) 2006 Elsevier B.V. All rights reserved.