An Aristotelian notion of size

The näıve idea of “size” for collections seems to obey both to Aristotle’s Principle: “the whole is greater than its parts” and to 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, halving instead 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 of further investigation. 2000 Mathematics Subject Classification: 03A05, 03E65, 03E10; 03C20, 03H15. During the preparation of this paper the authors were supported by MIUR PRIN grants “Metodi variazionali e topologici nello studio di fenomeni non lineari” and “Ominimalità, metodi e modelli non standard, linguaggi per la computabilità”.