Contributed Talks COUNTING INFINITE POINT-SETS

We axiomatize a notion of size for collections (numerosity) that satisfies the five com-mon notions of Euclid’s Elements, including the Aristotelian principle that “the whole is greater than the part”, under the natural Cantorian definitions of sum, product, and ordering (see [1, 2, 3]). These numerosities turn out to have a much better arithmetic than cardinalities, included as they are in the positive part of a discrete (partially) ordered ring. Therefore equinumerosity cannot be identified with equipotency. Focusing on finite dimensional point-sets, i.e. sets of tuples of points from a given “line” L, we only postulate that natural “geometric ” bijections are numerosity-preserving, inter alia all biunique “support preserving ” transformations of tuples. We thus obtain that the numerosities are a positive subsemiring N of the quotient ring R = Z[[T]]/p of the powerseries ring over a set T of indeterminates of size |L|, modulo a suitable prime ideal p. We name “Euclidean ” such ideals, as well as the corresponding ultrafilters over κ = |L|. The ring R is then isomorphic to a ring of hyperintegers, namely the ultrapower ZκUp, where Up is the ultrafilter corresponding to p. The existence of “Euclidean ” ultrafilters on an arbitrary cardinal κ is problematic