Numerosities of point sets over the real line

Abstract

We consider possibility of a notion of size for point sets, i.e. subsets of Euclidean spaces E d (R) of all d-tuples of real numbers, that satisfies fifth common notion of Euclid's Elements: the whole is larger than part. Clearly, such a notion of numerosity can agree with cardinality only for finite sets. We show that can be assigned to every point set in such a way that natural Cantorian definitions of arithmetical operations provide a very good algebraic structure. Contrasting with cardinal arithmetic, numerosities can be taken as (nonnegative) elements of a discretely ordered ring, where sums and products correspond to disjoint unions and Cartesian products, respectively. Actually, our numerosities form suitable semirings of hyperintegers of nonstandard Analysis. Under mild set-theoretic hypotheses (e.g. cov(B) = c < N ω ), we can also have natural ordering property that, given any two countable point sets, one is equinumerous to a subset of other. Extending this property to uncountable sets seems to be a difficult problem.