Numerosities of point sets over the real line

We consider the possibility of a notion of size for point sets, i.e. subsets of the Euclidean spaces $\mathbb{E}_{d}( \mathbb{R})$ of all $d$-tuples of real numbers, that satisfies the fifth common notion of Euclid's Elements: ``the whole is larger than the part''. Clearly, such a notion of ``numerosity'' can agree with cardinality only for finite sets. We show that ``numerosities'' can be assigned to every point set in such a way that the natural Cantorian definitions of the 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. $\textbf{cov}(\mathcal{B})=\mathfrak{c}< \aleph_{\omega}$), we can also have the natural ordering property that, given any two countable point sets, one is equinumerous to a subset of the other. Extending this property to uncountable sets seems to be a difficult problem.