Numerosities of point sets over the real line

• Published in: Transactions of the American Mathematical Society 2010-10, Vol.362 (10), p.5355-5371
• Main Authors: DI NASSO, MAURO, FORTI, MARCO
• Format: Article
• Language: English
• Subjects: Algebra; Approximation; Cardinal points; Cardinality; Continuum hypothesis; Differential geometry; Euclidean space; Exact sciences and technology; General algebraic systems; General mathematics; General, history and biography; Geometry; Mathematical analysis; Mathematical functions; Mathematical rings; Mathematical sets; Mathematics; Real functions; Real lines; Sciences and techniques of general use; Ultrapowers
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-2010-04919-0

We consider the possibility of a notion of size for \emph {point sets}, \emph {i.e.} subsets of the Euclidean spaces \mathbb {E}_{d}( \mathbb {R}) of all d-tuples of real numbers, that satisfies the \emph {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 \emph {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 (\emph {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.