Numerosity (mathematics)
The numerosity of an infinite set, as ininitally introduced by the Italian mathematician Vieri Benci and later on extended with the help of Mauro Di Nasso and Marco Forti, is a concept that develops Cantor’s notion of cardinality. While Cantor’s classical cardinality classifies sets based on the existence of a one-to-one correspondence with other sets, the idea of numerosity aims to provide an alternative viewpoint, linking to the common Euclidean notion that "the whole is greater than the part". All of this naturally leads to the hypernatural numbers.
In short, Benci and his collaborators propose associating with an infinite set a numerical value that more directly reflects its “number of elements”, without resorting solely to one-to-one correspondences. This approach uses tools from logic and analysis, seeking to give an operational meaning to the notion of “counting” even when dealing with infinite sets. Numerosity thus proves useful for the study of certain problems in discrete mathematics and is the subject of research within alternative theories to traditional Cantorian cardinality.
Main axioms
In simplified terms, to define a numerosity one assumes the following:- A set of “labelled” sets.
- An ordered set of “numbers”.
- A surjective map that assigns to each set its numerosity value, obeying four fundamental principles:
- Union Principle: if and and the domains of and are disjoint, then.
- Cartesian Product Principle: if and, then.
- Zermelo's Principle : if, then there exists a proper subset with.
- Asymptotic Principle : if for all the counting function of is less than or equal to that of, then.
Examples: countably infinite sets
A classic example is the set of positive natural numbers, which in this approach is associated with an “infinite number”, often denoted by :If one considers the set of even numbers, in Cantor’s theory this set is equipotent to, but in the numerosity approach of Vieri Benci and his collaborators it has the value, so that it is “half” of the naturals.
Naturally, is not a standard real number but an element of a non-Archimedean set that extends the naturals.
With similar considerations, we can obtain the numerosity of the following infinite sets
Connection with nonstandard analysis
The ideas underlying numerosity also connect with Robinson’s Nonstandard Analysis: one obtains numerical systems that include infinities and infinitesimals “coherent” with the operations of addition and multiplication. The infinity that expresses the numerosity of can be treated as a non-standard element, larger than all finite numbers, thus allowing proofs and methods typical of non-Archimedean analysis.Applications and ongoing research
Research on numerosity has been applied or discussed in:- Alternative classifications of set sizes in certain discrete or combinatorial contexts.
- Rigorous exploration of properties akin to measures, straddling the fields of measure theory and cardinal arithmetic.
- Investigations into the foundations of mathematics, particularly concerning the nature of infinity.
- Probability and the philosophy of science.