The numerosity of an infinite set, as initially 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 (defining, for example, ℵ 0 {\displaystyle \aleph _{0}} for countable sets, ℵ 1 {\displaystyle \aleph _{1}} and so on for larger infinities), 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 (or complementary) theories to traditional Cantorian cardinality.
Main axioms In simplified terms, to define a numerosity one assumes the following:
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