Natural numerosities of sets of tuples

TL;DR

This paper introduces a new notion of 'numerosity' for sets of tuples of natural numbers that aligns with Euclidean principles, forming a rich algebraic structure distinct from traditional cardinality.

Contribution

It axiomatizes a 'numerosity' concept satisfying Euclidean axioms, linking it to algebraic structures like ordered rings and hypernatural numbers.

Findings

Numerosities form a discretely ordered ring.

Special numerosities correspond to hypernatural numbers.

The approach diverges from classical cardinal arithmetic.

Abstract

We consider a notion of "numerosity" for sets of tuples of natural numbers, that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show that, contrasting to cardinal arithmetic, the natural "Cantorian" definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable ("gauge") ideal. In particular, special numerosities, called "natural", can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N.