Euclidean numbers and numerosities

TL;DR

This paper unifies various theories of numerosities within a consistent framework using Euclidean numbers, enabling new insights into ordinals and measure theory through a set of labels approach.

Contribution

It introduces a unified approach to numerosities via set of labels, connecting them with Euclidean numbers and extending their applications.

Findings

Established a connection between numerosities and Euclidean numbers.

Defined ordinals and natural operations using numerosities.

Introduced Lebesgue measure as a counting measure on reals.

Abstract

Several different versions of the theory of numerosities have been introduced in the literature. Here, we unify these approaches in a consistent frame through the notion of set of labels, relating numerosities with the Kiesler field of Euclidean numbers. This approach allows to easily introduce, by means of numerosities, ordinals and their natural operations, as well as the Lebesgue measure as a counting measure on the reals.