Non-Anomalous Semigroups and Real Numbers

TL;DR

This paper introduces non-anomalous semigroups as a foundation for defining real numbers, emphasizing their order properties and deriving the field structure through categorical relationships.

Contribution

It formulates non-anomalous semigroups based on physical intuition and defines real numbers as a terminal object in a related category, establishing a novel categorical perspective.

Findings

Real numbers are characterized as the terminal object in a category of non-anomalous semigroups.

A field structure on ℝ is derived from morphisms between non-anomalous semigroups.

The approach connects physical intuition with algebraic and categorical structures.

Abstract

Motivated by intuitive properties of physical quantities, the notion of a non-anomalous semigroup is formulated. These are totally ordered semigroups where there are no `infinitesimally close' elements. The real numbers are then defined as the terminal object in a closely related category. From this definition a field structure on $\mathbb R$ is derived, relating multiplication to morphisms between non-anomalous semigroups.