The Eudoxus Reals

TL;DR

This paper introduces a novel construction of the real numbers using near-endomorphisms of integers, demonstrating their structure as a complete ordered field and exploring extensions to other abelian groups.

Contribution

It presents a new approach to constructing the reals from integers via near-endomorphisms and extends the concept to other abelian groups, connecting to p-adic numbers and adeles.

Findings

Near-endomorphisms form a complete ordered field isomorphic to the reals.

Uncountably many near-endomorphisms exist independently of the reals.

Extensions to other abelian groups relate to p-adic numbers and adeles.

Abstract

We examine a unique construction of the real numbers which proceeds directly from the integers using approximately linear-endomorphisms with finite error, called near-endomorphisms. In this paper, we show that the set of near-endomorphisms forms a complete ordered field isomorphic to the reals. Moreover, we show that there are uncountably many near-endomorphisms without reference to the reals. We then investigate a natural extension of near-endomorphisms, which we call quasi-homomorphisms, to other abelian groups. Extending prior results about the construction of the $p$-adic numbers and the rational adele ring, we find the ring of near-endomorphisms of certain localizations of the integers, and suggest further directions for exploration.