The Eudoxus Reals
AJ Kumar, Reese Long, Andrew Tung, Ivan Wong

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 -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.
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Taxonomy
TopicsMathematical and Theoretical Analysis · Advanced Topology and Set Theory · advanced mathematical theories
