Another Proof of the Existence a Dedekind Complete Totally Ordered Field

TL;DR

This paper presents a new, simpler construction of Dedekind complete totally ordered fields using non-standard rational numbers, providing an alternative proof of the real numbers' axioms' consistency.

Contribution

It introduces a novel, more straightforward method for constructing Dedekind complete fields via non-standard analysis, differing from classical approaches.

Findings

Construction is simpler and shorter than classical methods

Provides an alternative proof of the consistency of real number axioms

Utilizes non-standard rational numbers for the construction

Abstract

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.