The reals as rational Cauchy filters

TL;DR

This paper provides an elementary construction of the real numbers using minimal Cauchy filters on rationals, offering a direct proof of their completeness and ordered field structure without advanced topological tools.

Contribution

It introduces a novel construction of the reals via minimal Cauchy filters, aligning with Bourbaki's approach and simplifying the understanding of real number completeness.

Findings

Real numbers are characterized as minimal Cauchy filters on rationals.

The construction is elementary and does not rely on uniform space techniques.

The resulting structure is a complete ordered field.

Abstract

We present a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in $\mathbb{Q}$ (where the Cauchy condition is defined in terms of the absolute value function on $\mathbb{Q}$) and are proven directly, without employing any of the techniques of uniform spaces, to form a complete ordered field. The construction can be seen as a variant of Bachmann's construction by means of nested rational intervals, allowing for a canonical choice of representatives.