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The reals as rational Cauchy filters (1503.04348v3)
Published 14 Mar 2015 in math.HO
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.