Revisiting the construction of the real line

Abstract

The most common constructions of the reals from the rationals attempt to “fill in the gaps,” using (equivalence classes of) Cauchy sequences of rationals or Dedekind cuts. Another technique, more familiar to logicians than to analysts, is to expand the set Q of rationals to the set QN of all sequences of rationals and then collapse back down in two stages. In the first stage, produce a structure X by identifying sequences that agree on a large set; in the second stage, further collapse X by removing its infinite elements, and collapsing its infinitesimals to 0. The latter approach, though less widely known, provides, in its intermediary construction X, an example of interesting pathologies. We show here that X is an example of a complete (even open-complete) ordered field that is not Archimedean. The field X showcases the significant role of the Archimedean property in the structure of R, as many of the fundamental properties of R are found to fail in X precisely because this property is absent, such as the Heine-Borel Theorem and the Bolzano-Weierstrass Theorem. Indeed, we show that the field X is not even metrizable. We conclude with a detailed proof, accessible to non-logicians, that the second stage of the construction succeeds in producing a complete Archimedean ordered field. Mathematics Subject Classification: 12J15, 26E35