Hahn’s Über die Nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them

The ordered field ℜ of real numbers is of course up to isomorphism the unique Dedekind continuous ordered field. Equally important, though apparently less well known, is the fact that ℜ is also up to isomorphism the unique Archimedean complete, Archimedean ordered field. The idea of an Archimedean complete ordered field was introduced by Hans Hahn in his celebrated investigation Uber die nichtarchimedischen Grossensysteme which was presented to the Royal Academy of Sciences in Vienna in 1907. It is a special case of his more general conception of an Archimedean complete, ordered abelian group, a conception that was motivated by, and substantially generalizes, the idea of ℜ as an Archimedean ordered field which admits no proper extension to an Archimedean ordered field; that is, the idea of an Archimedean ordered field which satisfies Hilbert’s Axiom of (arithmetic) Completeness (Hilbert 1900a, p. 183; 1903a, p. 16).

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