EXTENSIVE MEASUREMENT WITH UNRESTRICTED CONCATENATION AND NO MAXIMAL ELEMENTS

First-order predicate logic is consistently used here to prove the representation theorem for extensive measurement with unrestricted concatenation and no maximal elements stated by P. Suppes in his 1951 article "A set of independent axioms for extensive quantities." Suppes regarded as unnecessary to present a detailed proof of the theorem. He limited himself to showing that a function for any structure satisfying his axioms exists that satisfies the desired properties. He added that its proof follows along standard lines, as given, for instance, by O. Holder in 1901. Our proof here follows where feasible Holder's arguments and requires an Archimedean axiom which is slightly different from that provided by Suppes. Dedekind's theory of irrational numbers, simplified and described in great detail by E. Landau, is adopted because it is the most convenient for our purposes.