Measurement Without Archimedean Axioms

Abstract

Axiomatizations of measurement systems usually require an axiom—called an Archimedean axiom—that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot—except in the most trivial cases—be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of “Archimedean axioms” is given and it is shown that in all interesting cases “Archimedean axioms” are independent of other measurement axioms.