An Independent Axiom System for the Real Numbers

The reals, of course, already have an axiom system. They are a complete, linearly ordered field. This axiom system is even categorical, meaning that it completely characterizes the reals. Up to isomorphism, the reals are the only complete, linearly ordered field. Another property of axiom systems, considered to be particularly elegant ever since the birth of formal logic, is independence. In an independent axiom system no axiom is redundant; that is, no axiom may be proved from the remaining axioms. An early example of an independent axiom system is the set of axioms developed by David Hilbert for Euclidean geometry. In chapter two of [7], Hilbert carefully proved each of his axioms independent of the others by finding a model for the other axioms in which each chosen axiom was false. Since then, independent axioms systems have been found for such algebraic entities as fields [2] and vector spaces [9]. Hilbert’s axioms for Euclidean geometry, incidentally, are categorical as well as independent. The purpose of this paper is to describe an independent, categorical axiom system for the reals. A categorical axiom system has a maximum property. No new axiom can be consistently added to such a system that is not already a consequence of the other axioms. An independent axiom system has a minimum property. No axiom can be subtracted from such a system and still be a consequence of the remaining axioms. Thus, an independent, categorical axiom system is a rather special mathematical object, a kind-of max-min or min-max system. Hilbert’s axiom system for Euclidean geometry is not often used. Its special qualities make it unsuitable for teaching. Other systems—one developed by CUPM, for example—have supplanted it, although it still appears in some geometry classes. Likewise, it is unlikely that the axiom system presented here will supplant the concept of a complete, linearly ordered field. The latter system has many convenient features and is comparatively easy to teach. Still, it seems only fair that the real numbers, the foundation of so much mathematics, including all of analysis, should be accorded the same privilege as the (equally fundamental) theory of Euclidean geometry, and thus be given an independent, categorical axiomatization.