Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers

Abstract

It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear order structure on the geometric line. Such a dramatically limiting and thus harmful belief comes, unknown to equally many, from the similarly tacit acceptance of the ancient Archimedean Axiom in Euclid's Geometry. The consequence of that belief is a similar belief in the uniqueness of the field $\mathbb{C}$ of complex numbers, and therefore, of the various spaces, manifolds, etc., be they finite or infinite dimensional, constructed upon the real or complex numbers, including the Hilbert spaces used in Quantum Mechanics. An near total lack of awareness follows about the {\it rich self-similar} structure of various linearly ordered scalar fields obtained through the {\it ultrapower} construction which extend the usual field $\mathbb{R}$ of real numbers. Such {\it ultrapower field} extensions contain as a rather small subset the usual field $\mathbb{R}$ of real numbers. The rich self-similar structure of such field extensions is due to {\it infinitesimals}, and thus also of {\it infinitely large} elements in such fields, which make such fields {\it non-Archimedean}. With the concept of {\it walkable world}, which has highly intuitive and pragmatic algebraic and geometric meaning, the mentioned rich self-similar structure is illustrated. The ultrapower fields presented can have a wide ranging relevance in Physics, among others, for a proper treatment of what are usually called the "infinities in Physics". The ultrapower construction which gives such non-Archimedean fields is rather simple and elementary, requiring only 101 Algebra.