Axiomatics for the external numbers of nonstandard analysis

TL;DR

This paper develops an axiomatic framework for external numbers in nonstandard analysis, modeling orders of magnitude and extending ordered fields with weaker axioms, and demonstrates their consistency and structure.

Contribution

It introduces a new axiomatic system for external numbers, called complete arithmetical solids, and proves their properties and relation to nonstandard models.

Findings

External numbers form a complete arithmetical solid.

The set of precise elements models nonstandard rationals.

The axioms are consistent and extend ordered field properties.

Abstract

Neutrices are additive subgroups of a nonstandard model of the real numbers. An external number is the algebraic sum of a nonstandard real number and a neutrix. Due to the stability by some shifts, external numbers may be seen as mathematical models for orders of magnitude. The algebraic properties of external numbers gave rise to the so-called solids, which are extensions of ordered fields, having a restricted distributivity law. However, necessary and sufficient conditions can be given for distributivity to hold. In this article we develop an axiomatics for the external numbers. The axioms are similar to, but mostly somewhat weaker than the axioms for the real numbers and deal with algebraic rules, Dedekind completeness and the Archimedean property. A structure satisfying these axioms is called a complete arithmetical solid. We show that the external numbers form a complete arithmetical solid, implying the consistency of the axioms presented. We also show that the set of precise elements (elements with minimal magnitude) has a built-in nonstandard model of the rationals. Indeed the set of precise elements is situated between the nonstandard rationals and the nonstandard reals whereas the set of non-precise numbers is completely determined.