Axiomatics for the external numbers of nonstandard analysis

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.2010 Mathematics Subject Classification. 03H05, 03C65, 26E35.

On Flexible Sequences

“…Remark 5. 10. By similar arguments one shows that the conclusions in Propositions 5.8 and 5.9 remain true if one replaces ≤ by ≥.…”

“…Then (n, a) ∈ u, in contradiction with the fact that u is functional. We conclude that there exists a standard finite z ⊆ Y such that (10) u n = y∈z I y (n).…”

“…We denote by E the set of external numbers as constructed in [10,Section 6]. An external number α is always of the form α = a + A, where a is a (possibly nonstandard) real number and A is a neutrix, called the neutrix part of α.…”

“…We recall that the external numbers satisfy a generalized form of Dedekind Completeness [2,Cor. 4.25] (see also [10,Thm. 3.2]).…”

“…External numbers satisfy, to a large extent, the algebraic and analytic properties of the real numbers [17,18,9], including a Generalized Dedekind completeness property (see [2]). Structures with these properties were called complete arithmetical solids in [10](see also [11,12]). This means in particular that we may use the algebraic properties of neutrices to study properties of convergence.…”