Axiomatics for the external numbers of nonstandard analysis

Bruno Miguel Antunes Dinis, Imme Pieter Van den Berg


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.


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Journal of Logic and Analysis ISSN:  1759-9008