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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 aneutrix. Due to the stability by some shifts, external numbers may be seen asmathematical models for orders of magnitude. The algebraic properties ofexternal numbers gave rise to the so-called solids, which are extensions ofordered fields, having a restricted distributivity law. However, necessary andsufficient conditions can be given for distributivity to hold. In this articlewe 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 withalgebraic rules, Dedekind completeness and the Archimedean property. Astructure satisfying these axioms is called a complete arithmetical solid. Weshow that the external numbers form a complete arithmetical solid, implying theconsistency of the axioms presented. We also show that the set of preciseelements (elements with minimal magnitude) has a built-in nonstandard model ofthe rationals. Indeed the set of precise elements is situated between thenonstandard rationals and the nonstandard reals whereas the set of non-precisenumbers is completely determined.

Axiomatics for the external numbers of nonstandard analysis