Hahn Field Representation of A. Robinson's Asymptotic Numbers

Abstract

Let ${}^{\ast }\mathbb{R}$ be a nonstandard extension of $\mathbb{R}$ and $\rho$ be a positive infinitesimal in ${}^{\ast }\mathbb{R}$. We show how to create a variety of isomorphisms between A. Robinson's field of asymptotic numbers ${}^{\rho }\mathbb{R}$ and the Hahn field $\widehat{{}^{\rho }\mathbb{R}}(t^{\mathbb{R}})$, where $\widehat{{}^{\rho }\mathbb{R}}$ is the residue class field of ${}^{\rho }\mathbb{R}$. Then, assuming that ${}^{\ast }\mathbb{R}$ is fully saturated we show that $\widehat{{}^{\rho }\mathbb{R}}$ is isomorphic to ${}^{\ast }\mathbb{R}$ and so ${}^{\rho }\mathbb{R}$ contains a copy of ${}^{\ast }\mathbb{R}$. As a consequence (that is important for applications in non-linear theory of generalized functions) we show that every two fields of asymptotic numbers corresponding to different scales are isomorphic.

18 pages

Cited by in corpus (2)

• Completeness of Ordered Fields
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