Hahn Field Representation of A. Robinson's Asymptotic Numbers

Abstract

Let $^*\mathbb{R}$ be a nonstandard extension of $\mathbb{R}$ and $\rho$ be a positive infinitesimal in $^*\mathbb{R}$. We show how to create a variety of isomorphisms between A. Robinson's field of asymptotic numbers $^\rho\mathbb{R}$ and the Hahn field $\hat{^\rho\mathbb{R}}(t^\mathbb{R})$, where $\hat{^\rho\mathbb{R}}$ is the residue class field of $^\rho\mathbb{R}$. Then, assuming that $^*\mathbb{R}$ is fully saturated we show that $\hat{^\rho\mathbb{R}}$ is isomorphic to $^*\mathbb{R}$ and so $^\rho\mathbb{R}$ contains a copy of $^*\mathbb{R}$. 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.