On all numbers great and small (Topological fields of Conway's numbers and their completions)

Abstract

The proper Class $\bf{No}$ of all Conway's numbers $\cite{l3}$ is considered as a region of investigation. It turns out to be a total ordered Field (i.e., a field whose domain is a proper Class) and this totally, or linear ordered Class, containing the real numbers ${\mathbb R}$ and the ordinal numbers {\bf On}. For any subfield $F$ of $\bf{No}$, i.e., $F$ is a set nor proper class, considered with topology induced by a linear ordering on $F$ a completion $\tilde F$ is constructed; in particular, for $\zeta=\omega^{\omega^\mu}$, $0\leq\mu<\Omega$, and for a specially defined subfield $F={\mathbb P}_\zeta\subset{\bf No}$ a complete subfield ${\mathbb R}_\zeta\subset{\bf No}$ is defined as $\tilde {\mathbb P}_\zeta$. Fundamental (Cauchy) sequences $(x_\alpha)_{0\leq\alpha<\zeta}$ are considered in a subfield $F\subset {\mathbb P}_\zeta\subset{\bf No}$, where $\zeta$ is the smallest ordinal number which does not belong to $F$, and they are the main instrument in the paper. A fragment of Mathematical Analysis in ${\mathbb R}_\zeta$ is given and two of its non-trivial results are presented: every positive number $x\in{\mathbb R}_\zeta$ has a unique $n$-th root in ${\mathbb R}_\zeta$, for each positive integer $n$ and every odd-degree polynomial with coefficients in ${\mathbb R}_\zeta$ has a root in ${\mathbb R}_\zeta$. Hence so-called fundamental theorem of algebra: the ring ${\mathbb R}_\zeta[i]\stackrel{def}{=}{\mathbb C}_\zeta$ of all numbers of the form $x+iy$ ($x,y\in{\mathbb R}_\zeta$), $i^2=-1$, is an algebraically closed field.