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

Abstract

The proper Class $\mathbf{No}$ 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 $\mathbf{No}$, 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 $\zeta ={\omega }^{{\omega }^{\mu }}$, $0\le \mu <\Omega$, and for a specially defined subfield $F={\mathbb{P}}_{\zeta }\subset \mathbf{No}$ a complete subfield ${\mathbb{R}}_{\zeta }\subset \mathbf{No}$ is defined as ${\tilde{\mathbb{P}}}_{\zeta }$. Fundamental (Cauchy) sequences $(x_{\alpha }{)}_{0\le \alpha <\zeta }$ are considered in a subfield $F\subset {\mathbb{P}}_{\zeta }\subset \mathbf{No}$, 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.