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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.12482 |
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| _version_ | 1866913396654342144 |
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| author | Lisica, Ju. T. |
| author_facet | Lisica, Ju. T. |
| contents | 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 $ζ=ω^{ω^μ}$, $0\leqμ<Ω$, and for a specially defined subfield $F={\mathbb P}_ζ\subset{\bf No}$ a complete subfield ${\mathbb R}_ζ\subset{\bf No}$ is defined as $\tilde {\mathbb P}_ζ$.
Fundamental (Cauchy) sequences $(x_α)_{0\leqα<ζ}$ are considered in a subfield $F\subset {\mathbb P}_ζ\subset{\bf No}$, where $ζ$ 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 ${\mathbb R}_ζ$ is given and two of its non-trivial results are presented: every positive number $x\in{\mathbb R}_ζ$ has a unique $n$-th root in ${\mathbb R}_ζ$, for each positive integer $n$ and every odd-degree polynomial with coefficients in ${\mathbb R}_ζ$ has a root in ${\mathbb R}_ζ$. Hence so-called fundamental theorem of algebra: the ring ${\mathbb R}_ζ[i]\stackrel{def}{=}{\mathbb C}_ζ$ of all numbers of the form $x+iy$ ($x,y\in{\mathbb R}_ζ$), $i^2=-1$, is an algebraically closed field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On all numbers great and small (Topological fields of Conway's numbers and their completions) Lisica, Ju. T. Logic 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 $ζ=ω^{ω^μ}$, $0\leqμ<Ω$, and for a specially defined subfield $F={\mathbb P}_ζ\subset{\bf No}$ a complete subfield ${\mathbb R}_ζ\subset{\bf No}$ is defined as $\tilde {\mathbb P}_ζ$. Fundamental (Cauchy) sequences $(x_α)_{0\leqα<ζ}$ are considered in a subfield $F\subset {\mathbb P}_ζ\subset{\bf No}$, where $ζ$ 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 ${\mathbb R}_ζ$ is given and two of its non-trivial results are presented: every positive number $x\in{\mathbb R}_ζ$ has a unique $n$-th root in ${\mathbb R}_ζ$, for each positive integer $n$ and every odd-degree polynomial with coefficients in ${\mathbb R}_ζ$ has a root in ${\mathbb R}_ζ$. Hence so-called fundamental theorem of algebra: the ring ${\mathbb R}_ζ[i]\stackrel{def}{=}{\mathbb C}_ζ$ of all numbers of the form $x+iy$ ($x,y\in{\mathbb R}_ζ$), $i^2=-1$, is an algebraically closed field. |
| title | On all numbers great and small (Topological fields of Conway's numbers and their completions) |
| topic | Logic |
| url | https://arxiv.org/abs/2406.12482 |