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

The proper Class No of all Conway's numbers 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 R and the ordinal numbers On. For any subfield F of No, i. e. , F is a set nor proper class, considered with topology induced by a linear ordering on F a completion F is constructed; in particular, for =^^, 0<, and for a specially defined subfield F= P_ No a complete subfield R_ No is defined as P_. Fundamental (Cauchy) sequences (x_) ₀< are considered in a subfield F P_ No, where 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 R_ is given and two of its non-trivial results are presented: every positive number x R_ has a unique n-th root in R_, for each positive integer n and every odd-degree polynomial with coefficients in R_ has a root in R_. Hence so-called fundamental theorem of algebra: the ring R= C_ of all numbers of the form x+iy (x, y R_), i²=-1, is an algebraically closed field.