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 ... ... ordering on $F$ a completion $\tilde F$ is constructed; in particular, for $\zeta=\omega^{\omega^\mu}$, $0\leq\mu Jury Lisica jutlisica@yandex.ru