A characterization of pseudo complete finitely ramified valued fields through a Hahn-like construction

We give a characterization of finitely ramified $\omega$-pseudo complete valued fields of mixed characteristic $(0,p)$, with fixed residue field $k$ and value group $G$ of cardinality ${\mathrm{\aleph }}_1$, in terms of a Hahn-like construction over the Cohen field $C(k)$, modulo the Continuum Hypothesis. This is a generalization of results due to Ax and Kochen in '65 for formally $p$-adic fields and by Kochen in '74 for unramified valued fields with perfect residue field. We consider a more general context of finitely ramified valued fields of mixed characteristic with arbitrary residue field and a cross-section.