Pseudo completions and completions in stages of o-minimal structures

For an o-minimal expansion R of a real closed field and a set V of Th(R)-convex valuation rings, we construct a "pseudo completion" with respect to V. This is an elementary extension S of R generated by all completions of all the residue fields of the V epsilon V, when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings up to an R-isomorphism. For polynomially bounded R we can iterate the construction of the pseudo completion in order to get a "completion in stages" S of R with respect to V. S is the "smallest" extension of R such that all residue fields of the unique extensions of all V epsilon V to S are complete.