We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic.
For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KPn, as bounded metric structures.
We show that the class of (projective spaces over) metric valued fields is elementary, with theory MVF, and that the projective spaces Pn and are Pm biinterpretable for every n, m ≥ 1. The theory MVF admits a model completion ACMVF, the theory of algebraically closed metric valued fields (with a nontrivial valuation). This theory is strictly stable (even up to perturbation).
Similarly, we show that the theory of real closed metric valued fields, RCMVF, is the model companion of the theory of formally real metric valued fields, and that it is dependent.