MODEL THEORETIC PROPERTIES OF METRIC VALUED FIELDS ITAI BEN YAACOV Abstract. We study model theoretic properties of valued fields (equipped with a real-valued multi- plicative 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 MV F , and that the projective spaces Pn and Pm are biınterpretable for every n,m ≥ 1. The theory MV F admits a model completion ACMV F , the theory of algebraically closed metric valued fields (with a non trivial valuation). This theory is strictly stable, even up to perturbation. Similarly, we show that the theory of real closed metric valued fields, RCMV F , is the model companion of the theory of formally real metric valued fields, and that it is dependent. 1. The theory of metric valued fields Let us recall some terminology from [Ber90]. A semi-normed ring is a unital commutative ring R equipped with a mapping | · | : R? R≥0 such that (i) |1| = 1, (ii) |xy| ≤ |x||y|, (iii) |x+ y| ≤ |x|+ |y|.
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