Submission date: 22 August 2022

Abstract:

We study metric valued fields in continuous logic, following Ben Yaacov's approach, thus working in the metric space given by the projective line. As our main result, we obtain an approximate Ax-Kochen-Ershov principle in this framework, completely describing elementary equivalence in equicharacteristic 0 in terms of the residue field and value group. Moreover, we show that, in any characteristic, the theory of metric valued difference fields does not admit a model-companion. This answers a question of Ben Yaacov.

Mathematics Subject Classification: 03C66, 03C60 (Primary), 12L10 (Secondary)

Keywords and phrases:

Full text arXiv 2208.10186: pdf, ps.