An Approximate AKE Principle for Metric Valued Fields

TL;DR

This paper establishes an approximate Ax-Kochen-Ershov principle for metric valued fields in continuous logic, characterizing elementary equivalence in equicharacteristic 0, and shows the non-existence of a model companion for metric valued difference fields.

Contribution

It introduces an approximate Ax-Kochen-Ershov principle in the context of metric valued fields and addresses the model companion question for metric valued difference fields.

Findings

Complete description of elementary equivalence in equicharacteristic 0

Non-existence of a model companion in any characteristic

Extension of continuous logic methods to metric valued fields

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.