An approximate Ax-Kochen-Ershov principle for valued fields in continuous logic

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Abstract

In a paper from 2014, Itai Ben Yaacov considered complete valued fields, with value group in the reals, as structures in continuous logic, working with the projective line - a bounded metric space - instead of the valued field itself. He showed that the theory of algebraically closed metric non-trivially valued fields forms the model-companion of the corresponding theory. In the talk, I will present two recent results on metric valued fields of residue characteristic 0, namely a complete description of the elementary classes in terms of the residue field and value group (in a sense an approximate AKE principle), and the fact that, contrarily to the situation in discrete logic, the theory of metric valued fields with a distinguished isometric isomorphism does not admit a model-companion, answering a question of Ben Yaacov. This is joint work with Stefan Ludwig.

Speakers from the University of Münster

Hils, Martin

Professorship for Mathematical Logic (Prof. Hils)

Projects the talk is about

Geometric and Combinatorial Configurations in Model Theory (GeoMod) Duration: 01/01/2020 - 31/12/2024 Funded by: DFG - Individual Grants Programme Type of project: Individual project

EXC 2044 - A2: Groups, model theory and sets Duration: 01/01/2019 - 31/12/2025 | 1st Funding period Funded by: DFG - Cluster of Excellence Type of project: Subproject in DFG-joint project hosted at University of Münster

Publications referred to in the talk

An Approximate AKE Principle for Metric Valued Fields Hils, Martin; Ludwig, Stefan Marian (2022) In: arxiv.org:2208.10186. Research article in digital collection | Preprint | submitted / under review