Ordered henselian valued fields: definability and Borel sets

TL;DR

This paper proves that ordered henselian valued fields have quantifier elimination in an expanded language and that their definable sets are Borel, linking model theory with descriptive set theory.

Contribution

It establishes relative quantifier elimination for ordered henselian valued fields and shows definable sets are Borel, advancing understanding of their logical and topological properties.

Findings

Ordered henselian valued fields admit relative quantifier elimination.

Definable sets in these fields are Borel sets in the order topology.

Results relate to Shelah's classification conjecture of NIP fields.

Abstract

We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a dimensionality reduction theorem that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. Our results are contextualised within Shelah's classification conjecture of NIP fields and its connections to the study of definable henselian valuations and the Fundamental Theorem of Statistical Learning.