Ordered fields dense in their real closure and definable convex valuations

TL;DR

This paper investigates ordered fields dense in their real closure, focusing on definable valuations, and re-examines the Shelah-Hasson Conjecture with new valuation theoretic insights.

Contribution

It provides a systematic model-theoretic and valuation-theoretic analysis of such ordered fields and offers new perspectives on definable henselian valuations and the Shelah-Hasson Conjecture.

Findings

Characterization of ordered fields dense in their real closure

Identification of definable henselian valuations in ordered fields

Counterexample impacting the Shelah-Hasson Conjecture

Abstract

In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah-Hasson Conjecture (specialised to ordered fields) and provide an example limiting its valuation theoretic conclusions.