The surreal numbers as a universal H-field

TL;DR

This paper demonstrates that the surreal numbers, equipped with a specific derivation, serve as a universal structure for embedding various ordered differential fields, including transseries and Hardy fields.

Contribution

It establishes that the surreal numbers form a universal H-field into which many differential fields can be embedded, extending the understanding of their algebraic and model-theoretic properties.

Findings

Embedding of transseries into surreals is elementary.

Any Hardy field can be embedded into the surreal numbers with the Berarducci-Mantova derivation.

Surreal numbers serve as a universal H-field for these structures.

Abstract

We show that the natural embedding of the differential field of transseries into Conway's field of surreal numbers with the Berarducci-Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field of surreals with the Berarducci-Mantova derivation.