On numbers, germs, and transseries

TL;DR

This paper explores the model theory of $H$-fields, unifying germs, surreal numbers, and transseries within asymptotic differential algebra to deepen understanding of infinitesimal and infinite quantities.

Contribution

It provides new insights into the algebraic and model-theoretic structures connecting germs, surreal numbers, and transseries through the framework of $H$-fields.

Findings

Unified framework for germs, surreal numbers, and transseries

Advances in the model theory of $H$-fields

Progress in asymptotic differential algebra

Abstract

Germs of real-valued functions, surreal numbers, and transseries are three ways to enrich the real continuum by infinitesimal and infinite quantities. Each of these comes with naturally interacting notions of ordering and derivative. The category of $H$-fields provides a common framework for the relevant algebraic structures. We give an exposition of our results on the model theory of $H$-fields, and we report on recent progress in unifying germs, surreal numbers, and transseries from the point of view of asymptotic differential algebra.