Surreal numbers with derivation, Hardy fields and transseries: a survey

TL;DR

This survey explores surreal numbers, their structure as real closed exponential fields, recent developments in equipping them with derivations, and their relation to transseries and Hardy fields, highlighting their universal properties.

Contribution

It provides an overview of surreal numbers' structure, recent advances in their derivation, and their connection to transseries and Hardy fields, emphasizing their universality.

Findings

Surreal numbers form a universal real closed exponential field.

Recent work shows surreal numbers can be viewed as transseries.

The structure of H-fields provides an abstract framework for surreal numbers.

Abstract

The present article surveys surreal numbers with an informal approach, from their very first definition to their structure of universal real closed analytic and exponential field. Then we proceed to give an overview of the recent achievements on equipping them with a derivation, which is done by proving that surreal numbers can be seen as transseries and by finding the `simplest' structure of H-field, the abstract version of a Hardy field. All the latter notions and their context are also addressed, as well as the universality of the resulting structure for surreal numbers.