Surreal fields stable under exponential, logarithmic, derivative and anti-derivative functions

TL;DR

This paper investigates the stability of surreal number fields under exponential, logarithmic, derivation, and anti-derivation operations, providing conditions for such stability and applications to fields with smaller ordinal bounds.

Contribution

It establishes sufficient conditions for surreal fields to remain stable under key operations and introduces a field with a reduced ordinal complexity for computability considerations.

Findings

Surreal fields are stable under exponential, logarithmic, derivation, and anti-derivation operations.

A new field with only up to psilon_omega ordinal complexity is constructed.

The results have implications for computability and the structure of surreal number fields.

Abstract

The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficients, namely Hahn series with exponents in $\textbf{No}$ itself. Some years ago, Berarducci and Mantova considered derivation over the surreal numbers, seeing them as germs of functions, in correspondence to transseries. In this article, following our previous work, we exhibit a sufficient condition on the structure of a surreal field to be stable under all operations among exponential, logarithm, derivation and anti-derivation. Motivated, in the long term, by computability considerations, we also provide a non-trivial application of this theorem: the existence of a pretty reasonable field that only requires ordinals up to $\epsilon_\omega$, which is far smaller than $\omega_1^{CK}$ (resp. $\omega_1$), the first non-computable (resp. uncountable) ordinal.