Integration on the Surreals

TL;DR

This paper develops a framework for extending and integrating a broad class of functions, including many from practical applications, within Conway's surreal numbers, addressing longstanding foundational and definitional challenges.

Contribution

It demonstrates the existence of function extensions and integrals for most practical functions on the surreals, including resurgent functions and solutions to complex differential equations.

Findings

Extensions exist for a large subclass of resurgent functions.

Constructive methods allow work in NBG without the Axiom of Choice.

Obstructions arise for more general functions like smooth functions.

Abstract

Conway's real closed field $\mathbf{No}$ of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper we address this and related unresolved issues by showing that extensions to $\mathbf{No}$, and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the \emph{resurgent functions}, a subclass that contains the functions that at $\infty$ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations. By suitable changes of variables we deal with arbitrarily located singular points. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of $\mathbf{No}$ more generally and illustrate the result with structures familiar from the surreal literature. The extensions of functions and integrals that concern us are constructive in nature, which permits us to work in NBG less the Axiom of Choice (for both sets and proper classes). Following the completion of the positive portion of the paper, it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics.