Applying numerosity to surreal integration

TL;DR

This paper introduces a surreal-valued numerosity framework for measuring and integrating over discrete sets, satisfying Euclid's principle, and provides explicit formulas and properties for surreal integration.

Contribution

It develops a novel surreal numerosity-based approach to measure and integrate over sets, including infinite and infinitesimal parts, with explicit formulas and properties.

Findings

Explicit formula for surreal numerosity of sequences

Introduction of a surreal-valued function similar to Dirac Delta

New formulas for surreal integration connecting surreal numbers

Abstract

We present a novel framework for measuring the size of discrete subsets of using surreal-valued numerosity, which strictly satisfies Euclid's principle that "the whole is greater than a part". By mapping numerosities to surreal numbers via the canonical embedding of Hardy fields, we provide an explicit formula for the full numerosity of sequences, including their infinite, finite and infinitesimal parts, with examples. Then (interpreting germs at infinity as as Laplace transforms) we introduce a surreal-valued function that shares many properties with the Dirac Delta distribution and employ it to derive some integration properties of surreal values. Finally, we provide a formula for integrating a surreal-valued function over surreal domain, employing numerosity in a way, similar to Lebesgue measure and derive some novel formulas, connecting surreal numbers via integration. The suggested here method of surreal integration is different from methods suggested by other authors in dropping the property of linearity regarding infinite factors. In our opinion, the struggle to keep this rule intact is the reason for many failed attempts to define the surreal integration. For the suggested formulas, code in Wolfram Mathematica language is provided.