The hyperserial field of surreal numbers

TL;DR

This paper introduces hyperexponential and hyperlogarithm functions on surreal numbers, establishing that surreal numbers form a hyperserial field with extremely fast and slow growth functions at infinity.

Contribution

It defines hyperexponential and hyperlogarithm functions on surreal numbers and proves that surreal numbers form a hyperserial field under these definitions.

Findings

Defined hyperexponential functions $E_{\omega^{\alpha}}$ and hyperlogarithms $L_{\omega^{\alpha}}$ for surreal numbers.

Proved surreal numbers form a hyperserial field with these functions.

Established the functions as archetypes of fast and slow growth at infinity.

Abstract

For any ordinal $\alpha > 0$, we show how to define a hyperexponential $E_{\omega^{\alpha}}$ and a hyperlogarithm $L_{\omega^{\alpha}}$ on the class $\mathbf{No}^{>, \succ}$ of positive infinitely large surreal numbers. Such functions are archetypes of extremely fast and slowly growing functions at infinity. We also show that the surreal numbers form a so-called hyperserial field for our definition.