Higher arithmetic on the ordinals

TL;DR

This paper introduces a sequence of higher-order binary operations on ordinal numbers, extending standard ordinal arithmetic and exploring their relation to hyperoperations on natural numbers.

Contribution

It proposes a new hierarchy of ordinal operations that generalize ordinal addition and multiplication, linking them to hyperoperations.

Findings

Defined a sequence of higher ordinal operations

Established connections to hyperoperations on natural numbers

Explored properties and potential applications of these operations

Abstract

We motivate and study an infinite sequence of binary operations on the ordinal numbers, extending the standard arithmetic on the ordinals to higher degrees of iteration. Connections to the hyperoperations on the natural numbers are discussed.