An infinite natural sum

TL;DR

This paper explores a countably infinite natural sum of ordinals, providing an order-theoretic characterization, and demonstrating its relation to the usual ordinal sum with practical evaluation methods.

Contribution

It introduces a new order-theoretic characterization of the countable natural sum of ordinals and compares it to the traditional ordinal sum, including evaluation techniques.

Findings

The countable natural sum differs from the ordinal sum only at a finite initial segment.

The sum can be evaluated using finite natural sum computations.

Various infinite mixed sums of ordinals are discussed.

Abstract

As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a better alternative, since it shares most of the usual properties of the addition on the naturals. A countably infinite version of the natural sum has been used in a recent paper by V\"a\"an\"anen and Wang, with applications to infinitary logics. We provide an order theoretical characterization of this operation. We show that this countable natural sum differs from the more usual infinite ordinal sum only for an initial finite "head" and agrees on the remaining infinite "tail". We show how to evaluate the countable natural sum just by computing a finite natural sum. Various kinds of infinite mixed sums of ordinals are discussed.