Some transfinite natural sums

TL;DR

This paper explores a transfinite iteration of the ordinal natural sum, revealing its relation to ordinal sums, introducing permutation-invariant sums, and applying these concepts to define and characterize the size of well-founded trees.

Contribution

It introduces a new transfinite iteration of the natural sum, compares it with ordinal sums, and applies these sums to define and characterize the size of well-founded trees.

Findings

Iterated natural sum differs from ordinal sum only finitely many times.

Introduces permutation-invariant infinite natural sums that coincide in the countable case.

Provides an order-theoretical characterization of tree size using natural sums.

Abstract

We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages and show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a "mixed sum" (in an order-theoretical sense) of the ordinals in the sequence, in fact, it is the largest mixed sum which satisfies a finiteness condition, relative to the ordering of the sequence. We introduce other infinite natural sums which are invariant under permutations and show that they all coincide in the countable case. Finally, in the last section we use the above infinitary natural sums in order to provide a definition of size for a well-founded tree, together with an order-theoretical characterization in the countable case. The proof of this order-theoretical characterization is mostly independent from the rest of this paper.