# An infinite natural product

## TL;DR

This paper introduces a method to compute the infinite natural product of ordinals, revealing that in many cases it depends solely on the natural sum of the degrees of the factors, and provides an order-theoretical characterization.

## Contribution

It offers a new effective computation method for infinite natural products of ordinals and extends previous results on infinitary sums to products.

## Key findings

- The infinite natural product depends only on the natural sum of degrees in nontrivial cases.
- Provides an order-theoretical characterization merging Carruth's theorem and ordinal product descriptions.
- Lifts results from infinitary sums to infinitary products.

## Abstract

We study an infinite countable iteration of the natural product between ordinals. We present an "effective" way to compute this countable natural product, in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product, this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals.