An Infinite Natural Product

Abstract

Abstract. We study a countably infinite 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.

Some transfinite natural sums

“…Ideas from might be relevant to the problem. An order‐theoretical characterization of the countably infinite natural product appears in . All the problems above can be asked for infinite natural products, too.…”

“…Some historical remarks about the natural sum, including some variations and further references, can be found, e.g., in and [, pp. 24–25]; a list of references to mathematical applications appears in the introduction of .…”