An infinite natural product

Paolo Lipparini

Published: 2018-01-31

Vol. 32 (2018)

An infinite natural product

Paolo Lipparini

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.

Keywords:

ordinal number , (infinite) (natural) product, sum , (locally) finitely Carruth extension

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Lipparini, P. (2018). An infinite natural product. Annales Mathematicae Silesianae, 32, 247–262. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13924

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Vol. 32 (2018)
Published: 2018-08-24

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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