<abstract xmlns="http://www.w3.org/1999/xhtml"><p> 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.</p></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.

An Infinite Natural Product

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"An Infinite Natural Product"}

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...

An Infinite Natural Product

Published Online: Aug 24, 2018

Page range: 247 - 262

Received: Jun 14, 2017

Accepted: Nov 08, 2017

DOI: https://doi.org/10.1515/amsil-2017-0013

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

© 2018 Paolo Lipparini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

An Infinite Natural Product

Published Online: Aug 24, 2018

Page range: 247 - 262

Received: Jun 14, 2017

Accepted: Nov 08, 2017

DOI: https://doi.org/10.1515/amsil-2017-0013

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

© 2018 Paolo Lipparini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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