We study a countably infinite iteration of the natural product be- tween 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 pro- vide 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.

Lipparini, P. (2018). An Infinite Natural Product. ANNALES MATHEMATICAE SILESIANAE, 32(1), 247-262 [10.1515/amsil-2017-0013].

### An Infinite Natural Product

#### Abstract

We study a countably infinite iteration of the natural product be- tween 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 pro- vide 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2018
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/01 - LOGICA MATEMATICA
English
Senza Impact Factor ISI
ordinal number; (infinite) (natural) product, sum; (locally) finitely Carruth extension.
https://arxiv.org/abs/1703.06908
Lipparini, P. (2018). An Infinite Natural Product. ANNALES MATHEMATICAE SILESIANAE, 32(1), 247-262 [10.1515/amsil-2017-0013].
Lipparini, P
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2108/207926`