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
Lipparini, Paolo
2018-01-01
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.File | Dimensione | Formato | |
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