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.
Highlights
The usual additionand multiplicationbetween ordinals can be defined by transfinite recursion and have a clear order-theoretical meaning; for example, αβ is the order-type of a copy of α to which a copy of β is added at the top
In the present note we introduce and study the analogously defined infinitary product in the case of sequences of length ω. The computation of this infinite natural product can be reduced to the computation of some— possibly infinite—natural sum; in particular, we can directly transfer results from sums to products, rather than repeating essentially the same arguments
As we showed in [14], Carruth characterization of the finite natural sum cannot be generalized as it stands to the infinitary natural sum; see, in particular, the comments at the beginning of [14, Section 4]
Summary
The usual additionand multiplicationbetween ordinals can be defined by transfinite recursion and have a clear order-theoretical meaning; for example, αβ is the order-type of a copy of α to which a copy of β is added at the top. The particular case of the sum of a sequence of length ω has been used in Wang [25] and Väänänen and Wang [24] with applications to infinitary logics. In the present note we introduce and study the analogously defined infinitary product in the case of sequences of length ω The computation of this infinite natural product can be reduced to the computation of some— possibly infinite—natural sum; in particular, we can directly transfer results from sums to products, rather than repeating essentially the same arguments. If ď1 is a linear order on the finite-support-product of a sequence pαiqiăω of ordinals and ď1 is such that, for every element c, the set of the ď1 predecessors of c is built in a way similar to above, ď1 is a well-order of order type less than or equal to the infinite natural product of the αi’s.
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