Enumeration of the adjunctive hierarchy of hereditarily finite sets
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Publication:5262491
DOI10.1093/LOGCOM/EXU062zbMATH Open1348.03045arXiv1309.2512OpenAlexW2157750270MaRDI QIDQ5262491
Author name not available (Why is that?)
Publication date: 15 July 2015
Published in: (Search for Journal in Brave)
Abstract: Hereditarily finite sets (sets which are finite and have only hereditarily finite sets as members) are basic mathematical and computational objects, and also stand at the basis of some programming languages. This raises the need for efficient representation of such sets, for example by numbers. In 2008, Kirby proposed an adjunctive hierarchy of hereditarily finite sets, based on the fact that they can also be seen as built up from the empty set by repeated adjunction, that is, by the addition of a new single element drawn from the already existing sets to an already existing set. Determining the cardinality of each level of this hierarchy, problem crucial in establishing whether the natural adjunctive hierarchy leads to an efficient encoding by numbers, was left open. In this paper we solve this problem. Our results can be generalized to hereditarily finite sets with atoms, or can be further refined by imposing restrictions on rank, on cardinality, or on the maximum level from where the new adjoined element can be drawn. We also show that satisfies the asymptotic formula , for a constant , which is a too fast asymptotic growth for practical purposes. We thus propose a very natural variant of the adjunctive hierarchy, whose asymptotic behavior we prove to be . To our knowledge, this is the first result of this kind.
Full work available at URL: https://arxiv.org/abs/1309.2512
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