Enumeration of the adjunctive hierarchy of hereditarily finite sets

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. We solve an open problem proposed by Kirby in 2008 concerning a recurrence relation for the cardinality an of the n-th level of the adjunctive hierarchy of hereditarily finite sets; in this hierarchy, new sets are formed by the addition of a new single element drawn from the already existing sets to an already existing set. We also show that our results can be generalized to sets with atoms, or can be refined by rank, cardinality, or by the maximum level from where the new adjoined element is drawn. We also show that an satisfies the asymptotic formula an = C 2 n + O(C n−1 ), for a constant C ≈ 1.3399, 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 Θ(2n).