Enumeration of the adjunctive hierarchy of hereditarily finite sets

TL;DR

This paper analyzes the adjunctive hierarchy of hereditarily finite sets, providing exact cardinalities, asymptotic growth formulas, and proposing a modified hierarchy with more practical growth rate for efficient set encoding.

Contribution

It solves the open problem of determining the cardinality of each hierarchy level and introduces a variant with more manageable asymptotic growth.

Findings

Exact cardinality formula for hierarchy levels

Asymptotic growth of original hierarchy is exponential with base ~1.34

Modified hierarchy has linear exponential growth, making it more practical

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 $a_n$ 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 $a_n$ satisfies the asymptotic formula $a_n = C^{2^n} + O(C^{2^{n-1}})$, for a constant $C \approx 1.3399$, 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 $\Theta(2^n)$. To our knowledge, this is the first result of this kind.