On generalised Ackermann encodings – the basis issue

In this paper, a generalised version A𝛽 of the celebrated Ackermann encoding of the hereditarily finite sets, aimed at assigning a real number also to each hereditarily finite hyperset and multiset, is studied. Such a mapping establishes a significant link between real numbers and the theories of such generalised notions of set, so that performing set-theoretic operations can be translated into their number-theoretic equivalent. By appropriately choosing a parameter 𝛽, both the Ackermann encoding and the less known map R𝐴 arise as special cases; a bijective encoding of a subuniverse of hereditarily finite multisets occurs whenever this parameter is chosen among natural numbers, while if it is taken transcendental and within a peculiar interval of the real positive line, then the function is surmised to ensure an injective mapping of both the aforementioned universes.