The Ackermann encoding and its siblings

Abstract The celebrated Ackermann encoding of hereditarily finite sets is generalized to a parametric formula designed to map not only these sets but also hereditarily finite multisets and hypersets into the non-negative real numbers. This extension suggests a novel approach to the graph canonization problem by reducing it to a simple comparison of real values. By suitably varying the sole parameter of this formula, both the original Ackermann encoding and another previously studied map emerge as special cases. When the parameter is chosen from the natural numbers, the function yields a bijective encoding of a subuniverse of hereditarily finite multisets into the natural numbers. If, instead, the parameter is chosen to be transcendental and lies within a specific interval on the positive real line, the function is conjectured to provide an injective encoding of both multisets and hypersets.