Encoding Sets as Real Numbers (Extended version)

We study a variant of the Ackermann encoding $$\mathbb{N}$(x) := \sum_{y\in x}2^{$\mathbb{N}$(y)}$ of the hereditarily finite sets by the natural numbers, applicable to the larger collection ${\mathsf{H}\mathsf{F}}^{1\mathrm{/}2}$ of the hereditarily finite hypersets. The proposed variation is obtained by simply placing a `minus’ sign before each exponent in the definition of $\mathbb{N}$, resulting in the expression $\mathbb{R}(x):={\sum }_{y\in x}{2}^{-\mathbb{R}(y)}$. By a careful analysis, we prove that the encoding ${\mathbb{R}}_A$ is well-defined over the whole collection ${\mathsf{H}\mathsf{F}}^{1\mathrm{/}2}$, as it allows one to univocally assign a real-valued code to each hereditarily finite hyperset. We also address some preliminary cases of the injectivity problem for ${\mathbb{R}}_A$.