We study a variant of the Ackermann encoding $mathbb{N}(x) := sum_{yin x}2^{mathbb{N}(y)}$ of the hereditarily finite sets by the natural numbers, applicable to the larger collection $mathsf{HF}^{1/2}$ 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 $mathbb{R}(x) := sum_{yin x}2^{-mathbb{R}(y)}$. By a careful analysis, we prove that the encoding $mathbb{R}_{A}$ is well-defined over the whole collection $mathsf{HF}^{1/2}$, 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 $mathbb{R}_{A}$.
تحميل البحث