Encoding Sets as Real Numbers (Extended version)

TL;DR

This paper introduces a novel real-number encoding for hereditarily finite hypersets, extending Ackermann's set encoding to a broader class and proving its well-definedness and injectivity properties.

Contribution

It proposes a new hyperset encoding using negative exponents, extending Ackermann's encoding to the class of hereditarily finite hypersets and analyzing its properties.

Findings

Encoding is well-defined over all hereditarily finite hypersets.

The encoding assigns unique real numbers to each hyperset.

Preliminary results on injectivity of the encoding.

Abstract

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 $\mathsf{HF}^{1/2}$ 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 $\mathbb{R}(x) := \sum_{y\in x}2^{-\mathbb{R}(y)}$. By a careful analysis, we prove that the encoding $\mathbb{R}_{A}$ is well-defined over the whole collection $\mathsf{HF}^{1/2}$, 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 $\mathbb{R}_{A}$.