Encoding Sets as Real Numbers (Extended version)
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We study a variant of the Ackermann encoding N(x) := ∑_y∈ x2^N(y) of the hereditarily finite sets by the natural numbers, applicable to the larger collection 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 N, resulting in the expression R(x) := ∑_y∈ x2^-R(y). By a careful analysis, we prove that the encoding R_A is well-defined over the whole collection 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 R_A.
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