The Uncomputable Halting Probability and Cantor Set's Fractal Dimension — E8 Intelligence Research

Abstract

FINDING: Chaitin's Omega (Ω) is an algorithmically random, uncomputable real number encoding the halting probability of a prefix-free Turing machine, defined via the Cantor set's binary tree structure. | MATH: Ω = Σ_{p halts} 2^{-|p|}, where p ranges over prefix-free programs (no codeword is a prefix of another); Kraft inequality ensures Σ 2^{-|p|} ≤ 1, so Ω ∈ [0,1] and is a normal, Martin-Löf random number. The Cantor set C = {Σ aₙ/3ⁿ : aₙ ∈ {0,2}} has Hausdorff dimension log₂/log₃ ≈ 0.6309, Lebesgue measure 0, and is homeomorphic to {0,1}^ℕ — the same binary tree space where Ω lives. | CONNECTION: The prefix-free condition mirrors the golden-ratio-like self-similarity: the binary tree's branching ratio 1/2 per level, when summed, yields the geometric series Σ 2^{-n} = 1 — the same convergence structure as the golden ratio's continued fraction [1;1,1,...] = φ = 1.618. The Cantor set's ternary construction (removing middle thirds) has a self-similarity ratio 1/3, whose dimension log₂/l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com