Chaitin's Omega: The Algorithmically Random, Uncomputable Halting Probability — E8 Intelligence Research

FINDING: Chaitin's Omega is a real number in [0,1] defined as the halting probability of a universal prefix-free Turing machine; its binary expansion is algorithmically random (Martin-Löf random) and encodes the halting problem, making it uncomputable and transcendentally irreducible to any formal system. MATH: - Definition: \( \Omega = \sum_{p \in \text{Halting programs}} 2^{-|p|} \), where \( |p| \) is the length of program \( p \) in bits (prefix-free code). - Key property: \( \Omega \) is a dyadic rational only if the halting problem were decidable (impossible). Its binary expansion is a normal number in base 2 (Borel normality) and is Martin-Löf random: no computable test can distinguish it from a random sequence. - Measure-theoretic basis: The sum converges to a value in (0,1) because the Kraft inequality ensures \( \sum 2^{-|p|} \leq 1 \) for prefix-free codes. - Constants: The exact value depends on the choice of universal Turing machine; no closed-form expression exis Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com