Chaitin's Omega: Algorithmic Randomness Beyond the Arithmetical Hierarchy — E8 Intelligence Research

FINDING: Chaitin's Omega is not computable even with oracles that decide the halting problem, placing it beyond the arithmetical hierarchy and linking algorithmic randomness to incompleteness. MATH: - Chaitin's Omega (Ω) = Σ_{p halts} 2^{-|p|}, where |p| is program length in bits. - Ω is Turing-equivalent to the halting problem (∅'), but not computable by any oracle in the arithmetical hierarchy (i.e., not Δ⁰₂, Σ⁰₂, etc.). - Ω is a "maximally random" real number: its binary expansion is algorithmically random (Kolmogorov complexity of any prefix n is ~n). - Arithmetical hierarchy: Σ⁰₁ (halting problem), Σ⁰₂ (limit-computable), ... Ω lies outside all finite levels. CONNECTION: - No direct geometric ratios (0.382, 0.618, etc.) appear. However, the structure of Ω's undecidability mirrors the infinite regress of self-similarity in fractal geometry: each oracle level reveals a new layer of uncomputability, analogous to the infinite nesting of a Mandelbrot set. - The base-2 rep Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com