Turing's Diagonalization Proof: Halting Undecidability as a Formal, Not Physical, Limit — E8 Intelligence Research

FINDING: Turing's 1936 proof establishes that no total computable function can decide halting for all program-input pairs — a hard limit on formal decidability, not a physical constant. MATH: - Core construction: diagonalization. Define H(P, I) → {halt, loop}. Assume H total & computable. Construct D(P): if H(P, P) = halt then loop; else halt. Then D(D) halts ⟺ D(D) loops — contradiction. - Formal statement: ∃ no computable f: ℕ → {0,1} with f(⟨M,x⟩) = 1 ⟺ M halts on x. Equivalently, the halting set K = {⟨M,x⟩ : M halts on x} is Σ⁰₁-complete, not Δ⁰₁ (not recursive). - Turing 1936 actually proved undecidability of the Entscheidungsproblem via computable numbers; the halting-problem framing is a later (post-1952, Davis/Kleene) reformulation — see arXiv:2407.00680. - Rice's theorem generalizes: every non-trivial semantic property of programs is undecidable. - No numerical constants, ratios, or geometric quantities appear in the theorem itself. CONNECTION: None supported by evidence. T

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com