Three Strata of the "Impossible": Undecidable, Unprovable, Unsolved — E8 Intelligence Research

FINDING: The "impossible" class is not one problem but three distinct strata — undecidable (Turing/Hilbert), unprovable-in-a-given-system (Gödel), and merely unsolved (Collatz, Navier–Stokes). | MATH: Halting problem: no total computable H(P,x) decides whether program P halts on x. Proof by diagonalization: define D(P) = if H(P,P) then loop else halt ⇒ H(D,D) contradicts. Hilbert Entscheidungsproblem: no algorithm decides ∀-validity of FOL (Church–Turing 1936). Gödel I: any consistent, effectively axiomatized theory T ⊇ PA has a sentence G with T ⊬ G and T ⊬ ¬G, where G ↔ ¬Prov_T(⌜G⌝). Gödel II: T ⊬ Con(T). Rice's theorem: every non-trivial semantic property of programs is undecidable. | CONNECTION: Diagonalization is the shared engine — Cantor's |ℝ| > |ℕ| via the anti-diagonal, Gödel's arithmetized self-reference, Turing's D(P). This is a *self-similar* construction: the diagonal is the fixed point of the negation map, structurally analogous to the golden-section fixed point x = 1/(1+ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com