Transfinite Ladder of Undecidability: From Arithmetical to Ordinal-Time Hierarchies — E8 Intelligence Research

FINDING: The arithmetical hierarchy and oracle Turing machines define a transfinite ladder of undecidability, where each level (Σₙ, Πₙ) corresponds to quantifier alternation over the natural numbers, and infinite-time Turing machines (ITTM) extend this into ordinal time, revealing a non-collapsing hierarchy of hyperarithmetical and beyond. | MATH: Arithmetical hierarchy: Σₙ⁰ = ∃x₁∀x₂…Qxₙ φ(x₁,…,xₙ), Πₙ⁰ = ∀x₁∃x₂…Qxₙ φ; Σₙ⁰ ∪ Πₙ⁰ ⊂ Δₙ₊₁⁰ ⊂ Σₙ₊₁⁰ ∩ Πₙ₊₁⁰. Oracle machine: M^A with characteristic function χ_A; Turing jump: A' = {e : φ_e^A(e) halts}, yielding degrees 0 < 0' < 0'' < … < 0^(α) for recursive ordinals α. ITTM: computation in ordinal time, halting at limit ordinals via limsup/liminf of cell states; ITTM-decidable sets include Π¹₁ but not Σ¹₂ (by Hamkins–Lewis). | CONNECTION: The hierarchy's ordinal indexing (0^(α) for α < ω₁^CK) mirrors the recursive ordinals, which are order-types of computable well-orderings — these embed the golden ratio's continued fraction [1;1,1,…] as the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com