Undecidability as Structural Limit, Not Physical Barrier: A Hierarchy of Unsolvability — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Undecidability (halting problem, Gödel incompleteness) is a structural limit of formal systems, not a physical barrier — reducibility maps problems onto each other, revealing a hierarchy of unsolvability. | MATH: Halting problem: no total computable function H(p,i) decides if program p halts on input i. Gödel: for any consistent, recursively axiomatizable theory T ⊇ PA, ∃ sentence G with T⊬G and T⊬¬G. Reducibility: A ≤_m B (many-one) or A ≤_T B (Turing) — degrees form a partial order with 𝟘 (decidable) and 𝟘′ (halting set) as minimal non-zero jump. | CONNECTION: The Turing degree structure is a lattice — its partial order has joins/meets, echoing crystallographic root system lattices (e.g., A_n, D_n) in that both are discrete, ordered structures with symmetry under automorphisms. The jump operator (𝟘 → 𝟘′ → 𝟘″ …) is a recursive analogue of a symmetry group action — each jump is a "rotation" in the space of unsolvability. No golden ratio appears; the structure is ordinal, not m Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204133666
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