Undecidability as Structural Boundary: Truth Diverges from Provability — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Undecidability is a structural boundary of formal systems, not a computational inconvenience — Gödel's incompleteness and Turing's halting problem reveal that truth and provability diverge within any sufficiently expressive formal system. | MATH: Gödel numbering (bijection ℕ ↔ formulas); First Incompleteness Theorem: ∃G such that ⊬G and ⊬¬G (assuming ω-consistency); Halting Problem: no total computable function h(e,x) decides whether φ_e(x) halts — proof via diagonalization: define d(e) = 1 if φ_e(e) halts, else loop; contradiction. Reducibility: A ≤_m B implies if B decidable then A decidable; undecidability propagates via many-one reductions. | CONNECTION: The diagonalization argument mirrors the golden ratio's self-similarity — a recursive self-reference that produces a fixed point (Gödel sentence G ↔ ¬Prov(G) is a fixed point of the provability predicate, analogous to φ = 1/(1+φ) = 0.618…). The lattice of Turing degrees (≤_T) forms a partial order with no maximal element — 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: W7208833478
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