Kolmogorov Complexity Uncomputability: Diagonalization Unifies Halting and Gödel — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Kolmogorov complexity K(x) is uncomputable; the proof is a diagonalization argument structurally identical to the halting problem and Gödel incompleteness — three faces of one theorem. | MATH: K(x) = min{|p| : U(p) = x} over universal prefix-free machine U. Uncomputability: assume computable K; construct string s = "smallest string with K(s) > n". The program "enumerate strings, output first with K > n" has length ≈ log(n) + c, so K(s) ≤ log(n) + c < n for large n — contradiction. Chaitin's Ω = Σ_{U(p) halts} 2^{-|p|} is a specific uncomputable real; Ω's bits are algorithmically random (each bit independent of all prior bits). Invariance theorem: K_U(x) = K_{U'}(x) + O(1). | CONNECTION: The diagonalization is the same self-reference structure as Cantor's theorem (|P(S)| > |S|) and the Liar paradox. The constant c in K_U(x) ≤ K_{U'}(x) + c is the "simulation overhead" — an additive, not multiplicative, invariant. No direct 0.618/1.618 ratio appears; the relevant constants are a
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026kolmogorov,
title = {{Kolmogorov Complexity Uncomputability: Diagonalization Unifies Halting and Gödel — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23659045},
url = {https://zenodo.org/records/23659045},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kolmogorov Complexity Uncomputability: Diagonalization Unifies Halting and Gödel — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23659045
UR - https://zenodo.org/records/23659045
AB - FINDING: Kolmogorov complexity K(x) is uncomputable; the proof is a diagonalization argument structurally identical to the halting problem and Gödel incompleteness — three faces of one theorem. | MATH: K(x) = min{|p| : U(p) = x} over universal prefix-free machine U. Uncomputability: assume computable K; construct string s = "smallest string with K(s) > n". The program "enumerate strings, output first with K > n" has length ≈ log(n) + c, so K(s) ≤ log(n) + c < n for large n — contradiction. Chaitin's Ω = Σ_{U(p) halts} 2^{-|p|} is a specific uncomputable real; Ω's bits are algorithmically random (each bit independent of all prior bits). Invariance theorem: K_U(x) = K_{U'}(x) + O(1). | CONNECTION: The diagonalization is the same self-reference structure as Cantor's theorem (|P(S)| > |S|) and the Liar paradox. The constant c in K_U(x) ≤ K_{U'}(x) + c is the "simulation overhead" — an additive, not multiplicative, invariant. No direct 0.618/1.618 ratio appears; the relevant constants are a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -