Categorical Fixed-Point Theorem Unifies Diagonalization Across Logic — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Lawvere's categorical fixed-point theorem unifies Cantor, Turing, Tarski, and Gödel via a single diagonalization schema; substructural logics further generalize it. | MATH: Lawvere's theorem: If ∃ f: A → BA (exponential object) is surjective, then every endomorphism α: B → B has a fixed point. Contrapositive yields diagonal argument: no surjection A → BA if α lacks fixed points. Fixed-point combinator: Δ(x) = α(f(x)(x)), with d = f(a) giving d(d) = α(d(d)). Substructural weakening: replace cartesian closed with symmetric monoidal closed + relevant logic — fixed point exists if α is "pointed" (has a distinguished element). | CONNECTION: The diagonal map Δ: A → A × A is a self-pairing — structurally analogous to the golden ratio's self-similarity ϕ = 1 + 1/ϕ, where the fixed point equation x = 1 + 1/x yields ϕ = (1+√5)/2 ≈ 1.618. Th
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026categorical,
title = {{Categorical Fixed-Point Theorem Unifies Diagonalization Across Logic — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635367},
url = {https://zenodo.org/records/23635367},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Categorical Fixed-Point Theorem Unifies Diagonalization Across Logic — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635367
UR - https://zenodo.org/records/23635367
AB - FINDING: Lawvere's categorical fixed-point theorem unifies Cantor, Turing, Tarski, and Gödel via a single diagonalization schema; substructural logics further generalize it. | MATH: Lawvere's theorem: If ∃ f: A → B^A (exponential object) is surjective, then every endomorphism α: B → B has a fixed point. Contrapositive yields diagonal argument: no surjection A → B^A if α lacks fixed points. Fixed-point combinator: Δ(x) = α(f(x)(x)), with d = f(a) giving d(d) = α(d(d)). Substructural weakening: replace cartesian closed with symmetric monoidal closed + relevant logic — fixed point exists if α is "pointed" (has a distinguished element). | CONNECTION: The diagonal map Δ: A → A × A is a self-pairing — structurally analogous to the golden ratio's self-similarity ϕ = 1 + 1/ϕ, where the fixed point equation x = 1 + 1/x yields ϕ = (1+√5)/2 ≈ 1.618. Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -