Lawvere's Fixed-Point Theorem Unifies Cantor, Turing, Tarski, and Gödel — E8 Intelligence Research

FINDING: Diagonalization, self-reference, and fixed points are three faces of one theorem — Lawvere's fixed-point theorem — which subsumes Cantor, Turing, Tarski, and Gödel as instances. | MATH: Lawvere: if f: A → B^A is surjective (or has a right inverse / "weakly point-surjective"), then every g: B → B has a fixed point. Contrapositive: no surjection A ↠ B^A when |B| ≥ 2. Cantor: A = X, B = {0,1}, g = ¬ ⇒ no surjection X ↠ 2^X. Gödel/Tarski: A = formulas, B = truth values, g = negation of provability/truth ⇒ no definable truth predicate. Knaster–Tarski: monotone f: L → L on complete lattice L has least fixed point lfp(f) = ⋀{x : f(x) ≤ x} and greatest gfp(f) = ⋁{x : x ≤ f(x)}. | CONNECTION: Complete lattices are the algebraic skeleton of geometric harmony — the fixed-point interval [lfp, gfp] is a sublattice; the 0.618/1.618 pair arises as the unique positive fixed point of x ↦ 1 + 1/x (φ² = φ + 1) and x ↦ 1 − x (0.5, trivial), and as the attracting fixed point of the continued-fract Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com