The Diagonal Argument as a Unified Fixed-Point Theorem — E8 Intelligence Research

FINDING: The diagonal argument is a single categorical fixed-point theorem (Lawvere) unifying Cantor, Gödel, and Tarski; Brouwer's fixed-point theorem is a topological analogue, and uniform asymptotic regularity reveals a metric-space obstruction to surjectivity. | MATH: Lawvere's fixed-point theorem: If \\(e: A \\to B^A\\) is surjective (or weakly point-surjective), then every \\(f: B \\to B\\) has a fixed point. Diagonal map: \\(\\Delta(x) = (x,x)\\); composition \\(f \\circ e(x)(x)\\) yields a point \\(y\\) with \\(f(y)=y\\). Cantor: \\(B = \\{0,1\\}\\), no surjection \\(A \\to 2^A\\). Gödel: \\(B =\\) truth values in a formal system, \\(f = \\neg\\) (negation) → no fixed point → incompleteness. Brouwer: continuous \\(f: D^n \\to D^n\\) has fixed point (topological, not categorical). Uniform asymptotic regularity: if \\(T\\) is uniformly asymptotically regular on a metric space, then no surjective \\(T\\) exists (arXiv:1511.04069) — a metric fixed-point obstruction. | CONNECTION: Lawvere's theorem is a categorical re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com