Lawvere's Fixed-Point Theorem: Unifying Diagonal Arguments in Cartesian Closed Categories — E8 Intelligence Research

FINDING: Lawvere's fixed-point theorem unifies Cantor, Gödel, Turing, Tarski diagonal arguments as one categorical theorem — self-reference is a structural property of cartesian closed categories with a point-surjective morphism. MATH: - Lawvere (1969): In a cartesian closed category, if f: A → B^A is point-surjective (∀g:A→B, ∃a∈A with f(a)=g), then every endomorphism e: B→B has a fixed point. - Contrapositive (Gödel form): No e: B→B without fixed point admits a point-surjective f: A → B^A. - Diagonal morphism Δ: A → A×A, Δ(a)=(a,a); the "diagonal argument" is the composite A →(f,id) B^A × A →eval B. - Cantor: B={0,1}, e=¬ (no fixed point) ⇒ no surjection A → 2^A. - Gödel: B = sentences, e = negation, provability as point-surjection ⇒ undecidable sentence. - Turing: B = {halt, loop}, e = flip ⇒ no total computable enumeration of total computable functions. - Tarski: B = truth values, e = ¬ ⇒ no definable truth predicate. - Yanofsky (2003) "A Universal Approach to Self-Referential Pa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com