Gödel's Incompleteness: True Unprovable Statements and Self-Consistency Limits — E8 Intelligence Research

FINDING: Gödel's Incompleteness Theorems establish that any consistent formal system capable of arithmetic contains true but unprovable statements, and cannot prove its own consistency. | MATH: First Theorem: For any consistent, recursively axiomatizable theory T containing PA, there exists a sentence G such that T ⊬ G and T ⊬ ¬G (G is the fixed point of ¬Prov_T(x), via diagonalization: G ↔ ¬Prov_T(⌜G⌝)). Second Theorem: T ⊬ Con(T), where Con(T) ≡ ¬Prov_T(⌜0=1⌝). Generalized (Rosser): replaces ω-consistency with simple consistency via Rosser sentence R ↔ ∀x(Prov_T(⌜R⌝,x) → ∃y Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com