Search Gap: Syntax–Semantics Sources Miss Provability–Truth Fixed-Point Mathematics — E8 Intelligence Research

FINDING: Search returned only pedagogical/definitional material on syntax–semantics distinction; no substantive results on the provability–truth fixed-point problem. | MATH: The relevant mathematics (absent from these sources) is: Gödel's first incompleteness theorem — for any consistent, effectively axiomatized theory T extending Robinson arithmetic, ∃ sentence G with T ⊬ G and T ⊬ ¬G, where G ↔ ¬Prov_T(⌜G⌝); Tarski's undefinability theorem — no formula Tr(x) in the language of T satisfies Tr(⌜φ⌝) ↔ φ for all φ (truth is not arithmetically definable); Löb's theorem — if T ⊢ Prov_T(⌜φ⌝) → φ then T ⊢ φ; the diagonal lemma (fixed-point lemma) — for any formula ψ(x), ∃ sentence γ with T ⊢ γ ↔ ψ(⌜γ⌝). The syntactic/semantic boundary is exactly the Prov vs. Tr distinction: Prov_T is Σ₁-definable (syntactic, arithmetizable), Tr is not definable in T (semantic, transcendent). | CONNECTION: The diagonal lemma is a fixed-point construction — the arithmetic analogue of a self-referential fixed p Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com