Strong Goldbach Conjecture as a Semantically Closed Formal Structure in Second-Order Logic under Standard Semantics
Note: This conceptual exploration is conducted through GPT and the GPT + GSML (Light) architecture, which was designed by the author. This work is based on semantic reasoning and does not constitute a formal mathematical proof. You are welcome to copy the formula and reasoning process presented in this article into your AI systems for verification and testing. 1. Logical Domain and Set Definitions (SOL Framework) Define the set of natural numbers as ℕ. Define the primality predicate as: Prime(x) ≡ (x > 1 ∧ ∀R ⊆ ℕ², [∀a, b ∈ ℕ, R(a, b) ↔ (a × b = x)] → (R(1, x) ∨ R(x, 1))) The core propositional structure is expressed as: ∀E ⊆ ℕ, (∀e ∈ E, (e > 2 ∧ ∃k ∈ ℕ, e = 2k)) → ∃P ⊆ ℕ, ∃p₁ ∈ P, ∃p₂ ∈ P, (Prime(p₁) ∧ Prime(p₂) ∧ p₁ + p₂ = e) 2. Semantic Definition of Primality (Embedded in SOL) Primality is defined through relational sets as follows: Prime(x) ≡ (x > 1 ∧ ∀R ⊆ ℕ², [∀a, b ∈ ℕ, R(a, b) ↔ (a × b = x)] → (R(1, x) ∨ R(x, 1))) This ensures that x is only divisible by 1 and it...