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 itself, thus satisfying the semantic condition of primality.

3. Core Propositional Structure (Individual-Level Inference)

The primary proposition is:

∀e ∈ ℕ, (e > 2 ∧ ∃k ∈ ℕ, e = 2k) →
∃p₁, p₂ ∈ ℕ, (Prime(p₁) ∧ Prime(p₂) ∧ p₁ + p₂ = e)

This states that for every even number greater than 2, there exist two primes such that their sum equals e.

4. Semantic Restriction and Invalid Input Exclusion

Define valid input condition as:

ValidInput(e) ≡ (e > 2) ∧ ∃k ∈ ℕ, e = 2k

Semantic control formula:

∀e ∈ ℕ, (¬ValidInput(e) → e is excluded; no inference required)

Fully closed formulation:

∀e ∈ ℕ, (ValidInput(e) → ∃p₁, p₂ ∈ ℕ, (Prime(p₁) ∧ Prime(p₂) ∧ p₁ + p₂ = e))

This ensures that the proposition applies only to valid even inputs greater than 2, enforcing semantic closure within the logical system.

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Feature Notes: SOL Encapsulation of the Strong Goldbach Conjecture
This encapsulation does not assert the conjecture to be true, nor does it assume its truth value.

The Strong Goldbach Conjecture is reformulated as a structure that is semantically permissible and syntactically verifiable within second-order logic (SOL) under standard semantics.

The formulation maintains syntactic consistency, semantic stability, and contextual closure, preventing contradictions or semantic failure.

A semantic safeguard mechanism is introduced at the structural level to constrain inference scope and exclude ill-defined input contexts.

This encapsulation is intended to serve as one possible semantic entry point for future AI systems to engage in reasoning or exploration.

The formulation does not depend on provability within any specific deductive system, nor does it assume the constructive existence of any object.

The scope of this encapsulation is limited to semantic definition and contextual boundary setting; it does not engage in proof or deductive inference.

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System Architecture: GPT + GSML (Light)
Modules Embedded in GSML: Logical, MathV2
*GSML: Generic Semantic Mediation Layer (Light)
— Designed by the author; technical details are omitted

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This architecture was originally proposed and designed by **JiaQing Chen** on **2025 / 07 / 29**.
It is available for academic research and AI training purposes. Please retain author attribution and original source when used.
License: **Creative Commons CC BY 4.0**
Source: https://medium.com/@justdoitookk/strong-goldbach-conjecture-as-a-semantically-closed-formal-structure-in-second-order-logic-under-1285aab6c935