2018-11-05 17:33:46 UTC
A world class expert provided some coaching. They have published very much in the field of Incompleteness and many related fields.
They changed my formulation of a correct simplification of Gödel's 1931 Incompleteness Theorem:
∀F ∈ Formal_Systems (∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G)))
L(F) means the language of formal system F.
∀F (F ∈ Formal_Systems & Q ⊆ F) → ∃G ∈ L(F) (G ↔ ~(F ⊢ G))
Q here is Robinson Arithmetic (the theorem fails for some weaker formal systems)
As long as any simplification of the original theorem sufficiently captures the essence of the original theorem, any refutation of this simplification applies equally to the original theorem by analogy.
Since the following G is neither Provable nor Refutable in F it forms a Gödel sentence in F.
∃F ∈ Formal_Systems ⊇ Q (∃G ∈ L(F) (G ↔ ~(F ⊢ G)))
If the above expression evaluates to False it refutes every Gödel sentence in every F ⊇ Q.
Copyright 2018 Pete Olcott