**So the correct formalization would be something like:**

∀F((F ∈ Formal_Systems & Q ⊆ F & Consistent(F)) →

∃G ∈ L(F) (G ↔ ~(F ⊢ G)))

**English meaning of the above logic sentence:**

For all consistent formal systems that are a superset of Robinson Arithmetic

there is a sentence of the language of this formal system that is materially

equivalent to a statement of its own unprovability.

**Its rebuttal would be formalized as this:
**∀F ∈ Formal_Systems (~∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))

**English meaning of the above logic sentence:**

For all formal systems there is no sentence of the language of this formal system that is materially equivalent to a statement of its own unprovability.

If G was Provable in F this contradicts its assertion: G is not Provable in F.

If ~G was Provable in F this contradicts its assertion: G is Provable in F.

Since G is neither Provable nor Refutable in F it forms a Gödel sentence in F.

∴ There does not exist any sentence of any formal language that is materially equivalent to a statement of its own unprovability.

Kurt Gödel, Allan Turing and Alfred Tarski misinterpreted these erroneous expressions of language proving undecidability as fundamental limitations of Math, Computer Science and Logic.

http://mathworld.wolfram.com/Undecidable.html

Not decidable as a result of being neither formally provable nor unprovable.

Copyright 2018 Pete Olcott