So, you are presuming (INCORRECTLY) that the x in this formula is an
epistemological antinomy in Godel's Proof
It isn't.
But since you don't seem to be able to understand what Godel's G is, and
are too arogent to learn, you are doomed to just being ignorantly wrong.
Note, G is NOT the statement, even in "effect", that G asserts that it
can not be proven.
G is the statement that there does not exist a Natural Number g, that
meets a specifically defined Primitive Recursive Relationship.
And that is all that G is in the field F.
Such a statement MUST be a "Truth Bearer" as either a number g exists
that meets the requirement or it doesn't. That is a basic fact of the
mathematics of Natural Numbers, either numbers exist that meet a
computable property, or they don't, there is no "fuzzy" state between or
outside.
The key point of the proof, is that the specific PRR was built in a
meta-F that has the enumeration of all the axioms of F, assigning them
to numbers, and an encoding system that can express ANY statement, or
series of statements in F as a number, and the PRR is constructed so
that a number that satisfies it WILL be an encoding of a proof of the
statement G, and any proof that might exist in F, will have a number.
And the complicated paper is the proof that such a PRR can be constructed.
Given that we can construct such a PRR, and ask about a number
satisfying it, we can then show in meta-F that the existance of a number
that satisfies the PRR has an identical truth value to the provability
of the statement G in F. Thus the existance of the number g has the same
truth value as the provability of G in F, or the non-existance of the
number g has the same truth value as the unprovability of G in F.
Thus since G asserts that there is no number g, that means we can
logically derive from G the statements that G is true if, and only if, G
is unprovable, thus it is this DERIVED statement that is the statement
of a statement that asserts its own unprovability, and this statement is
in meta-F, not F.
Despite what you try to claim, this is NOT the statement G, but a
statement provable to have a logical equivalence (in meta-F), and since
G (in F) was a truth bearer, so must this derived statement.
Note, that the form of this equivalent statement has a similar mophology
to the liar, the liar is L asserts that L is not True, while this one is
that G asserts that G is not Provable in F. Same form, but different
predicate function referred to. This is what Godel was refering to,
given any epistemological antinomy, with a similar change of predicate,
you could do a similar derivation to find a PRR that is its equivalent.
Note, this means the epistemological antinomy itself, was never used as
a premise of any logical deduction, so the "non-sense" of them never
mattered. The morphical transformation turns that "non-sense" into a
Truth Bearer that can show that some statements are not provable in any
system rich enough to perform the proof in, which just requires a number
of the basic properties of the Natural Numbers.