I'll keep this short and sweet.
Construct Axiomatic System A in which we can do math.
Gödel Sentence G = G is unprovable in A.
Gödel's Argument (I)
If G is provable then there's proof that G has no proof. Contradiction!
If G is unprovable then no (auto-)contradiction.
Either G is provable or G is unprovable.
G is unprovable (in A)
Hypothetical Argument (K)
Argument I has to be outside A for Gödel's theorems to be true.
Argument I uses logic (even though it doesn't use any of the axioms in A).
Axiomatic system A also uses logic.
How can something that employs logic (argument I) be outside something that too uses logic (axiomatic system A). At the very least they overlap in utilizing logic.
Steel manning: Gödel has used axiomatic system A to prove G that claims it does not use axiomatic system A. Contradiction!
Is argument K any good? Have an awesome day.
EDIT 1 START
(A big thank you to Conifold and Scott Rowe)
Note that G is true in(side) axiomatic system A. Argument I (Gödel's argument) is outside axiomatic system A (re Conifold's comment) then. How is this possible? I'm stumped. Gödel didn't use any of the axioms of A and proved that there's a true statement in A that isn't provable from the axioms of A.
How does one define as being in(side) axiomatic system A? I would say, naïvely perhaps, that for a theorem or proposition to be in(side) axiomatic system A, it would need to be derived/proved from the axioms of A.
If G is in(side) A (G has to be inside axiomatic system A for Gödel's proof) then it has to be provable from the axioms of A. That's what being in(side) an axiomatic system means, conventionally.
But Gödel's argument G claims it is not provable from the axioms of the axiomatic system A.
This is a contradiction.
EDIT 1 END
EDIT 2 START
The Gödel sentence G has to be
But, analogoulsy, E = Aliens exist, is also constructible, but this has no aletheic import. E can be true/false. Likewise G can be true or false. So much for being constructible.
According to articles that I read on Gödel's eponymous theorems, the only "proof" we have that G is true is G itself: G asserts G is true. Isn't this a circulos in probando akin to the Bible Fallacy: The Bible is true "because" The Bible asserts The Bible is true??
EDIT 2 END