If so then there is at least one x such that x = "x is unprovable":
1) x = "x is unprovable"
It must now be true that:
2) x is provable IFF "x is unprovable" is provable
Since only true sentences are provable simplification gets a contradiction:
3) x is provable IFF x is unprovable
And we must conclude that there is no statement claiming itself to be unprovable. (QED)
As to formal systems: They may perhaps produce statements saying that they are unprovable in the formal system ... which is true if the system is consistent.
That is a restricted concept of unprovability.
It has been said that the term "unprovable" is vague.
That it is relative and needs a qualification such as "Unprovable by ruler and compass" ...
But I think there is a basic meaning that can be defined:
(definition) x is unprovable IFF there can be no proof of x.