The thing is, that Godel's theorem is provable. So by your definition, it is true.

The usual definition of false is not "I can't prove it's true" as that is pretty hard to decide. Suppose I have a statement S that I can't prove. Is it false, or am I just not clever enough to prove that it is true?

The normal definition of S being false is that the negation of S is true.

Part of what the incompleteness theorem says is that in any system of logic that doesn't contradict itself, there will be statements that are neither provably true nor provably false. Thus you can take these statements to be true OR false as an axiom and it won't lead to contradictions.