The incompleteness theorem doesn't say that there are statements which are unprovable in any absolute sense. What it says is that given a formal system, there will always be statements which that particular formal system can't prove. But in fact as part of the proof, Godel proves this statement, just not by deriving it in the formal system in question (obviously, since that's what he's proving is impossible).

The way this is done is by using a "metalanguage" to talk about the formal theory in question. In this case it's a kind of ambient set theory. Of course, the proof also implies that if this ambient metalanguage is formalized then there will be sentences which it can't prove either, but these in general will be different sentences for each formalized theory.