This is not what Gödel says, and in fact your statement is true in a trivial way: you can prove 1+1=2, and Not(Not(1+1=2)), and Not(Not(Not(Not(1+1=2)))), etc.

This is not what Gödel says, and in fact your statement is true in a trivial way: you can prove 1+1=2, and Not(Not(1+1=2)), and Not(Not(Not(Not(1+1=2)))), etc. ad infinitum. This is an infinite space of provable true statements that can be exhausted by a ten-line Python script. IMO, this is why we actually need mathematics -- as a field in which to learn what it means to know what you're talking about.