To a certain degree it depends on what you mean by logic. In particular the incompleteness results depend on a particular kind of logic that uses axioms assumptions basically, and the provability of any particular statement is depend on the axioms you choose. So if you have a statement you want to prove, and you can't under your current logical system, you can always change the system in such a way that the statement becomes provable (or disprovable).

I guess the way I've always thought about this is by picturing myself standing in a dark room with a flashlight. Whatever logical system you're using determines where you stand in the room, and from that spot you can use the flashlight to study things in the room. However, the flashlight will also cast shadows, so to see what lies in the shadows, you have to move (ie change your logical system).