Much of the math in Goedel, Escher & Bach went over my head or has since leaked out, but I think I remember that Goedel's contribution was to prove you can write "this is false" in any sufficiently expressive language, which shattered the hope of any complete and perfect number system. Does that sound right?
A good question is never answered. It is not a bolt to be tightened into place but a seed to be planted and to bear more seed toward the hope of greening the landscape of the idea. John Ciardi