Such constructions are interesting! However, they are often done with PA instead of ZFC. For an interesting discussion, I recommend Torkel Franzén's book *Inexhaustibility: a non-exhaustive treatment* (Lecture Notes in Logic 16, ASL, 2004). You can also read this excellent <a href="http://xorshammer.com/2009/03/23/what-happens-when-you-iterate-godels-theorem/">blog article</a> by Mike O'Connor.