Godel was working under a philosophical system that assumed that everything could be deconstructed into first order logic. He disagreed and essentially wrote "this statement is false" as a proof that it can not be done. However, he also points out in his proof that there is an infinite number of such statements and that they do not have to be self-referential. Nor was what he was doing exploiting an interesting property of the integers but exploiting an interesting property of the system that creates the integers. This decoder can tell if something is an integer and tell one what integer it is, therefore it is able to evaluate everything needed for Godels proof, therefore it must have not only one but an infinite number of cases for which it hangs.