The proof of the incompleteness theorem only shows you that recursively-enumerable arithmetic systems are always missing something, because the construction alw

The proof of the incompleteness theorem only shows you that recursively-enumerable arithmetic systems are always missing something, because the construction always holds. It doesn't make any grand statements about the structure or nature of things that are true and without proof. The second incompleteness theorem gives an example of something undecidable that's probably more interesting that the uselessly paradoxical statement in the first: it says that no system can consistently prove its own consistency.