Ultimately this proof rests on whether human beings can truly understand what God is and that its concept cannot be derived simply as an extension or ultimate regression of something which exists in fact. In which case, God doesn't really exist in your mind, only some superficial idea you are generically referring to as God. But obviously, humans can't completely conceive of God, or maybe they can...
That's why we have the Bible. In the Bible, God Himself explains Himself in ways that we can understand. And He includes enough info so that we can figure out many things about God that we couldn't if He hadn't revealed what He had. We don't really need the hard to understand Gödel information. In other words, we have the miracle of the Bible that we can understand, to back up the Gödel information that we have a difficult time understanding.
