The problem is that Gödel builds self-reference. It's not like Peano arithmetic has recursion or self-reference operators. To put it another way, in order to

The problem is that Gödel builds self-reference. It's not like Peano arithmetic has recursion or self-reference operators. To put it another way, in order to disqualify self-reference, you would need to employ a higher order proof system--which gives the game away anyway.