Big Red Dog

I'm not sure because of this notation but if this is induction you're referring to, one can build consistent and complete system with induction. One more remark: self-reference: you might have noted that Gödel's key set of assertions (those "referencing" themselves) are not referencing directly themselves but rather a property about themselves. Indeed, the "provable" predicate used by Gödel applies to numbers, not to predicate. As to the self references in axioms or mathematical assertions, they are part of the game: x = 2x is just another assertion (say in Arithmetic) that states that x = 0