Existence and uniqueness of God Over lunch, my math professor teasingly gave this argument

God by definition is perfect. Non-existence would be an imperfection, therefore God exists. Non-uniqueness would be an imperfection, therefore God is unique.

I have thought about it, please critique from mathematical/logical point of view on


*

*Why does/doesn't this argument fall through? Does it violate any logical deduction rules?  

*Can this statement be altered in a way that it belongs to ZF + something? What about any axiomatic system?  

*Is it positive to make mathematically precise the notion of "perfect"? 


Existence is not a predicate. You may want to read Gödel's onthological proof, which you can find on Wikipedia. 
Equally good is the claim that uniqueness is imperfection, since something which is perfect cannot be scarce and unique. Therefore God is inconsistent..?
