Gödel's ontological proof and the incompleteness theorem

Question

In Gödel's ontological proof, he concludes:

"necessarily, God exists" (Theorem 4)

Does Gödel's second incompleteness theorem apply to this proof?

Answer 1

Definition 1 and Axiom 3 in the Wiki article together seem to have something of a Diagonalization flair to them. G(x) is defined to be a property of x such that for every positive property ɸ, ɸ(x). G is then affirmed axiomatically to itself be a positive property. This is rather similar to the argument invoked in the first incompleteness theorem that the numerical predicates representable in numerical syntax (e.g. ɸ(x)) can themselves be used to define a representable self-application sentence, γ iff ɸ([γ]).

I think this might be where the similarities end, but it actually looks quite promising that the argument will eventually fall into contradiction for reasons more like the Liar paradox than the Incompleteness theorem, if it ends up reasoning that there are properties that God does not have, that he must therefore have their negations as properties and, using the necessary implication hook, that God must have an impossible property.

Answer 2

The incompleteness theorem isn't really relevant. One can use modal logic to investigate provability (and thereby prove incompleteness theorems), and Godel used modal logic to construct his "proof" for God also.

But that's about as far as the connection goes. Incompleteness theorems rely crucially on the inability of a system to form a complete proof-system within itself. Godel's ontological argument doesn't touch proof-systems at all.

Also, the ontological argument isn't really worth paying attention to except as a logical exercise. Although the logic is sound, the premises are bizarre and undemonstrated. (For example, "if a property is positive, it is necessarily positive" means that whether something is positive cannot depend whatsoever on the contents of the world, which is a radical departure from the colloquial understanding of what a positive property is.)

Answer 3

His incompleteness theorems are about elementary arithmetic, which seems to me to be independent of the particular modal logic implicit in various versions of his proof. You don't need arithmetic to do modal logic, and you don't need quantification over possible worlds to do arithmetic.

The two can be developed simultaneously, of course. If that were the case, then if the arithmetic were sufficiently expressive, the hybrid system would also suffer from incompleteness. But even then, no harm would be done to the ontological proof.

Consider an analogous case. Let's say someone presents a proof that number k is prime. What application can Goedel's incompleteness have in this context? Arithmetic is incomplete, fine. But does it tell us anything about this particular proof of the primality of k? Whether k is prime or not, is independent of the fact that arithmetic is incomplete. The same is true of the proof of God case.