they do necessarily. As to Axiom 5, necessary existence is certainly compatible with
perfection, and perfect being reasoning suggests that it is necessary for it.
There remains the modal question, of whether a concept of possibility and necessity such
that being syntactically consistent (entailing no explicit contradiction) entails being in
this way possible also conforms to the Brouwer axiom. Syntactic consistency amounts to
“logical possibility,” in one sense of the term. But not all that is possible in this narrow
logical sense is really or metaphysically possible: there is no formal, explicit
contradiction in the claim that something is red and green all over at once, and yet this
claim is not metaphysically possible. So there is a gap between what Gödel establishes
and its being metaphysically possible that a divine being exist. And it's a substantive
question whether the Brouwer axiom governs real metaphysical possibility. We can
describe coherently a set of possible worlds in which the Brouwer axiom doesn't hold,
and in which, while it's possibly necessary that God exists, God does not exist. We need
only two worlds to do so, in fact:
Suppose that W2 is actual, and W1 is possible relative to W2 but not vice versa. Then
were W2 actual, W1 would be possible. As we're supposing that there are only these two
worlds, a God who exists in W1 exists in every world possible relative to W1, if W2 is
not possible relative to W1. So in W1, God exists necessarily (and W2 is impossible).
Thus, since W1 is possible relative to W2, in this setup, God is possibly necessary and
yet does not exist.
Gödel's argument (as emended) shows us that the concepts of a perfect being and of
divinity are consistent, given a reasonable concept of perfection. But the gap between
consistency and metaphysical possibility and the need to establish that the logic of
metaphysical possibility includes the Brouwer axiom stand between it and the Holy Grail
of proving God's existence. As well, as a modal argument, Gödel's faces the epistemic
problems we've observed: the portion of the argument that contends that possibly a divine
being exists may admit of significant parody. On the other hand, consistency is evidence
for possibility, though defeasibly so, and if I've assessed Proslogion 2 correctly, that
argument is promising and does not require us to deal with the epistemic problems the
modal argument faces. There is (I think) little good to be said for Descartes' argument.
But the Pros. 2 argument appears to survive objections; to accept its premise (1a) we
needn't have more reason to believe in God's possibility than in Zod's; and we do have
evidence that possibly God exists. So while there is of course much more to be said here,
perhaps Anselm's argument has a future.