Gödel's proof of the existence of God
Gödel did indeed produce such a proof, an argument of the type known
as ontological. The proof can be found in the Collected
Works, vol. III, p.403. General information about ontological arguments
for the existence of God is to be found in the on-line Stanford
encyclopedia of philosophy. Christopher Small has an extensive presentation
and discussion of Gödel's ontological argument.
Gödel's ontological argument doesn't have anything to do with the
incompleteness theorem or its proof. It shouldn't be assumed that the
God referred to in the argument has much to do with God as conceived
in any theistic religion. Gödel was a great admirer of Leibniz (who
also had an ontological proof) and believed in the possibility of a
"rational theology".
