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Gödel's ontological proof and the incompleteness theorem

In Gödel's ontological proof, he concludes: "necessarily, God exists" (Theorem 4) Does Gödel's second incompleteness theorem apply to this proof?
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Is there a finitistic set theory, and if there is, is it provably consistent?

ZFC is the mainstream set theory. It has an axiom of infinity which claims that there is at least one infinite set. Now suppose like Aristotle, we object and say that there are no actually infinites ...