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Apparently Kurt Gödel believed that his incompleteness theorems have some kind of religious implications. Despite Gödel's belief in a personal God, this was still somewhat surprising to me. Discussions and theories about weird (i.e. outside of mathematics) consequences of his theorems are all over the internet, and are often labeled as misunderstandings or "crank" interpretations of his work. But Gödel himself seemed to think that there are indeed legitimate applications of his work to religion.

I recall reading the quote below a while ago. My memory is a bit fuzzy, but I believe it was in response to Kurt Gödel having heard from his mother that a religious magazine or journal of some sort printed an article describing a simplified account of his incompleteness theorems for a general audience. The article then discussed some religious implications.

The actual quote from Gödel is:

It was something to be expected that sooner or later my proof will be
made useful for religion, since that is doubtless also justified in a
certain sense.

The quote can be viewed on page 125 of Reflections on Kurt Gödel
by Hao Wang, on Google Books as a preview. The context I described above is not there in the preview exactly as I remember, so I'm pretty sure I read it somewhere else (or am going insane). I do not have a copy of Wang's book either, so if anyone else wants to provide additional context beyond the preview or from other sources that is great.

My question is: What religious implications did Kurt Gödel think his incompleteness theorems have, and why?

My question is mainly about Gödel's own thoughts, but if anyone wants to speculate or "connect the dots" based on any other information they might have about Gödel's writing or thinking on the matter, this is more than welcome too.

Gödel's theism is discussed by Franzen in Gödel’s Theorem: An Incomplete Guideto Its Use and Abuse. He penned a version of the ontological argument, and in 1961 ranked the worldviews “according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion)... Skepticism, materialism, and positivism stand on one side; spiritualism, idealism, and theology on the other”. Idealism "in its pantheistic form” is dismissed as as “a weakened form of theology in the proper sense”. Nonetheless, he did not attempt to draw theistic conclusions from the incompleteness theorem:

"Gödel sometimes described himself as a theist and believed in the possibility
of a “rational theology,” although he did not belong to any church. In
[Wang 87] he is quoted as remarking that “I believe that there is much
more reason in religion, though not in the churches, that one commonly
believes...” Among his unpublished papers was a version of St. Anselm’s ontological proof of the existence of God. More precisely, the conclusion of the argument is that there is a God-like individual, where x is defined to be God-like if every
essential property of x is positive and x has every positive property as an
essential property. As this explanation of “God-like” should make clear,
Godel’s idea of a rational theology was not of an evangelical character,
and Oskar Morgenstern relates ([Dawson 97, p. 237]) that he hesitated to
publish the proof “for fear that a belief in God might be ascribed to him,
whereas, he said, it was undertaken as a purely logical investigation, to
demonstrate that such a proof could be carried out on the basis of accepted
principles of formal logic.” Although Gödel was thus not at all averse to theological reasoning, he did not attempt to draw any theological conclusions from the incompleteness theorem."

This did not stop others from doing just that, or even ascribing it to Gödel. Much of it is also discussed by Franzen: there can be no "theory of everything", existence of truths which can not be mechanically derived imply the existence of God, for ultimate truth is beyond reason, methodology of science cannot be based upon science only, scientists must rely on faith as much as non-scientists, finite beings can never answer all the questions they seek after, etc., etc. Related, although not exatly theological, is the Penrose-Lucas argument that "consciousness" surpasses Turing machines. For a recent sampler, see e.g. Goldman's God of Mathematicians:

"At twenty-five he ruined the positivist hope of making mathematics into a self-contained formal system with his incompleteness theorems, implying, as he noted, that machines never will be able to think, and computer algorithms never will replace intuition. To Gödel this implies that we cannot give a credible account of reality without God.

[...] Whether or not we believe, as did Gödel, in Leibniz’s God, we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.

Other attempted drawings of implications suffer from similar reasoning by loose association, they are not so much implications as vague analogies. And while it is not clear that Gödel's God was Leibniz's God exactly (as opposed to, say, Anselm's), it is true that Gödel was quite preoccupied with Leibniz himself, see Why did Gödel believe that there was a conspiracy to suppress Leibniz's works? He even told Hao Wang:"My theory is a monadology with a central monad [namely God]. It is like the monadology by Leibniz in its general structure". Unfortunately, Gödel's surviving writings on this theory, and theology generally, are very scarce. His notes on philosophy, known as Max Phil (Maximen Philosophie), occasionally touch on theological issues, Ternullo in Gödel’s Cantorianism discusses Gödel’s views of the "absolute infinite", which Cantor associated with God.

This answer is taken from my answer to an earlier question, "What is god for religious people?"

God is the Completeness Theorem, in contrast to the Incompleteness
Theorem. Gödel's first Incompleteness Theorem says, loosely, that any
system like mathematics will always have some problem that cannot be
resolved given the existing set of axioms; one additional axiom is
always necessary.

God is the Final Axiom. God is that axiom which resolves all existing
problems and will further resolve all future problems. The content of
that axiom is in perennial dispute, but its simple existence is more
generally agreed upon.

I am not saying that I agree with this conception of god. However, in many religions the idea of a supreme being functions as an “Anti-Incompleteness Theorem”: that set of ideas which, taken together, are sufficient to explain all events in the world.