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 Among his unpublished papers was a version of St. Anselm’s ontological proof of the existence of God.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.
