Gödel's work actually echoes Christianity quite deeply. A key theme of Christianity is that humans are incapable of being good (much less of achieving perfection) on their own strength--no more than Hilbert could axiomatize all of mathematics with a consistent, computable set of axioms.

"Do not be overly righteous, and do not make yourself too wise." (Ecclesiastes 7:16)

"There is no distinction, for all have sinned and fall short of the glory of God, and are justified freely by His grace through the redemption that is in Christ Jesus." (Romans 3:22-23)

It would be interesting to see an analysis where the most fundamental results in mathematics are looked at under the light of their discoverers' ideological beliefs and psychologies.

I wonder whether Gödel would have pursued the Proof if he did not believe in an omnipotent god.

To echo this, Godel was influenced by Cantor and Leibniz, both definitive Christian theists. Godel himself was also a theist.

Cantor believed that he had been chosen as a messenger by God to be the recipient of what was to become set theory.