GÃ¶del's incompleteness theorem
Mathematician Kurt Gödel formulated a famous theorem which can be summarized as "Given any sufficiently-powerful logic system, there exists at least one statement which is true within that logic system, but cannot be proven (derived from the axioms of the system)."
Philosophy of intelligence
Many scholars have debated over what Gödel's incompleteness theorem implies about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.
Some philosophers claim that Gödel's theorem conclusively prove that Artificial Intelligence is inherently impossible, because a purely program-based "intelligence" would be subject to Gödel's theorem whereas organic intelligence is not.
Others might take this argument one step further, claiming that intelligence cannot be a product of purely physical processes (such as evolution). Intelligence requires a special nonphysical "spark" of some sort. This is the premise of dualism.
Many of the anti-mechanistic arguments used to prove that minds are not Turing Machines point out that minds can prove anything. Yet these arguments are flawed because they do not claim that such minds are consistent. If minds are allowed to be inconsistent, then they conform to Gödel's theorem rather than violate it.
This does not prove that minds truly are Turing Machines, but it does demonstrate that it is still very much an open question in philosophy.
Application to God
This has been used as a proof of the nonexistence of God: "God is defined as being omniscient. But by the incompleteness theorem, there is a statement that is true, but which God doesn't know. Therefore, God is not omniscient. A being which is not omniscient is not God, therefore God does not exist."
This argument only works if it can be assumed that the mind of God behaves like a Turing Machine. As noted in the previous section, this is not known for human minds, let alone for a supernatural mind. Without this crucial assumption, it is far from clear that "God believes X" or "God knows X" is comparable to "X can be derived from axioms."