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)."
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 is fallacious. For one thing, Gödel's Incompleteness theorem applies to mathematical systems called first-order logic systems, and no one claims that God is a first-order logic system. For another, it is far from clear that "God believes X" or "God knows X" is comparable to "X can be derived from axioms." In short, this argument compares apples and oranges.