Gödel's incompleteness theorem
- Given any sufficiently powerful logical system (one that can prove simple arithmetical statements and cannot "prove" a contradiction), there exists at least one statement which is true within the system, but cannot be proven within it (i.e., derived as a theorem from the axioms of the system).
Another way of stating this is:
- No formal system (of the type described above) can be both complete and consistent.
This result has profound implications for mathematical logic in particular, and by extension all of mathematics. As with Einstein's theory of relativity, however, the theorem has been overgeneralized and used to support arguments in areas that have nothing to do with the type of logical systems addressed by the theorem.
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 theorem 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."
On the other hand, it has also been used as a proof of the existence of God: "Science is a formal axiomatic system. Therefore, if science aptly describes the universe, the universe itself acts like an axiomatic system. Therefore, it needs something analogous to an axiom outside of itself to be consistent and complete. So god exists." (See http://www.perrymarshall.com/10043/godels-incompleteness-theorem-the-universe-mathematics-and-god/)