Editing Metamathematics (section)

===Gödel's incompleteness theorem ===
{{main|Gödel's incompleteness theorem}}
Gödel's incompleteness theorems are two [[theorem]]s of [[mathematical logic]] that establish inherent limitations of all but the most trivial [[axiomatic system]]s capable of doing [[arithmetic]].  The theorems, proven by [[Kurt Gödel]] in 1931, are important both in mathematical logic and in the [[philosophy of mathematics]]. The two results are widely, but not universally, interpreted as showing that [[Hilbert's program]] to find a complete and consistent set of [[axiom]]s for all [[mathematics]] is impossible, giving a negative answer to [[Hilbert's second problem]].

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "[[effective procedure]]" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the [[natural numbers]] ([[arithmetic]]).  For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.