or Gödel's theorem, n. (Logic) the crucial result that, in an axiomatic formal calculus of the complexity of number theory (Peano arithmetic), it is impossible to prove consistency without using methods from outside the system. Gödel showed this by proving that validity corresponds to a property of Gödel numbers, describing the construction of the Gödel number corresponding to an assertion that the formula with that number is not provable, and then proving that were arithmetic complete, that statement would have that property. It follows from his theorem that Hilbert's program of devising a decision algorithm for all of mathematics is unattainable, and that the logicist doctrine of the deducibility of all of mathematics from the axioms of logic is false.