In logic and mathematics, a statement—an assertion within the formal language of a theory--is decidable if it can be shown to be either true or false. In other words, a statement is decidable if either it or its negation can be proved within the theory. For example, the negation of the statement "There exists some whole number that evenly divides the number 43" can be proven within formal arithmetic, which means that the statement is decidable. If a statement or well-formed formula within a theory is undecidable, then the theory is called incomplete.

In the early 1900s, German mathematician David Hilbert brought up the question of whether mathematics itself was complete, that is, whether a formal system could be created that would generate all the true mathematical statements that exist and none others. "Hilbert's programme," as his project was called, would create its formal system on the...