Gödel's 1931 proof demonstrated that any consistent formal system containing elementary arithmetic contains true statements that cannot be proved within that system.

In a landmark paper in 1931 Kurt Gödel proved that any consistent formal system that contains elementary arithmetic is fundamentally incomplete in the sense that it contains true statements that cannot be proved within the system. In a philosophical context this implies that the semantics of a formal system rich enough to contain elementary mathematics cannot be defined in terms of mathematical functions within the system, i.e., there are statements that contain semantic information about the sy