Principle of the foundations of mathematics .
One of the most important discoveries of 20th-century mathematics, it states the impossibility of defining a complete system of axioms that is also consistent (does not give rise to contradictions). Any formal system (e.g., a computer program or a set of mathematical rules and axioms) powerful enough to generate meaningful statements can generate statements that are true but that cannot be proven or derived within the system. As a consequence, mathematics cannot be placed on an entirely rigorous basis. Named for Kurt Godel , who published his proof in 1931, it immediately had consequences for philosophy (particularly logic) and other areas. Its ramifications continue to be debated.