gödel's theorem

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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

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Gödel's incompleteness theorem
A theorem in mathematical logic that states that no consistent system can be used to prove its own consistency
gödel's theorem