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Gödel's first incompleteness theorem shows that no axiomatic theory can prove all mathematical truths, while Gödel's second incompleteness theorem shows that a specific mathematical result is unprovable. A famous mathematician of the time, David Hilbert, had asked for a proof that an important axiomatic theory was consistent, and Godel showed that such a proof could not be carried out within the axiomatic theory itself, and presumably could therefore not be established in a convincing way outside of the theory either.
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