# Logic | Mathematical Logic

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## Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic . The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics . The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem . The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an " effective procedure " (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers ( arithmetic ). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation . Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies , and continues to be used in that alternate sense. A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions .