High Quality Content by WIKIPEDIA articles! Godel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Godel in 1929. A first-order formula is called logically valid if it is true in every structure for its language. The completeness theorem shows that if a formula is logically valid then there is a finite deduction (a formal proof) of the formula. The deduction is a finite object that can be verified by hand or computer. This relationship between truth and provability establishes a close link between model theory and proof theory in mathematical logic. An important consequence of the completeness theorem is that it is possible to enumerate the logical consequences of any effective first-order theory, by enumerating all the correct deductions using axioms from the theory.

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