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Gödel's completeness theorem is a theorem in model theory proved by Kurt Gödel in 1929. It states that in first-order predicate calculus every universally valid statement can be proven. |
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Gödel's completeness theorem is a fundamental theorem in Mathematical logic proved by Kurt Gödel in 1929. It states, in its most familiar form, that in first-order predicate calculus every universally valid formula can be proved. A logical formula is called universally valid if it is true in every possible domain and with every possible interpretation, inside that domain, of non-constant symbols used in the formula. To say that it can be proved means that there exists a formal proof of that formula which uses only the logical axioms and rules of inference adopted in some particular formalisation of first-order predicate calculus. The branch of mathematical logic that deals with what is true in different domains and under different interpretations is Model theory; the branch that deals with what can be formally proved is [Proof theory]?. The completeness theorem, therefore, establishes a fundamental connection between what can be proved and what is (universally) true; between model theory and proof theory; between semantics and syntax in mathematical logic. It should not, however, be misinterpreted as obliterating the difference between these two concepts; in fact, another celebrated result by the same author, Goedels Incompleteness Theorem, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics. [/Original Proof]? Some of the older text that is to be integrated when more is added to the above: |
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* Kurt Gödel, "Die Vollständigkeit der Axiome des logischen Funktionen-kalküls", Monatshefte für Mathematik und Physik 37 (1930), 349-360. This article contains the same material as the doctoral dissertation, in a rewritten and shortened form. The proofs are more brief, the explanations more succint, and the lengthy introduction has been omitted. |
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* Kurt Gödel, "Die Vollständigkeit der Axiome des logischen Funktionen-kalküls", Monatshefte für Mathematik und Physik 37 (1930), 349-360. This article contains the same material as the doctoral dissertation, in a rewritten and shortened form. The proofs are more brief, the explanations more succinct, and the lengthy introduction has been omitted. |
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