: Mathematics: Gödel's completeness theorem

mathematical logic > Gödel's completeness theorem

Gödel's completeness theorem

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable". This does not contradict Gödel's incompleteness theorem, which shows that some formula φ_u is unprovable although true in the natural numbers, which are a particular model of a first-order theory describing them — φ_u is just false in some other model of the first-order theory being considered (such as a non-standard model of arithmetic for Peano arithmetic). This kind of failure of consistency between a standard and non-standard model is also called Omega Inconsistency.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem)

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