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mathematical logic > Gödel's completeness theorem

Terme préférentiel

Gödel's completeness theorem  

Définition(s)

  • 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 φu is unprovable although true in the natural numbers, which are a particular model of a first-order theory describing them — φu 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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RDF/XML TURTLE JSON-LD Dernière modif. 12/10/2023