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- Complete_theory abstract "In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent. For theories in logics which contain classical propositional logic, this is equivalent to asking that for every sentence φ in the language of the theory it contains either φ itself or its negation ¬φ.Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's incompleteness theorem.This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness.Complete theories are closed under a number of conditions internally modelling the T-schema:For a set if and only if and ,For a set if and only if or .Maximal consistent sets are a fundamental tool in the model theory of classical logic and modal logic. Their existence in a given case is usually a straightforward consequence of Zorn's lemma, based on the idea that a contradiction involves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent sets extending a theory T (closed under the necessitation rule) can be given the structure of a model of T, called the canonical model.".
- Complete_theory wikiPageID "10086335".
- Complete_theory wikiPageRevisionID "563847204".
- Complete_theory hasPhotoCollection Complete_theory.
- Complete_theory subject Category:Mathematical_logic.
- Complete_theory subject Category:Model_theory.
- Complete_theory comment "In mathematical logic, a theory is complete if it is a maximal consistent set of sentences, i.e., if it is consistent, and none of its proper extensions is consistent.".
- Complete_theory label "Complete theory".
- Complete_theory label "Teoria completa".
- Complete_theory label "Théorie complète".
- Complete_theory sameAs Úplná_teorie.
- Complete_theory sameAs Théorie_complète.
- Complete_theory sameAs Teoria_completa.
- Complete_theory sameAs m.02q1bj2.
- Complete_theory sameAs Q3508260.
- Complete_theory sameAs Q3508260.
- Complete_theory wasDerivedFrom Complete_theory?oldid=563847204.
- Complete_theory isPrimaryTopicOf Complete_theory.