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- Consistency abstract "In classical deductive logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.If these semantic and syntactic definitions are equivalent for a particular deductive logic, the logic is complete.[citation needed] The completeness of the sentential calculus was proved by Paul Bernays in 1918[citation needed] and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.".
- Consistency wikiPageExternalLink mathematics-inconsistent.
- Consistency wikiPageID "75802".
- Consistency wikiPageRevisionID "603572641".
- Consistency date "May 2012".
- Consistency hasPhotoCollection Consistency.
- Consistency reason "which notion of cmpleteness is this?".
- Consistency subject Category:Hilbert's_problems.
- Consistency subject Category:Metalogic.
- Consistency subject Category:Proof_theory.
- Consistency comment "In classical deductive logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.".
- Consistency label "Coerenza (logica matematica)".
- Consistency label "Cohérence (logique)".
- Consistency label "Consistencia (lógica)".
- Consistency label "Consistency".
- Consistency label "Consistentie (logica)".
- Consistency label "Niesprzeczność".
- Consistency label "Widerspruchsfreiheit".
- Consistency label "Непротиворечивость".
- Consistency label "تناسقية".
- Consistency label "一致性 (邏輯)".
- Consistency sameAs Bezesporná_teorie.
- Consistency sameAs Widerspruchsfreiheit.
- Consistency sameAs Consistencia_(lógica).
- Consistency sameAs Cohérence_(logique).
- Consistency sameAs Coerenza_(logica_matematica).
- Consistency sameAs Consistentie_(logica).
- Consistency sameAs Niesprzeczność.
- Consistency sameAs m.0k1ky.
- Consistency sameAs Q1319773.
- Consistency sameAs Q1319773.
- Consistency wasDerivedFrom Consistency?oldid=603572641.
- Consistency isPrimaryTopicOf Consistency.