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sentences provable from under some (specified, possibly implicitly) formal deductive system. The set of axioms is consistent when there is no formula such that and . A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory

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