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of that theory. More formally, if is a Tarski-style consequence relation, then is closed under (and so each of its theorems is a logical consequence of its axioms) if and only if, for all sentences in the language of the theory , if , then ; or, equivalently, if is a finite subset of (possibly the set of axioms of in the case of finitely axiomatizable theories) and , then , and therefore .

Consistency and completeness

A syntactically consistent theory is a theory from

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