Consistency

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. A theory ${\displaystyle T}$ is consistent if and only if there is no formula ${\displaystyle \phi }$ such that both ${\displaystyle \phi }$ and its negation ${\displaystyle \lnot \phi }$ are elements of the set ${\displaystyle T}$. Let ${\displaystyle A}$ be set of closed sentences (informally "axioms") and ${\displaystyle \langle A\rangle }$ the set of closed sentences provable from ${\displaystyle A}$ under a meta-theoretical deductive system such as informal mathematics. The set of axioms ${\displaystyle A}$ is consistent when ${\displaystyle \langle A\rangle }$ is.

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