Consistency proof

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 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 some (specified, possibly implicitly) formal deductive system. The set of axioms ${\displaystyle A}$ is consistent when ${\displaystyle \langle A\rangle }$ is.

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