Axiom independence

Statement

An axiom $P$ is independent of a theory $T$, i.e., a collection of sentences, if and only if neither $P$ nor its negation $\neg P$ is provable from $T$. Equivalently, $P$ and $\neg P$ are each consistent with $T$, i.e., some models of $T$ make $P$ true and others make $\neg P$ true. Proving the independence of $P$ from $T$ is usually done by demonstrating the existence of such models.