Given a first-order theory $T$, and a sentence $\varphi$ in the language of $T$, we say that $\varphi$ is independent of $T$ if there is no proof of $\varphi$ or $\lnot\varphi$ from $T$.
Common methods for proving the independence of a sentence $\varphi$ in set theory are forcing and inner models, which are used to construct models of set theory in which statements can be prescribed to hold or fail, and this way we can show that $\sf ZFC$ (or $\sf ZF$ at times) cannot prove nor disprove $\varphi$.