This tag is for questions about proving that some statement is independent from a theory, meaning it is neither provable nor refutable from that theory. Common examples are the continuum hypothesis from the axioms of ZFC, and the axiom of choice from the axioms of ZF.

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 $T$.

In set theory independence results make a large portion of modern set theory, and many of the modern research can be shown in light of independence results.

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 $\varphi$.