A Note on the Decidability of the Necessity of Axioms

A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement $\varphi$ in some axiomatic system $T$, one looks for minimal subsystems of $T$ that allow deriving $\varphi$. In particular, one asks whether, given some system $T+\psi$, $T$ alone suffices to prove $\varphi$. We show that this problem is undecidable unless $T+\mathrm{\neg }\psi$ is decidable.