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

Logic