A Note on the Decidability of the Necessity of Axioms
Merlin Carl · arXiv (Cornell University) · 2014
A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement $ϕ$ in some axiomatic system $T$, one looks for minimal subsystems of $T$ that allow deriving $ϕ$. In particular, one asks whether, given some system $T+ψ$, $T$ alone suffices to prove $ϕ$. We show that this problem is undecidable unless $T+ egψ$ is decidable.