  We point out that some questions in quantum field theory are undecidable in a
precise mathematical sense. More concretely, it will be demonstrated that there
is no algorithm answering whether a given 2d supersymmetric Lagrangian theory
breaks supersymmetry or not. It will also be shown that there is a specific 2d
supersymmetric Lagrangian theory which breaks supersymmetry if and only if the
standard Zermelo-Fraenkel set theory with the axiom of choice is consistent,
which can never be proved or disproved as the consequence of G\"odel's second
incompleteness theorem. The article includes a brief and informal introduction
to the phenomenon of undecidability and its previous appearances in theoretical
physics.
