In this setting, the adversary seeks to find a deduction $\phi_0, \dots, \phi_n$ of $P \wedge \neg P$ quickly. If ZFC, for example, is inconsistent, there exists such a deduction and hence there exists a (constant time) adversary, which simply publishes $\phi$.

In order to have a zero-knowledge proof problem, one needs a family of problems, for which the adversary's task becomes increasingly hard as $n \rightarrow \infty$. With just one theory, such as ZFC, this does not happen.