I don’t think so. ZFC is either consistent or it isn’t. That is, a Turing machine enumerating all theorems of ZFC either produces 0=1 or it doesn’t. This is, however, independent of ZFC (if the latter is consistent) by the second incompleteness theorem. See https://mathoverflow.net/a/332266, in particular Nik Weaver’s comment:
> I think the strongest argument in favor of the definiteness of ℕ, and against the idea that PA, or any other axiomatization, is the be-all end-all, is the evident fact that we have an open-ended ability to go beyond any computable set of axioms, for instance by affirming their consistency. If you accept PA you should accept Con(PA), and the process doesn't stop there: you can then accept Con(PA + Con(PA)), and so on. This goes on to transfinite levels. If our understanding of ℕ really were fully captured by some particular set of axioms then we would not feel we had a right to strengthen those axioms in any special way; the fact that we do feel we have this right shows that our understanding is not captured by any particular set of axioms.