Let us say that someone was able to prove that $0=1$ using ZFC, thereby proving it inconsistent. What impact would this have on the study of meta-mathematics?
Most mathematicians would just move onto a different set theory, since most mathematics is not sensitive to the exact axioms being used.
Meta-mathematics, on the other hand, is. In particular, I’m talking about model theory, set theory, proof theory, etc… What results would become meaningless, and which could be salvaged. What other set theories could be used instead?
People will start paying more attention to results like Patey-Yokoyama on Ramsey phenomena, which show that a lot more mathematics than was thought earlier can done conservatively relative to a finitistic framework.
Another issue is the possible impact of a discovery of an inconsistency in ZFC on traditional beliefs in the existence of an intended model/intended interpretation of ZFC. Namely, in such a hypothetical situation of ZFC having turned out to be inconsistent, how will such beliefs evolve and what strategies will be developed to deflect the question “what is this supposed to have been an intended model of exactly”.