Question

We adopt axioms not because we can prove their consistency, but because we believe that they accurately describe something that we want to study. A proof from these axioms will have value in that it shows how the proposition (which may be surprising or complicated) follows from things that we already believe and are simple. If we someday prove an inconsistency using a given set of axioms, this shows that our possibly naïve intuition for reasonable axioms was incorrect. (e.g. Russel's paradox showing that unrestricted comprehension is a bad idea)