TITLE: What would happen if ZFC were found to be inconsistent?
QUESTION [20 upvotes]: If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that would enable mathematicians to continue pursuing mathematics, should the unthinkable occur?

REPLY [2 votes]: Note that an inconsistency implies that any theorem and its negation can be proven. The fact such a thing hasn't happened (no one has ever proven the negation of a theorem in mainstream math) means that wherever ZFC fails, it is almost certain that a more restricted set of axioms can be found where most math can still be done. 
To see an example of what could happen you have to look no farther than Russell's Paradox. It shows that naive set theory makes no sense, but the example is far removed from the regular sets (with "actual elements") that we all usually use, and axioms where set theory works were found.