TITLE: What if Current Foundations of Mathematics are Inconsistent?
QUESTION [97 upvotes]: The title of the question is also the title of a talk by Vladimir Voevodsky, available here.  
Had this kind of opinion been expressed before?
EDIT. Thanks to all answerers, commentators, voters, and viewers! --- Here are three more links: 
Question arising from Voevodsky's talk on inconsistency by John Stillwell, 
Nelson's program to show inconsistency of ZF, by Andreas Thom, 
Pierre Colmez, La logique c’est pas logique !
EDIT. Here the link to the FOM list discussing these themes.

REPLY [8 votes]: Mathematics is too big to fail.