This talk was given as part of a celebration of the 80th anniversary of the founding of the Institute for Advanced Study in Princeton. As you might guess there were quite a few very well-known mathematicians and physicists in the audience. (To name just a few, Jack Milnor, Freeman Dyson, Jean Bourgain, Robert Langlands and Frank Wilczek, and, all of whom also spoke during the weekend.)

The talk was a gem, and what did come as a surprise, at least to me, was that towards the end of his talk Voevodsky let on that he hoped that someone did find an inconsistency---and that by that time there was no audible gasp from the audience. There was of course a very lively discussion after the talk, and nobody seemed willing to say they felt that the "Current Foundations" (whatever they are) are definitely consistent. Of course Voevodsky was NOT saying that he felt that the body of theorems making up the "classic mathematics" that we normally deal with might be inconsistent, that is quite a different matter.

What we should keep in mind is that a hundred years ago an earlier generation of mathematicians were quite surprised by not one, but several "antinomies", like Russell's Paradox, The Burali - Forti Paradox, etc., (and that was followed by the greatest century in the history of Mathematics).

Of course this opinion has been expressed before, but perhaps not so forcefully or in such a high-level forum. One person who has been expressing such ideas in recent years is Ed Nelson, who was also in the audience. (You can see his ideas in a recent paper:

It seems the general consensus amongst mathematicians is that it is perfectly acceptable to question the limits of mathematics, even outside of ZFC.