What if the current foundations of Mathematics are inconsistent?
Had this kind of opinion been expressed before?
The opinion that the Peano Arithmetic is likely to be inconsistent is not uncommon, along with ideas on how to deal with this (targeting the "what if" question). Wikipedia has an article about that, and MathOverflow has a question. These have links to works by Nelson, and to a paper by Sazonov, which among others refer to Parikh (1971) and Yessenin-Volpin (1959). These things have been discussed also in a paper by Rashevski (1973) and a few years ago also (quite extensively, with a number of additional references) at the FOM mailing list.
An implicit question is "What do you think of Vladimir Voevodsky's talk?"
His message is obviously: "Guys, your Peano Arithemetic is something not to be taken too seriously. Which is a good reason to be a bit more serious about Voevodsky's univalent foundations!" I hear this message, in particular, when he speaks of "reliable proofs", and in fact it does resonate with me. His subsequent talk about the univalent foundations is much more substantial; having a separate copy of the "slides" helps to follow the video.