TIL: Voevodsky thought that first-order arithmetic may be inconsistent.
In his talk "What if the current foundations of mathematics are inconsistent?", he says that he doesn't find the common arguments (1) for the consistency of first-order arithmetic very convincing (2).
(1) Not "arguments" in the sense of a part of an actual proof, but more like philosophical/intuitive arguments supporting the case.
(2) If I understand it correctly, his objection is that these employ some non-construtive/non-computable methods for which it is not clear whether they reflect "reality".
Here is the video of the talk: https://youtu.be/O45LaFsaqMA?t=25m04s (link to the part where he starts to talk about these objections)
Here are the slides from the talk: https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2010_09_25_slides.pdf (the discussion of the aforementioned issues start on slide 13)
