It sort of depends on what you mean by false. Syntactically, inconsistency would mean that I can derive anything, including proofs of impossibility of completeness, proofs of completeness etc.
Taking the lecture literally, I take it that Voevodsky considers it wholly possible that PA will cough up mutually incompatible statements at some point and we will no longer have a good reason to suppose that it is inconsistent.
I get the impression that he is likely a constructivist who does not accept current consistency proofs of PA, and he is to some degree leveraging his position in order to push for a streamlined, automated constructivist foundations for future mathematics research. An effort that I support 100%.
It sort of depends on what you mean by false. Syntactically, inconsistency would mean that I can derive anything, including proofs of impossibility of completeness, proofs of completeness etc.
Taking the lecture literally, I take it that Voevodsky considers it wholly possible that PA will cough up mutually incompatible statements at some point and we will no longer have a good reason to suppose that it is inconsistent.
Another possibility is that he wanted to to point out that this is not strictly speaking ruled out so you should pay attention to his work on a nice alternative that just so happens to be in line with his goal of streamlining automated theorem proving and proof verification ;).
I get the impression that he is likely a constructivist who does not accept current consistency proofs of PA, and he is to some degree leveraging his position in order to push for a streamlined, automated constructivist foundations for future mathematics research. An effort that I support 100%.