In this video njwildberger argues that the idea that we can uniquely factor any integer into primes is problematic. His central argument appears to involve skepticism that we can say anything meaningful about very large numbers even if we can define them in some sense.
It seems crazy to doubt such a core part of mathematics. But is there some self-consistent world where his argument about very large numbers makes sense? Or is there another flaw in his reasoning?
His argument seems inexact. Rather than "The number $10\Delta 10 + 23$ does not have a prime factorization", his argument seems to be "The prime factorization of $10\Delta 10 + 23$ is not computable." He makes a fairly good case for the latter, and the former does hold when interpreted in certain logical realms such as Ultrafinitism. The first statement does not hold when interpreted classical logic and the ordinary understanding of prime factorization.