What he means when he says "unprovable" is "not provably guaranteed to run in O(x)" since the entire discussion is about complexity. A "provable" algorithm, the

What he means when he says "unprovable" is "not provably guaranteed to run in O(x)" since the entire discussion is about complexity. A "provable" algorithm, then, is one where the running time can be calculated. In a deterministic algorithm, this bound will always exist. In a non-deterministic algorithm (e.g. one that uses a random walk), this bound will not. The funny thing is that in the real world (and for big data) the latter algorithms generally run faster than the former. If you're wondering why a non-deterministic algorithm can't have bounds, it's simple! Because it would range from a trivial best case of I get really lucky and randomly hit every "good" factor in constant time O(1) to the trivial worst cast of I get really unlucky and keep guessing the wrong numbers ad infinitum.