This guy raises the stupid Liar Paradox multiple times, in different contexts, while making his explanations, and every time he states that "philosophers have worried about this for thousands of years".
Well, I've never seen a true philosopher dedicate a long time to this question, or even to consider it a question or a problem or an inconsistency in the world we must solve -- I can only see this guy and his fellow mathematicians do that.
Nor it is true that Gödel's theorem is a form of Liar Paradox, it is not. The fact that these mathematicians keep bringing the Liar Paradox to explain Gödel's theorem to the public makes it seem like they are terrified by it, and actually attribute an enormous value to it.