This is Math's Fatal FlawSo this discovery was one of the greatest accomplishments of the XX century, and yet it is a "fatal flaw"?
Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer.
The video is actually pretty good, but I object to all the explanations that say that Mathematics is somehow deficient because the consistency of the axioms cannot be proved from the axioms.
Nobody would ever want the consistency to be provable from the axioms anyway. Such a proof would mean nothing. Inconsistent systems allow such proofs, and nobody wants that.
It would be nice to have an algorithm to determine whether a given math statement is true or false. The above discovery shows that it is not possible. But again, this is not a fatal flaw. It is what makes Math interesting.