> In 1940 the famous logician Kurt Gödel proved that, under the commonly accepted rules of set theory, it’s impossible to prove that an infinity exists between that of the natural numbers and that of the reals. That might seem like a big step toward proving that the continuum hypothesis is true, but two decades later the mathematician Paul Cohen proved that it’s impossible to prove that such an infinity doesn’t exist! It turns out the continuum hypothesis can’t be proved one way or the other.

This, to me, is the exciting bit of the whole article.

There are questions that cannot be answered definitively. There is no way to say that it's true or false. Not simply that we haven't figured it out yet, but that no one, no matter how clever, will ever be able to prove the answer one way or another. If we meet an extraterrestrial alien civilization that is a million years of math theory ahead of us, they will not have an answer.