"It is not possible to prove that a set is both smaller than the reals and larger than the rationals, because such a set would disprove the continuum hypothesis."
Sure, even without much of a mathematical background, people generally take it for granted. Which is why it's disappointing that a suggestion of overturning it isn't fulfilled.
Suggestion of overturning it? It's a proof. That article would be headlined "disproving the independence of the continuum hypothesis" or some such; it would be huge news, not somebody's fun blog post.
Sure, even without much of a mathematical background, people generally take it for granted. Which is why it's disappointing that a suggestion of overturning it isn't fulfilled.