Proof that it is unsolvable whether there's an infinity between countable and uncountable?

Question

I have recently watched a video by "Undefined Behavior", explaining countable and uncountable infinities, and showing why uncountable infinity is larger than countable infinity. He then stated that a question had been asked if there is some infinity that's in-between the two (larger than countable, but smaller than uncountable), and that it has been proven that this question in unsolvable. However, he did not mention any name of the theorem proving this, and also did not show any proof. What is the proof that this problem is unsolvable?

Answer 1

That person was talking about the continuum hyphothesis. It was proved (by Kurt Gödel and Paul Cohen) that, assuming that set theory is consistent, neither the continuum hyphothesis nor its negation can be proved from the standard set theory axioms (the Zermelo-Fraenkel axioms).