I don't know how many people know this, but often it's mindblowing when you learn, that some infinities are bigger than others. [...] The number of counting numbers...so 1, 2, 3, up to infinity...the numbers you would use to count things, that's infinite. The number of irrational numbers...so the numbers you cannot represent as a fraction, okay, there are more of those than there are counting numbers, by far. So these are orders of infinity. Then there are more transcendental numbers than there are irrational numbers. So that's a number you'll never find as a solution to an algebraic equation. So pi is a transcendental number. e is a transcendental number. These are magic numbers that show up in mathematics. And it turns out there's an even bigger infinity of those than there is of these other two classes of numbers. And they use the Hebrew letter aleph in ranking. So it's aleph-1, aleph-2, aleph-3, aleph-4. I think there are five levels of infinity.

This is mostly confused and wrong, and something that Tyson hasn't, as he's claimed, learned: there are not more transcendental numbers than irrational numbers. The transcendental numbers are not magic numbers that show up in maths. And there are not a mere five alephs (there are more alephs than we have an aleph to measure).

Why is Tyson venturing into territory where he is ignorant in his capacity as a science communicator? I really like the guy, but it just leaves me wondering how often he is overstepping the bounds of his expertise. If I read his popular science work, how do I distinguish the bits where he's talking competently and the bits where he's got it completely wrong because he doesn't even have a bachelor's level understanding of a topic? It breaks a bond of trust I expect from a science communicator, and makes me ever more reluctant to trust claims from popularisers of science. I dare say, it makes me feel suspicious of self-proclaimed "experts".

I'd level this criticism at Lawrence Krauss as well. I consider it pretty irresponsible coming from people who are presenting as the public face of science.

Go to 17m50.