Posted: Mar 01, 2017 4:10 pm
JoeB wrote:The idea of different / bigger types of infinity is nothing strange, I'm not sure what the OP is objecting against.

The OP is very familiar with the subject area (as are a number of other mathematicians on the boards). The objection is to the numerous false statements Tyson made.

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.

So throughout this Tyson is speaking informally, and well outside his area of expertise. He's actually talking about the cardinality of numbers in standard set theory with the Axiom of Choice.

 The transcendentals are a strict subset of the irrationals. There cannot possibly be more transcendentals than irrationals because every transcendental is an irrational.
 In fact both sets have the same cardinality. https://www.quora.com/What-is-the-cardi ... al-numbers
 There are infinitely many. There are infinitely many infinitely manies of levels of infinity. https://en.wikipedia.org/wiki/Cardinal_number