[moelf]

two sets can both be infinite and yet one is "larger" than the other (c.f. https://en.wikipedia.org/wiki/Aleph_number).

You can establish two infinite sets are as large as one another by finding a bijection between them. These two sets would have the same "cardinality"

We know the real numbers has larger cardinality than natural numbers, but we don't know if there's anything in between -- can you construct an infinite set that has natural numbers < X < real numbers in terms of cardinality?

[jmilloy]

OP understands that, the quote says that the "the infinitude of real numbers is next largest to that of natural numbers", which makes it sound like it is next largest as in second largest as in less large as in lower cardinality. So it seems backwards. Maybe the quote means "next largest" as in larger.

[rvbissell]

Thank you. It seems absurd that 'next largest' could mean 'larger than the largest'. Hence my (OP's) interpretation.

[moelf]

I see. yeah I can see why it's confusing now -- we use "next-to-leading order" to mean a smaller effect than the leading order, but here, "next largest" means the one larger than natural numbers. Language is hard