[andrewla]

Tangential to your point about the poor phrasing is that having "infinite ... numbers between just 0 and 1" is not evidence of a higher infinity.

Specifically, there are infinite rational numbers between 0 and 1, but rationals have the same cardinality as the natural numbers.

[rvbissell]

That's an interesting statement about rationals. My intuition would be to consider them a subset of the reals, and for natural numbers to be a subset of the rationals. How is my intuition failing me, in this case?

EDIT: found the answer: with Cantor's diagonalization, you can count all the rationals -- effectively mapping each one to a natural number. Since this mapping is demonstrably possible, they have the same cardinality.