[IronJack]

Exactly. "Infinity" isn't a number. It's not like you can add infinity plus one. Mathematically, it just means "unbounded."

So the idea of comparing "infinities" is in itself absurd. How can one "forever" be longer than another?

It is correct to say the real numbers is an unbounded (infinite) set. It is also correct to say that the integers are an unbounded set. It is not correct to say that the set of real numbers is greater than the set of integers.

[coloradan]

Integers are countably infinite, real numbers are uncountably infinite. They are provably different, e.g. by Cantor’s diagonal proof. The number of rational numbers is also countable.