[abnry]

If you take the power set of an infinite set, you /always/ get a cardinality bigger than the original set. So there are an infinite number of infinities.

[Others]

This is a good point. But to be super pedantic, there isn't a size of infinity large enough to describe how many sizes of infinity there are.

[fooker]

Why not? The aleph numbers have a bijection with naturals.

[Others]

See this answer on the mathematics stack exchange (it's really a book excerpt, but I don't know where else to find it.) https://math.stackexchange.com/a/5390/220797