Hilbert's Hotel has an infinite number of rooms, and yet, even when it's full, it can still fit more people in!
Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. At best you're likely to come away with a headache, at worse the firm belief that 1 = 0. This article discusses. . . .
This article gives a proof of the uncountability of the Cantor set.