The concept of infinity is a fascinating one, albeit a hard one to grasp. There are several intertwined mathematical notions of infinity, each with its practical usefulness and axiomatic framework. The set theoretical notion which allows a distinction between countably and uncountably infinite sets is a remarkable idea which pushes the boundaries of logic. Hilbert's Grand Hotel is a clever way to convey the subtleties of a countable infinity in everyday's terms. It would be nice to develop the analogy further in order to explain the difference between a countable and an uncountable infinity.