This video from Veritasium is a nice explanation of the mathematician David Hilbert's paradox of the Grand Hotel, which illustrates that a hotel with an infinite number of rooms can still accommodate new guests even when it's full. Until it can't, that is. See also Steven Strogatz's explanation of Hilbert's infinite hotel and how Georg Cantor's discovery of different types of infinities complicated the hotel's hospitality. (via digg)