In order to understand the difficulties that arise when dealing with infinity, David Hilbert (1862-1943) came up with the following example, which illustrates how little infinity matches our finite intuition.








Welcome to the Infinite Hotel: a hotel with infinitely many rooms, numbered 1, 2, 3, 4, 5, 6, ... one room for each natural number.


You can read a very nice version of the story here.