Hilbert owns a hotel with an infinite number of rooms numbered . The rooms are all occupied by guests who are numbered just like the rooms they occupy, that is, guest is in room .
A taxi arrives with new guests labeled . How can the manager give them rooms in the already full hotel? But there is always room at the Hilbert hotel, no matter how many people show up! To make room for them, the manager moves all of the original guests over by rooms.
A plane arrives with rows of seats, each row seating an infinite number of fliers. Label the fliers in the row . To make room for all of them, the manager shifts the original guests in the hotel from room to room .
A ship arrives with an infinite number of decks, each with an infinite number of sailors. Label the sailors on the deck . To fit them all into the hotel, the hotel manager and the captain of the ship free the first deck by shifting all the sailors by one deck. They move the original hotel guests to the first deck. The hotel is empty. The manager arranges the room numbers diagonally in a table. The original guests and the sailors go to the room assigned to them by the table. The person on the deck goes to the room that is in the table at row and column .