The infinite hotel paradox

The infinite hotel paradox or The Hilbert's paradox of the Grand Hotel is a paradox or a sort of riddle that tells us about how a hotel with infinite rooms contains vacant rooms even if it is filled with an infinite crew. 

Imagine there's a hotel with an infinite number of rooms. On one night, the Hotel is completely full with no rooms left for any upcoming guests. Suddenly, a man comes and asks for a room. But the hotel is full, still, the manager assures the man that he would get a room. But how??? The manager asks the guest in room 1 to move to room 2, the guest in room 2 to room 3, and so on. Every guest moves from room 'x ' to room 'x+1 '. (because there is an infinite number of rooms).  

Now an infinitely large bus with countably infinite people arrives at the hotel. Now the manager askes people in room 1 to move to room 2, people in room 2 to room 4, and people in room 6 to room 12. Every guest moves from room 'x ' to room '2x '. Only the infinite even number of rooms are filled now. So the guests in the bus take all rooms of odd numbers. 

The hotel becomes famous. People from all over the planet come to the hotel. One night, The manager sees an infinite line of infinitely long buses with countably infinite guests. What can the manager do now? He remembers that there is an infinite amount of prime numbers. Now the manager gives every existing guest room with the room number of the first prime number- 2 to the power of their room number. Now the manager asks guests in bus 1 to go to a room of the next prime number- 3 to the power of their seat number in the bus. Now, the person in seat no. 4 in the first bus goes to room 3 to the power 4 ie. room 81. Each next bus follows the power of 5, 7, 11, and so on. 

These ideas are possible because they deal with the 'countable infinity '. Its lowest level of infinity. For more advanced calculation, this hotel won't be enough. So our hotel also haves integer, Rational and Decimal rooms in the basement !!! An amazing try to show how complicated is the 'Infinity '.