Once upon a time, there was a hotel. It had a spacious pool, a high-quality spa, a buffet with every type of food and most importantly, a countable infinity number of rooms. One night, the hotel was full, totally booked up with an infinite number of guests.
At that moment, a man walked into the hotel and asked for a room. The hotel manager Hilbert suddenly had a strange idea. Rather then turning him down, he decided to make room for him. How? He asked the guests to move 1 room larger than the current room. Since there is an infinite number of rooms, there is a new room for each existing guest. And room 1 is available for the new guest. He also found out that this method can be used for any finite number of guests, and served more customers.
But then, an infinite large bus with a countably infinite number of customers appeared to rent rooms. This perplexed Hilbert at first, but then he asked the guests in the hotel to move to double the number of the current room, filling up only the infinite even rooms--and this leaved all the odd rooms for the new customers. He also realized that this can be applied to a finite number of infinite buses.
All guests were satisfied, and one might be tempted to think the hotel's business was booming more than ever. But it was actually booming same as before--banking an infinite number of money every day. Words spread about this fabulous hotel. Guests poured in from far and wide. But one night, the unthinkable happened...
That night, Hilbert looked outside and saw an infinite line of infinitely large buses, each with a countably infinite number of customers. What could he do? If he failed to find rooms for them, the hotel would lose an infinitely number of money and he would surely be fired. He sat on his chair for a while, and decided to assign all guests to move to 2 raised to the power of the current room number. Then, he filled the customers on the first bus to move to 3 raised to the power of their seat numbers. After that he repeated this with primes and filled all customers in the hotel.
Although there were many empty rooms since they weren't powers of primes, his boss was fortunately bad at math (and only good at business), so Hilbert's job was safe... for now. What happened two weeks later sealed his doom, but also the hotel's.
Two weeks later, a strange bus with the same number of customers as all the numbers from 0 to 1 passed the hotel. Hilbert thought it was interesting, and it would give him a promotion if he accomodated all of them, so he decided to find rooms for them. He tried the methods he used before, and arranged rooms for them in 3 hours. Hilbert was tired and thought the job was done, so he headed back to his office for a nap.
But 4 hours later he received a phone call from his boss. The boss asked him angrily, 'Why there are customers outside?'
'I...I don't know. I thought the customers were...' Hilbert answered.
'All accomdated? Are you blind? There are lots of customers outside!'
'How come this happen? Anyway, I need to investigate--May I have a call with the customers?'
The boss redirected the call to the customer counter. A customer told him, 'My seat number's first digit isn't the same as the first digit of the first customer.'
'Then?' Hilbert questioned, 'it may be the same as any guest, right?'
'Not really, manager. My seat number's second digit isn't the same as the second digit of the next customer, my third digit is different from the third--My seat number isn't the same as anyone in the hotel. I don't have a room!' The customer yelled at him.
'Me too! Me too! The hotel is not infinite!' The other customers protested.
Hilbert's face turned pale. He found out it was impossible to fill all customers into rooms, and apologised to the customers. His boss was furious, and fired him.
But as stated before, the only person who knew math was fired now. The boss couldn't even deal with a customer when facing the full hotel! He was forced to turn down many customers, and rumours spread that the infinite hotel was a false claim. After several months, all the guests left without paying, and the hotel was shut down.
And that's the end of the (countably) Infinite Hotel.
Note: This imaginary story is based on Hilbert's infinite hotel paradox and information from DGS math team. Credits to them, and wish you enjoy the journey of the infinites.
