The Hilbert’s Hotel Paradox

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Hotel rooms

Ever been to a hotel with infinite room? Allow me to take you to the Hilbert Hotel.

The latin roots of the word Paradox mean “distinct from”(para) and “your opinion”(dox). A paradox is basically a logical statement which either contradicts itself or your expectations. Some easy examples might be, save money by spending it, or the famous one by Oscar Wilde – “I can resist anything but temptation”. You might come across many paradoxes which are invalid arguments and you might think, why waste time on it but any kind of paradox is good for a little brain exercise. We’ll soon write a detailed article where we’ll explain what a paradox is in more depth and also different types of paradoxes. Till then let’s look at the Hilbert’s hotel paradox. 

David Hibert was a 19th century German mathematician who, among other things, created 23 “unsolved” mathematical problems. Infinity is a very complex concept and one can’t grasp it easily. So to explain the complexity of the concept of infinity, countably infinite sets to be precise, Hilbert created a thought experiment known as the “Hilbert hotel paradox”. So the premise of this paradox is that there is a hotel with countably infinite rooms and all the rooms are occupied. Can it accommodate any more guests? Some of you might be tempted to say no. But the answer is yes. Yes! So let us see how we can accommodate finitely many, infinitely many, and guests coming in infinite busses with infinite people in the already full hotel with infinite room.

Can we accommodate finitely many more guests in the already full hotel?

So let’s say the hotel is at it’s full occupancy and one more guest arrives. So space for that one person can be made by shifting everyone to the next room, i.e., we’ll shift the person in room 1 to room 2, the person in room 2 to room 3, person in room 3 to room 4 and so on. So we are shifting the person in room n to room n+1. So the first room has become empty now and the new person can be accommodated there. So in general, if finitely many guests, say k, arrive at the hotel then they can be accommodated by shifting the person in room n to the room n+k. 

Can we accommodate infinitely many more guests in the already full hotel?

But let’s say this time infinitely many guests arrive. Can they be accommodated too? Yes. This can be easily done by sending the person in room 1 to room 2, the person in room 2 to room 4, the person in room 3 to room 6 and so on. So we are shifting the person in room n to room 2n. So all the odd numbered rooms would become empty and since there are infinitely many odd and even numbers these infinitely many new guests can also be accommodated in those infinitely many odd number rooms. 

Can we accommodate guest who are coming in infinite busses with infinite people?

But let’s make things a little more crazy. Let’s say this time an infinite number of busses arrive with an infinite number of passengers in all of the busses. Guess what? All these people can still be accommodated. While there are a lot of ways of doing this but I state just one of them and you can think of the others as an exercise. So we know that there are an infinite number of prime numbers. And if there are two prime numbers, say p and q, p^m can never be equal to q^n for any m,n where m and n are non zero positive integers. So we can assign each person already in the hotel to a new room, i.e., 2^(their current room number) and then the people from the first bus can take the rooms 3^(their seat numbers), and then people from the next bus can take the rooms 5^(their seat number) and so on. 

This paradox tries to explain the complexity of the concept of infinity. The hotel was always fully occupied but no matter how many more guests came we could still accommodate them. 

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