Hilbert imagines a hypothetical hotel with rooms numbered 1, 2, 3, and so on. The hotel is full and new guest arrives wanting a room. I can not move any guest. So i say to new guest wait 1 sec a new room must become available as Infinite guests one must leaving at any time.
is this good answer?
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No. Your answer has absolutely nothing to do with infinity nor is it paradoxical.
The paradox is saying that for every number N there is an occupied room in the infinite hotel. But that there is still vacancy.
That’s it.
We take every guest in their room N, and ask them to move to room N+1, and now room 1 is vacant.
So the infinite hotel is always full but always has vacancy. That’s what makes it a paradox.
Why not ask them to move to room 2N+1? Then you could accommodate an infinite number of new guests (all the rooms 2N are now empty) before having to ask anyone to move again.
As I understand, that works too, and leads down that path to exploring the multiple ‘levels’ of infinity.
That’s how you handle an infinite number of people showing up and needing rooms.
It was probably 50 years ago but I loved Hotel Infinity movie.
The paradox is saying that for every number N there is an occupied room in the infinite hotel. But that there is still vacancy. The paradox
wikipedia - Hilbert’s paradox of the Grand Hotel Hilbert imagines a hypothetical hotel with rooms numbered 1, 2, 3, and so on with no upper limit. This is called a countably infinite number of rooms. Initially every room is occupied, and yet new visitors arrive, each expecting their own room. A normal, finite hotel could not accommodate new guests once every room is full. However, it can be shown that the existing guests and newcomers – even an infinite number of them – can each have their own room in the infinite hotel.
As for the one i wrote what do you think?
The hotel with infinite rooms is a thought experiment meant to show how counter-intuitive the concept of infinity is. It’s not a riddle to be solved.
That being said, the idea that an infinite number of guests must be leaving the hotel at any point in time in certainly insightful and creative.
thanks. the Hotel part is not focused on musch.
The important feature of Hilbert’s example is not the hotel, or the guests, but the strange consequence of Cantor’s method for ‘counting’ an infinite collection of objects. William Dunham’s Journey through genius does an excellent job with Cantor in his final two chapters. I’m paraphrasing him.
Suppose you want to know if you have more than 5 raspberries. You could count them, or you could stick them on the ends of your fingers. If you have berries left over, there were more than five. If all the berries fit all the fingers, you have put your fingers in a 1-to-1 correspondence with the berries, and concluded that the numbers of each are the same. That’s how small children count, and Cantor’s genius was to extend this idea to sets of infinite objects.
If you can find a 1-to-1 correspondence between the counting numbers (1,2,3,…) and some other set of objects, then the size of the two sets must be equal (in the business, size of sets is called the ‘cardinality’). It is traditional to denote the counting numbers by ‘n’… Hilberts ‘paradox’ arises from Cantors claim that there are as many counting numbers as even numbers. The correspondence is n->2n. For every counting number, I can double it to find every even number. That’s weird.
You can just as easily show that there are as many odd numbers as counting numbers. The scheme is more complicated, but you can also find a 1-to-1 correspondence between the counting numbers and all the fractions between 0 and 1. It seems like the counting numbers can count everything! But that’s not the whole story. The big shock is that there are more numbers between 0 and 1 than counting numbers. That is Cantors ‘nondenumerability of the continuum’. I highly recommend Dunhams book. The proof of the nondeumerabilty theorem is so beautiful and accessible (like accessible to a ten year old).
your comment is more about infinites then the problem to be solved. this is just me having fun.
No, because if you have a room empty, you now have to identify the vacated room, which could take an infinite amount of time
Also, it would take up to an infinite amount of distance to walk to or from the front desk, so we just be dealing with immortal guests who have no time sensitivity at all
Furthermore, the room must be made ready and guests tend to check out at a certain time of day, so you would have to decentralize everything while accounting for light speed limiting the speed of information
Some of these problems are solvable if you add more constraints to the puzzle. Like, if you have a rate of check-in to checkout you could do something with statistics and discrete math to divide the infinity into an infinite number of sub hotels, and in doing so possibly guarantee availability for the guest in a long but linear amount of time
It would be an interesting programming/math puzzle. My intuition tells me it is solvable, but it might not be
Thanks very helpful :)
I’m glad. It was an entertaining thought puzzle
I hope you keep asking interesting questions
Those would all also a problem in the original and are never relevant. For the bus with infinite guests, every current guest has to move. So some of the now vacant rooms are infinitely far away and also none of them are ever prepared by staff etc.
The problem with OPs take is the wrong assumption that a room will be ensured to become available.
I wouldn’t call it a wrong assumption, it’s just ambiguous. We don’t know the constraints
An infinite hotel would need either infinite housecleaners or none at all, so it’s a relatively minor part of the problem
OPs assumption is: “there are infinite guests, so at any point someone will be checking out at that exact moment.”
Under the information given by the problem, making that assumption is wrong. In fact, assuming any of the guests will ever check out is already wrong. The problem never makes any mention if checkout at all. All guests checking out a millisecond after checking is just as likely of a scenario as them never checking out until the death if the universe.
No. What about: “All guests are here for the festival. Nobody is going to check out before that.”
Your “infinite guests” -> “infinite different itineraries, including one where somebody checks out right this second” is the same wronh assumption as the “infinite universes”->“there must exists a universe where x…”.
Just because there are infinite rational numbers between 1 and 2, does not mean any of them is 3.
You can have infinite guests and nobody checking out for an infinite amount if time.
I’ve seen the paradox mentioned a lot but I don’t see it as paradox at all.
A hotel with infinite rooms has infinite size.
Just idea of asking guests to move would take infinite time (i.e will never happen).
if each guest wait for the his new room to vacate, no one would move at all.
infinite guests with infinite different leave times. so one room most be free.
If they all step out and into the new rooms at the same time, it’s possible.
It’s not a mathematical paradox, only against intuition.
But how would they all know when to step out and into new room?
They need some kind of a signal (bell, phone call, flashing light, speaker instructing them to move, …)
Fastest of these communication is light which has maximum speed of 300,000 km/s
The light will reach first room instantly but since number of rooms is infinite, the light need infinite time to reach that last room
It’s an illustration of infinities, not an actual hotel.
