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.
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 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.
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.