Originally posted by borissaIf each room is full, then surely there must be a finite number of rooms. And also if you put an infinite number of people in an infinite number of rooms then there are infinity/infinity persons per room which = 0 persons per room. so you have all of your rooms empty for the next infinite coach load of people. 😛
Imagine a hotel with an infinite number of rooms. The rooms are simply numbered 1,2,3...ad infinitum. Each room will fit only 1 person in, and each room is full.
Now, what happens if one more person comes? Which room do you put them in?
When I saw this, I remembered how to do the single extra person, from second year set theory. The infinitely large coach with an infinite number of people on, that's fine too.
But I definitely remember there being a way to handle an infinite number of coaches with infinitely many people on each turning up, too. Argh, that's annoying me now... Anyone?
[Edit: of course, since posting this, Wikipedia furnished me with an answer... I'll leave it an open question though 😉 ]
Doesn't seem to be logical question to me. The way I see it, if the hotel has infinite number of rooms (ignoring the fact that that is impossible in itself, because surely there is limit to space?), then it can never be full. To me the word 'infinite' means there is no end, no limit, i.e. there is always another rooms available. Which necessarily mean it can accommodates ANY number of guests.
Mathematicians have plenty of definitions of infinity to bandy around; the one you want here is aleph-null, if it helps - it's defined as the size of the set of whole numbers...
It doesn't work in the real world, it's an abstract maths problem, but that's just how these things go.
Originally posted by jimbobulatorIf an infinite number of coaches turn up each with an infinite number of people on board, you simply use some bijection f mapping
When I saw this, I remembered how to do the single extra person, from second year set theory. The infinitely large coach with an infinite number of people on, that's fine too.
But I definitely remember there being a way to handle an infinite number of coaches with infinitely many people on each turning up, too. Argh, that's annoying me now... Anyone? ...[text shortened]... ing this, Wikipedia furnished me with an answer... I'll leave it an open question though 😉 ]
N x N to N
where N is the set of natural numbers 0,1,2,3,...
An example of such a map is
f(n,m) = 2^n * (2m+1) - 1.
So you'd put the nth person on the mth bus in the f(n,m)th room.
All this works because as you say N and N x N have the same cardinality.
Originally posted by FabianFnasIn the 30 second time frame mentioned, at the speed of light, assuming 20 feet per room, you can't notify more than 1.47 billion people and even in one year you can't get to more than 1.5 E15 dudes. Thats 1,500 trillion (american trillion). Not quite infinite. So like he says, it would take an inifinte amount of time and too much time even for the close in people.
An interesting aspect of this 'paradox' is the following:
You have a hotel with infinite number of rooms and they're all filled up. You rent the rooms night after night.
A new guest comes. You place every guest in room #n in room #(n+1). That's the solution, right?
But the room #1 is not free before all guest move one room upwards. This takes inf ...[text shortened]... inite time at your disposal - the 'paradox' has not a solution. It simply can't be done.
