This evening I had a discussion with a friend of my about a mathematical riddle and the concept of 'infinite'
The riddle
Imagine a hotel with an infinite amount of rooms, and all of the rooms are occupied.
At that moment a bus arrives with an infinite amount of people who all want a room in that hotel. Possible or not, and how?
Possible answer (according to friend)
The owner of the hotel lets everyone who is currently sitting in a room, move to a room further. In that way the first chamber will be available.
To be honest I am not really knowledgable at math, but something tells me that this is equivalent to shifting the problem. Thanks to this 'solution' there is always somebody without a room, or not?
After all, if someone moves to the room further one will never find an empty room. The person after him also has to move and the person after him also etc. etc.
I think there can never be more people in that hotel as all (infinite) rooms were already filled with an infinite number of people. (infinite + 1 is impossible, right?)
Is it because of my limited understanding of the term 'infinity' and am I missing something, or is something wrong with the riddle?
*Sorry for the English, I hope it's clear. *
Best Answer
The first room is clearly empty: there's nobody moving into it.
The only way a person could fail to move into the next room is if they were in the last room. But then there would only be finitely many rooms, so clearly this particular hotel doesn't suffer from this problem.
The main point of this example is to vividly demonstrate one of the ways in which infinite collections differ from finite ones.
In this case, we're adding ordinal numbers. And you can add them. The ordinal number describing the hotel rooms is called $\omega$. It is essentially just the sequence of natural numbers.
When you add two ordinal numbers, you essentially just place one after the other. So if we draw a picture of 1:
and a picture of $\omega$
then to get $1 + \omega$, we place 1 first, then $\omega$ next:
Looks the same, doesn't it? That's what happens in the hotel. And that's because $1 + \omega = \omega$.
Incidentally, if we add the other way, $\omega + 1$, we get an ordinal number that's bigger than $\omega$. One way to draw it is
The pipe (|) is a decoration to indicate that the .... really refers to an infinite sequence of asterisks (*), and they all are located to the left of the pipe. This is so it's not confused with something like
in which the ... would usually be interpreted as a finite number of asterisks that we were too lazy to write out.
One particularly important thing to note about $\omega + 1$ is that last asterisk doesn't have an immediate predecessor. That's another unusual feature that infinite ordinal numbers (except for $\omega$ have: they can have elements that have infinitely many things before it, but none of them are immediately before it.