# Understanding the concept of infinity

## Main Question or Discussion Point

In Hilbert infinity hotel, all the rooms were occupied. Then how did the occupant were able to shift to their adjoining room?? Here I understand, by full mean, ALL the infinite room has a corresponding occupant.

I also understand some infinity number are greater because it can be proove when they are placed side-to-side.

Here, if they are able to shift, then it is possible when one room is vacant, say the last one. But again in infinity, there is no end.

Thanks.

fresh_42
Mentor
I'm not sure I understand your question. As infinity isn't something real and Hilbert's hotel only a metaphor, you cannot expect a one-to-one correspondence. However, this leaves me with the question: What do you understand by infinity? And be aware of the fact, that we do not discuss it on a philosophical level, i.e. we require a definition so that we can talk about the same.

Then what does Hilbert actually want to show from his example.

Nugatory
Mentor
In Hilbert infinity hotel, all the rooms were occupied. Then how did the occupant were able to shift to their adjoining room?? Here I understand, by full mean, ALL the infinite room has a corresponding occupant.
At the stroke of midnight everyone steps out of their room into the hall, walks down the hall and stands in front of the door of the next higher-numbered room. Now all the rooms are empty because everyone is standing in the hall, so everyone can open the door they're standing next to and enter an empty room.

Dale
Mentor
At the stroke of midnight everyone steps out of their room into the hall, walks down the hall and stands in front of the door of the next higher-numbered room. Now all the rooms are empty because everyone is standing in the hall, so everyone can open the door they're standing next to and enter an empty room.
That is how I understand it also

At the stroke of midnight everyone steps out of their room into the hall, walks down the hall and stands in front of the door of the next higher-numbered room. Now all the rooms are empty because everyone is standing in the hall, so everyone can open the door they're standing next to and enter an empty room.
Thanks a lot. I understand in infinity there is just no last room. If there is any last room, then the rooms are not infinite but instead consist of finite rooms.

Since there are no last room, one can simply shift to the next room.

mfb
Mentor
Right.

Svein
You are talking about countable infinity, but there are other kinds of infinity too. A neat mapping of complex infinity is the Riemann Sphere:

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lavinia
Gold Member
If after each person leaves his room, how many ways are there for each to enter a new room?

pinball1970
Gold Member
If after each person leaves his room, how many ways are there for each to enter a new room?
I am tempted to say infinity, an infinite number of ways since there infinite doors and people.

jbriggs444
Homework Helper
2019 Award
I am tempted to say infinity, an infinite number of ways since there infinite doors and people.
Yes, but there is more that can be said.

Suppose that one tried to make a list of all the ways (permutations) in which an infinite row of numbered people could each step into a numbered room. The first list entry goes into the first room. The second list entry goes into the second room and so on.

Could one manage to fit all the permutations, one permutation per room without running out of rooms? Cantor has an argument about that.

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From lattice theoretic perspective, one can think of infinity as the $1$ of a certain bounded infinite lattice. It is possible to study such structures, so why not?

One can take an infinite semigroup (countable or uncountable) that is not a monoid and adjoin to it an extra element that is not already contained in the set and define multiplication such that this new element acts as the multiplicative identity. A philosopher would ask - well, how do you know you can add this extra element to the set? How do you know it exists?

Mathematics doesn't concern itself with whether something exists (not to be confused with a statement such as $\exists x P(x)$). Real numbers certainly don't exist. Neither does a parametric curve that traces the devil's horns on a surface homeomorphic to a feather on an angel's wing. It doesn't matter. We play these games, supposing for the purpose of discussion that we have certain tools to work with. What can we learn?

Mark44
Mentor
Mentor note: A number of off-topic posts and replies to them have been removed. Let's keep the focus on what the OP asked.

pinball1970
Gold Member
Yes, but there is more that can be said.

Suppose that one tried to make a list of all the ways (permutations) in which an infinite row of numbered people could each step into a numbered room. The first list entry goes into the first room. The second list entry goes into the second room and so on.

Could one manage to fit all the permutations, one permutation per room without running out of rooms? Cantor has an argument about that.
I googles this and its too difficult to get into the Cantor stuff, infinite infinities? Can you explain the question/answer above please? We are still in the hotel so its not off topic

jbriggs444
Homework Helper
2019 Award
I googles this and its too difficult to get into the Cantor stuff, infinite infinities? Can you explain the question/answer above please? We are still in the hotel so its not off topic
Are you OK with the idea of making a list of all the ways to arrange an infinite number of people into an infinite number of rooms? We start with one arrangement, write it down (on an infinite piece of paper) and stick it into the first room. Then we write down the next arrangement and stick it into the second room.

Since there are infinitely many rooms, we can never run out, so it seems reasonable that we can fit all of these pieces of paper into separate rooms. But there is an argument that shows that no such attempt can succeed. It is a "diagonalization" argument. It is a proof that shows that any purported list of all possible arrangements must be missing something.

The argument starts by supposing that someone has created a list of arrangements and filled the rooms with slips of paper, each describing such an arrangement, one slip of paper per room. We need not assume that this list of arrangements is complete.

One proceeds to put together a slip of paper describing an arrangement that is guaranteed to be different from every arrangement on the slips of paper in every room.

If the arrangement described by the piece of paper in room n has person n staying in room n, we have person n stay elsewhere.
If the arrangement described by the piece of paper in room n has person n staying elsewhere, we have person n stay in room n.

[The tricky part is demonstrating that one can do this without conflict]

Whatever arrangement we end up with on our slip of paper, it will not match any arrangement on any slip of paper in any room. The list of arrangements on those slips of paper was not complete.

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pinball1970
Gold Member
Whatever arrangement we end up with on our slip of paper, it will not match any arrangement on any slip of paper in any room. The list of arrangements on those slips of paper was not complete.
If I am following ok so far I need to play with some combinations to try and get this- thanks for your patience

lavinia
Gold Member
Other ways to change rooms are everyone moves over two rooms or three or any number of rooms.

Mark44
Mentor
As an extension of the problem, a bus with an infinite number of passengers arrives. After some thought, the hotel manager decides on this scheme:
At precisely 4:00pm, each current guest exits his or her room, and goes to the room whose number is twice that of the room he or she is leaving. This will leave rooms 1, 3, 5, ..., 2n + 1, ... vacant, and the new arrivals can be given these rooms.

fresh_42
Mentor
I found a beautiful quotation from cardinal Nicholas of Cusa (1401-1464) about infinity. I think it's worth noting as it demonstrates a knowledge six centuries ago, which could already answer a lot of infinity questions posted on PF:

"If an infinite line were composed of infinite stretches of one foot in length, another of infinitely many stretches of two feet in length, they nevertheless would be the same, since the infinite can not be greater than infinite."

pinball1970
Gold Member
One proceeds to put together a slip of paper describing an arrangement that is guaranteed to be different from every arrangement on the slips of paper in every room.
I am struggling with this part, is this not deliberately setting yourself up for a fail? We are already compiling an infinite list, it would not be possible to put together slip that is different to every other list, it would just be another permutation in the list?

jbriggs444
Homework Helper
2019 Award
I am struggling with this part, is this not deliberately setting yourself up for a fail? We are already compiling an infinite list, it would not be possible to put together slip that is different to every other list, it would just be another permutation in the list?
It cannot already be in the list. The construction guarantees that.

The thrust of the argument is that the set of all possible permutations is "too large" to fit into the rooms in the hotel, even though there are infinitely many rooms. It is a higher order of infinity.

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Svein
As an extension of the problem, a bus with an infinite number of passengers arrives. After some thought, the hotel manager decides on this scheme:
At precisely 4:00pm, each current guest exits his or her room, and goes to the room whose number is twice that of the room he or she is leaving. This will leave rooms 1, 3, 5, ..., 2n + 1, ... vacant, and the new arrivals can be given these rooms.
Yes. What is not said, is that the walk from room n to room 2n will take n times (the distance between rooms) divided by the walking speed - which approaches infinity along with n.

Happily that does not cause any conflict, since the walk from the lobby to room n also takes n times (the distance between rooms) divided by the walking speed. The downside is that the room shuffling will take an infinite amount of time.

pinball1970
Gold Member
It cannot already be in the list. The construction guarantees that.

The thrust of the argument is that the set of all possible permutations is "too large" to fit into the rooms in the hotel, even though there are infinitely many rooms. It is a higher order of infinity.
Can you send me some more links on this? I have the Hilbert hotel wiki stuff but any books/links greatly appreciated.