# Hilbert's grand hotel as infinite number of pairs

by Sissoev
Tags: grand, hilbert, hotel, infinite, infinity, number, pairs
 P: 13 Can we think of Hilbert's Grand Hotel problem as of infinite number of pairs; room/guest pairs? If we pair infinite number of left shoes with infinite number of right shoes, we will have infinite number of shoe pairs. In such pairing there will not be left any unpaired shoes, because that would brake the infinity in both the sets. That would mean that there is no way to add one left shoe and get new pair of shoes in the shoe pair infinity. Same way, it would not be possible to add even one only guest in Hilbert's Grand Hotel. Should we not treat Hilbert's hotel as two paired infinite sets, where adding to one of the pair sides will brake the infinity in both sides?
P: 834
 Quote by Sissoev If we pair infinite number of left shoes with infinite number of right shoes, we will have infinite number of shoe pairs. In such pairing there will not be left any unpaired shoes, because that would brake the infinity in both the sets. That would mean that there is no way to add one left shoe and get new pair of shoes in the shoe pair infinity.
No, if I take away a left shoe from you, you still have an infinite number of left shoes and an infinite number of right shoes left over. Just one of which is unpaired. And then, if you want, you can pair them up again if you want.
If you have an infinite number of something and take away a finite number, you still have an infinite number left over.
Similarly, if you have an infinite number of something and add a finite number, you still have an infinite number.

I also have no idea what you mean by the words "break infinity".

 Should we not treat Hilbert's hotel as two paired infinite sets,
Sure if that's your model. The Hotel is now a subset of ##\mathbb{N} \times \mathbb{Z}##.
The first coordinate is room number, second coordinate guest number. I chose the integers here because I want "old guests" as positives and "newly arrived guests" as negatives.

The Hotel starts with ##\{(1,1), (2,2), (3,3), \ldots\}##.

A new guest arrives, whose name is "-1". The clerk sends the guest in room 1 to room 2 etc. Guest "-1" goes to Room 1. We now have a Hotel that looks like ##\{(1,-1), (2,1), (3,2), \ldots\}##.
P: 13
 Quote by pwsnafu No, if I take away a left shoe from you, you still have an infinite number of left shoes and an infinite number of right shoes left over. Just one of which is unpaired. And then, if you want, you can pair them up again if you want. If you have an infinite number of something and take away a finite number, you still have an infinite number left over. Similarly, if you have an infinite number of something and add a finite number, you still have an infinite number. I also have no idea what you mean by the words "break infinity".
To break infinity...
Well, not exactly to break it, but in a meaning that it wasn't infinity in a first place.

We know that infinity+1=infinity, but that's not the case with pairs, I think.
Why?
In the OP I said:
 If we pair infinite number of left shoes with infinite number of right shoes, we will have infinite number of shoe pairs. In such pairing there will not be left any unpaired shoes, because that would brake the infinity in both the sets.
In other words; if we pair two infinite sets, and we are left with one unpaired object, that would tell us that we weren't dealing with infinite sets.
I think that we have to keep that in mind when we work with paired sets.
Right and left shoes are in pairs, which brings equation in the two paired infinite sets.
If you take one out from one of the sets, the equation breaks, from which the infinity "breaks".
(One of the sets becomes with 1 greater than the other set, which results in one object not having its pair. And that is clearly not infinity.)

Following the above we can create a rule: two infinite sets can be paired only once.
Breaking that rule would create paradoxes.

P: 834
Hilbert's grand hotel as infinite number of pairs

 Quote by Sissoev In other words; if we pair two infinite sets, and we are left with one unpaired object, that would tell us that we weren't dealing with infinite sets.
This does not follow. You cannot pair the natural numbers with the reals because the latter is uncountable. Doesn't change the fact that the naturals and reals are infinite to begin with.

 I think that we have to keep that in mind when we work with paired sets. Right and left shoes are in pairs, which brings equation in the two paired infinite sets. If you take one out from one of the sets, the equation brakes, from which the infinity "brakes".
No it does not. All it does is a relabel.

 (One of the sets becomes with 1 greater than the other set, which results in one object not having its pair. And that is clearly not infinity.)
"One of the sets becomes with 1 greater than the other set" is a false statement. They still have the exact same number. That is what being an infinite set means: there is an embedding into a proper subset of itself.

 Following the above we can create a rule: two infinite sets can be paired only once. Braking that rule would create paradoxes.
That's nonsense. We are free to relabel any number of times. Every real function* you learned in school does this. In other words you are claiming that taking a function ##f(x)=x## and defining a new function ##g(x) = f(x)+1## is an invalid procedure.
*well, every real function that is a bijection.

Now if you mean, define a function, and then claim it is something else? Yeah, that's not allowed. But that is also true for finite sets as well. But Hilbert's Hotel doesn't do that. It defines a different function (in the function model), or changes state (in the dynamical system model). In the pairs model I gave above the Hotel has changed from one infinte set to another set, but those two sets have the same cardinality because it's clear there is a bijection between them.

Edit: To add to this, you are using the word "pair" in two ways.
On one hand you are talking about a set of pairs, that is the Cartesian product of two sets.
On the other you are talking about pairing two sets, which means to define a function (specifically a bijection) between them.
Those are two separate things.
 P: 13 Lets keep it simple for you, pwsnafu; imagine that two guests came in the hotel and in order to accommodate them the manager orders all guests to get out of their rooms and move in front of the room with two numbers up. Then №1 moves in front of room №3, #2 in front of room №4 and so on. Then one of the new guests goes in front of room №2, but the other guest suddenly decides that the hotel is too dirty and leaves it. Room №1 remains without guest in front of it. Now the empty rooms ready to accommodate the guests waiting in front of them are with one more, which makes the infinity false. To avoid such stupid situations, two infinite sets can be paired only once
 Emeritus Sci Advisor PF Gold P: 4,500 Sissoev, at that point the hotel manager could, if he wanted to, ask all the guests to move one room back, making all the rooms matched up with guests again. It is true that given two countably infinite sets, you can make a function from one to the other that does not create a perfect pairing, but that doesn't change the fact that you can also pair them up perfectly. It doesn't even make sense to say "can only be paired once'. A pairing mathematically is just a function $f:X\to Y$ which is both one-to-one and onto. I can come up with tons of functions $\mathbb{N} \to \mathbb{N}$ which are pairings, and you could cycle through and apply them however you want.
P: 13
 Quote by Office_Shredder Sissoev, at that point the hotel manager could, if he wanted to, ask all the guests to move one room back, making all the rooms matched up with guests again.
Sure, but that wouldn't make the set infinite again.
That would just make the last room empty ;)
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P: 4,500
 Quote by Sissoev Sure, but that wouldn't make the set infinite again.
Which set? The set of people and the set of rooms are always infinite.

 That would just make the last room empty ;)
There is no last room, just as there is no last natural number.
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P: 21,304
 Quote by Sissoev Sure, but that wouldn't make the set infinite again. That would just make the last room empty ;)
For an infinite number of rooms (or an infinite number of anything), there is no "last" room.
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P: 21,304
 Quote by Sissoev To break infinity... Well, not exactly to break it, but in a meaning that it wasn't infinity in a first place.
I think you have a basic misunderstanding of what infinity means. An infinite set doesn't become finite just by removing one element from it.
 Quote by Sissoev We know that infinity+1=infinity, but that's not the case with pairs, I think. Why? In the OP I said: In other words; if we pair two infinite sets, and we are left with one unpaired object, that would tell us that we weren't dealing with infinite sets.
No, not at all.

For a simple example, assume that the hotel is completely booked, with a person in each room. We have a pairing between the set of hotel patrons and rooms, both of which are infinite sets. If the manager asks each person to move to a room whose number is one higher, then room 1 will be empty. In no way does this somehow imply that the set of rooms is now finite. What we have now is a one-to-one pairing between the hotel patrons and rooms 2, 3, 4, ..., N, ...
 Quote by Sissoev I think that we have to keep that in mind when we work with paired sets. Right and left shoes are in pairs, which brings equation in the two paired infinite sets. If you take one out from one of the sets, the equation breaks, from which the infinity "breaks". (One of the sets becomes with 1 greater than the other set, which results in one object not having its pair. And that is clearly not infinity.)
It's clear that you don't understand...
 Quote by Sissoev Following the above we can create a rule: two infinite sets can be paired only once. Breaking that rule would create paradoxes.
P: 13
Come on, guys!
I know that there is no last room in an infinity set, but you don't read carefully.
Both the rooms and the guests are infinite as a pairs.
Once you take the guests out of the rooms, and try to pair them again, you loose the infinity, and that's why it is wrong to do it.
The rooms became with one more than the guests, the moment we moved them out with two numbers up.
Remember?
 Now the empty rooms ready to accommodate the guests waiting in front of them are with one more, which makes the infinity false.
So, as long as there is no infinity, we can have last room.
 Sci Advisor PF Gold P: 2,226 The point is that the definition of an infinite set is one for which there is a bijection from itself to a proper subset. You can't get around the fact there's always a way to make more room as it's trivial from the assumption that the number of guests and the number of rooms are both |N|.
 Emeritus Sci Advisor PF Gold P: 4,500 Sissoev, so your claim is that as soon as the guests shift around, the number of rooms in the hotel changes from being infinite to finite?
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P: 21,304
 Quote by Sissoev Come on, guys! I know that there is no last room in an infinity set, but you don't read carefully. Both the rooms and the guests are infinite as a pairs. Once you take the guests out of the rooms, and try to pair them again, you loose the infinity, and that's why it is wrong to do it.
No, you don't "lose the infinity" so it's not wrong.
 Quote by Sissoev The rooms became with one more than the guests, the moment we moved them out with two numbers up. Remember?
There's a big difference between two finite sets and two infinite sets that you're missing. Consider these finite sets: A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Clearly there is one more element in set A than in set B, so set A is larger.

With infinite sets things don't work this way. Suppose C = {1, 2, 3, 4, ...} and D = {2, 3, 4, ... }. If we pair 2 in C with 2 in D, and 3 in C with 3 in D, and so forth, it would appear that C has more elements than D, but this is not true. With infinite sets, all you have to do is to show that there is a one-to-one mapping between the two sets. If there is, the sets have the same cardinality.
For my example, the mapping from set C to set D is 1 --> 2, 2 --> 3, 3 --> 4, and so on. If you tell me any number in set C, I can tell you what it maps to in set D. Conversely, if you tell me a number in set D, I can tell you the corresponding element in set C.
 Quote by Sissoev So, as long as there is no infinity, we can have last room.
P: 13
 Quote by Office_Shredder Sissoev, so your claim is that as soon as the guests shift around, the number of rooms in the hotel changes from being infinite to finite?
Hahaha
No, Shredder, my claim is that two infinite sets can be paired only once, because if you try to do second pairing, you'll fall in to paradoxes like rooms appearing from nowhere and breaking the infinity.
How about, emptying all the odd number rooms?
We'll get infinite occupied rooms + infinite empty rooms from two sets which were perfectly paired.
Don't you find it disturbing?
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P: 4,500
 Quote by Sissoev Hahaha No, Shredder, my claim is that two infinite sets can be paired only once, because if you try to do second pairing, you'll fall in to paradoxes like rooms appearing from nowhere and breaking the infinity.
When you say things like
 So, as long as there is no infinity, we can have last room.
it sounds like you think the number of rooms is finite.

 How about, emptying all the odd number rooms? We'll get infinite occupied rooms + infinite empty rooms from two sets which were perfectly paired. Don't you find it disturbing?

Let's stop talking about the hotel problem, because it is an analogy that apparently causing you trouble. The point is that infinite sets are kind of weird if you don't think about them the right way. Consider the set of all natural numbers, and the set of all even numbers. On the one hand, the natural numbers are "obviously" bigger. But now suppose that I have a set X, and I multiply every element in X by 2 to get a new set Y. All I've done is re-label the elements in my set X, so "obviously" X has the same size as Y. So from one perspective it is obvious that the evens and natural numbers are a different size, from another perspective they are the same size. Which is correct?

It turns out they are the same size. The best mathematical way of comparing two infinite sets is to figure out if there is a function from one to the other that is one-to-one and onto, i.e. a perfect pairing between them. It is well known that if two sets have a perfect pairing, there are also lots of ways to give a non-perfect pairing (in fact, a set is infinite if and only if it has a non-perfect pairing, i.e. a one-to-one but not onto function, with itself). A lot of people have trouble wrapping their heads around this but it is the fundamental tool in set theory for comparing sets, and leads to a very rich theory of sets.

In the hotel example, the guests are the natural numers, and the rooms are also the set of natural numbers, and describing how guests stay in rooms is the same as describing a function from the natural numbers to the natural numbers. Sometimes these functions are not onto, sometimes they are not one-to-one, but it doesn't change the fact that both copies of the natural numbers are the same size and can be paired up in a lot of ways.
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P: 21,304
 Quote by Sissoev Hahaha No, Shredder, my claim is that two infinite sets can be paired only once, because if you try to do second pairing, you'll fall in to paradoxes like rooms appearing from nowhere and breaking the infinity.
There is no requirement that a pairing be unique.
 Quote by Sissoev How about, emptying all the odd number rooms? We'll get infinite occupied rooms + infinite empty rooms from two sets which were perfectly paired.
Again, this is what makes infinite sets fundamentally different from finite sets.
 Quote by Sissoev Don't you find it disturbing?
No, I don't.

If you read further about the Hilbert Grand Hotel, you'll see that the story continues when a very large bus (containing an infinite number of passengers) arrives.

The hotel is already full, but the manager comes up with a plan to accommodate the new arrivals. He asks each of the current guests to move from their room to the room whose number is two times their old room number. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on. This frees up rooms 1, 3, 5, ..., and the new arrivals can be given the odd-numbered rooms.