# Hilbert's grand hotel as infinite number of pairs

by Sissoev
Tags: grand, hilbert, hotel, infinite, infinity, number, pairs
P: 13
 Quote by Mark44 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.
I understand, Mark, but in the example I gave you, I mapped the rooms with the guests in front of them, and we still had one empty room.
You'll tell me to move the guests one room back, in order to fix the infinity, but I'll say NO, I've already mapped them and I have one extra room.
Who is right then, you or I?
Mentor
P: 21,256
 Quote by Sissoev I understand, Mark, but in the example I gave you, I mapped the rooms with the guests in front of them, and we still had one empty room. You'll tell me to move the guests one room back, in order to fix the infinity, but I'll say NO, I've already mapped them and I have one extra room.
Infinity is not being "broken" so it doesn't need to be "fixed."
 Quote by Sissoev Who is right then, you or I?
In this thread, pwsnafu, Office_Shredder, jcsd, and I are all telling you that you're wrong. We are also telling you why you are wrong. Please go back and carefully reread what we have been saying.
PF Gold
P: 6,276
 Quote by Sissoev I understand, Mark, but in the example I gave you, I mapped the rooms with the guests in front of them, and we still had one empty room. You'll tell me to move the guests one room back, in order to fix the infinity, but I'll say NO, I've already mapped them and I have one extra room. Who is right then, you or I?
Sounds to me like you are still insisting, despite having been told repeatedly that it's wrong, that infinity plus 1 is not the same as infinity and infinity minus one is not the same as infinity, whereas they are both exactly the same.
 P: 13 OK, I understand and thank you all Now my question is: Why Hilbert didn't give the simplest answer; unpair the set, add as much as you want and pair it again?
Mentor
P: 21,256
 Quote by Sissoev OK, I understand and thank you all Now my question is: Why Hilbert didn't give the simplest answer; unpair the set, add as much as you want and pair it again?
That's exactly what happens. Each time there is a pairing between the occupants and the hotel rooms.

Person1 --> Room 1
Person2 --> Room 2
...
PersonN --> Room N
etc.
There is a one-to-one pairing between occupants and rooms.

Another guest arrives. Where should he go? The hotel manager asks each person who already has a room to move to the next higher numbered room. The pairing is now like so:
New person --> Room 1
Person1 --> Room 2
Person2 --> Room 3
...
PersonN --> Room N+1
etc.
This is also a one-to-one pairing between occupants and rooms. For each person I can say what room he or she is in, and for each room, I can say who is in the room.

Now for the sake of simplicity in the next step, let's renumber the people: Person1, Person2, etc.

At this time a bus with an infinite number of people arrives. What to do? As already mentioned, the manager asks each room occupant to move to the room whose number is two times his/her current room number. This frees up rooms 1, 3, 5, ..., all the odd-numbered rooms. The people from the bus file in and each person is assigned one of the odd-numbered rooms.

The pairing is now (BP - person from bus, P = person already present):
BP1 --> Room 1
P1 --> Room 2
BP2 --> Room 3
P2 --> Room 4
...
BP_N --> Room 2N-1
P_N --> Room 2N
etc.
This too is a one-to-one pairing. For each person I can say what room he or she is in, and for each room, I can say who is in the room.
P: 13
 Quote by Mark44 That's exactly what happens. Each time there is a pairing between the occupants and the hotel rooms.
Thanks Mark!
Very good explanation.

Now, I understand that the result of infinity minus any number is still infinity, but I still cannot wrap my mind around subtracting from paired sets.
Although infinity as number has infinite value, when paired it's bound to the other side as equal value (size).
Let's take Galileo's argument that S = {1,4,9,16,25,...} is the same size as N = {1,2,3,4,5,...} because there is a one-to-one correspondence: 1 ⇔ 1, 2 ⇔ 4, 3 ⇔ 9, 4 ⇔ 16, 5 ⇔ 25, ...
If we subtract 1,2,3,4,5 from N it will still be infinite on its own, but it will be with 5 less than S (smaller size), which will brake the pairing.
I wonder how to deal with that
Mentor
P: 21,256
 Quote by Sissoev Thanks Mark! Very good explanation. Now, I understand that the result of infinity minus any number is still infinity, but I still cannot wrap my mind around subtracting from paired sets. Although infinity as number has infinite value, when paired it's bound to the other side as equal value (size). Let's take Galileo's argument that S = {1,4,9,16,25,...} is the same size as N = {1,2,3,4,5,...} because there is a one-to-one correspondence: 1 ⇔ 1, 2 ⇔ 4, 3 ⇔ 9, 4 ⇔ 16, 5 ⇔ 25, ... If we subtract 1,2,3,4,5 from N it will still be infinite on its own, but it will be with 5 less than S (smaller size), which will brake the pairing.
No, it won't be a smaller set with 5 numbers removed. You're still thinking along the lines of finite sets.

It doesn't matter if we break a pairing when we remove or add some elements to one of the sets. All that we need to do is to establish a new pairing that is also one-to-one. Here's a pairing of the set {6, 7, 8, 9, ...} with {1, 4, 9, 16, ...}.
6 ⇔ 1
7 ⇔ 4
8 ⇔ 9
9 ⇔ 16
....
n ⇔ (n - 5)2
...
 Quote by Sissoev I wonder how to deal with that
P: 13
 Quote by Mark44 No, it won't be a smaller set with 5 numbers removed. You're still thinking along the lines of finite sets.
I see paradoxes as broken logic which occurs by mixing two different properties.
Zeno's paradox is a result from mixing time with distance, not taking in account the speed.
Hilbert's paradox is a result from mixing pairs with pair parts.
Left and right as parts of one pair are bound together and you should not be able to create new pair by adding only left or only right part to it.
An infinite paired set is infinite in number but its parts are limited to each other (each left shoe has as a pair a right shoe, and there are no singles available, otherwise the set wouldn't be called paired set)
Regardless whether the number is limited or infinite, the pairs in the row are created and cannot be increased by adding only to one side of the pair.
One would argue that we don't increase infinity, because its value cannot be increased or decreased by adding or subtracting, but by not taking in account the properties of the paired infinite set we have a moment between the pairings when one of the sets is with number greater than the other.
Again, "with number greater than the other" is not correct use for infinity, so we rather call it unpaired number (one of the sides contains unpaired number of shoes). That unpaired number implies limit to both sides when we look at them as pair parts.
That's why I say that a pairing should occur only once if we don't want to create a paradox.
In Hilbert's case we have complete infinite number of pairs (no singles available) and any adding to one of the sides will break that completeness.
P: 13
 An infinite paired set is infinite in number but its parts are limited to each other (each left shoe has as a pair a right shoe, and there are no singles available, otherwise the set wouldn't be called paired set) Regardless whether the number is limited or infinite, the pairs in the row are created and cannot be increased by adding only to one side of the pair.
Let me add to the above hoping that it'll show my point in more understandable way.
A pair has a property which should not be confused with the properties of its parts.
My opinion is that we should think of pair's property as of the result from 1 liter of water paired with 1 kg of cement.
So, if we pair 1 liter of water with 1 kg of cement we'll get +/- 2 kg of concrete.
Now, think of the rooms as of buckets of water, and the guests as of 1 kg of cement.
You can add as much as you want to each infinite set before the pairing, but nothing you can do after you put the cement in to the water.
Second pairing is impossible.
P: 827
 Quote by Sissoev Left and right as parts of one pair are bound together and you should not be able to create new pair by adding only left or only right part to it.
With all due respect, we don't care about what you think "should" or "should not" be doable in mathematics.* It is doable in standard mathematics, and we have demonstrated this many times in this thread.

You can argue this is a veridical paradox, in that you have to reject your intuition of finite sets when dealing infinite sets. That's the entire point of Hilbert when he created this: to teach his students intuition of countable infinity. But it is certainly not a falsidical paradox.

Edit: * this probably is a bit too harsh. The problem is that what you are claiming is whether something is an allowed operation or not. The problem is that the definition of "infinite set" is what allows us to do this, so your argument distills down to rejecting the axiom of infinity. And that is a philosophy question, not a mathematics one.
P: 13
 Quote by pwsnafu With all due respect, we don't care about what you think "should" or "should not" be doable in mathematics. It is doable in standard mathematics, and we have demonstrated this many times in this thread. You can argue this is a veridical paradox, in that you have to reject your intuition of finite sets when dealing infinite sets. That's the entire point of Hilbert when he created this: to teach his students intuition of countable infinity. But it is certainly not a falsidical paradox.
I'm not trying to change mathematics, neither am I challenging your knowledge to prove you wrong.
Don't get frustrated if you cannot answer some of my questions and logical points.
Most probably that is because my points are so lame that you cannot make sense of them
But still, if the authorities were never questioned, the science wouldn't go that far
P: 13
 Quote by pwsnafu Edit: * this probably is a bit too harsh. The problem is that what you are claiming is whether something is an allowed operation or not. The problem is that the definition of "infinite set" is what allows us to do this, so your argument distills down to rejecting the axiom of infinity. And that is a philosophy question, not a mathematics one.
Don't worry, no offence taken, pwsnafu
I didn't question the axiom of infinity, but the way we deal with it.
I think that the "concrete" example clarified my point.
Mentor
P: 21,256
 Quote by Sissoev Left and right as parts of one pair are bound together and you should not be able to create new pair by adding only left or only right part to it.
You really need to get past this notion that the pairing has to be unique. It is keeping you from understanding this concept.
 Quote by Sissoev An infinite paired set is infinite in number but its parts are limited to each other (each left shoe has as a pair a right shoe, and there are no singles available, otherwise the set wouldn't be called paired set)
This "paired set" idea that you are fixated on is no help to you. You are missing the main idea -- a set is countably infinite if there is a one-to-one mapping between the elements of the set and the set of positive integers. Period.
 Quote by Sissoev Regardless whether the number is limited or infinite, the pairs in the row are created and cannot be increased by adding only to one side of the pair.
Baloney.
 Quote by Sissoev One would argue that we don't increase infinity, because its value cannot be increased or decreased by adding or subtracting, but by not taking in account the properties of the paired infinite set we have a moment between the pairings when one of the sets is with number greater than the other.
No, this isn't true. You not grasping the idea that an infinite set is fundamentally different from a finite set.
 Quote by Sissoev Again, "with number greater than the other" is not correct use for infinity, so we rather call it unpaired number (one of the sides contains unpaired number of shoes). That unpaired number implies limit to both sides when we look at them as pair parts.
Unpaired numbers would be relevant if we were dealing with finite sets, but once we start talking about infinite sets, it doesn't matter in the slightest that there are some numbers that aren't included. What does matter is that we can establish a one-to-one pairing with a countably infinite set (such as the positive integers). In every example I gave, I showed you the pairing.
 Quote by Sissoev That's why I say that a pairing should occur only once if we don't want to create a paradox. In Hilbert's case we have complete infinite number of pairs (no singles available) and any adding to one of the sides will break that completeness. Hence the paradox.
Like pwsnafu said, "we don't care about what you think "should" or "should not" be doable in mathematics."
 Quote by Sissoev I'm not trying to change mathematics, neither am I challenging your knowledge to prove you wrong. Don't get frustrated if you cannot answer some of my questions and logical points.
We have answered every one of your questions and have refuted all of your logical points. What is frustrating, is that you are unable to let go of your incorrect ideas about infinite sets, despite being shown that they are faulty.
 Quote by Sissoev Most probably that is because my points are so lame that you cannot make sense of them