# Hilbert's Hotel: new Guest arrives (Infinite number of Guests)

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## Summary:

Hilberts Hotel, proof me that there is room 1 empty.

## Main Question or Discussion Point

Hilberts Hotel has infinity numbers of rooms and in every room is exactly one guest.

On Wikipedia Hilberts Hotel gets described as well:
Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests.

1) How can you proof, that you can move every guest from his current room to the next room (n+1).

2) Remeber the Text says: "After this, room 1 is empty". But then you have to proof first, that "before" really every guest moved into the next room.

My opinion: There is no proof that a single new guest can be moved in room 1. It can not be proofed, that every guest can move into the next room. As long you can't proof that ever guest moved from his current room to room n+1 you can not say room 1 is empty.

What do you think, do you think i am completely wrong?

mfb
Mentor
1) How can you proof, that you can move every guest from his current room to the next room (n+1).
No room should be occupied by more than one guest, and every guest should have a room. This is satisfied for every room and guest. The question how to tell an infinite number of guests that they should change their room is not considered here.
2) Remeber the Text says: "After this, room 1 is empty". But then you have to proof first, that "before" really every guest moved into the next room.
Have to prove what?

QuantumQuest
fresh_42
Mentor
Not completely, just a bit. You are right, it cannot be proven. It is an axiom (look up Peano). In this case the second: every natural number has a successor. This allows us to move the guests into the room labelled as successor.

Last edited:
Klystron, QuantumQuest, Jarvis323 and 1 other person
PeroK
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Summary:: Hilberts Hotel, proof me that there is room 1 empty.

Hilberts Hotel has infinity numbers of rooms and in every room is exactly one guest.

On Wikipedia Hilberts Hotel gets described as well:
Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests.

1) How can you proof, that you can move every guest from his current room to the next room (n+1).

2) Remeber the Text says: "After this, room 1 is empty". But then you have to proof first, that "before" really every guest moved into the next room.

My opinion: There is no proof that a single new guest can be moved in room 1. It can not be proofed, that every guest can move into the next room. As long you can't proof that ever guest moved from his current room to room n+1 you can not say room 1 is empty.

What do you think, do you think i am completely wrong?
Hilbert's Hotel is not physically possible. But, it's mathematically possible.

pinball1970
etotheipi
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Hilbert's Hotel is not physically possible. But, it's mathematically possible.
I mean how would they cater for the breakfast buffet? That's a lot of orange juice...

Klystron, phinds, Math_QED and 1 other person
Mark44
Mentor
@dakiprae, if you have trouble understanding how the hotel could accommodate a single new guest, you're really going to have trouble when that bus with an infinite number of passengers arrives.

In that case, any current guest in room N will be asked to move to room 2N, thereby freeing up all of the odd-numbered rooms.
1) How can you proof, that you can move every guest from his current room to the next room (n+1).
The guests are assumed to be very compliant. The guest in room 1 moved to room 2, the guest in room 2 moves to room 3, and so on, ad infinitum.
2) Remeber the Text says: "After this, room 1 is empty". But then you have to proof first, that "before" really every guest moved into the next room.
No, "before," the guests haven't moved yet. "After," the current guests are in a new room next to (higher room number than) their old room.
My opinion: There is no proof that a single new guest can be moved in room 1. It can not be proofed, that every guest can move into the next room. As long you can't proof that ever guest moved from his current room to room n+1 you can not say room 1 is empty.

What do you think, do you think i am completely wrong?
Yes.

pinball1970 and Math_QED
Hilbert's Hotel is not physically possible. But, it's mathematically possible.
I can think about it, so it may be physically possible. Maybe there is a god who can create a universe with infinity space and this hotel. Or maybe there is a universe, which is already infinity years old and has infinity space and has this hotel.

PeroK
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I can think about it, so it may be physically possible. Maybe there is a god who can create a universe with infinity space and this hotel. Or maybe there is a universe, which is already infinity years old and has infinity space and has this hotel.
I doubt that. As @etotheipi pointed out, there would be a problem with breakfast. Not to mention the infinite laundry bill.

pinball1970, etotheipi and fresh_42
fresh_42
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I doubt that. As @etotheipi pointed out, there would be a problem with breakfast. Not to mention the infinite laundry bill.
Interesting physics question here: The communication of the order to move is of finite speed. Whereas this doesn't seem to be a problem for the first billion rooms, will it work out at infinity?

etotheipi
PeroK
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Interesting physics question here: The communication of the order to move is of finite speed. Whereas this doesn't seem to be a problem for the first billion rooms, will it work out at infinity?
The other option is to have every room half the size of the previous one. That has its practical problem too!

etotheipi
@dakiprae, if you have trouble understanding how the hotel could accommodate a single new guest, you're really going to have trouble when that bus with an infinite number of passengers arrives.

In that case, any current guest in room N will be asked to move to room 2N, thereby freeing up all of the odd-numbered rooms.
The guests are assumed to be very compliant. The guest in room 1 moved to room 2, the guest in room 2 moves to room 3, and so on, ad infinitum.
No, "before," the guests haven't moved yet. "After," the current guests are in a new room next to (higher room number than) their old room.
Yes.
I have no trouble to think about an infinite number of buses, no problem. But my point is different, I asked for a proof, that every guest will find a room, when only 1 guest arrives.

1) Yes I can ad infinitum, but I did not see the proof, that all guest find a hotel room.

2) Same thing, how can you proof that every guest will find a room.

My point is, if there exists a hotel with infinite number of rooms and in every room is one guest, how can you proof that a new guest arriving will find a room in the hotel.

PeroK
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I have no trouble to think about an infinite number of buses, no problem. But my point is different, I asked for a proof, that every guest will find a room, when only 1 guest arrives.

1) Yes I can ad infinitum, but I did not see the proof, that all guest find a hotel room.

2) Same thing, how can you proof that every guest will find a room.

My point is, if there exists a hotel with infinite number of rooms and in every room is one guest, how can you proof that a new guest arriving will find a room in the hotel.
Hilbert's hotel is about numbers; it has nothing to do with hotels, really.

You can think about an infinite number of buses, but you can't have them. Even the number of buses on Oxford Street is finite.

Hilbert's hotel is about numbers; it has nothing to do with hotels, really.

You can think about an infinite number of buses, but you can't have them. Even the number of buses on Oxford Street is finite.
Why I can not have an infinite number of buses? We do not know if we can have an infinite number of buses. Some people belief in an almighty god, where god can create an infinite number of buses. Also people not believe in god can think that an infinite number of buses could exists, but that's another topic.

weirdoguy
fresh_42
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Hilbert's hotel is about numbers; it has nothing to do with hotels, really.

You can think about an infinite number of buses, but you can't have them. Even the number of buses on Oxford Street is finite.
I answered this in post #3.

Not completely, just a bit. You are right, it cannot be proven. It is an axiom (look up Peano). In this case the second: every natural number has a successor. This allows us to move the guests in to the room labelled as successor.
What is the successor for infinity?

etotheipi
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What is the successor for infinity?
Infinity is not a number, it doesn't make sense to ask what its successor is. But for any natural number $n$ you can name, I can always do you one better with $n+1$.

QuantumQuest
fresh_42
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What is the successor for infinity?
There is no infinity involved. Only a for all quantifier. Every single room is labelled with a natural (and finite) number. So every room has a successor.

PeroK
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What is the successor for infinity?
That's God's room!

pinball1970
There is no infinity involved. Only a for all quantifier. Every single room is labelled with a natural (and finite) number. So every room has a successor.
So infinite numbers has an succesor, if every natural number has a succesor right? But we said in every succesor is already a guest. So there will be no free room and there is no proof that all guest will find a room, if one more guest arrives?

fresh_42
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All rooms are occupied.
All guests step out at the same time.
All guests step in front of the next door.
All guest enter the new room.

Where do you see an infinite number?

You may equally require that only room number one is occupied and the arriving guests demands room number one. So the concierge uses the successor property to move the first guest in room number two. This is the same principle, just less funny.

Klystron and etotheipi
All rooms are occupied.
All guests step out at the same time.
All guests step in front of the next door.
All guest enter the new room.

Where do you see an infinite number?

You may equally require that only room number one is occupied and the arriving guests demands room number one. So the concierge uses the successor property to move the first guest in room number two. This is the same principle, just less funny.

If there exists an infinite number of rooms, how can you proof that ever single guest can enter next door? If there is no last guest, how can we proof every guest entered the new room?

fresh_42
Mentor
Every single guest has a finite room number. And the procedure only takes care about this room and the next one. Infinity is hidden in our predicate logic, which allows us to quantify "for all numbers". This way the logic applies to any finite number as long as we do not pose any restrictions on this number.

If you alter the logic, not the example, then we have another discussion. But in our common logical system we have an all quantifier and are allowed to handle all instances at once.

Edit: It is the same as if I claimed: every second number is divisible by two. This is true, although nobody ever divided all even numbers.

Klystron and QuantumQuest
Mark44
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So infinite numbers has an succesor, if every natural number has a succesor right?
No, not right. It makes no sense to say "infinite numbers has a successor." Every finite number N has a successor N + 1.
But we said in every succesor is already a guest. So there will be no free room and there is no proof that all guest will find a room, if one more guest arrives?
At the time all guests switch rooms, all of the current guests step outside and move to the next higher numbered room. The free room is room #1.

Klystron and QuantumQuest
No, not right. It makes no sense to say "infinite numbers has a successor." Every finite number N has a successor N + 1.
At the time all guests switch rooms, all of the current guests step outside and move to the next higher numbered room. The free room is room #1.
I got you point, but for me, still there is no proof that all guest switch rooms. If there is no last guest, there is no proof that all guest moved.

fresh_42
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I got you point, but for me, still there is no proof that all guest switch rooms. If there is no last guest, there is no proof that all guest moved.
You can't handle an infinite number of anything by imagination of a physical procedure. If we say "all guests move into the room on their right", then this happens at once and costs no time. It "happens" within the for-all-quantifier. Again, there is no way to prove that there are infinitely many even numbers by dividing all numbers and check.

jbriggs444