A.
Hilberts Hotel
Hilbert’s Hotel is full. Guest X arrives. Every guest n moves intro room n+1 synchronously. Guest X moves into room 1.
X and all other guests are in a room.
In B should happen the same than in A (if a is true).
B.
It does not matter if an infinite amount of moves taken simultaneously or an infinite amount of moves taken one after another.
Hilberts Hotel is full. Guest X arrives. X moves intro room 1 and guest 1 moves out of room 1 simultaneously. Guest 1 moves into room 2 and guest 2 to moves out of room 2 simultaneously… After an infinite amount of moves taken*, every guest n is in room n+1.
In A. every guest n moved in the exactly the same room than in B.
A. and B. both don’t have a last guest.
X and all other guests are in a room.
In C. should happen the same than in B..
C.
C. is like B. but with a red hat.
Hilberts Hotel is full. Guest X arrives. Guest X has a red hat and all guests in the hotel have no hat. X moves intro room 1, gives guest 1 the red hat and guest 1 moves out of room 1 simultaneously. Guest 1 moves into room 2, gives guest 2 the red hat and guest 2 to moves out of room 2 simultaneously… After an infinite amount of moves taken*, ever guest n is in room n+1.
All guest are in a room.
Conclusion:
X and all other guests are in a room.
But there is no room with a guest, which has the red hat. What happened with the red hat?
*After an infinite amount of moves taken: Just an example how guest could move. Guest 1 moves in 1 minute, guest 2 in 1/2 minute, guest 3 in 1/4 minute, guest 4 in 1/8 minute and so on. After two minutes every guest n moved from room n to n+1. There are an infinite amount of smaller time periods needed to reach the 2 minutes (in this example). Each of the infinite amount of smaller timestemps is needed to reach the 2-minute mark. So in real life, we reach 2 minutes.
OK, that sounds weird, the guest needs more time?
To be sure, every single guest has enough time, you can give every guest an extra hour on top. So guest 1 moves in 1 hour and 1 minute, guest 2 in 1 hour and 1/2 minute… If we can take an infinite amount of timestemps, then it should also be possible to add to each timestemp an extra time.
Hilberts Hotel
Hilbert’s Hotel is full. Guest X arrives. Every guest n moves intro room n+1 synchronously. Guest X moves into room 1.
X and all other guests are in a room.
In B should happen the same than in A (if a is true).
B.
It does not matter if an infinite amount of moves taken simultaneously or an infinite amount of moves taken one after another.
Hilberts Hotel is full. Guest X arrives. X moves intro room 1 and guest 1 moves out of room 1 simultaneously. Guest 1 moves into room 2 and guest 2 to moves out of room 2 simultaneously… After an infinite amount of moves taken*, every guest n is in room n+1.
In A. every guest n moved in the exactly the same room than in B.
A. and B. both don’t have a last guest.
X and all other guests are in a room.
In C. should happen the same than in B..
C.
C. is like B. but with a red hat.
Hilberts Hotel is full. Guest X arrives. Guest X has a red hat and all guests in the hotel have no hat. X moves intro room 1, gives guest 1 the red hat and guest 1 moves out of room 1 simultaneously. Guest 1 moves into room 2, gives guest 2 the red hat and guest 2 to moves out of room 2 simultaneously… After an infinite amount of moves taken*, ever guest n is in room n+1.
All guest are in a room.
Conclusion:
X and all other guests are in a room.
But there is no room with a guest, which has the red hat. What happened with the red hat?
*After an infinite amount of moves taken: Just an example how guest could move. Guest 1 moves in 1 minute, guest 2 in 1/2 minute, guest 3 in 1/4 minute, guest 4 in 1/8 minute and so on. After two minutes every guest n moved from room n to n+1. There are an infinite amount of smaller time periods needed to reach the 2 minutes (in this example). Each of the infinite amount of smaller timestemps is needed to reach the 2-minute mark. So in real life, we reach 2 minutes.
OK, that sounds weird, the guest needs more time?
To be sure, every single guest has enough time, you can give every guest an extra hour on top. So guest 1 moves in 1 hour and 1 minute, guest 2 in 1 hour and 1/2 minute… If we can take an infinite amount of timestemps, then it should also be possible to add to each timestemp an extra time.
