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.
If A is true, then B 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, ever guest n is in room n+1.
X and all other guests are in a room.
If B is true, then C is true.
C.
It does not matter if guests moving in a chain or X takes all the moves by himself.
Hilberts Hotel is full. Guest X arrives. X moves into room 1 and guest 1 moves out of room 1 simultaneously. Then they switch back, guests 1 moves back into room 1 and X moves out of room 1 simultaneously. X moves into room 2 and guest 2 moves out of room 2 simultaneously. Then they switch back, guests 2 moves back into room 2 and X moves out of room 1 simultaneously… After an infinite amount of moves taken, every guest n is in room n. X is in a room, too.
X and all other guests are in a room.
If C is true, then D is true.
D.
Example C with X and no other guests
Hilberts Hotel is empty. Guest X arrives. X moves into room 1. X moves into room 2… After an infinite amount of moves taken, X is in a room and all rooms n are empty.
X is in a room. All rooms n are empty in Hilberts Hotel.
Edit:
So some of you point out, that C is false.
It seems like to matter, if I can point out the moving guest. Am I getting it right?
Let’s try again.
If A is true, then E is true.
E.
E. is like B. but with a red hut.
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 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, ever guest n is in room n+1.
X with his red hat is in a room and all other guests are in a room.
Conclusion: If A is true, then E is true.
F.
F. is like E. but the red hut is moving.
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 and guest 1 moves out of room 1 simultaneously. X gives guest 1 the red hat. Guest 1 moves into room 2 and guest 2 to moves out of room 2 simultaneously. Guest 1 gives guest 2 the red hat… After an infinite amount of moves taken, ever guest n is in room n+1.
X is in a room and all other guests are in a room. There is no room with a guest, which has the red hat.
Is then the Conclusion: If E. is true, F. don’t have to be true?
Every guest in E. is moving exactly in the same room as in F. But in F. there is always a guest with a red hat moving, who can never find a room?
Then it seems like, if you can track the last guest somehow, then Hilberts Hotel is not working but if you don’t track the last guy it is working?
There is an essential difference between $B$ and $C$.
In $B$, every guest gets a precise order: "vacate your room and move to room $n$", with $n$ a specific natural number. No one has any problem.
In $C$, the guest $X$ is never done. Whatever room they are in, they can't stay there. If there was a room they were in, it would have a room number, and that room number would be a natural number (since all room numbers are) but every single room with a natural number is already taken by a guest.
This shows that transfinite numbers are not something we should use our intuition for; rather we should rely on formal definitions and formal logic derivations.
The difference between $B$ and $C$ shows that $1 + \omega = \omega \neq \omega + 1$.