Hilbert imagines a hypothetical hotel with rooms numbered 1, 2, 3, and so on. The hotel is full and new guest arrives wanting a room. I can not move any guest. So i say to new guest wait 1 sec a new room must become available as Infinite guests one must leaving at any time.
is this good answer?
If they all step out and into the new rooms at the same time, it’s possible.
It’s not a mathematical paradox, only against intuition.
But how would they all know when to step out and into new room?
They need some kind of a signal (bell, phone call, flashing light, speaker instructing them to move, …)
Fastest of these communication is light which has maximum speed of 300,000 km/s
The light will reach first room instantly but since number of rooms is infinite, the light need infinite time to reach that last room
It’s an illustration of infinities, not an actual hotel.