A well-known anecdote of the mathematician David Hilbert (1881-1966), who was a follower of Cantor (which was not at all mathematicians at that time!), Is an excellent example of the concept of "countable infinite".
A normal hotel can be fully occupied, so that a just arrived guest finds all rooms occupied. This is a problem that does not apply to Hilbert's hotel. Hilbert's hotel has countless hotel rooms. Each of the hotel rooms is numbered, so there are rooms 1, 2, 3, 4, 5, 6, 7, 8, 9, ... .. and so on. Now, a new guest arrives at the front desk asking for a room, but all the rooms are occupied.
An exception in any other hotel, but not in this hotel.
All guests are asked to change the room and move on to a room, which means the current number of rooms, For example the number n +1. The rooms are very clean and the staff is very friendly.Now our new guest can move into the free room 1.
If 100 new guests arrive, this would be just as little a problem as we would leave the guests in the occupied rooms instead of a number around 100, and the problem would be solved.
But now the scenario that the usual imaginative ability of infinity breaks:
An extraordinary train with infinitely many passengers stops in front of the hotel. Infinitely many guests looking for a room in Hilbert's fully booked hotel. Again, this is no problem. The new rooms number of guests who have already been consulted can be determined by taking the current room number n x 2, or by adding them together. For the room numbers 1, the new guests receive the new number 2, for room 2 the new number 4, for 3 the new number 6 ... to infinity, and look for their new room on the endless corridor while the now vacated rooms, All of which have odd room numbers, can be occupied by the countless countless new guests.
