Hilbert's paradox of the Grand Hotel

The German mathematician David Hilbert (1862 - 1943) presented the following paradox about infinity.

The paradox of the Grand Hotel

In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1.

It is even possible to make place for an infinite (countable) number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests.

If an infinite (countable) number of coaches arrive, each with an infinite (countable) number of passengers, you can even deal with that: first empty the odd numbered rooms as above, then put the first coach's load in rooms 3ⁿ for n = 1, 2, 3, ..., the second coach's load in rooms 5ⁿ for n = 1, 2, ... and so on; for coach number i we use the rooms pⁿ where p is the i+1-st prime number. You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers (if the seats are not numbered, number them). Regard the hotel as coach #0. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The guest in room number 1729 moves to room 1070209. The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.

This state of affairs is not really paradoxical but just profoundly counterintuitive. It is difficult to come to grips with infinite 'collections of things', as their properties are quite different from the properties of ordinary 'collections of things'. In an ordinary hotel, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel the 'number' of odd-numbered rooms is as 'large' as the total 'number' of rooms. In mathematical terms, this would be expressed as follows: the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets this cardinality is called <math>\aleph_0</math>.

An even stranger story regarding this hotel shows that mathematical induction only works in one direction. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room.