Hilbert's Hotel, also known as Hilbert's Infinite Hotel, is a thought experiment proposed by the German mathematician David Hilbert. It is used to illustrate some counterintuitive concepts related to infinity, particularly in the field of set theory.
The Problem:
Imagine a hotel with an infinite number of rooms, labeled with positive integers 1, 2, 3, and so on, stretching infinitely in both directions along a single corridor. Each room is already occupied by a guest.
Now, suppose a new guest arrives at the hotel and wants to check-in. Surprisingly, the hotel manager claims they can accommodate the new guest, even though every room is taken.
Solution:
The hotel manager asks each current guest to move to the room number with a label that is one more than their current room. For example, the guest in Room 1 moves to Room 2, the guest in Room 2 moves to Room 3, and so on.
This process continues indefinitely, and as a result, every guest moves to the room number that is one unit higher than their current room. Now, there is an empty Room 1 available for the new guest to check-in.
But that's not all! The hotel manager can handle an infinite number of new guests, one for each room number. For instance, if an infinite group of new guests shows up, the manager can simply ask the current guests to move to Room number n+1, where n is their original room number, and the new guests can then occupy the vacated Room 1, Room 2, Room 3, and so on.
The counterintuitive part of this thought experiment lies in the idea that even though the hotel was already fully occupied, the hotel manager was still able to accommodate an infinite number of new guests without turning anyone away.
This concept has implications in set theory and various areas of mathematics, where dealing with infinite sets leads to fascinating results and paradoxes. Hilbert's Hotel problem helps to illustrate the non-intuitive nature of infinite sets and some of the unusual properties associated with them.
