Hilbert’s Hotel: The Hotel with infinite rooms.Redooc Blog
Let’s keep moving through the beauty of infinity. This time we look at travels. The least organized among us would have surely experienced to arrive in one city and to find no available rooms in any hotel. Well, I want to tell you about a hotel where this problem doesn’t exist. It’s the Hilbert Hotel, from the name of the great German mathematician who used this example in the form of paradox, to represent the great difference existing between finite and infinite sets.
Hilbert Hotel is a hotel with infinite rooms, all occupied. Suppose that a new guest arrives: how do we accomodate him at the hotel? Our friend Hilbert tells us that all we have to do, is just to move each guest from his/ her room to the following one (e.g. the guest who is in room number 10 is moved to room number 11): with an infinite number of rooms, it is in fact always possible to go from n to n+1, and thus to settle in the newcomer in the room number 1 which is now available.
Now, let’s suppose that an infinite number of new people arrives at the hotel: is it possible to put up all of them at Hilbert Hotel ? Of course, we can. Hilbert suggests to move each guest from room number n to room number 2n (only even rooms will thus be occupied) and place the newcomers in the odd-numbered rooms, now all available… this is possible because also odd and even numbers are both infinite… that’s wonderful!
Finally, we come to the most complex, but interesting, example. In the city there are infinite hotels all with infinite rooms, all occupied. Let’s imagine that all but Hilbert Hotel must close and that all their guests are knocking on the doors of Hotel Hilbert. Is it possible to accomodate all these people at Hilbert Hotel? Of course! And Hilbert shows us how to do it! Let’s mark each person with a pair of numbers (m, n) where m represents his/her previous hotel (m=1 for Hilbert Hotel guests) and n instead is his/her room number at previous hotel. We thus have a matrix with infinite rows and columns. By zigzagging through this matrix, it is possible to accomodate each person in a Hotel Hilbert room: for example (1,1) in room 1, (1.2) in room 2, (2.1) in room 3, (3, 1) in room 4, (2.2) in room 5, (1.3) in room 6, etc… don’t you think this is really wonderful? I really do!
If you have started thinking Hilbert Hotel is the panacea for all travelers’ problems, I’m very sorry to have to immediately stop your enthusiasm. Hilbert Hotel can solve all problems when we deal with a countable infinity: it can’t instead solve them when we deal with an higher order infinity (e.g. real numbers), as you can read in this funny article.
What we have considered in these last two posts are just two examples of properties of infinity. I just want now to wish you a good journey through the world of infinity: bring with you just a lot of curiosity and of elasticity and you will remain fascinated!