The University of Gottingen, at the dawn of the 20th century, was a leading research center for math. David Hilbert, a mathematician, was a professor at the University of Gottingen. During the winter semester of 1924-25, he delivered lectures on the infinite in mathematics, physics, and astronomy. Springer-Verlag has published these and other Hilbert classes in book form. The IAS library has the book in both translation and original German. In one of his lectures, he gave an example that helped explain the difference between finite sets and infinite ones: in a hotel with a limited number of rooms, there’s no room for any new guests if they are all occupied. In a hotel with infinite rooms, this problem is solved: If all the rooms are complete and a guest arrives, the old guests can be moved one room to the left, thus leaving the first empty room for the new guest. The same argument allows us to accommodate an infinite number of newly arrived guests.
George Gamow, the author of the famous Alpher-Bethe-Gamow physical cosmology paper, a few decades after these lectures, was a postdoctoral student at the University of Gottingen and likely learned of Hilbert’s infinite hotel example there. In his 1947 book One, Two, Three…Infinity Facts and Speculations in Science popularized the idea.
Let’s go back to Hilbert’s Hotel. Let’s say the infinitely many hotel rooms have numbers 1, 2, 3, and 5 to make it neat. . . . The next night, all the rooms are occupied. A new guest comes in. We move the guests from rooms 1 and 2 to 3 and 4.
Say that twenty guests instead of one arrive. The trick that was used previously works as well. Move the guest from room 1 to room 21. Move the guest from room 2 to room 22. And in general, move the guest out of the room and into room and+20. This will make twenty rooms available for the new guests.
What if an infinite number of new passengers arrive on a bus that never ends? This argument can be modified to work for the situation. We will space out all the guests in the Hotel so they only occupy the other rooms. Mathematically, you can move the guest from room n into room 2 n so that the rooms with even numbers are all occupied. The (infinitely) many other rooms are now empty. This leaves every other room (infinitely numerous!) The bus will be arriving with many people. If sitting in seat number n, move to the n or odd-numbered room. This is the n– 1.
What happens if 99 infinite buses arrive at the Hotel? Then, move the guests of the Hotel to rooms 100,200,300, etc., and the bus passengers into rooms 1,101,201, etc. Then, do the same for the other buses. The Hotel will be fully occupied, and guests will have a room. If the bus passengers were also numbered 1, 2, 3, or 4, then the Hotel would be complete. . . The first 100 rooms in the Hotel would be filled with passenger 1, the following hundred rooms with passenger 2, etc.
Next, you will have to deal with an infinite number of buses (each with an endless number of passengers). First, get everyone to leave the Hotel, then organize them in a grid-like form on paper or in the parking area. The original guests (a.k.a. passengers of bus 0) should be arranged in a row, from left to right. The passengers of the first bus should form a second row directly below; the passengers of the double bus, a third row, etc. Line up the rows so that the first row of passengers from each bus is in a line, the second row to the right, etc. Let’s fill up the hotel rooms in order: 1, 2, 3, 4. . . We will always need more time to finish the grid with people in the first row. The same is true if we start with the column. Visualizing diagonal lines running from the bottom-left up to the top-right in the grid is essential. This diagonal line only hits one person at the top left (bus 0, passenger 1), so put them in room 1. Following the diagonal line: bus 1, passenger 1, and Bus 0, passenger 2: place these two people in rooms 2 and 3. Next, a diagonal line will hit three people. Please place them in the following empty rooms: 4, 5, 6. Continue this pattern, and we will assign rooms to everyone standing patiently in the parking lot.
Can we get deeper into the infinite than the infinite number of endless buses? Imagine a parking lot right next to Hilbert’s Hotel. The first floor is right outside the front door of the Hotel. There are the infinitely infinite buses we know and love. The garage, however, has endless feet, each with numerous endless buses. Can the Hilbert Hotel handle this additional layer of infinity? Yes, it is possible! Imagine using the method described above to create a single line of passengers in each parking garage floor and then telling each single line to get into an infinite bus. We have now reduced the problem to endless buses that can fit in the Hotel.
What if there was another layer of infinite? What if, for example, there were infinitely large parking garages, with each having infinitely high floors? Each floor would have infinitely numerous buses. And each bus would have infinitely great numbers of passengers? Four layers of infinity, and still, the answer is yes! The answer is still yes, even if you add four thousand layers to infinity. Does it ever stop? Hilbert’s Hotel never fails to welcome new guests. Hilbert’s Hotel can only accommodate a limited number of guests.
There is. Hilbert’s Hotel could not accommodate all of those people if there were infinitely many layers. What is going on? All the infinities up until this one are the same size. {That size is called 0 (aleph naught), the size of the set N = This size is called (aleph zero), the size of set N = 1,2,3,4. . .} Hilbert’s Hotel has how many rooms? Georg Cantor introduced 1874 the idea of comparing the sizes of infinities and demonstrated that there are infinities of various sizes. Several mathematicians, including Poincare, Kronecker, and Weyl, opposed Cantor. Some theologians also argued that Cantor’s ideas questioned the absolute infinity of God. Hilbert supported and defended Cantor.
It is no different than comparing the sizes of finite and infinite sets. To know if there are more chairs in a classroom, counting chairs and people is unnecessary. If you look around the room, you can see whether there are empty seats (more chairs than people) or if people are standing (more people than there are chairs). If every person is sitting in a seat and there aren’t any empty chairs, the sets of chairs and people are the same size. If every passenger in the bus gets a room at Hilbert’s Hotel and there is no empty room, then the total number of passengers will be infinitely the same size as Hilbert’s Hotel, 0. Cantor used the idea to show that R is a set of numbers more significant than N. His beautiful argument was known as Cantor’s diagonal. Cantor also conjectured – and tried to prove, but failed – the Continuum Hypothesis, that there is not an infinite set that is larger than N but smaller than R. Hilbert added this problem to his famous list of 23 issues that he presented to the 1900 International Congress of Mathematicians, in Paris This hypothesis is undecidable. It cannot be proven false, nor can it be proven true, as Cohen, in the 1960s, has shown.
Hilbert said in a famous quote about Cantor’s ideas on infinity and the new mathematics they created: “Nobody shall expel us out of the paradise Cantor has made.”
