These are among the marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites.
The Hilbert Hotel, Count von Count, and Cookie Monster
Below are two stories. The first is well known within the mathematics community; the second is presented with two classic characters from children's television. In both cases, the stories highlight that our intuition about counting elements and determining the size of finite sets does not always transfer well to the setting of infinite sets.
The Hilbert Hotel. David Hilbert was a German mathematician who contributed transformative results in a number of areas of mathematics and provided valuable vision and leadership to the mathematics community. Because he introduced the following example in the 1920s, it is often described as the Hilbert Hotel. The Hilbert Hotel has many rooms. In fact, each room is numbered with a natural number, and there are just as many rooms as there are natural numbers. Thus there is a room #1, a room #2, …, a room #n, … and so on.
The Hilbert Hotel has acquired a nice reputation among tour group bus drivers, since it always seems to be able to accommodate very large tour groups. For instance, a bus drove up one evening that held infinitely many passengers: every passenger's shirt was labelled with a natural number, and every natural number appeared on just one shirt. This did not pose a problem for Hilbert, of course, because he made a single, precise announcement over his intercom system that told everyone where to go: “Go to the room whose number corresponds to the number on your shirt.” So passenger #1 went to room #1, passenger #2 went to room #2, and so on.
Later that evening, a single person drove up in a car and asked for a room. Hilbert couldn't send the person to the first vacant room, since there were no vacancies. Instead, Hilbert simply made the following announcement over the intercom to everyone who already had a room: “I apologize for the inconvenience, but in order for us to accommodate a new guest, we need you to change rooms. If you are in room #k, please move to room #(k + 1) for the night.