Is there no end to our fascination with infinity?
That’s Maths: the concept of infinity leads to some interesting paradoxes
Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realise there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number – the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results.
In ancient Greek times, Zeno of Elea devised several paradoxes involving infinity. They are still debated today. Essentially, he argued that one cannot travel from A to B: to do so, one must first travel half the distance, then half of the remaining half, then half the remainder, and so on forever. He concluded that motion is logically impossible. Today we understand that an infinite sum may converge to a finite limit, but this understanding took thousands of years to emerge. Bertrand Russell described Zeno’s paradoxes as “immeasurably subtle and profound”.
Transfinite numbersinfinite sets
Cantor used this approach to compare infinite sets: if there is a bijection between them, two such sets are considered to be the same “size” (technically they have the same cardinality). Cantor built up an entire theory of infinity based on this idea. He showed that for any infinite set there exists another with a higher order of infinity, and he defined the system of transfinite cardinals, an infinite hierarchy of infinities.
But Cantor’s method leads to paradoxical results. This had been realised much earlier by Galileo, who saw that the counting numbers 1, 2, 3 . . . and the even numbers 2, 4, 6 . . . could be matched one to one: we simply map each number n to its double, 2n. Thus, there are as many even numbers as counting numbers!
The German mathematician David Hilbert constructed an amusing metaphor to illustrate the enigmatic properties of infinity. He imagined a hotel with an infinite number of rooms. Even if all rooms are occupied, there is always space for an additional guest. Simply, and simultaneously, move guest 1 to room 2, guest 2 to room 3 and so on, thereby vacating the first room. Indeed, an infinite number of new arrivals can be accommodated: for all rooms n, move the guest in room n to room 2n and, magically, all the odd-numbered rooms become vacant. Surely, Galileo would have smiled at Hilbert’s Grand Hotel.