Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that 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 [TM110 or search for “thatsmaths” at irishtimes.com].
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”.
Systematic mathematical study of infinite sets began around 1875 when Georg Cantor developed a theory of transfinite numbers. He reasoned that the method of comparing the sizes of finite sets could be carried over to infinite ones. If two finite sets, for example the cards in a deck and the weeks in a year, can be matched up one to one, each card corresponding to a particular week and each week to a specific card, they must have the same number of elements. Mathematicians call this correspondence a bijection.
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! Cantor exploited this counter-intuitive idea, using it to define an infinite set: a set is infinite if there is a bijection from the set to part of itself. When part of an infinite set is removed, what remains can have the same “size” as the original set. Galileo’s paradox exemplifies this idea.
Hilbert’s Grand Hotel.
Leading 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.
* * *
The collection That’s Maths, with 100 articles, has just been published by Gill Books. Available from