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At its most basic, a set is a collection of things. Defining this construct such made it quite flexible – a necessary trait for the purposes that mathematicians had for it. Sets are the building blocks of all mathematical fields, and integral to the development of more complex mathematics.

A set can contain letters, numbers, symbols, words – really, any element that the user so desires. A = {chocolate, vanilla, strawberry} is as much a set as B = {3, 6, 9}. Sets can even have elements that are sets themselves, such as = {1, {2, 3}, {4, 5, 6}}.

Let’s say we have a set that contains all elements that are not in A. Thus, D does not have chocolatevanilla, or strawberry, but it does have all of the elements of B and C, as well as all the non-chocolate/vanilla/strawberry elements of all other sets in the universe. As D itself does not have the elements of A, it must be an element of itself, right? (D \in  D.) This concept is part of the groundbreaking idea now known as Russell’s paradox.

Bertrand Russell
Bertrand Russell (Photo by John Drysdale)

In the spring of 1901, Bertrand Russell (1872-1970), a British earl who wore many hats, including philosopher and mathematician, introduced the following puzzle: For the set that includes all sets that do not include themselves as elements, i.e.

F = {X : X is a set and\notin },

is F an element of F?

The problem is that if F was not an element of itself, by definition it belongs to the set F. However, if F is an element of itself, then it violates its own definition as the set of all sets that do not contain themselves, because then would contain itself as an element. F must contain itself and not contain itself at the same time.

For a more concrete example, consider the barber’s paradox, a derivation of Russell’s paradox. In a town, there lives a barber who only shaves those who do not shave themselves. The question arises: Does the barber shave himself? If the barber does not shave himself, then he must shave himself, because he shaves all those who do not shave themselves. If the barber does shave himself, then he cannot be the barber who shaves only those who do not shave themselves. Thus, he simultaneously does shave and does not shave himself.

barber
Comic by Geoff Draper

(This example is not as fine-tuned as the original paradox, as one might solve it by either concluding that such a barber a) does not exist, b) cannot grow a beard, or 3) is female. However, it still helps with basic understanding.)

To iron out the kinks of the naïve set theory of German mathematician Georg Cantor, mathematicians Ernst Zermelo and Abraham Fraenkel proposed the Zermelo-Fraenkel set theory – a collection of axioms that seek to avoid such mishaps as Russell’s paradox.

And set theory lives on.