What is a set?
In mathematics, a set is a collection of objects, such as the set F={apple, orange, pear} or the set of prime numbers P={2, 3, 5, 7, 11, 13, …}. An important property of a set is that there always needs to be a clear “yes/no” answer to the question “is the element x a member of the set A?” For example, “orange” is an element of the set F but “banana” is not, and 17 is an element of the set P but 18 is not. Does any collection of objects qualify as a set? Is there a universal set, meaning a set of all sets? In 1901, the philosopher Bertrand Russell (pictured above) showed that the answer to these questions is no: there is no “set of all sets”.
The proof that there is no set of all sets is a famous argument, which goes as follows. The basic idea is that if S is a set, then we should be able to create a subset T by selecting the elements of S that satisfy a certain property. For example, if we accept that the set of natural numbers N={1, 2, 3, 4, 5, …} is a set, then we can select those elements of N that are prime. Doing this forms another set, the set of prime numbers P.
Let’s pretend for the moment that the “set of all sets” actually exists, and let’s denote it by U. We should then be able to form a new set, B, by selecting those elements of U that are not elements of themselves. The elements of the set B, shown in the formula above, are precisely the sets that are not elements of themselves. The key question is: is B an element of itself, or not? If B is an element of itself, then it fails the criterion for membership in B, so B is not an element of itself. But if B is not an element of itself, then it satisfies the criterion for membership in B, so B is an element of itself. This is Russell’s paradox, which forces us to conclude that there is no “set of all sets” in the first place.
Russell’s paradox is similar in spirit to the liar paradox, which was known to the ancient Greeks. The statement “I am lying” cannot be true, because that would make it false, and it cannot be false, because that would make it true. The same applies to the statement “this statement is false”.
The “moral” of Russell’s paradox is that we have to be careful about what we choose to accept as a set, or we run into inconsistencies very quickly. The usual modern way to avoid these problems is to use Zermelo–Fraenkel set theory (“ZF” for short) which was developed by Ernst Zermelo and Abraham Fraenkel between 1908 and 1922. The theory provides a list of axioms governing what sets can and cannot do. For example, the axioms show that the empty set is a set, that the natural numbers are a set, and that sets are never allowed to be elements of themselves.
An axiom that is often included alongside the ZF axioms is the Axiom of Choice. The picture above shows a collection of infinitely many non-overlapping sets, in this case, jars of marbles. The Axiom of Choice states that there exists a set that contains one marble from each jar, illustrated in the picture by the set of non-grey marbles. Although it seems “obviously true” that this can be done, it is not provable or disprovable from the axioms of ZF, so mathematicians are free to treat the Axiom of Choice as either true or false. The Axiom of Choice is very useful for proving things, so mathematicians usually treat it as true.
One drawback of the Axiom of Choice is that it has some bizarre consequences, such as the Banach–Tarski paradox, illustrated above. The paradox is that it is possible, if we assume that the Axiom of Choice is true, to decompose a solid three-dimensional ball into a finite number of pieces, and reassemble them into two copies of the original ball. It is even possible to reassemble the pieces by moving them around and rotating them in such a way that they never collide. The catch is that the pieces themselves are not reasonable “chunks” of the original sphere with measurable volumes, but rather dust-like infinite scatterings of points. The standard joke that mathematicians tell about the paradox is the following.
Question: What’s an anagram of “Banach–Tarski”?
Answer: “Banach–Tarski Banach–Tarski.”
Picture credits and relevant links
Bertrand Russell (1872–1970) is mostly known as a mathematical philosopher, which makes it even more surprising that he was awarded the Nobel Prize for Literature in 1950. The photograph of Russell (taken in 1957) comes from the Wikipedia page about him.
The graphic of the set complement is by kismalac.
The graphic of the jars is by Fschwarzentruber and appears on the Wikipedia page on the Axiom of Choice.
The graphic of the Banach–Tarski paradox is by Benjamin D. Esham and appears on Wikipedia.
Wikipedia’s page on Zermelo–Fraenkel set theory discusses both the original version of the theory (known as ZF) and the version with the Axiom of Choice included (known as ZFC).
The other graphics are my own work.
Wikipedia also has a page about the liar paradox.