Discovered in 1901 by Bertrand Russell, this paradox is an extremely intriguing contradiction which concerns the constraints of naïve set theory, and what we can define as a set. It can be represented informally as follows:
A set can be defined as a collection of distinct objects. Hence, there is nothing to say that a set which includes itself cannot exist. Thus, we can split all sets into two categories, normal and abnormal, where a normal set is one which does not include itself as an element, and an abnormal set is one which does.
Let C be the set of all normal sets. Suppose C is normal. Therefore, C must be included in the set of all normal sets, since it is normal itself. Hence C must be included within C. Therefore C must be abnormal.
Suppose C is abnormal. Therefore, since C is abnormal, it cannot be included in the set of all normal sets. Therefore, C cannot include itself. Therefore C is normal.
Therefore, if we assume C to be normal, it must be abnormal, and vice versa. So what is C? A set cannot be neither normal nor abnormal, since that means that the set neither contains itself, nor does not contain itself. In a similar fashion one can conclude that a set cannot be both normal and abnormal.
This paradox in fact lead to the change from naïve set theory, to the set theory we use today, known as Zermelo-Fraenkel set theory, or ZFC. In a nutshell, ZFC does not assume that for any given property, there exists a set of all sets which satisfies the property. And hence, we cannot assume such a set C exists.