Let us call a set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself. Clearly every set must be either normal or abnormal. The set of squares in the plane is normal. In contrast, the complementary set that contains everything which is not a square in the plane is itself not a square in the plane, and so it is one of its own members and is therefore abnormal. Now we consider the set of all normal sets, R, and try to determine whether R is normal or abnormal. If R were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox. Formal presentation The term "naive set theory" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a first-order language with a binary non-logical predicate , and that includes the Axiom of extensionality: and the axiom schema of unrestricted comprehension: for any formula with the variable as a free variable inside . Substitute for to get Then by existential instantiation (reusing the symbol ) and universal instantiation we have a contradiction. Therefore, this naive set theory is inconsistent. Philosophical implications Prior to Russell's paradox (and to other similar paradoxes discovered around the time, such as the Burali-Forti paradox), a common conception of the idea of set was the "extensional concept of set", as recounted by von Neumann and Morgenstern: In particular, there was no distinction between sets and proper classes as collections of objects. Additionally, the existence of each of the elements of a collection was seen as sufficient for the existence of the set of said elements. However, paradoxes such as Russell's and Burali-Forti's showed the impossibility of this conception of set, by examples of collections of objects that do not form sets, despite all said objects being existent. Set-theoretic responses From the principle of explosion of classical logic, any proposition can be proved from a contradiction. Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proven true it destroys the