axiom of extensionality

Moreover, by the axiom of extensionality, this set X is uniquely determined by ϕ(x). A flaw in Frege’s system was uncovered by Russell, who pointed out some obvious contradictions involving sets that contain themselves as elements—e.g., by taking ϕ(x) to be ¬(x ∊ x). Russell illustrated this by…

…principle is known as the principle of extensionality. A class with no members, such as the class of atheistic popes, is said to be null. Since the membership of all such classes is the same, there is only one null class, which is therefore usually called the null class (or…

…by what is called the principle of extension—a set is determined by its members rather than by any particular way of describing the set. Thus, sets A and B are equal if and only if every element in A is also in B and every element in B is in…

…several much more restrictive axioms:

The ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is determined solely by its members. It should be noted that this is not merely a logically necessary property of equality but an assumption about the membership relation as well.