Axiom of union

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x. Together with the axiom of pairing this implies that for any two sets, there is a set that contains exactly the elements of both.

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:

$\forall A, \exist B, \forall C: C \in B \iff (\exist D: C \in D \and D \in A)$

or in words:

Given any set A, there is a set B such that, given any set C, C is a member of B if and only if there is a set D such that C is a member of D which is a member of A and .

Thus, what the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the members of the members of A. By the axiom of extensionality this set B is unique and it is called the union of A, and denoted ∪A. Thus the essence of the axiom is:

The union of a set is a set.

The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.

Note that there is no corresponding axiom of intersection. In the case where A is the empty set, there is no intersection of A in Zermelo-Fraenkel set theory. On the other hand, if A has some member B, then we can form the intersection ∩A as {C in B : for all D in A, C is in D} using the axiom schema of specification.