In the formal language of the ZermeloFraenkel axioms, the axiom reads:
To understand this axiom, note that the clause involving D in the symbolic statement above states that C is a member of some member of A. 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. We can use the axiom of extension to show that this set B is unique. We call the set B the union of A, and denote it ∪A. Thus the essence of the axiom is:
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 ZermeloFraenkel 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.
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