Axiom of Union - Interpretation

Interpretation

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 . 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. If A is a nonempty set containing E, then we can form the intersection using the axiom schema of specification as

{c in E: for all D in A, c is in D},

so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as

{c: for all D in A, c is in D}

is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of a universal set is antithetical to Zermelo–Fraenkel set theory.)