How does the axiom of union combine the members of a set of sets?

In axiomatic set theory, such as the Zermelo-Fraenkel (ZF) system, the axiom of union provides a formal method for constructing new sets from existing ones.

If you have a set $S$ that contains other sets as its members (a "set of sets"), the axiom of union asserts that there exists a set containing all and only the elements that are members of the sets contained in $S$. Essentially, it "flattens" the collection by gathering every individual element found within the inner sets into one single, unified set.12

This axiom is fundamental to the "iterative" conception of set theory, allowing mathematicians to build complex structures from simpler ones. While naive set theory often introduces the union operation intuitively for two sets, the axiom of union provides the rigorous foundation necessary to perform this operation on arbitrary collections of sets within an axiomatic framework.1342