Why is the axiom of foundation used to rule out infinite descending membership chains?

The axiom of foundation, also known as the axiom of restriction, is used to rule out infinite descending membership chains—sequences such as $x_2\in x_1$, $x_3\in x_2$, and so on—because such structures do not exist within the standard iterative hierarchy of sets.1

This hierarchy is constructed by starting from the empty set and iteratively applying operations like the power set and union to build up the universe of sets. By adopting the axiom of foundation, mathematicians ensure that the models of set theory align with this intuitive, iterative view of the set-theoretic universe, where every set is built from simpler sets at lower levels. While this axiom provides a cleaner definition of ordinal numbers, it is important to note that there is no universal consensus on its necessity, as it has not been shown to have specific mathematical applications, and some set theorists argue there is no compelling reason to exclude sets that might exist outside of this hierarchy.21