Why does the axiom of foundation prevent a set from containing itself?

The axiom of foundation, also known as the axiom of restriction, prevents a set from containing itself by prohibiting the existence of infinite descending $\in$-chains (sequences such as $x_1\ni x_2\ni x_3\dots$).1

If a set $A$ were to contain itself ($A\in A$), it would create a sequence where $x_1=A$, $x_2=A$, $x_3=A$, and so on. This infinite repetition constitutes a descending $\in$-chain, which the axiom of foundation explicitly eliminates from the models of set theory.1

By ruling out these chains, the axiom ensures that all sets exist within an iterative hierarchy, where each set is constructed from elements that appear at lower levels of the hierarchy. This structure aligns with the view that the universe of sets is built up step-by-step, rather than allowing for circular membership where a set could be a member of itself.12