In material set theory, a pure set is called transitive if implies that . Note that this does not mean that is a transitive binary relation on itself. (In fact, assuming the axiom of foundation, is a transitive relation on precisely when is a von Neumann ordinal number.)
Transitive sets are a natural place to look for inner model?s of a material set theory: that is, sets such that together with the restriction of to satisfies some or all of the axioms of the set theory. This is especially so because of Mostowski's collapsing lemma that any extensional well-founded relation is isomorphic to a transitive set.
In ZFC, one can prove that every pure set is contained in a least transitive pure set, called its transitive closure. This can be defined as the set of all such that there is a chain for some natural number . The proof that this set exists requires the axiom of replacement and the property of induction for unbounded formulas (which follows from the axiom of separation). In set theories that are too weak to prove the existence of transitive closures in this way, their existence is sometimes assumed explicitly, as the axiom of transitive closure.
Given the existence of transitive closures, pure sets can be identified with subsets of transitive sets, and hence (given Mostowski’s lemma) with subsets of extensional well-founded relations. This latter characterization is completely structural, and thus can be used to model pure sets in structural set theory.