the chain has length for any . The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation on a class that is extensional, there exists a class such that is isomorphic to . Reflexivity A relation is said to be reflexive if holds for every in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order ≤, we have . To avoid these trivial descending sequences, when working with a partial order ≤, it is common to apply the definition of well foundedness (perhaps implicitly) to the alternate relation < defined such that if and only if and . More generally, when working with a preorder ≤, it is common to use the relation < defined such that if and only if and . In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of a well-founded relation is changed from the definition above to include these conventions. References Just, Winfried and Weese, Martin (1998) Discovering Modern Set Theory. I, American Mathematical Society . Karel Hrbáček & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, "Well-founded relations", pages 251–5, Marcel Dekker Binary relations