structural:reformatting, stylistic:tone
Other properties

If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers with a new element ω that is bigger than any integer. Then X is a well-founded set, butthere are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2,..., 2, 1 has length n for any n.

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X that is extensional, there exists a class C such that (X, R) is isomorphic to (C, ∈).