Editing Well-founded relation (section)

==Other properties==

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

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