An

order (

total order or

partial order) is said to be

*well ordered* (or a

*well ordering*, sometimes even a

*well order*, irregardless of the irreparable

damage done to the

English language)

iff **every** set of elements has a

minimal element.

For instance, the natural numbers are well ordered, but the rational numbers, real numbers, and integers are not. For partial orders, a hierarchy given by a (rooted) tree is a well order (since an element closest to the root in some set is minimal), even if the tree is infinite.

Having a well ordered set, certain induction-like techniques can be performed on it. Unfortunately, the only way to get a well ordering of a general set is to invoke the Axiom of Choice. So you get a very "unnatural" order, which will have virtually no properties in common with the problem domain. Nonetheless, it's better than nothing, and sometimes used.