Well-order

A well-order (or well-ordering) on a set a S is a total order on S with the property that every nonempty subset of S has a least element in this ordering. For example, the standard ordering of the natural numbers is a well-ordering, but the standard ordering of the integers is not.

In a well-ordered set, every element, unless it is the overall largest, has a unique successor: the smallest element that is larger than it. However, not every element need to have a predecessor. As an example, consider two copies of the natural numbers, ordered in such a way that every element of the second copy is bigger than every element of the first copy. Within each copy, the normal order is used. This is a well-ordered set and is usually denoted by ω + ω. Note that while every element has a successor (there is no largest element), two elements lack a predecessor: the zero from copy number one (the overall smallest element) and the zero from copy number two.

If a set is well-ordered, the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

The [well-ordering principle]? states that every set can be well-ordered.

See also Ordinal.

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