Well-order

In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order is then called a well-ordered set.

Examples:

• The standard ordering of the natural numbers is a well-ordering.
• The standard ordering of the integers, is not a well-ordering, since, for example, the set of negative integers does not contain a least element.
• The standard ordering of the positive real numbers, is not a well-ordering, since, for example, the open interval (0, 1) does not contain a least element.

In a well-ordered set, there cannot exist any infinitely long descending chains. Using the axiom of choice, one can show that this property is in fact equivalent to the well-order property; it is also clearly equivalent to the Kuratowski-Zorn lemma.

The set of negative integers is not well-ordered by the ordinary comparison operator <, or 'less than', but it is possible to define a different relationship that does well-order the negative integers. For instance, the following definition well-orders the integers: x <_z y iff |x| < |y| or (|x| = |y| and x < y).

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 needs 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 theorem, which is equivalent to the axiom of choice, states that every set is well-orderable.

See also Ordinal number, Well-founded set, Well partial order