Well-order

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.

For example, the standard ordering of the natural numbers is a well-ordering, but neither the standard ordering of the integers nor the standard ordering of the positive real numbers is a well-ordering.

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.

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 principle, which is equivalent to the axiom of choice, states that every set can be well-ordered.