For example, the standard ordering of the natural numbers is a wellordering, but neither the standard ordering of the integers nor the standard ordering of the positive real numbers is a wellordering.
In a wellordered 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 wellorder property; it is also clearly equivalent to the KuratowskiZorn lemma.
In a wellordered 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 wellordered 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 wellordered, the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.
The wellordering principle, which is equivalent to the axiom of choice, states that every set can be wellordered.
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