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Theoretic principle in mathematics stating every set can be well ordered. In mathematics, the well ordering theorem, also known as Zermelo's theorem, states that every set can be well ordered. A set X is well ordered by a strict total order if every non empty subset of X has a least element under the ordering. The well ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also ). Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well ordering theorem. One can conclude from the well ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. However, it is considered difficult or even impossible to visualize a well ordering of ; such a visualization would have to incorporate the axiom of choice. In 1904, Gyula Kőnig claimed to have proven that such a well ordering cannot exist. A few weeks later, Felix Hausdorff found a mistake in the proof. It turned out, though, that in first order logic the well ordering theorem is equivalent to the axiom of choice, in the sense that the Zermelo–Fraenkel axioms with the axiom of choice included are sufficient to prove the well ordering theorem, and conversely, the Zermelo–Fraenkel axioms without the axiom of choice but with the well ordering theorem included are sufficient to prove the axiom of choice. (The same applies to Zorn's lemma.) In second order logic, however, the well ordering theorem is strictly stronger than the axiom of choice: from the well ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well ordering theorem. There is a well known joke about the three statements, and their relative amenability to intuition:The axiom of choice is obviously true, the well ordering principle obviously false, and who can tell about Zorn's lemma?

The well ordering theorem follows from the axiom of choice as follows. Let the set we are trying to well order be , and let be a choice function for the family of non empty subsets of For every ordinal , define an element that is in by setting if this complement is nonempty, or leave undefined if it is. That is, is chosen from the set of elements of that have not yet been assigned a place in the ordering (or undefined if the entirety of has been successfully enumerated). Then the order on defined by if and only if (in the usual well order of the ordinals) is a well order of as desired, of order type .

The axiom of choice can be proven from the well ordering theorem as follows. To make a choice function for a collection of non empty sets, , take the union of the sets in and call it There exists a well ordering of ; let be such an ordering. The function that to each set of associates the smallest element of , as ordered by (the restriction to of) , is a choice function for the collection
An essential point of this proof is that it involves only a single arbitrary choice, that of ; applying the well ordering theorem to each member of separately would not work, since the theorem only asserts the existence of a well ordering, and choosing for each a well ordering would require just as many choices as simply choosing an element from each Particularly, if contains uncountably many sets, making all uncountably many choices is not allowed under the axioms of Zermelo Fraenkel set theory without the axiom of choice.