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 Axiom of choice § Equivalents).
[1][2] Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.
[3] 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.
[3] One famous consequence of the theorem is the Banach–Tarski paradox.
Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought".
[4] 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.
[5] 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.
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?
[8] The well-ordering theorem follows from the axiom of choice as follows.
be a choice function for the family of non-empty subsets of
that have not yet been assigned a place in the ordering (or undefined if the entirety of
.The axiom of choice can be proven from the well-ordering theorem as follows.
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
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
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.