Well-ordering theorem
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In mathematics, the well-ordering 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. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice.[1][2] Ernst Zermelo introduced the Axiom of Choice as an "unobjectionable logical principle" to prove the well-ordering theorem. This is important because it makes every set susceptible to the powerful technique of transfinite induction. The well-ordering theorem has consequences that may seem paradoxical, such as the Banach–Tarski paradox.
History
Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought." Most mathematicians however find it difficult to visualize a well-ordering of, for example, the set R of real numbers. 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.[3] It turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo–Fraenkel axioms is sufficient to prove the other, in first order logic (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.[4]
Statement and proof
For every set X, there exists a well-ordering with domain X.
The well-ordering theorem follows from Zorn's Lemma. Take the set A of all well-orderings of subsets of X: an element of A is an ordered pair (a,b) where a is a subset of X and b is a well-ordering of a. A can be partially ordered by continuation. That means, define E ≤ F if E is an initial segment of F and the ordering of the members in E is the same as their ordering in F. If E is a chain in A, then the union of the sets in E can be ordered in a way that makes it a continuation of any set in E; this ordering is a well-ordering, and therefore, an upper bound of E in A. We may therefore apply Zorn's Lemma to conclude that A has a maximal element, say (M,R). The set M must be equal to X, for if X has an element x not in M, then the set M∪{x} has a well-ordering that restricts to R on M, and for which x is larger than all elements of M. This well ordered set is a continuation of (M,R), contradicting its maximality, therefore M = X. Now R is a well-ordering of X.[5]
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, E, take the union of the sets in E and call it X. There exists a well-ordering of X; let R be such an ordering. The function that to each set S of E associates the smallest element of S, as ordered by (the restriction to S of) R, is a choice function for the collection E. An essential point of this proof is that it involves only a single arbitrary choice, that of R; applying the well-ordering theorem to each member S of E separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each S a well-ordering would not be easier than choosing an element.
See also
Notes
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