Zorn's lemma, the well-ordering principle and the axiom of choice are three equivalent propositions. It has been said that from appearances, the axiom of choice has to be true, the well-ordering principle has to be false and Zorn's lemma is too confusing to figure out. Zermelo Frankel formulated the axiom of choice in 1904 in an attempt to solve the first problem on Hilbert's famous list, the continuum hypothesis. The axiom states that given any collection of mutually disjoint sets there is a set that contains one element from each of the given sets. Zermelo proved that this axiom is equivalent to the well-ordering principle. His axiom was controversial at the time. Many hoped that it was unnecessary. Paul Cohen dashed these hopes when he proved (in 1963) that the axiom of choice is independent of Zermelo-Fraenkel set theory.
To explain these notions, a little set theory is necessary. A partially ordered set is a set X with a relation which satisfies two properties: 1. If x, y and z are in X and x y and y x then x = y. 2. If x y and y z then x z. A totally ordering is just a partially ordering such that every pair of elements is comparable, i.e. if x and y are in X then either x y or y x. The real numbers are totally ordered, for example. For any two pairs of real number (x, y) and (w, z), let's say that (x, y) is less than or equal to (w, z) if x is less than or equal to w and y is less than or equal to z. Under this order, the set of (ordered) pairs of real numbers is partially ordered by not totally ordered. A set is well-ordered if it is totally ordered and every subset has a smallest element. For example the smallest element of {1,2,3,4,5,...} is 1 but the smallest element of the rational numbers does not exist so neither the rationals nor the real numbers are well-ordered. The well-ordering principle states that any set can be well-ordered, i.e. a well-ordering of the set exists. Zermelo proved that if the well-ordering principle is true then given any two sets X and Y, either the cardinality of X is less than, equal to, or greater than that of Y. This is sometimes called the trichotomy law.
If S is a partially ordered set then a subset of S that is totally ordered under the given ordering is called a chain. An upper bound of a chain is an element which is greater than or equal to all elements in the chain. A maximal element of S in an element which is greater than or equal to all other elements of S. Zorn's lemma states if S is a partially ordered set with the property that every chain has an upper bound, then S has a maximal element. Zorn's lemma is sometimes stated as transfinite induction, a method of proof that is like finite induction except that the induction variable is assumed only to be contained in a well-ordered set.
Today, Zorn's lemma is used by mathematicians of all fields. However, many make a special note signifying the reliance of their methods on Zorn's lemma and then try to find a way to avoid its use.
This is the complete article, containing 553 words
(approx. 2 pages at 300 words per page).
