Editing Set (mathematics) (section)

=== Zorn's lemma ===
{{main|Zorn's lemma}}

Zorn's lemma is an assertion that is equivalent to the axiom of choice under the other axioms of set theory, and is easier to use in usual mathematics. 

Let {{tmath|S}} be a partial ordered set. A [[chain (order theory)|chain]] in {{tmath|S}} is a subset that is [[total order|totally ordered]] under the induced order. Zorn's lemma states that, if every chain in {{tmath|S}} has an [[upper bound]] in {{tmath|S}}, then {{tmath|S}} has (at least) a [[maximal element]], that is, an element that is not smaller than another element of {{tmath|S}}.

In most uses of Zorn's lemma, {{tmath|S}} is a set of sets, the order is set inclusion, and the upperbound of a chain is taken as the union of its members.

An example of use of Zorn's lemma, is the proof that every [[vector space]] has a [[Hamel basis|basis]]. Here the elements of {{tmath|S}} are [[linearly independent]] subsets of the vector space. The union of a chain of elements of {{tmath|S}} is linearly independent, since an infinite set is linearly independent if and only if each finite subset is, and every finite subset of the union of a chain must be included in a member of the chain. So, there exist a maximal linearly independent set. This linearly independant set must span the vector space because of maximality, and is therefore a basis.

Another classical use of Zorn's lemma is the proof that every proper [[ideal (ring theory)|ideal]]{{mdash}}that is, an ideal that is not the whole ring{{mdash}}of a [[ring (mathematics)|ring]] is contained in a [[maximal ideal]]. Here, {{tmath|S}} is the set of the proper ideals containing the given ideal. The union of chain of ideals is an ideal, since the axioms of an ideal involve a finite number of elements. The union of a chain of proper ideals is a proper ideal, since otherwise {{tmath|1}} would belong to the union, and this implies that it would belong to a member of the chain.