- Also known as:
- Kuratowski-Zorn lemma
- Originally called:
- maximum principle
- Related Topics:
- set theory
- axiom of choice
Zorn’s lemma
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What is Zorn’s lemma?
Zorn's lemma is a statement in set theory equivalent to the axiom of choice. It is often applied to prove the existence of a mathematical object when it cannot be explicitly produced.
Formally, Zorn's lemma states that if a set S is partially ordered, and every subset of S in which each pair of elements is related by "less than," "equal to," or "greater than" has an upper bound in S, then S contains at least one element that has no greater element in S.
In 1935, Max Zorn proposed adding the maximum principle to set theory axioms. Zorn demonstrated the formulation's usefulness in applications, especially in algebra and analysis. He stated that the maximum principle, the axiom of choice, and Ernst Zermelo’s well-ordering principle were equivalent.
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Why is Zorn’s lemma important in mathematics?
Zorn's lemma is a significant statement in set theory, equivalent to the axiom of choice, with broad applications in mathematics. It provides that if a set S is partially ordered and every subset is related by "less than", "equal to", or "greater than" having an upper bound in S, then S contains at least one element for which there is no greater element in S.
Zorn's lemma is often used to prove the existence of a mathematical object when it cannot be explicitly produced. For example, Zorn's lemma can be applied in algebra to prove that any vector space V has a basis and that S is closed under unions of chains.
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What does it mean for a set to be partially ordered?
In set theory, a set is partially ordered if some or all of its elements are connected by a relation that is reflexive, transitive, and antisymmetric.
Zorn's lemma applies to partially ordered sets: if a set S is partially ordered and every subset of S for which each pair of elements is related by "less than," "equal to," or "greater than" has an upper bound in S, then S contains at least one element for which there is no greater element in S.
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How does Zorn’s lemma relate to the idea of a maximal element?
Zorn's lemma, equivalent to the axiom of choice in set theory, is often used to prove the existence of mathematical objects that cannot be explicitly produced.
Formally, Zorn’s lemma states that if a collection ( S ) of sets is "closed under unions of chains," then ( S ) contains a maximal member. Here, "closed under unions of chains" means that whenever a chain ( C ) (where each pair of members is a subset of the other) is included in ( S ), its union also belongs to ( S ). A maximal member of ( S ) is one that is not a subset of any other member of ( S ).
In essence, if you have a collection of sets where you can always form a bigger set by taking the union of any chain of sets within the collection, then there must be a "biggest" set in the collection that cannot be extended further.
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What is the connection between Zorn’s lemma and the Axiom of Choice?
Zorn's lemma and the axiom of choice are equivalent in set theory. The axiom of choice allows forming sets by choosing one element from each set in an infinite collection, even without a selection algorithm. Zorn's lemma states that if a set S is partially ordered, and every subset has an upper bound in S, then S contains an element with no greater element in S.
Any one of three assumptions—the axiom of choice, the well-ordering principle, or Zorn’s lemma—enables one to prove the other two; that is to say, all three are mathematically equivalent.
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How do mathematicians use Zorn’s lemma to prove the existence of certain objects, like a basis for every vector space?
Zorn's lemma, equivalent to the axiom of choice, is used to prove the existence of mathematical objects when they cannot be explicitly produced.
For example, consider the proof that any vector space V has a basis. Let S be the collection of all linearly independent sets of vectors in V. It can be shown that S is closed under unions of chains. Then by Zorn's lemma there exists a maximal linearly independent set of vectors, which by definition must be a basis for V.
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Zorn’s lemma, statement in the language of set theory, equivalent to the axiom of choice, that is often used to prove the existence of a mathematical object when it cannot be explicitly produced.
In 1935 the German-born American mathematician Max Zorn proposed adding the maximum principle to the standard axioms of set theory (see the Click Here to see full-size table
table). (Informally, a closed collection of sets contains a maximal member—a set that cannot be contained in any other set in the collection.) Although it is now known that Zorn was not the first to suggest the maximum principle (the Polish mathematician Kazimierz Kuratowski discovered it in 1922), he demonstrated how useful this particular formulation could be in applications, particularly in algebra and analysis. He also stated, but did not prove, that the maximum principle, the axiom of choice, and German mathematician Ernst Zermelo’s well-ordering principle were equivalent; that is, accepting any one of them enables the other two to be proved. See also set theory: Axioms for infinite and ordered sets.
A formal definition of Zorn’s lemma requires some preliminary definitions. A collection C of sets is called a chain if, for each pair of members of C (Ci and Cj), one is a subset of the other (Ci ⊆ Cj). A collection S of sets is said to be “closed under unions of chains” if whenever a chain C is included in S (i.e., C ⊆ S), then its union belongs to S (i.e., ∪ Ck ∊ S). A member of S is said to be maximal if it is not a subset of any other member of S. Zorn’s lemma is the statement: Any collection of sets closed under unions of chains contains a maximal member.
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As an example of an application of Zorn’s lemma in algebra, consider the proof that any vector space V has a basis (a linearly independent subset that spans the vector space; informally, a subset of vectors that can be combined to obtain any other element in the space). Taking S to be the collection of all linearly independent sets of vectors in V, it can be shown that S is closed under unions of chains. Then by Zorn’s lemma there exists a maximal linearly independent set of vectors, which by definition must be a basis for V. (It is known that, without the axiom of choice, it is possible for there to be a vector space without a basis.)
An informal argument for Zorn’s lemma can be given as follows: Assume that S is closed under unions of chains. Then the empty set Ø, being the union of the empty chain, is in S. If it is not a maximal member, then some other member that includes it is chosen. This last step is then iterated for a very long time (i.e., transfinitely, by using ordinal numbers to index the stages in the construction). Whenever (at limit ordinal stages) a long chain of larger and larger sets has been formed, the union of that chain is taken and used to continue. Because S is a set (and not a proper class like the class of ordinal numbers), this construction ultimately must stop with a maximal member of S.