Zorn’s lemma

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Zorn’s lemma

mathematics

Also known as: Kuratowski-Zorn lemma, maximum principle

Written by Herbert Enderton

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Also known as:: Kuratowski-Zorn lemma

Originally called:: maximum principle

Related Topics:: set theory; axiom of choice

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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 Zermelo-Fraenkel axioms 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 (C_i and C_j), one is a subset of the other (C_i ⊆ C_j). 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., ∪ C_k ∊ 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.

Herbert Enderton

set theory

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set theory

mathematics

Written by Herbert Enderton, Robert R. Stoll•All

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Last Updated: Apr 19, 2025 • Article History

Key People:: Georg Cantor; Paul Erdős; John von Neumann; Saul Kripke; Stanislaw Ulam

Related Topics:: axiom schema of replacement; principle of abstraction; generalized continuum hypothesis; axiom of extensionality; axiom of infinity

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set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.

Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements or members of the set.

The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinity of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophers and violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.

At just that time, however, several contradictions in so-called naive set theory were discovered. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the Nicolas Bourbaki Éléments de mathématique (begun 1939; “Elements of Mathematics”): “Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets.”

Introduction to naive set theory

Fundamental set concepts

In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set may be defined by a membership rule (formula) or by listing its members within braces. For example, the set given by the rule “prime numbers less than 10” can also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates that the list of natural numbers ℕ goes on forever. The empty (or void, or null) set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the status of being a set.

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A set A is called a subset of a set B (symbolized by A ⊆ B) if all the members of A are also members of B. For example, any set is a subset of itself, and Ø is a subset of any set. If both A ⊆ B and B ⊆ A, then A and B have exactly the same members. Part of the set concept is that in this case A = B; that is, A and B are the same set.