Zorn's lemma
A set theory lemma equivalent to the axiom of choice.
Zorn's lemma is a statement in set theory, also called the Kuratowski–Zorn lemma. It says that if every chain (a totally ordered subset) in a partially ordered set has an upper bound within that set, then the set must contain at least one maximal element. Within Zermelo–Fraenkel set theory without the axiom of choice (ZF), Zorn's lemma is equivalent to both the well-ordering theorem and the axiom of choice: any one of these three can prove the other two.
The lemma was proven by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935, both assuming the axiom of choice. It appears in proofs of many fundamental results, such as the Hahn–Banach theorem in functional analysis, the existence of a basis for every vector space, Tychonoff's theorem that any product of compact spaces is compact, and theorems in abstract algebra like every proper ideal in a ring with identity lies inside a maximal ideal, and every field has an algebraic closure. An earlier version is the Hausdorff maximal principle, which states that any totally ordered subset of a partially ordered set is contained in a maximal totally ordered subset.
To prove the existence of an object that can be seen as a maximal element in some partially ordered set, one might assume no such element exists and use transfinite induction to reach a contradiction. Zorn's lemma streamlines this process: instead of repeating the transfinite induction argument each time, mathematicians only need to verify that the conditions of the lemma hold. If you are constructing an object in stages, and you never finish even after infinitely many stages, with no apparent obstacle to continuing, Zorn's lemma can often help.
A partially ordered set is a set P with a binary relation ≤ that is reflexive, antisymmetric, and transitive. Not every pair of elements needs to be comparable; if every pair is comparable, the set is totally ordered. A chain is a subset of P that is totally ordered under the inherited relation. A maximal element m in P has no other element greater than it (no s in P with s ≠ m and m ≤ s). A totally ordered set can have at most one maximal element. An upper bound u of a subset S is an element of P that is greater than or equal to every element of S; u need not be in S, and S need not be a chain.
Formally, Zorn's lemma can be stated as: if every chain in a nonempty partially ordered set P has an upp
- field
- Set theory
- known_for
- Zorn's lemma (Kuratowski–Zorn lemma)
- proven_by
- Kazimierz Kuratowski (1922) and Max Zorn (1935)
Lore & Background
Zorn's lemma was proven (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. An earlier formulation is the Hausdorff maximal principle, which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set. The lemma occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology, and theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.
Reader's Guide
Zorn's lemma is a fundamental tool in mathematics, particularly in set theory and its applications across various fields. It allows mathematicians to prove the existence of maximal elements in partially ordered sets without performing transfinite induction manually each time. The lemma is equivalent to the axiom of choice and the well-ordering theorem within ZF set theory, meaning that any one of these three statements can prove the other two. Its practical use is illustrated in the proof that every vector space has a basis: one considers the set of all linearly independent subsets, partially ordered by inclusion, and applies Zorn's lemma to obtain a maximal such subset, which then forms a basis. The lemma also appears in proofs of the Hahn–Banach theorem, Tychonoff's theorem, and the existence of maximal ideals and algebraic closures. However, Zorn's lemma can fail for a partially ordered class that is not a set, as shown by the class of all ordinals, which has no maximal element. The lemma's significance lies in its ability to unify and simplify existence proofs across many branches of mathematics.
Did You Know?
- Zorn's lemma is also known as the Kuratowski–Zorn lemma.
- It was proven by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935.
- Zorn's lemma is equivalent to the well-ordering theorem and to the axiom of choice within ZF set theory.
- An earlier formulation is the Hausdorff maximal principle.
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