Zorn's lemma

Statement

Assume the axiom of choice. Let $(P, \le)$ be a nonempty partially ordered set in which each chain has an upper bound in $P$. Then $P$ has a maximal element.

Remarks

Over ZF, Zorn's lemma is equivalent to the axiom of choice. If one requires an upper bound also for the empty chain, the hypothesis "$P$ is nonempty" is not needed. Applications: each vector space has a basis, and each nonzero ring with unity has a maximal ideal.

History

Kazimierz Kuratowski proved the lemma in 1922, and Max Zorn proved it independently in 1935. The Hausdorff maximal principle of 1914 is an earlier statement of the same kind.

Explanations

L1 human

Zorn's lemma is a tool to show that a maximal object exists. You check that each increasing chain of objects has an object above it. Then the lemma gives an object that you cannot enlarge any more.