Zorn's Lemma states that if a partially ordered set has the property that every chain, meaning every totally ordered subset, has an upper bound within the set, then the set contains at least one maximal element. It is logically equivalent to the axiom of choice and to the well-ordering theorem, and it is invoked throughout algebra and analysis to prove the existence of objects, such as maximal ideals or bases of vector spaces, that cannot be constructed explicitly.
Facts
StatementA partially ordered set in which every chain has an upper bound contains at least one maximal element. 1 Classification
Statement Form Statement Form Connections
Sources
1. Zorn's Lemma (Wikipedia)
Wikimedia FoundationLead section, second paragraph
The lemma was proven (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935.
Lead section
It states that a partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.
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