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Every set can be given a well-order.
Well-ordering theorem: For every set , there exists a relation such that is a well-ordered set.
Over ZF, the well-ordering theorem is equivalent to the Axiom of Choice and to Zorn's lemma.
In axiomatic set theory, set and the membership relation are primitive notions. The axiom of extensionality determines equality by membership: for sets ,
A relation from a set to a set is a set with
where is the Cartesian product. If , one often writes .
A well-ordered set is a set equipped with a total order such that every nonempty subset has a least element; that is, there exists with for all .
Axiom of Choice: For every indexed family of nonempty sets, there exists a function such that for all .
Zorn's lemma: Let be a partially ordered set. If every chain has an upper bound in , then has a maximal element.
Here a chain means a subset on which the restriction of is a total order, and an element is maximal if there is no with (i.e., and ).