Well-ordering theorem

The well-ordering theorem states that every set admits a well-order, in which every nonempty subset has a least element. In Zermelo–Fraenkel set theory, it is equivalent to the axiom of choice.

Axiom of choice: The principle that every family of nonempty sets has a choice function. Choosing one element from each subset yields a well-ordering of any set.
Linear order: An order relation in which any two elements are comparable. A well-order is a linear order with an additional least-element condition.
Ernst Zermelo: A German mathematician who developed axiomatic set theory and proved the well-ordering theorem in 1904. His proof established the theorem using a principle that became the axiom of choice.
Banach–Tarski paradox: A theorem that a solid ball can be partitioned into finitely many pieces and reassembled into two identical balls. Its proof uses choice, whose equivalent well-ordering principle underwrites highly nonconstructive selections.
Zorn's lemma: A principle stating that a partially ordered set with upper bounds for all chains has a maximal element. It is equivalent to the well-ordering theorem in standard set theory.
Least element: An element of a partially ordered set that is below every other element in that set. Every nonempty subset must contain one for an order to be a well-order.
Georg Cantor: A German mathematician who founded set theory and developed the theory of transfinite numbers. His work on sets and ordinals supplied the setting in which well-ordering became central.
Hamel basis: A basis of a vector space over a field, with every vector a finite linear combination of basis vectors. Choice guarantees such a basis for every vector space, via the equivalent well-ordering theorem.
Hausdorff maximal principle: The principle that every partially ordered set has a maximal linearly ordered subset. Its maximal-chain principle is another equivalent form of choice.
Well-founded relation: A relation is well-founded if every nonempty subset has a minimal element. Well-orders are well-founded, though well-founded relations need not be linear.

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