Well-order

A well-order is a total order in which every nonempty subset has a least element. Unlike an arbitrary total order, it rules out infinite descending sequences.

Total order: A relation that orders every pair of elements, consistently with transitivity and antisymmetry. Every well-order is total, but totality alone does not guarantee least elements for all subsets.
Transfinite induction: A proof method that establishes a claim along a well-order by using its truth at all earlier elements. A least counterexample would contradict the induction step.
Natural number order: The usual order on the natural numbers, in which every nonempty subset has a least number. It is the standard infinite example of a well-order.
Ernst Zermelo: A German mathematician who proved the well-ordering theorem and helped develop axiomatic set theory. His 1904 theorem established the claim that every set can be well-ordered, assuming choice.
Least element: An element of a set that is no greater than any other element in that set. The existence of a least element in every nonempty subset is the defining condition.
Transfinite recursion: A method for defining objects successively along a well-order, using earlier values at each stage. The order ensures each stage depends only on values already defined.
Lexicographic order: An order on sequences or tuples determined by the first coordinate at which they differ. On finite sequences over a well-ordered alphabet, lexicographic order can produce well-orders.
Well-ordering theorem: The theorem that every set admits a well-ordering, equivalent to the axiom of choice in standard set theory. It extends well-ordering from familiar examples to arbitrary sets.
Well-founded relation: A relation is well-founded if every nonempty subset of its domain has a minimal element. Well-foundedness supplies the subset-minimum condition; adding totality yields a well-order.
Descending chain condition: A condition forbidding infinite strictly descending sequences in a partially ordered set. For total orders, the absence of infinite descending sequences characterizes well-ordering.

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