welgeordend

Welgeordend is a term used in order theory to describe a set that is well-ordered with respect to a given relation. Specifically, a set S equipped with a relation ≤ is called welgeordend if ≤ is a total order on S and every non-empty subset of S has a least element under ≤. In Dutch mathematical language, the phrase welgeordende verzameling is common; the adjective welgeordend is used to modify a set, a relation, or an order.

A well-ordered set has two key properties: comparability of any two elements (totality) and the well-foundedness property that guarantees a minimal element in every non-empty subset. Because of these properties, finite totally ordered sets are always welgeordend, and the set of natural numbers with the standard order is a canonical example. The integers with their usual order are not welgeordend, since the subset of negative integers lacks a least element. The real numbers with the standard order likewise are not well-ordered.

Well-ordering supports mathematical induction and recursive definitions in a broad sense: many induction principles rely on the ability to select least elements from subsets. In set theory, the Well-Ordering Theorem states that every set can be well-ordered, a result equivalent to the Axiom of Choice. The concept also underpins ordinal numbers, transfinite induction, and the study of order types.

In summary, welgeordend denotes a robust form of order where every non-empty subset has a least element, enabling a wide range of foundational and constructive methods in mathematics.