In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered.
The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers {…, −2, −1, 0, 1, 2, 3, …} contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.
Depending on the framework in which the natural numbers are introduced, this (second order) property of the set of natural numbers is either an axiom or a provable theorem. For example:
In the second sense, the phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set S, assume the contrary and infer the existence of a (non-zero) smallest counterexample. Then show either that there must be a still smaller counterexample or that the smallest counterexample is not a counter example, producing a contradiction. This mode of argument bears the same relation to proof by mathematical induction that "If not B then not A" (the style of modus tollens) bears to "If A then B" (the style of modus ponens). It is known light-heartedly as the "minimal criminal" method and is similar in its nature to Fermat's method of "infinite descent".
Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic; i.e., it cannot be deduced from the algebraic properties of the integers (which form an ordered integral domain).