Well-ordering principle

Sometimes the phrase "well-ordering principle" (or the axiom of choice) is taken to be synonymous with "well-ordering theorem". On other occasions the phrase is taken to mean the proposition that the set of natural numbers {1, 2, 3, ....} is well-ordered, i.e., each of its non-empty subsets has a smallest member. In the second sense, the phrase is used when that proposition is relied on for the purpose of justifying proof 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 smallest counterexample. Then show that there must be a still smaller counterexample, getting a contradiction. This mode of argument bears the same relation to proof by mathematical induction that "If not B then not A" bears to "If A then B".