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 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 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" (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 it nature to Fermat's method of "infinite descent".