another application of such chain conditions is in noetherian induction—also known as well-founded induction—which is a generalization of mathematical induction. it frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. suppose that "s" is a partially ordered set. one way of proving a statement about the objects of "s" is to assume the existence of a counterexample and deduce a contradiction, thereby proving the contrapositive of the original statement. the basic premise of noetherian induction is that every non-empty subset of "s" contains a minimal element. in particular, the set of all counterexamples contains a minimal element, the "minimal counterexample". in order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: for any counter-example, there is a smaller counter-example.