Well-ordering principle

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.