Editing Well-ordering principle (section)

{{Short description|Statement that all non empty subsets of positive numbers contains a least element}}
{{distinguish|Well-ordering theorem}}
{{more citations needed|date=July 2008}}

In [[mathematics]], the '''well-ordering principle''' states that every non-empty subset of nonnegative integers contains a [[least element]].<ref>{{cite book |title=Introduction to Analytic Number Theory |last=Apostol |first=Tom |author-link=Tom M. Apostol |year=1976 |publisher=Springer-Verlag |location=New York |isbn=0-387-90163-9 |pages=[https://archive.org/details/introductiontoan00apos_0/page/13 13] |url-access=registration |url=https://archive.org/details/introductiontoan00apos_0/page/13 }}</ref> In other words, the set of nonnegative integers is [[well-order]]ed by its "natural" or "magnitude" order in which <math>x</math> precedes <math>y</math> if and only if <math>y</math> is either <math>x</math> or the sum of <math>x</math> and some nonnegative integer (other orderings include the ordering <math>2, 4, 6, ...</math>; and <math>1, 3, 5, ...</math>).

The phrase "well-ordering principle" is sometimes taken to be synonymous with the "[[well-ordering theorem]]", according to which every set can be well-ordered. On other occasions it is understood to be the proposition that the set of [[integers]] <math>\{\ldots, -2, -1, 0, 1, 2, 3, \ldots \}</math> contains a [[well-order]]ed subset, called the [[natural numbers]], in which every nonempty subset contains a least element.