# Well-ordering principle

## Statement that all non empty subsets of positive numbers contains a least element / From Wikipedia, the free encyclopedia

Not to be confused with Well-ordering theorem.

In mathematics, the **well-ordering principle** states that every non-empty subset of positive integers contains a least element.^{[1]} In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which $x$ precedes $y$ if and only if $y$ is either $x$ or the sum of $x$ and some positive integer (other orderings include the ordering $2,4,6,...$; and $1,3,5,...$).

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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 $\{\ldots ,-2,-1,0,1,2,3,\ldots \}$ contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.