Well-ordering principle

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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 ${\displaystyle x}$ precedes ${\displaystyle y}$ if and only if ${\displaystyle y}$ is either ${\displaystyle x}$ or the sum of ${\displaystyle x}$ and some positive integer (other orderings include the ordering ${\displaystyle 2,4,6,...}$; and ${\displaystyle 1,3,5,...}$).

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