Well-ordering principle

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In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. In other words, the set of nonnegative integers is well-ordered by its "natural" or "magnitude" order in which x {\displaystyle x} precedes y {\displaystyle y} if and only if y {\displaystyle y} is either x {\displaystyle x} or the sum of x {\displaystyle x} and some nonnegative integer (other orderings include the ordering 2 , 4 , 6 , . . . {\displaystyle 2,4,6,...} ; and 1 , 3 , 5 , . . . {\displaystyle 1,3,5,...} ). 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 { … , − 2 , − 1 , 0 , 1 , 2 , 3 , … } {\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.