The well-ordering principle states that every non-empty subset of the nonnegative integers contains a least element, meaning the nonnegative integers are well-ordered under their natural magnitude order. The phrase is sometimes used as a synonym for the well-ordering theorem, which states that every set can be well-ordered, and on other occasions is understood as the proposition that the integers contain the natural numbers as a well-ordered subset in which every nonempty subset has a least element.
Facts
Partially Attested
Origin YearDates Peano's formal publication of the induction axiom (his fifth axiom), which the atlas's own cited Wikipedia source states the well-ordering principle for the natural numbers is derived from in Peano arithmetic. Not the principle's earliest informal use: Fermat's infinite descent already relied on it in the 17th century, before any formal axiomatization. Connections
Associated With
Sets, Concepts Source Well-Ordering Principle (Wikipedia)
In Branch
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsBiographies/Peano page, paragraph beginning "In 1889 Peano published his famous axioms"Quote, Biographies/Peano page, paragraph beginning "In 1889 Peano published his famous axioms"
In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets.
Well-Ordering Principle (Wikipedia)
Wikimedia FoundationLead section
In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element.
Associated With: Sets, Lead section, first sentence
the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element
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