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- Well-ordering_principle abstract "In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered.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 {…, −2, −1, 0, 1, 2, 3, …} contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.Depending on the framework in which the natural numbers are introduced, this (second order) property of the set of natural numbers is either an axiom or a provable theorem. For example: In Peano Arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic. Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set A of natural numbers has an infimum, say a*. We can now find an integer n* such that a* lies in the half-open interval (n*−1, n*], and can then show that we must have a* = n*, and n* in A. In axiomatic set theory, the natural numbers are defined as the smallest inductive set (i.e., set containing 0 and closed under the successor operation). One can (even without invoking the regularity axiom) show that the set of all natural numbers n such that \"{0, …, n} is well-ordered\" is inductive, and must therefore contain all natural numbers; from this property one can conclude that the set of all natural numbers is also well-ordered. In the second sense, the phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set S, assume the contrary and infer the existence of a (non-zero) smallest counterexample. Then show either that there must be a still smaller counterexample or that the smallest counterexample is not a counter example, producing a contradiction. This mode of argument bears the same relation to proof by mathematical induction that \"If not B then not A\" (the style of modus tollens) bears to \"If A then B\" (the style of modus ponens). It is known light-heartedly as the \"minimal criminal\" method and is similar in its nature to Fermat's method of \"infinite descent\".Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic; i.e., it cannot be deduced from the algebraic properties of the integers (which form an ordered integral domain).".
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- Well-ordering_principle wikiPageLength "3703".
- Well-ordering_principle wikiPageOutDegree "26".
- Well-ordering_principle wikiPageRevisionID "677318578".
- Well-ordering_principle wikiPageWikiLink Axiom.
- Well-ordering_principle wikiPageWikiLink Axiom_of_infinity.
- Well-ordering_principle wikiPageWikiLink Axiom_of_regularity.
- Well-ordering_principle wikiPageWikiLink Category:Mathematical_principles.
- Well-ordering_principle wikiPageWikiLink Category:Wellfoundedness.
- Well-ordering_principle wikiPageWikiLink Completeness_of_the_real_numbers.
- Well-ordering_principle wikiPageWikiLink Garrett_Birkhoff.
- Well-ordering_principle wikiPageWikiLink Greatest_element.
- Well-ordering_principle wikiPageWikiLink Integer.
- Well-ordering_principle wikiPageWikiLink Integral_domain.
- Well-ordering_principle wikiPageWikiLink Mathematical_induction.
- Well-ordering_principle wikiPageWikiLink Mathematics.
- Well-ordering_principle wikiPageWikiLink Minimal_counterexample.
- Well-ordering_principle wikiPageWikiLink Modus_ponens.
- Well-ordering_principle wikiPageWikiLink Modus_tollens.
- Well-ordering_principle wikiPageWikiLink Natural_number.
- Well-ordering_principle wikiPageWikiLink Peano_axioms.
- Well-ordering_principle wikiPageWikiLink Pierre_de_Fermat.
- Well-ordering_principle wikiPageWikiLink Proof_by_infinite_descent.
- Well-ordering_principle wikiPageWikiLink Saunders_Mac_Lane.
- Well-ordering_principle wikiPageWikiLink Second-order_arithmetic.
- Well-ordering_principle wikiPageWikiLink Set_theory.
- Well-ordering_principle wikiPageWikiLink Well-order.
- Well-ordering_principle wikiPageWikiLink Well-ordering_theorem.
- Well-ordering_principle wikiPageWikiLinkText "Well-ordering principle".
- Well-ordering_principle wikiPageWikiLinkText "this is impossible in the set of natural numbers".
- Well-ordering_principle wikiPageWikiLinkText "well-ordering principle".
- Well-ordering_principle wikiPageUsesTemplate Template:Distinguish.
- Well-ordering_principle wikiPageUsesTemplate Template:Refimprove.
- Well-ordering_principle wikiPageUsesTemplate Template:Reflist.
- Well-ordering_principle subject Category:Mathematical_principles.
- Well-ordering_principle subject Category:Wellfoundedness.
- Well-ordering_principle type Concept.
- Well-ordering_principle type Redirect.
- Well-ordering_principle type Thing.
- Well-ordering_principle comment "In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered.The phrase \"well-ordering principle\" is sometimes taken to be synonymous with the \"well-ordering theorem\".".
- Well-ordering_principle label "Well-ordering principle".
- Well-ordering_principle differentFrom Well-ordering_theorem.
- Well-ordering_principle sameAs Q2488476.
- Well-ordering_principle sameAs Αρχή_της_καλής_διάταξης.
- Well-ordering_principle sameAs Principio_de_buena_ordenación.
- Well-ordering_principle sameAs اصل_خوشترتیبی.
- Well-ordering_principle sameAs Hyvän_järjestyksen_periaate.
- Well-ordering_principle sameAs עקרון_הסדר_הטוב.
- Well-ordering_principle sameAs Principio_del_buon_ordinamento.
- Well-ordering_principle sameAs 자연수의_정렬성.
- Well-ordering_principle sameAs Zasada_dobrego_uporządkowania.
- Well-ordering_principle sameAs Princípio_da_boa_ordenação.
- Well-ordering_principle sameAs m.016m2x.
- Well-ordering_principle sameAs Välordningsaxiomet.
- Well-ordering_principle sameAs Q2488476.
- Well-ordering_principle wasDerivedFrom Well-ordering_principle?oldid=677318578.
- Well-ordering_principle isPrimaryTopicOf Well-ordering_principle.