Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Well-ordering_principle> ?p ?o. }
Showing items 1 to 32 of
32
with 100 items per page.
- Well-ordering_principle abstract "In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element.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).".
- Well-ordering_principle wikiPageID "168864".
- Well-ordering_principle wikiPageRevisionID "602788792".
- Well-ordering_principle hasPhotoCollection Well-ordering_principle.
- Well-ordering_principle subject Category:Mathematical_principles.
- Well-ordering_principle subject Category:Wellfoundedness.
- Well-ordering_principle type Abstraction100002137.
- Well-ordering_principle type Cognition100023271.
- Well-ordering_principle type Content105809192.
- Well-ordering_principle type Generalization105913275.
- Well-ordering_principle type Idea105833840.
- Well-ordering_principle type MathematicalPrinciples.
- Well-ordering_principle type Principle105913538.
- Well-ordering_principle type PsychologicalFeature100023100.
- Well-ordering_principle comment "In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element.The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem".".
- Well-ordering_principle label "Principio de buena ordenación".
- Well-ordering_principle label "Principio del buon ordinamento".
- Well-ordering_principle label "Princípio da boa ordenação".
- Well-ordering_principle label "Well-ordering principle".
- Well-ordering_principle label "Zasada dobrego uporządkowania".
- Well-ordering_principle sameAs Αρχή_της_καλής_διάταξης.
- Well-ordering_principle sameAs Principio_de_buena_ordenación.
- 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 Q2488476.
- Well-ordering_principle sameAs Q2488476.
- Well-ordering_principle sameAs Well-ordering_principle.
- Well-ordering_principle wasDerivedFrom Well-ordering_principle?oldid=602788792.
- Well-ordering_principle isPrimaryTopicOf Well-ordering_principle.