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- Axiom_of_regularity abstract "In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic the axiom reads: .The axiom implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.The axiom of regularity was introduced by von Neumann (1925); it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo (1930). Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent.In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.".
- Axiom_of_regularity wikiPageExternalLink CzechMathJ_07-1957-3_1.pdf.
- Axiom_of_regularity wikiPageExternalLink fm1615.pdf.
- Axiom_of_regularity wikiPageExternalLink view?rid=ensmat-001:1917:19::9&id=hitlist.
- Axiom_of_regularity wikiPageExternalLink sets.html.
- Axiom_of_regularity wikiPageExternalLink russelle.pdf.
- Axiom_of_regularity wikiPageID "2113".
- Axiom_of_regularity wikiPageRevisionID "604471219".
- Axiom_of_regularity authorLink "Herbert Enderton".
- Axiom_of_regularity authorlink "Dana Scott".
- Axiom_of_regularity first "Dana".
- Axiom_of_regularity first "Herbert".
- Axiom_of_regularity hasPhotoCollection Axiom_of_regularity.
- Axiom_of_regularity id "3485".
- Axiom_of_regularity last "Enderton".
- Axiom_of_regularity last "Scott".
- Axiom_of_regularity loc "p. 206".
- Axiom_of_regularity title "Axiom of Foundation".
- Axiom_of_regularity year "1974".
- Axiom_of_regularity year "1977".
- Axiom_of_regularity subject Category:Axioms_of_set_theory.
- Axiom_of_regularity subject Category:Wellfoundedness.
- Axiom_of_regularity type Abstraction100002137.
- Axiom_of_regularity type AuditoryCommunication107109019.
- Axiom_of_regularity type AxiomsOfSetTheory.
- Axiom_of_regularity type Communication100033020.
- Axiom_of_regularity type Maxim107152948.
- Axiom_of_regularity type Saying107151380.
- Axiom_of_regularity type Speech107109196.
- Axiom_of_regularity comment "In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic the axiom reads: .The axiom implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i.".
- Axiom_of_regularity label "Aksjomat regularności".
- Axiom_of_regularity label "Assioma di regolarità".
- Axiom_of_regularity label "Axiom of regularity".
- Axiom_of_regularity label "Axioma da regularidade".
- Axiom_of_regularity label "Axioma de regularidad".
- Axiom_of_regularity label "Axiome de fondation".
- Axiom_of_regularity label "Fundierungsaxiom".
- Axiom_of_regularity label "Аксиома регулярности".
- Axiom_of_regularity label "正则性公理".
- Axiom_of_regularity label "正則性公理".
- Axiom_of_regularity sameAs Fundierungsaxiom.
- Axiom_of_regularity sameAs Axioma_de_regularidad.
- Axiom_of_regularity sameAs Axiome_de_fondation.
- Axiom_of_regularity sameAs Assioma_di_regolarità.
- Axiom_of_regularity sameAs 正則性公理.
- Axiom_of_regularity sameAs Aksjomat_regularności.
- Axiom_of_regularity sameAs Axioma_da_regularidade.
- Axiom_of_regularity sameAs m.0ww8.
- Axiom_of_regularity sameAs Q470981.
- Axiom_of_regularity sameAs Q470981.
- Axiom_of_regularity sameAs Axiom_of_regularity.
- Axiom_of_regularity wasDerivedFrom Axiom_of_regularity?oldid=604471219.
- Axiom_of_regularity isPrimaryTopicOf Axiom_of_regularity.