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Statements

Subject Item: dbr:Cardinality_of_the_continuum
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Subject Item: dbr:Amorphous_set
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Subject Item: dbr:Preorder
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Subject Item: dbr:Prewellordering
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Subject Item: dbr:Principia_Mathematica
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Subject Item: dbr:Quine–Putnam_indispensability_argument
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Subject Item: dbr:Robert_M._Solovay
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Subject Item: dbr:Elementary_definition
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Subject Item: dbr:List_of_axioms
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Subject Item: dbr:List_of_first-order_theories
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Subject Item: dbr:Morita_conjectures
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Subject Item: dbr:Mostowski_collapse_lemma
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Subject Item: dbr:Mouse_(set_theory)
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Subject Item: dbr:Multiverse_(set_theory)
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Subject Item: dbr:New_Foundations
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Subject Item: dbr:Lévy_hierarchy
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Subject Item: dbr:Mereology
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Subject Item: dbr:Metamathematics
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Subject Item: dbr:Scott's_trick
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Subject Item: dbr:Determinacy
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Subject Item: dbr:Culture_of_Israel
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Subject Item: dbr:Curry's_paradox
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Subject Item: dbr:Ultrafilter_(set_theory)
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Subject Item: dbr:Von_Neumann–Bernays–Gödel_set_theory
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Subject Item: dbr:Dedekind-infinite_set
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Subject Item: dbr:Definable_real_number
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Subject Item: dbr:Intuitionism
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Subject Item: dbr:Kőnig's_lemma
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Subject Item: dbr:L(R)
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Subject Item: dbr:Timeline_of_category_theory_and_related_mathematics
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Subject Item: dbr:Consistency
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Subject Item: dbr:Russell's_paradox
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Subject Item: dbr:Saharon_Shelah
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Subject Item: dbr:General_topology
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Subject Item: dbr:Nice_name
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Subject Item: dbr:Tarski–Grothendieck_set_theory
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Subject Item: dbr:Class_(set_theory)
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Subject Item: dbr:Enumeration
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Subject Item: dbr:Georg_Cantor
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Subject Item: dbr:George_Boolos
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Subject Item: dbr:Gimel_function
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Subject Item: dbr:Bounded_quantifier
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Subject Item: dbr:Naive_set_theory
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Subject Item: dbr:Conglomerate_(mathematics)
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Subject Item: dbr:Conservative_extension
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Subject Item: dbr:Constructive_set_theory
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Subject Item: dbr:Continuum_hypothesis
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Subject Item: dbr:Criticism_of_nonstandard_analysis
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Subject Item: dbr:Theorem
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Subject Item: dbr:Epsilon-induction
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Subject Item: dbr:Equinumerosity
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Subject Item: dbr:Equivalent_definitions_of_mathematical_structures
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Subject Item: dbr:Equivalents_of_the_Axiom_of_Choice
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Subject Item: dbr:Ramified_forcing
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Subject Item: dbr:Ordinal_analysis
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Subject Item: dbr:Andrzej_Trybulec
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Subject Item: dbr:Arnold_Oberschelp
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Subject Item: dbr:Berkeley_cardinal
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Subject Item: dbr:Leonidas_Alaoglu
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Subject Item: dbr:Lp_space
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Subject Item: dbr:Zermelo–Fraenkel_set_theory
rdf:type: yago:Instrumentality103575240 yago:Whole100003553 dbo:Place yago:Artifact100021939 yago:Object100002684 yago:System104377057 yago:PhysicalEntity100001930 yago:WikicatSystemsOfSetTheory
rdfs:label: Zermelo-Fraenkel-Mengenlehre 策梅洛-弗兰克尔集合论 Система Цермело — Френкеля نظرية المجموعات حسب تسيرميلو-فرانكل Théorie des ensembles de Zermelo-Fraenkel Zermelova–Fraenkelova teorie množin Teoria degli insiemi di Zermelo-Fraenkel Zermelo-Fraenkel-verzamelingenleer Axiomas de Zermelo-Fraenkel Aksjomaty Zermela-Fraenkla ツェルメロ＝フレンケル集合論 체르멜로-프렝켈 집합론 Axiomas de Zermelo-Fraenkel ZFC Zermelo–Fraenkel set theory Теорія множин Цермело — Френкеля Zermelo–Fraenkels mängdteori
rdfs:comment: 策梅洛-弗兰克尔集合论（英語：Zermelo-Fraenkel Set Theory），含选择公理時常简写为ZFC，是在数学基础中最常用形式的公理化集合论，不含選擇公理的則簡寫為ZF。它是二十世纪早期为了建构一个不会导致类似罗素悖论的矛盾的集合理论所提出的一个公理系统。 La Teoria de conjunts de Zermelo-Fraenkel (ZFC) és el conjunt d'axiomes canònic de la teoria de conjunts. El seu nom es deu als matemàtics que la van desenvolupar: Ernst Zermelo i Abraham Fraenkel i la C per la inclusió de l'axioma d'elecció (Choice en anglès). Existeixen altres conjunts d'axiomes de la Teoria de Conjunts com el NBG (von Neumann, Bernays, Gödel), el TG (Tarski, Grothendieck) i el MK , però són extensions conservadora, no conservadora i pròpia, respectivament, de ZFC. في نظرية المجموعات، نظرية المجموعات حسب تسيرميلوءفراينكل (بالإنجليزية: Zermelo–Fraenkel set theory)‏ هي نظرية طورها عالما الرياضيات إرنست تسيرميلو وابراهام فرانكل في بداية القرن العشرين، من أجل إعطاء نظرية للمجموعات خالية من التناقضات و المفارقات، مفارقة راسل مثالا. In de verzamelingenleer, een deelgebied van de wiskunde, is de Zermelo-Fraenkel-verzamelingenleer, vernoemd naar de wiskundigen Ernst Zermelo en Abraham Fraenkel en vaak afgekort tot ZF, een van de verschillende axiomatische systemen, die in het begin van de twintigste eeuw werden voorgesteld om een verzamelingenleer te formuleren, zonder de paradoxen van de naïeve verzamelingenleer, zoals de paradox van Russell. In het bijzonder bevat ZF niet het , maar slechts een beperkte variant ervan. Daardoor is het in ZF niet voor elke eigenschap mogelijk een verzameling te vormen van alle objecten die deze eigenschap hebben. Zermelova-Fraenkelova teorie množin (ZF) je nejrozšířenější axiomatickou soustavou teorie množin, která je sama o sobě nebo v některých mírných modifikacích používána jako základ pro většinu dalších odvětví matematiky včetně algebry a matematické analýzy. ZF teorie může být například doplněna o axiom výběru - v takovém případě je označována jako ZFC (písmeno C značí výběr, z anglického choice). In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. En mathématiques, la théorie des ensembles de Zermelo-Fraenkel, abrégée en ZF, est une axiomatisation en logique du premier ordre de la théorie des ensembles telle qu'elle avait été développée dans le dernier quart du XIXe siècle par Georg Cantor. L'axiomatisation a été élaborée au début du XXe siècle par plusieurs mathématiciens dont Ernst Zermelo et Abraham Fraenkel mais aussi Thoralf Skolem. En raison de son statut particulier, on considère en général que l'axiome du choix ne fait pas partie de la définition de ZF et on note ZFC la théorie obtenue en ajoutant celui-ci. * * Систе́ма аксио́м Це́рмело — Фре́нкеля (ZF) — наиболее широко используемый вариант аксиоматической теории множеств, являющийся фактическим стандартом для оснований математики. Сформулирована Эрнстом Цермело в 1908 году как средство преодоления парадоксов теории множеств, и уточнена Абрахамом Френкелем в 1921 году. К этой системе аксиом часто добавляют аксиому выбора, и называют системой Цермело — Френкеля с аксиомой выбора (ZFC, англ. Zermelo—Fraenkel set theory with the axiom of Choice). 集合論において、ツェルメロ＝フレンケル集合論 (英: Zermelo-Fraenkel set theory) とは、ラッセルのパラドックスなどのパラドックスのない集合論を定式化するために20世紀初頭に提案された公理系である。名前は数学者のツェルメロとフレンケルにちなむ。歴史的に議論を呼んだ選択公理 (AC) を含むツェルメロ＝フレンケル集合論は公理的集合論の標準形式であり、今日では最も一般的な数学の基礎となっている。選択公理を含むツェルメロ＝フレンケル集合論はZFCと略される。Cは選択 (Choice) 公理を、 ZFは選択公理を除いたツェルメロ (Zermelo)＝フレンケル (Fraenkel) 集合論の公理を表す。 Aksjomaty Zermela-Fraenkla, aksjomatyka Zermela-Fraenkla – układ aksjomatów teorii mnogości zaproponowany przez Ernsta Zermela w 1904 roku i później uzupełniony przez Abrahama Fraenkla. Tym, co w istocie Fraenkel dodał do teorii Zermela, były funkcje. Dla aksjomatyki Zermela-Fraenkla stosuje się często wygodną symbolikę ZF. Ze względu na specyfikę jednego z jej aksjomatów zwanego aksjomatem wyboru, stosuje się także obok ZF oznaczenie ZFC dla zaznaczenia, że dowód jakiegoś twierdzenia wymaga lub nie wymaga zastosowania aksjomatu wyboru. 수학에서 체르멜로-프렝켈 집합론(영어: Zermelo-Fraenkel set theory, 약자 ZF)은 공리적 집합론 체계의 하나이다. 일반적으로 여기에 선택 공리를 추가해 사용하며 이를 선택 공리를 추가한 체르멜로-프렝켈 집합론(영어: Zermelo–Fraenkel set theory with the axiom of choice, 약자 ZFC)이라고 한다. ZF와 ZFC는 현대 수학의 표준적인 수학기초론으로 사용된다. Zermelo-Fraenkels mängdteori med urvalsaxiomet (förkortat ZFC) är ett axiomatiskt system för mängder, formaliserat i första ordningens logik med hjälp av ett språk som består av en icke-logisk symbol som betecknar elementrelationen, . ZFC betraktas allmänt som en adekvat axiomatisk grund för i stort sett all matematik. Två intressanta delteorier till ZFC är ZF och Z. Teorin är uppkallad efter matematikerna och . En lógica y matemáticas, los axiomas de Zermelo-Fraenkel, formulados por Ernst Zermelo y Adolf Fraenkel, son un sistema axiomático concebido para formular la teoría de conjuntos. Normalmente se abrevian como ZF o en su forma más común, complementados por el axioma de elección (axiom of choice), como ZFC. Na matemática, a teoria dos conjuntos de Zermelo-Fraenkel com o axioma da escolha, nomeada em homenagem aos matemáticos Ernst Zermelo e Abraham Fraenkel e comumente abreviada como ZFC, é um dos muitos sistemas axiomáticos que foram propostos no início do século XX para promover uma teoria dos conjuntos sem os paradoxos da teoria ingênua dos conjuntos, como o paradoxo de Russell. Especificamente, a ZFC não permite o axioma da compreensão. Atualmente, a ZFC é a forma padrão da teoria axiomática dos conjuntos, sendo o fundamento matemático mais comum.A ZFC deve formalizar uma única noção primitiva de um bem-fundado, para que cada indivíduo no domínio de discurso seja um conjunto. Desta forma, os axiomas da ZFC se referem apenas a conjuntos, e não urelementos (elementos de conjuntos que não s In matematica, e in particolare in logica matematica, la teoria degli insiemi di Zermelo-Fraenkel comprende gli assiomi standard della teoria assiomatica degli insiemi su cui, insieme con l'assioma di scelta, si basa tutta la matematica ordinaria secondo formulazioni moderne. Sono indicati come assiomi Zermelo–Fraenkel della teoria degli insiemi o sistema di assiomi di Zermelo-Fraenkel, e abbreviati con ZF. Теорія множин Цермело — Френкеля з аксіомою вибору (позначається ZFC) — найпоширеніша аксіоматика теорії множин, і, через це, найпоширеніша . ZFC містить єдине примітивне онтологічне поняття — множина, та єдине онтологічне припущення, що всі об'єкти в досліджуваному просторі (наприклад, всі математичні об'єкти) є множинами. Вводиться єдине бінарне відношення — приналежність до множини; позначає що множина є елементом множини , та записується як . Die Zermelo-Fraenkel-Mengenlehre ist eine verbreitete axiomatische Mengenlehre, die nach Ernst Zermelo und Abraham Adolf Fraenkel benannt ist. Sie ist heute Grundlage fast aller Zweige der Mathematik. Die Zermelo-Fraenkel-Mengenlehre ohne Auswahlaxiom wird durch ZF abgekürzt, mit Auswahlaxiom durch ZFC (wobei das C für das engl. Wort choice, also Auswahl oder Wahl steht).
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dbp:id: Zermelo-FraenkelSetTheory p/z130100
dbp:title: Zermelo-Fraenkel Set Theory ZFC
dbo:abstract: En mathématiques, la théorie des ensembles de Zermelo-Fraenkel, abrégée en ZF, est une axiomatisation en logique du premier ordre de la théorie des ensembles telle qu'elle avait été développée dans le dernier quart du XIXe siècle par Georg Cantor. L'axiomatisation a été élaborée au début du XXe siècle par plusieurs mathématiciens dont Ernst Zermelo et Abraham Fraenkel mais aussi Thoralf Skolem. Cette axiomatisation échappe aux paradoxes d'une théorie trop naïve des ensembles, comme le paradoxe de Russell, en écartant le schéma de compréhension non restreint (le fait que toute propriété puisse définir un ensemble, celui des objets ayant cette propriété) pour n'en conserver que certains cas particuliers utiles. De ce fait il existe des classes, des collections d’objets mathématiques définies par une propriété partagée par tous leurs membres, qui ne sont pas des ensembles. Dans la théorie ZF et ses extensions, ces classes dites classes propres ne correspondent pas à des objets de la théorie et ne peuvent être traitées qu'indirectement, à la différence de la très voisine théorie des classes de von Neumann-Bernays-Gödel (NBG). En raison de son statut particulier, on considère en général que l'axiome du choix ne fait pas partie de la définition de ZF et on note ZFC la théorie obtenue en ajoutant celui-ci. Les mathématiques usuelles peuvent être théoriquement développées entièrement dans le cadre de la théorie ZFC, éventuellement en ajoutant des axiomes, comme les axiomes de grands cardinaux, pour certains développements (ceux de la théorie des catégories par exemple). En ce sens il s'agit d'une théorie des fondements des mathématiques. En 1963 Paul Cohen utilise la théorie ZFC pour répondre à la question posée par Cantor de l'hypothèse du continu, en montrant qu'elle n'était pas conséquence des axiomes de cette théorie, et que l'axiome du choix n'était pas conséquence de la théorie ZF. La méthode qu'il développe, le forcing, est à l'origine de nombreux développements de la théorie des ensembles. La très grande majorité des travaux des théoriciens des ensembles depuis au moins cette époque se situent dans le cadre de la théorie ZF, de ses extensions, ou parfois de ses restrictions. La constructibilité, une méthode développée par Kurt Gödel en 1936 dans le cadre de la théorie NBG pour montrer que l'hypothèse du continu et l'axiome du choix n'étaient pas en contradiction avec les autres axiomes de la théorie des ensembles, s'adapte immédiatement à la théorie ZF. * Ernst Zermelo c. 1900 * Adolf Abraham Halevi Fraenkel Теорія множин Цермело — Френкеля з аксіомою вибору (позначається ZFC) — найпоширеніша аксіоматика теорії множин, і, через це, найпоширеніша . ZFC містить єдине примітивне онтологічне поняття — множина, та єдине онтологічне припущення, що всі об'єкти в досліджуваному просторі (наприклад, всі математичні об'єкти) є множинами. Вводиться єдине бінарне відношення — приналежність до множини; позначає що множина є елементом множини , та записується як . ZFC є теорією першого порядку; в ZFC містяться аксіоми, в яких використовується логіка першого порядку. Ці аксіоми описують: порівняння, існування, побудову та впорядкування множин. 策梅洛-弗兰克尔集合论（英語：Zermelo-Fraenkel Set Theory），含选择公理時常简写为ZFC，是在数学基础中最常用形式的公理化集合论，不含選擇公理的則簡寫為ZF。它是二十世纪早期为了建构一个不会导致类似罗素悖论的矛盾的集合理论所提出的一个公理系统。 في نظرية المجموعات، نظرية المجموعات حسب تسيرميلوءفراينكل (بالإنجليزية: Zermelo–Fraenkel set theory)‏ هي نظرية طورها عالما الرياضيات إرنست تسيرميلو وابراهام فرانكل في بداية القرن العشرين، من أجل إعطاء نظرية للمجموعات خالية من التناقضات و المفارقات، مفارقة راسل مثالا. Aksjomaty Zermela-Fraenkla, aksjomatyka Zermela-Fraenkla – układ aksjomatów teorii mnogości zaproponowany przez Ernsta Zermela w 1904 roku i później uzupełniony przez Abrahama Fraenkla. Tym, co w istocie Fraenkel dodał do teorii Zermela, były funkcje. Dla aksjomatyki Zermela-Fraenkla stosuje się często wygodną symbolikę ZF. Ze względu na specyfikę jednego z jej aksjomatów zwanego aksjomatem wyboru, stosuje się także obok ZF oznaczenie ZFC dla zaznaczenia, że dowód jakiegoś twierdzenia wymaga lub nie wymaga zastosowania aksjomatu wyboru. In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes. There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets and there is a new set containing exactly and . Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, intended to formalize set membership, which is usually denoted . The formula means that the set is a member of the set (which is also read, " is an element of " or " is in "). The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms (see Axiom of choice § Independence) and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem. Na matemática, a teoria dos conjuntos de Zermelo-Fraenkel com o axioma da escolha, nomeada em homenagem aos matemáticos Ernst Zermelo e Abraham Fraenkel e comumente abreviada como ZFC, é um dos muitos sistemas axiomáticos que foram propostos no início do século XX para promover uma teoria dos conjuntos sem os paradoxos da teoria ingênua dos conjuntos, como o paradoxo de Russell. Especificamente, a ZFC não permite o axioma da compreensão. Atualmente, a ZFC é a forma padrão da teoria axiomática dos conjuntos, sendo o fundamento matemático mais comum.A ZFC deve formalizar uma única noção primitiva de um bem-fundado, para que cada indivíduo no domínio de discurso seja um conjunto. Desta forma, os axiomas da ZFC se referem apenas a conjuntos, e não urelementos (elementos de conjuntos que não são conjuntos) ou classes (coleções de objetos matemáticos definidos por uma propriedade em comum de seus membros). Os axiomas da ZFC previnem seus modelos de possuírem urelementos, e classes próprias só podem ser tratadas indiretamente.Formalmente, a ZFC é uma teoria de estruturas de lógica de primeira ordem. A assinatura possui igualdade e uma única relação binária primitiva, que é a pertinência, normalmente denotada por ∈. A fórmula a ∈ b significa que o conjunto a é membro do conjunto b (que também é lido como "a é elemento de b" ou "a está em b").Existem várias formulações equivalentes dos axiomas da ZFC. A maioria de seus axiomas formulam a existência de conjuntos particulares definidos a partir de outros conjuntos. Por exemplo, o axioma do par diz que, dados dois conjuntos quaisquer a e b, existe um novo conjunto {a, b} contendo exatamente a e b. Outros axiomas descrevem propriedades da pertinência de conjuntos. Um dos objetivos dos axiomas da ZFC é que cada axioma deve ser verdade se interpretado como uma afirmação sobre a coleção de todos os conjuntos do universo de von Neumann (também conhecido como a hierarquia cumulativa).A metamatemática da ZFC foi estudada extensivamente. Resultados marcantes dessa área estabeleceram a independência da hipótese do contínuo da ZFC, e o axioma da escolha dos axiomas restantes da ZFC. Zermelo-Fraenkels mängdteori med urvalsaxiomet (förkortat ZFC) är ett axiomatiskt system för mängder, formaliserat i första ordningens logik med hjälp av ett språk som består av en icke-logisk symbol som betecknar elementrelationen, . ZFC betraktas allmänt som en adekvat axiomatisk grund för i stort sett all matematik. Två intressanta delteorier till ZFC är ZF och Z. Teorin är uppkallad efter matematikerna och . In matematica, e in particolare in logica matematica, la teoria degli insiemi di Zermelo-Fraenkel comprende gli assiomi standard della teoria assiomatica degli insiemi su cui, insieme con l'assioma di scelta, si basa tutta la matematica ordinaria secondo formulazioni moderne. Sono indicati come assiomi Zermelo–Fraenkel della teoria degli insiemi o sistema di assiomi di Zermelo-Fraenkel, e abbreviati con ZF. Gli assiomi sono il risultato del lavoro di Thoralf Skolem del 1922, basato su lavori precedenti di Abraham Fraenkel nello stesso anno, che si basa sul sistema assiomatico sviluppato da Ernst Zermelo nel 1908 (teoria degli insiemi di Zermelo). Il sistema assiomatico è scritto mediante un linguaggio del primo ordine; ha un numero infinito di assiomi poiché viene usato uno schema di assiomi. Un sistema alternativo finito viene dato dagli assiomi di von Neumann-Bernays-Gödel, che aggiungono il concetto di una classe in aggiunta a quello di un insieme; esso è "equivalente" nel senso che qualsiasi teorema riguardo agli insiemi che può essere provato in un sistema può essere provato nell'altro. Si indica con la sigla ZFC il sistema formale dato dagli assiomi di Zermelo - Fraenkel con l'aggiunta dell'assioma della scelta: data una famiglia non vuota di insiemi non vuoti esiste una funzione che ad ogni insieme della famiglia fa corrispondere un suo elemento. La "C" nella sigla è l'iniziale di choice (scelta in inglese): per lo stesso motivo, l'assioma della scelta viene spesso abbreviato con le lettere AC (la "A" sta per "axiom"). En lógica y matemáticas, los axiomas de Zermelo-Fraenkel, formulados por Ernst Zermelo y Adolf Fraenkel, son un sistema axiomático concebido para formular la teoría de conjuntos. Normalmente se abrevian como ZF o en su forma más común, complementados por el axioma de elección (axiom of choice), como ZFC. Durante el siglo XIX algunos matemáticos trataron de llevar a cabo un proceso de formalización de la matemática a partir de la teoría de conjuntos. Gottlob Frege intentó culminar este proceso creando una axiomática de la teoría de conjuntos. Bertrand Russell descubrió en 1901 una contradicción, la llamada paradoja de Russell. Consecuentemente, a principios del siglo XX se realizaron varios intentos alternativos y hoy en día ZFC se ha convertido en el estándar de las teorías axiomáticas de conjuntos. La Teoria de conjunts de Zermelo-Fraenkel (ZFC) és el conjunt d'axiomes canònic de la teoria de conjunts. El seu nom es deu als matemàtics que la van desenvolupar: Ernst Zermelo i Abraham Fraenkel i la C per la inclusió de l'axioma d'elecció (Choice en anglès). Existeixen altres conjunts d'axiomes de la Teoria de Conjunts com el NBG (von Neumann, Bernays, Gödel), el TG (Tarski, Grothendieck) i el MK , però són extensions conservadora, no conservadora i pròpia, respectivament, de ZFC. Zermelova-Fraenkelova teorie množin (ZF) je nejrozšířenější axiomatickou soustavou teorie množin, která je sama o sobě nebo v některých mírných modifikacích používána jako základ pro většinu dalších odvětví matematiky včetně algebry a matematické analýzy. ZF teorie může být například doplněna o axiom výběru - v takovém případě je označována jako ZFC (písmeno C značí výběr, z anglického choice). Principem ZF je postupná konstrukce množin - objektů množinového univerza - z několika základních axiomů tak, aby vzniklá teorie byla dostatečně bohatá (je třeba umožnit existenci nespočetných množin typu reálných čísel, existenci shora neomezené řady nekonečných kardinalit), ale zároveň neumožňovala existenci množin použitých v paradoxech klasické intuitivně pojaté teorie množin (např. Russellův paradox, Buraliův-Fortiho paradox). Систе́ма аксио́м Це́рмело — Фре́нкеля (ZF) — наиболее широко используемый вариант аксиоматической теории множеств, являющийся фактическим стандартом для оснований математики. Сформулирована Эрнстом Цермело в 1908 году как средство преодоления парадоксов теории множеств, и уточнена Абрахамом Френкелем в 1921 году. К этой системе аксиом часто добавляют аксиому выбора, и называют системой Цермело — Френкеля с аксиомой выбора (ZFC, англ. Zermelo—Fraenkel set theory with the axiom of Choice). Эта система аксиом записана на языке логики первого порядка. Существуют и другие системы; например, система аксиом фон Неймана — Бернайса — Гёделя (NBG) наряду с множествами рассматривает так называемые классы объектов, при этом она равносильна ZF в том смысле, что любая теорема о множествах (то есть не упоминающая о классах), доказуемая в одной системе, также доказуема и в другой. 수학에서 체르멜로-프렝켈 집합론(영어: Zermelo-Fraenkel set theory, 약자 ZF)은 공리적 집합론 체계의 하나이다. 일반적으로 여기에 선택 공리를 추가해 사용하며 이를 선택 공리를 추가한 체르멜로-프렝켈 집합론(영어: Zermelo–Fraenkel set theory with the axiom of choice, 약자 ZFC)이라고 한다. ZF와 ZFC는 현대 수학의 표준적인 수학기초론으로 사용된다. 集合論において、ツェルメロ＝フレンケル集合論 (英: Zermelo-Fraenkel set theory) とは、ラッセルのパラドックスなどのパラドックスのない集合論を定式化するために20世紀初頭に提案された公理系である。名前は数学者のツェルメロとフレンケルにちなむ。歴史的に議論を呼んだ選択公理 (AC) を含むツェルメロ＝フレンケル集合論は公理的集合論の標準形式であり、今日では最も一般的な数学の基礎となっている。選択公理を含むツェルメロ＝フレンケル集合論はZFCと略される。Cは選択 (Choice) 公理を、 ZFは選択公理を除いたツェルメロ (Zermelo)＝フレンケル (Fraenkel) 集合論の公理を表す。 Die Zermelo-Fraenkel-Mengenlehre ist eine verbreitete axiomatische Mengenlehre, die nach Ernst Zermelo und Abraham Adolf Fraenkel benannt ist. Sie ist heute Grundlage fast aller Zweige der Mathematik. Die Zermelo-Fraenkel-Mengenlehre ohne Auswahlaxiom wird durch ZF abgekürzt, mit Auswahlaxiom durch ZFC (wobei das C für das engl. Wort choice, also Auswahl oder Wahl steht). In de verzamelingenleer, een deelgebied van de wiskunde, is de Zermelo-Fraenkel-verzamelingenleer, vernoemd naar de wiskundigen Ernst Zermelo en Abraham Fraenkel en vaak afgekort tot ZF, een van de verschillende axiomatische systemen, die in het begin van de twintigste eeuw werden voorgesteld om een verzamelingenleer te formuleren, zonder de paradoxen van de naïeve verzamelingenleer, zoals de paradox van Russell. In het bijzonder bevat ZF niet het , maar slechts een beperkte variant ervan. Daardoor is het in ZF niet voor elke eigenschap mogelijk een verzameling te vormen van alle objecten die deze eigenschap hebben. Vandaag de dag is de Zermelo-Fraenkel-verzamelingenleer met keuzeaxioma (afgekort tot ZFC, waarbij de C voor het Engelse Choice staat) de standaardvorm van de axiomatische verzamelingenleer en als zodanig het meest gebruikelijke fundament van de wiskunde.
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Subject Item: dbr:Finitist_set_theory
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Subject Item: dbr:Richard's_paradox
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Subject Item: dbr:Naive_Set_Theory_(book)
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Subject Item: dbr:Principle_of_explosion
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Subject Item: dbr:Scott–Potter_set_theory
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Subject Item: dbr:Morse–Kelley_set_theory
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Subject Item: dbr:Urelement
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Subject Item: dbr:Set-builder_notation
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Subject Item: dbr:Set-theoretic_definition_of_natural_numbers
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Subject Item: dbr:Set-theoretic_topology
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Subject Item: dbr:Nonstandard_calculus
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Subject Item: dbr:S_(set_theory)
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Subject Item: dbr:Ultrafinitism
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Subject Item: dbr:Philosophy_of_logic
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Subject Item: dbr:Teichmüller–Tukey_lemma
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Subject Item: dbr:S_and_L_spaces
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Subject Item: dbr:Reinhardt_cardinal
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Subject Item: dbr:Tarski's_theorem_about_choice
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Subject Item: dbr:Tarski's_undefinability_theorem
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Subject Item: dbr:Outline_of_logic
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Subject Item: dbr:Zermelo_set_theory
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Subject Item: dbr:Vitali_set
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Subject Item: dbr:Uncountable_set
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Subject Item: dbr:The_Higher_Infinite
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Subject Item: dbr:Wetzel's_problem
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Subject Item: dbr:Worldly_cardinal
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Subject Item: dbr:Axioms_of_ZF
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Subject Item: dbr:Takeuti–Feferman–Buchholz_ordinal
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Subject Item: dbr:ZFD
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Subject Item: dbr:Zermelo–Fraenkel_axioms
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Subject Item: dbr:Zfc
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Subject Item: dbr:ZFC_Set_Theory
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Subject Item: dbr:ZFC_Set_theory
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Subject Item: dbr:ZFC_set
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Subject Item: dbr:ZFC_set_theory
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Subject Item: dbr:ZF_axioms
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Subject Item: dbr:ZF_set_theory
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Subject Item: dbr:Zermelo-Fraenkel
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Subject Item: dbr:Zermelo-Fraenkel-Skolem_set_theory
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Subject Item: dbr:Zermelo-Fraenkel_axiom
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Subject Item: dbr:Zermelo-Fraenkel_axiomatization
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Subject Item: dbr:Zermelo-Fraenkel_axioms
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Subject Item: dbr:Zermelo-Fraenkel_framework
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Subject Item: dbr:Zermelo-Frankel
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Subject Item: dbr:Zermelo-Frankel_axiom
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Subject Item: dbr:Zermelo-Frankel_axioms
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Subject Item: dbr:Zermelo-Frankel_set_theory
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Subject Item: dbr:Zermelo-Fränkel
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Subject Item: dbr:Zermelo-Fränkel_set_theory
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Subject Item: dbr:Zermelo-frankel
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Subject Item: dbr:Zermelo_Fraenkel_set_theory
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Subject Item: dbr:Zermelo–Fraenkel
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Subject Item: dbr:Zermelo–Fraenkel_axiom
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Subject Item: dbr:Zermelo–Fraenkel_axiomatization
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Subject Item: dbr:Zermelo–Fraenkel_framework
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Subject Item: dbr:Zermelo–Fraenkel_set_theory_with_the_axiom_of_choice
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Subject Item: dbr:Zermelo–Frankel_set_theory
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Subject Item: dbr:Zermelo−Fraenkel_set_theory
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Subject Item: wikipedia-en:Zermelo–Fraenkel_set_theory
foaf:primaryTopic: dbr:Zermelo–Fraenkel_set_theory