@prefix rdf: . @prefix dbpedia: . @prefix opencyc: . dbpedia:Set_theory rdf:type opencyc:Mx4rvVi3_JwpEbGdrcN5Y29ycA . @prefix owl: . dbpedia:Set_theory owl:sameAs opencyc:Mx4rv36W7ZwpEbGdrcN5Y29ycA . @prefix ns4: . dbpedia:Set_theory owl:sameAs ns4:Mx4rv36W7ZwpEbGdrcN5Y29ycA , . @prefix foaf: . @prefix ns6: . dbpedia:Set_theory foaf:page ns6:Set_theory . @prefix dbpprop: . dbpedia:Set_theory dbpprop:reference . @prefix rdfs: . dbpedia:Set_theory rdfs:label "Teoria de conjunts"@ca , "Halmazelm\u00E9let"@hu , "Teorie mno\u017Ein"@cs , "\u0422\u0435\u043E\u0440\u0456\u044F \u043C\u043D\u043E\u0436\u0438\u043D"@uk , "Teoria degli insiemi"@it , "Mengdel\u00E6re"@no , "Teoria mnogo\u015Bci"@pl , "\u0422\u0435\u043E\u0440\u0438\u044F \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432"@ru , "M\u00E4ngdteori"@sv , "Teor\u00EDa de conjuntos"@es , "Th\u00E9orie des ensembles"@fr , "\u96C6\u5408\u8AD6"@ja , "Mengenlehre"@de , "Verzamelingenleer"@nl , "\u0410\u043B\u0433\u0435\u0431\u0440\u0430 \u043C\u043D\u043E\u0436\u0438\u043D"@uk , "Set theory"@en , "\u96C6\u5408\u8BBA"@zh , "Joukko-oppi"@fi , "Teoria dos conjuntos"@pt ; dbpprop:abstract "\u0422\u0435\u043E\u0301\u0440\u0438\u044F \u043C\u043D\u043E\u0301\u0436\u0435\u0441\u0442\u0432\u00A0\u2014 \u0440\u0430\u0437\u0434\u0435\u043B \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438, \u0432 \u043A\u043E\u0442\u043E\u0440\u043E\u043C \u0438\u0437\u0443\u0447\u0430\u044E\u0442\u0441\u044F \u043E\u0431\u0449\u0438\u0435 \u0441\u0432\u043E\u0439\u0441\u0442\u0432\u0430 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432. \u0422\u0435\u043E\u0440\u0438\u044F \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432 \u043B\u0435\u0436\u0438\u0442 \u0432 \u043E\u0441\u043D\u043E\u0432\u0435 \u0431\u043E\u043B\u044C\u0448\u0438\u043D\u0441\u0442\u0432\u0430 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u0435\u0441\u043A\u0438\u0445 \u0434\u0438\u0441\u0446\u0438\u043F\u043B\u0438\u043D; \u043E\u043D\u0430 \u043E\u043A\u0430\u0437\u0430\u043B\u0430 \u0433\u043B\u0443\u0431\u043E\u043A\u043E\u0435 \u0432\u043B\u0438\u044F\u043D\u0438\u0435 \u043D\u0430 \u043F\u043E\u043D\u0438\u043C\u0430\u043D\u0438\u0435 \u043F\u0440\u0435\u0434\u043C\u0435\u0442\u0430 \u0441\u0430\u043C\u043E\u0439 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438."@ru , "La teoria degli insiemi svolge un ruolo importante per i fondamenti della matematica e attualmente si colloca nell'ambito della logica matematica. Prima della met\u00E0 del XIX secolo la nozione di insieme veniva considerata solo come qualcosa di intuitivo e generico. Essa \u00E8 stata inizialmente sviluppata nella seconda met\u00E0 del XIX secolo dal matematico tedesco Georg Cantor, \u00E8 stata al centro dei dibattiti sui fondamenti dal 1890 al 1930 ed ha ricevuto le prime sistemazioni assiomatiche per merito di Ernst Zermelo, Adolf Fraenkel, Paul Bernays, Kurt G\u00F6del, John von Neumann e Thoralf Skolem. In questo periodo si sono assestati due sistemi di assiomi chiamati rispettivamente sistema assiomatico di Zermelo-Fraenkel e sistema assiomatico di Von Neumann-Bernays-G\u00F6del. Successivamente si sono affrontate le tematiche riguardanti il problema della completezza dei sistemi di assiomi, i rapporti con la teoria della calcolabilit\u00E0 e la compatibilit\u00E0 dei sistemi di assiomi con l'assioma della scelta e con assiomi equivalenti o simili. Attualmente si dispone di differenti consolidate teorie formali degli insiemi. Inoltre, per chi non ha tempo e gusto per affrontare le astrattezze delle teorie assiomatiche (cosa che sembra accada anche alla maggior parte dei matematici), sono disponibili, e molto utili, esposizioni pi\u00F9 intuitive che costituiscono la cosiddetta teoria na\u00EFve degli insiemi. Elenchiamo le entit\u00E0 principali della teoria degli insiemi."@it , "Joukko-oppi on joukkoja tutkiva matematiikan haara. Joukko-opin perustaja Georg Cantor m\u00E4\u00E4ritteli joukon olevan toisistaan erotettavien objektien (olioiden) yhdistelm\u00E4. Intuitiivisesti t\u00E4m\u00E4 m\u00E4\u00E4ritelm\u00E4 toimii useimmiten edelleen, vaikka joukon formaali m\u00E4\u00E4ritteleminen osoittautui my\u00F6hemmin monimutkaisemmaksi. N\u00E4it\u00E4 objekteja kutsutaan joukon alkioiksi ja ne voivat olla ihmisen havaintoon tai ajatukseen perustuvia. Oleellista on tiet\u00E4\u00E4 mist\u00E4 tahansa objektista kuuluuko se tiettyyn joukkoon vai ei. Joukko-oppi kuuluu logiikan kanssa matematiikan perusteisiin, josta pyrit\u00E4\u00E4n johtamaan formaalisti muu matematiikka. Sit\u00E4 pidet\u00E4\u00E4n siis yleismaailmallisena modernin tieteellisen matematiikan esitt\u00E4mismuotona. Yleiskieless\u00E4 joukko-opilla viitataan usein 1970-luvun huonomaineiseen koulutuskokeiluun, jossa joukko-oppi tuotiin uutena metodina matematiikan perusteiden opetukseen."@fi , "A halmazelm\u00E9let a matematika egyik alapvet\u0151 tudom\u00E1ny\u00E1ga, mely a halmaz fogalm\u00E1n kereszt\u00FCl a v\u00E9gtelen sok elem\u0171 matematikai \u00F6sszess\u00E9gekkel, illetve a logika matematikai vizsg\u00E1lat\u00E1val foglalkozik. A halmazelm\u00E9let alapvet\u0151 jelent\u0151ss\u00E9g\u00E9t az mutatja, hogy a matematikai \u00E9s logikai fogalmak d\u00F6nt\u0151 t\u00F6bbs\u00E9ge kezelhet\u0151 a halmaz fogalma seg\u00EDts\u00E9g\u00E9vel. Sok\u00E1ig uralkod\u00F3 volt az a n\u00E9zet, hogy a teljes matematika megalapozhat\u00F3 a halmazelm\u00E9let seg\u00EDts\u00E9g\u00E9vel . A halmazelm\u00E9let megalkot\u00F3ja Georg Cantor n\u00E9met matematikus, aki a v\u00E9gtelen halmazokra \u00E9s a halmazok sz\u00E1moss\u00E1gaira vonatkoz\u00F3 \u00FAtt\u00F6r\u0151 kutat\u00E1saival nemcsak a halmazelm\u00E9letet ind\u00EDtotta \u00FAtj\u00E1ra, hanem alapvet\u0151en, drasztikusan megv\u00E1ltoztatta a matematika eg\u00E9sz arculat\u00E1t. Elm\u00E9lete, az ut\u00F3bb ellentmond\u00E1sosnak bizonyult naiv halmazelm\u00E9let, megreform\u00E1l\u00E1sra szorult ugyan, de alapkoncepci\u00F3i be\u00E9p\u00FCltek a matematika minden szeglet\u00E9be. Az 20. sz\u00E1zad elej\u00E9n Zermelo, Fraenkel, Neumann \u00E9s G\u00F6del munk\u00E1ss\u00E1ga r\u00E9v\u00E9n siker\u00FClt axiomatikus alapokra hozni a hamazelm\u00E9letet ."@hu , "Teoria dos conjuntos \u00E9 a teoria matem\u00E1tica que trata das propriedades dos conjuntos. Ela tem sua origem nos trabalhos do matem\u00E1tico russo Georg Cantor, e se baseia na ideia de definir conjunto como uma no\u00E7\u00E3o primitiva. Tamb\u00E9m chamada de teoria ing\u00EAnua ou intuitiva devido \u00E0 descoberta de v\u00E1rias antinomias relacionadas \u00E0 defini\u00E7\u00E3o de conjunto. Estas antinomias na teoria dos conjuntos conduziram a matem\u00E1tica a axiomatizar as teorias matem\u00E1ticas, com influ\u00EAncias profundas sobre a l\u00F3gica e os fundamentos da matem\u00E1tica."@pt , "La th\u00E9orie des ensembles est une branche des math\u00E9matiques, cr\u00E9\u00E9e par le math\u00E9maticien allemand Georg Cantor \u00E0 la fin du XIX si\u00E8cle. La th\u00E9orie des ensembles se donne comme primitives les notions d'ensemble et d'appartenance, \u00E0 partir desquelles elle reconstruit les objets usuels des math\u00E9matiques : fonctions, relations, entiers naturels, relatifs, rationnels, nombres r\u00E9els, complexes... C'est pourquoi la th\u00E9orie des ensembles est consid\u00E9r\u00E9e comme une th\u00E9orie fondamentale dont Hilbert a pu dire qu'elle \u00E9tait un \u00AB paradis \u00BB cr\u00E9\u00E9 par Cantor pour les math\u00E9maticiens. En plus de proposer un fondement aux math\u00E9matiques, Cantor introduisait avec la th\u00E9orie des ensembles des concepts radicalement nouveaux, et notamment l'id\u00E9e qu'il existe plusieurs types d'infini que l'on peut mesurer et comparer au moyen de nouveaux nombres. \u00C0 cause de sa modernit\u00E9, la th\u00E9orie des ensembles fut \u00E2prement controvers\u00E9e. Cantor ne l'avait pas vraiment formalis\u00E9e, et au d\u00E9but du XX\u00E8me si\u00E8cle, la d\u00E9couverte de paradoxes tels que le paradoxe de Russell semblait en remettre en cause les principes. Pour r\u00E9soudre ces probl\u00E8mes, on adopta une approche formelle qui conduisit \u00E0 plusieurs syst\u00E8mes axiomatiques, le plus connu \u00E9tant les axiomes de ZF, mais \u00E9galement la th\u00E9orie des classes de von Neumann ou la th\u00E9orie des types de Russell."@fr , "\u0410\u043B\u0433\u0435\u0431\u0440\u0430 \u043C\u043D\u043E\u0436\u0438\u043D \u2014 \u0440\u043E\u0437\u0434\u0456\u043B \u0442\u0435\u043E\u0440\u0456\u0457 \u043C\u043D\u043E\u0436\u0438\u043D, \u044F\u043A\u0438\u0439 \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454 \u0437\u0430\u043A\u043E\u043D\u0438 \u043A\u043E\u043C\u043F\u043E\u0437\u0438\u0446\u0456\u0457 \u043C\u043D\u043E\u0436\u0438\u043D, \u0432\u0438\u0445\u043E\u0434\u044F\u0447\u0438 \u0437 \u043E\u0441\u043D\u043E\u0432\u043D\u0438\u0445 \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0435\u0439 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0439 \u043D\u0430\u0434 \u043D\u0438\u043C\u0438, \u0430 \u0442\u0430\u043A\u043E\u0436 \u043F\u0440\u043E\u043F\u043E\u043D\u0443\u0454 \u043F\u0435\u0432\u043D\u0443 \u0441\u0438\u0441\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0443 \u043F\u0440\u043E\u0446\u0435\u0434\u0443\u0440\u0443 \u0434\u043B\u044F \u043E\u0431\u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044F \u0442\u0435\u043E\u0440\u0435\u0442\u0438\u043A\u043E-\u043C\u043D\u043E\u0436\u0438\u043D\u043D\u0438\u0445 \u0440\u0456\u0432\u043D\u044F\u043D\u044C \u0442\u0430 \u0441\u043F\u0456\u0432\u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u044C. \u0417 \u0442\u043E\u0447\u043A\u0438 \u0437\u043E\u0440\u0443 \u0430\u0431\u0441\u0442\u0440\u0430\u043A\u0442\u043D\u043E\u0457 \u0430\u043B\u0433\u0435\u0431\u0440\u0438 \u0430\u043B\u0433\u0435\u0431\u0440\u0430 \u043C\u043D\u043E\u0436\u0438\u043D \u2014 \u0446\u0435 \u043A\u0456\u043B\u044C\u0446\u0435 K \u043F\u0456\u0434\u043C\u043D\u043E\u0436\u0438\u043D \u043C\u043D\u043E\u0436\u0438\u043D\u0438 R, \u0449\u043E \u043C\u0456\u0441\u0442\u0438\u0442\u044C R."@uk , "La teoria de conjunts \u00E9s la branca de les matem\u00E0tiques que estudia els conjunts. El primer estudi formal sobre el tema va ser realitzat pel matem\u00E0tic alemany Georg Cantor el segle XIX."@ca , "For the musical concepts, see Set theory (music). \"Nested set\" redirects here. For nested set models, see there. Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo\u2013Fraenkel axioms, with the axiom of choice, are the best-known. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects. Elementary operations such as set union and intersection can be studied in this context. More advanced concepts such as cardinality are a standard part of the undergraduate mathematics curriculum. Set theory, formalized using first-order logic, is the most common foundational system for mathematics. Beyond its use as a foundational system, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals."@en , "Teorie mno\u017Ein je matematick\u00E1 teorie, kter\u00E1 se zab\u00FDv\u00E1 mno\u017Einami. Z form\u00E1ln\u00EDho pohledu jsou ve\u0161ker\u00E9 objekty modern\u00ED matematiky mno\u017Einy \u2013 \u010D\u00EDslo je form\u00E1ln\u011B mno\u017Eina, zobrazen\u00ED je form\u00E1ln\u011B mno\u017Eina, stejn\u011B jako v\u0161echny ostatn\u00ED relace. Zjednodu\u0161en\u011B by se dalo \u0159\u00EDci, \u017Ee zat\u00EDmco matematick\u00E1 logika poskytuje modern\u00ED matematice n\u00E1stroje, jak pracovat \u2013 jazyk matematick\u00FDch v\u011Bt a d\u016Fkaz\u016F, teorie mno\u017Ein j\u00ED poskytuje potravu - sv\u011Bt objekt\u016F, na kter\u00FDch m\u016F\u017Ee t\u00EDmto n\u00E1strojem (jazykem) pracovat. Po\u017Eadavkem samoz\u0159ejm\u011B je, aby tento sv\u011Bt byl natolik obs\u00E1hl\u00FD a rozmanit\u00FD, ale z\u00E1rove\u0148 \u201Elogick\u00FD\u201C, aby v n\u011Bm mohly ostatn\u00ED matematick\u00E9 teorie smyslupln\u011B existovat, a z\u00E1rove\u0148 vnit\u0159n\u011B bezesporn\u00FD z pohledu form\u00E1ln\u00ED logiky. P\u0159\u00EDkladem mohou b\u00FDt p\u0159irozen\u00E1 \u010D\u00EDsla. Z intuitivn\u00EDho pohledu b\u011B\u017En\u00E9 \u0161kolsk\u00E9 matematiky se jedn\u00E1 o \u201Epo\u010Dty\u201C objekt\u016F v jejich kone\u010Dn\u00FDch (tedy v re\u00E1ln\u00E9m vesm\u00EDru vlastn\u011B v\u0161ech mysliteln\u00FDch) souborech. Na t\u011Bchto po\u010Dtech si pak m\u016F\u017Eu zav\u00E9st nejr\u016Fzn\u011Bj\u0161\u00ED dal\u0161\u00ED pojmy a struktury, kter\u00E9 m\u011B zaj\u00EDmaj\u00ED \u2013 se\u0159adit si je podle velikosti, zav\u00E9st operace s\u010D\u00EDt\u00E1n\u00ED, od\u010D\u00EDt\u00E1n\u00ED, zav\u00E9st pojem d\u011Blitelnost, kongruence a tak d\u00E1le. Z pohledu formalizovan\u00E9 teorie mno\u017Ein jsou p\u0159irozen\u00E1 \u010D\u00EDsla mno\u017Einy \u2013 a to mno\u017Einy, kter\u00E9 maj\u00ED takovou strukturu a vz\u00E1jemn\u00E9 vztahy, aby na nich bylo mo\u017En\u00E9 modelovat v\u0161echny vlastnosti, o kter\u00E9 se zaj\u00EDm\u00E1 v\u00FD\u0161e uveden\u00E1 b\u011B\u017En\u00E1 teorie p\u0159irozen\u00FDch \u010D\u00EDsel. \u00DAlohou teorie mno\u017Ein je tedy zajistit, aby v pokud mo\u017Eno bezesporn\u00E9 form\u011B existovala ve vesm\u00EDru matematiky struktura mno\u017Ein modeluj\u00EDc\u00EDch intuitivn\u00ED chov\u00E1n\u00ED p\u0159irozen\u00FDch \u010D\u00EDsel \u2013 jejich konkr\u00E9tn\u00ED vlastnosti jako je d\u011Blitelnost pak u\u017E p\u0159enech\u00E1v\u00E1 konkr\u00E9tn\u00ED matematick\u00E9 discipl\u00EDn\u011B. Dodejme je\u0161t\u011B, \u017Ee ji\u017E na \u00FArovni zd\u00E1nliv\u011B jednoduch\u00E9 struktury, jako jsou p\u0159irozen\u00E1 \u010D\u00EDsla, klade teorie mno\u017Ein n\u011Bkter\u00E9 netrivi\u00E1ln\u00ED filosofick\u00E9 ot\u00E1zky \u2013 nap\u0159\u00EDklad m\u00E1 smysl existence (nekone\u010Dn\u00E9) mno\u017Einy, kter\u00E1 obsahuje v\u0161echna p\u0159irozen\u00E1 \u010D\u00EDsla? V\u011Bt\u0161inov\u00FD n\u00E1zor reprezentovan\u00FD Zermelo-Fraenkelovou teori\u00ED mno\u017Ein odpov\u00EDd\u00E1 kladn\u011B t\u00EDm, \u017Ee obsahuje axiom nekone\u010Dna. Jin\u00E9 soustavy teorie se s t\u00EDmto probl\u00E9mem vypo\u0159\u00E1d\u00E1vaj\u00ED mnohem opatrn\u011Bji a d\u00EDky tomu tak\u00E9 obt\u00ED\u017En\u011Bji \u2013 viz nap\u0159\u00EDklad Vop\u011Bnkova Alternativn\u00ED teorie mno\u017Ein. O tom, \u017Ee se jedn\u00E1 o hodn\u011B zapeklit\u00FD probl\u00E9m, se lze p\u0159esv\u011Bd\u010Dit nap\u0159\u00EDklad v G\u00F6delov\u00FDch v\u011Bt\u00E1ch o ne\u00FAplnosti."@cs , "De verzamelingenleer vormt sinds het begin van de twintigste eeuw een van de grondslagen van de wiskunde. De verzamelingenleer betreft de bestudering en formalisering van het begrip verzameling, en ondersteunt daarmee de axiomatische onderbouwing van andere deelgebieden van de wiskunde."@nl , "\u0422\u0435\u043E\u0301\u0440\u0456\u044F \u043C\u043D\u043E\u0436\u0438\u0301\u043D \u2014 \u0440\u043E\u0437\u0434\u0456\u043B \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438, \u0432 \u044F\u043A\u043E\u043C\u0443 \u0432\u0438\u0432\u0447\u0430\u044E\u0442\u044C\u0441\u044F \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0456 \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0456 \u043C\u043D\u043E\u0436\u0438\u043D. \u0422\u0435\u043E\u0440\u0456\u044F \u043C\u043D\u043E\u0436\u0438\u043D \u043B\u0435\u0436\u0438\u0442\u044C \u0432 \u043E\u0441\u043D\u043E\u0432\u0456 \u0431\u0456\u043B\u044C\u0448\u043E\u0441\u0442\u0456 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0438\u0445 \u0434\u0438\u0441\u0446\u0438\u043F\u043B\u0456\u043D; \u0432\u043E\u043D\u0430 \u0437\u0440\u043E\u0431\u0438\u043B\u0430 \u0433\u043B\u0438\u0431\u043E\u043A\u0438\u0439 \u0432\u043F\u043B\u0438\u0432 \u043D\u0430 \u0440\u043E\u0437\u0443\u043C\u0456\u043D\u043D\u044F \u043F\u0440\u0435\u0434\u043C\u0435\u0442\u0443 \u0441\u0430\u043C\u043E\u0457 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438."@uk , "La teor\u00EDa de conjuntos es una divisi\u00F3n de las matem\u00E1ticas que estudia los conjuntos. El primer estudio formal sobre el tema fue realizado por el matem\u00E1tico alem\u00E1n Georg Cantor, Gottlob Frege y Julius Wilhelm Richard Dedekind en el Siglo XIX y m\u00E1s tarde reformulada por Zermelo. El concepto de conjunto es intuitivo y se podr\u00EDa definir como una \"colecci\u00F3n de objetos\"; as\u00ED, se puede hablar de un conjunto de personas, ciudades, gafas, lapiceros o del conjunto de objetos que hay en un momento dado encima de una mesa. Un conjunto est\u00E1 bien definido si se sabe si un determinado elemento pertenece o no al conjunto. El conjunto de los bol\u00EDgrafos azules est\u00E1 bien definido, porque a la vista de un bol\u00EDgrafo se puede saber si es azul o no. El conjunto de las personas altas no est\u00E1 bien definido, porque a la vista de una persona, no siempre se podr\u00E1 decir si es alta o no, o puede haber distintas personas, que opinen si esa persona es alta o no lo es. En el siglo XIX, seg\u00FAn Frege, los elementos de un conjunto se defin\u00EDan s\u00F3lo por tal o cual propiedad. Actualmente la teor\u00EDa de conjuntos est\u00E1 bien definida por el sistema ZFC. Sin embargo, sigue siendo c\u00E9lebre la definici\u00F3n que public\u00F3 Cantor"@es , "Teoria mnogo\u015Bci \u2013 dzia\u0142 matematyki a zarazem logiki matematycznej zapocz\u0105tkowany przez niemieckiego matematyka Georga Cantora pod koniec XIX wieku. Pocz\u0105tkowo wzbudza\u0142a wiele kontrowersji, jednak wraz z post\u0119pem matematyki zacz\u0119\u0142a ona pe\u0142ni\u0107 rol\u0119 fundamentu, na kt\u00F3rym opiera si\u0119 wi\u0119kszo\u015B\u0107 matematycznych rozwa\u017Ca\u0144. Na przestrzeni lat j\u0119zyk i metody teorii mnogo\u015Bci przenikn\u0119\u0142y do wielu innych dzia\u0142\u00F3w matematyki (na przyk\u0142ad w algebrze rozwa\u017Ca si\u0119 obiekty teoriomnogo\u015Bciowe zwane ultrafiltrami). Teoria mnogo\u015Bci rozwijana jest tak\u017Ce jako samodzielna dyscyplina."@pl , "\u96C6\u5408\u8AD6\u662F\u4E00\u9580\u7814\u7A76\u96C6\u5408\uFF08\u7531\u4E00\u5806\u62BD\u8C61\u7269\u4EF6\u69CB\u6210\u7684\u6574\u9AD4\uFF09\u7684\u6578\u5B78\u7406\u8AD6\uFF0C\u5305\u542B\u96C6\u5408\u3001\u5143\u7D20\u548C\u6210\u54E1\u95DC\u4FC2\u7B49\u6578\u5B78\u4E2D\u6700\u57FA\u672C\u7684\u6982\u5FF5\u3002\u5728\u5927\u591A\u6578\u73FE\u4EE3\u6578\u5B78\u7684\u516C\u5F0F\u5316\u4E2D\uFF0C\u96C6\u5408\u8AD6\u63D0\u4F9B\u4E86\u8981\u5982\u4F55\u63CF\u8FF0\u6578\u5B78\u7269\u4EF6\u7684\u8A9E\u8A00\u3002\u96C6\u5408\u8AD6\u548C\u908F\u8F2F\u8207\u4E00\u968E\u908F\u8F2F\u5171\u540C\u69CB\u6210\u4E86\u6578\u5B78\u7684\u516C\u7406\u5316\u57FA\u790E\uFF0C\u4EE5\u672A\u5B9A\u7FA9\u7684\u300C\u96C6\u5408\u300D\u8207\u300C\u96C6\u5408\u6210\u54E1\u300D\u7B49\u8853\u8A9E\u4F86\u5F62\u5F0F\u5316\u5730\u5EFA\u69CB\u6578\u5B78\u7269\u4EF6\u3002 \u5728\u6A38\u7D20\u96C6\u5408\u8AD6\u4E2D\uFF0C\u96C6\u5408\u662F\u88AB\u7576\u505A\u4E00\u5806\u7269\u4EF6\u69CB\u6210\u7684\u6574\u9AD4\u4E4B\u985E\u7684\u81EA\u8B49\u6982\u5FF5\u3002 \u5728\u516C\u7406\u5316\u96C6\u5408\u8AD6\u4E2D\uFF0C\u96C6\u5408\u548C\u96C6\u5408\u6210\u54E1\u4E26\u4E0D\u76F4\u63A5\u88AB\u5B9A\u7FA9\uFF0C\u800C\u662F\u5148\u898F\u7BC4\u53EF\u4EE5\u63CF\u8FF0\u5176\u6027\u8CEA\u7684\u4E00\u4E9B\u516C\u7406\u3002\u5728\u6B64\u4E00\u60F3\u6CD5\u4E4B\u4E0B\uFF0C\u96C6\u5408\u548C\u96C6\u5408\u6210\u54E1\u662F\u6709\u5982\u5728\u6B50\u5F0F\u5E7E\u4F55\u4E2D\u7684\u9EDE\u548C\u7DDA\uFF0C\u800C\u4E0D\u88AB\u76F4\u63A5\u5B9A\u7FA9\u3002"@zh , "Mengdel\u00E6re er den matematiske teorien om mengder, som representerer samlinger av abstrakte objekter. Den omfatter hverdagslige begreper som barn arbeider med i barneskolen, om samlinger av objekter, og elementene i og medlemskap i, slike samlinger. I det meste av moderne matematisk formalisme, er mengdel\u00E6re spr\u00E5ket som brukes for \u00E5 beskrive matematiske objekter. Det er (sammen med logikk og predikatkalkulus et av de aksiomatiske grunnlag for matematikk, som gj\u00F8r at matematiske objekter kan konstrueres formelt fra de udefinerte termene \"mengde\" og \"element i mengde\". Den er selv en gren av matematikken og et aktivt felt for matematisk forskning. I naiv mengdel\u00E6re er mengder introdusert og forst\u00E5tt ved hjelp av begrepet mengder som samlinger av objekter sett p\u00E5 som et hele. I aksiomatisk mengdel\u00E6re, blir begrepene mengde og element i mengde definert indirekte ved f\u00F8rst \u00E5 postulere visse aksiomer som spesifiserer deres egenskaper. I denne versjonen er mengder og elementer fundamentale begreper som punkt og linje i euklidsk geometri, og er ikke selv direkte definert."@no , "M\u00E4ngdteori har en allm\u00E4n och en specifik betydelse."@sv , "Die Mengenlehre ist ein grundlegendes Teilgebiet der Mathematik. Zahlreiche mathematische Disziplinen werden heute auf der Mengenlehre aufgebaut, darunter Algebra, Analysis, Ma\u00DFtheorie, Stochastik und Topologie."@de , "\u96C6\u5408\u8AD6\uFF08\u3057\u3085\u3046\u3054\u3046\u308D\u3093\u3001\u72EC: Mengenlehre\u3001\u82F1: set theory\uFF09\u306F\u3001\u96C6\u5408\u3068\u3088\u3070\u308C\u308B\u6570\u5B66\u7684\u5BFE\u8C61\u3092\u3042\u3064\u304B\u3046\u6570\u5B66\u7406\u8AD6\u3067\u3042\u308B\u3002"@ja ; rdfs:comment ""@zh , "M\u00E4ngdteori har en allm\u00E4n och en specifik betydelse."@sv , "\u0410\u043B\u0433\u0435\u0431\u0440\u0430 \u043C\u043D\u043E\u0436\u0438\u043D \u2014 \u0440\u043E\u0437\u0434\u0456\u043B \u0442\u0435\u043E\u0440\u0456\u0457 \u043C\u043D\u043E\u0436\u0438\u043D, \u044F\u043A\u0438\u0439 \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454 \u0437\u0430\u043A\u043E\u043D\u0438 \u043A\u043E\u043C\u043F\u043E\u0437\u0438\u0446\u0456\u0457 \u043C\u043D\u043E\u0436\u0438\u043D, \u0432\u0438\u0445\u043E\u0434\u044F\u0447\u0438 \u0437 \u043E\u0441\u043D\u043E\u0432\u043D\u0438\u0445 \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0435\u0439 \u043E\u043F\u0435\u0440\u0430\u0446\u0456\u0439 \u043D\u0430\u0434 \u043D\u0438\u043C\u0438, \u0430 \u0442\u0430\u043A\u043E\u0436 \u043F\u0440\u043E\u043F\u043E\u043D\u0443\u0454 \u043F\u0435\u0432\u043D\u0443 \u0441\u0438\u0441\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0443 \u043F\u0440\u043E\u0446\u0435\u0434\u0443\u0440\u0443 \u0434\u043B\u044F \u043E\u0431\u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044F \u0442\u0435\u043E\u0440\u0435\u0442\u0438\u043A\u043E-\u043C\u043D\u043E\u0436\u0438\u043D\u043D\u0438\u0445 \u0440\u0456\u0432\u043D\u044F\u043D\u044C \u0442\u0430 \u0441\u043F\u0456\u0432\u0432\u0456\u0434\u043D\u043E\u0448\u0435\u043D\u044C."@uk , "For the musical concepts, see Set theory (music). \"Nested set\" redirects here. For nested set models, see there. Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Cantor and Dedekind in the 1870s."@en , "Joukko-oppi on joukkoja tutkiva matematiikan haara. Joukko-opin perustaja Georg Cantor m\u00E4\u00E4ritteli joukon olevan toisistaan erotettavien objektien (olioiden) yhdistelm\u00E4. Intuitiivisesti t\u00E4m\u00E4 m\u00E4\u00E4ritelm\u00E4 toimii useimmiten edelleen, vaikka joukon formaali m\u00E4\u00E4ritteleminen osoittautui my\u00F6hemmin monimutkaisemmaksi. N\u00E4it\u00E4 objekteja kutsutaan joukon alkioiksi ja ne voivat olla ihmisen havaintoon tai ajatukseen perustuvia."@fi , "La teoria degli insiemi svolge un ruolo importante per i fondamenti della matematica e attualmente si colloca nell'ambito della logica matematica. Prima della met\u00E0 del XIX secolo la nozione di insieme veniva considerata solo come qualcosa di intuitivo e generico."@it , "Teoria dos conjuntos \u00E9 a teoria matem\u00E1tica que trata das propriedades dos conjuntos. Ela tem sua origem nos trabalhos do matem\u00E1tico russo Georg Cantor, e se baseia na ideia de definir conjunto como uma no\u00E7\u00E3o primitiva. Tamb\u00E9m chamada de teoria ing\u00EAnua ou intuitiva devido \u00E0 descoberta de v\u00E1rias antinomias relacionadas \u00E0 defini\u00E7\u00E3o de conjunto."@pt , "La teor\u00EDa de conjuntos es una divisi\u00F3n de las matem\u00E1ticas que estudia los conjuntos. El primer estudio formal sobre el tema fue realizado por el matem\u00E1tico alem\u00E1n Georg Cantor, Gottlob Frege y Julius Wilhelm Richard Dedekind en el Siglo XIX y m\u00E1s tarde reformulada por Zermelo."@es , "\u0422\u0435\u043E\u0301\u0440\u0438\u044F \u043C\u043D\u043E\u0301\u0436\u0435\u0441\u0442\u0432\u00A0\u2014 \u0440\u0430\u0437\u0434\u0435\u043B \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438, \u0432 \u043A\u043E\u0442\u043E\u0440\u043E\u043C \u0438\u0437\u0443\u0447\u0430\u044E\u0442\u0441\u044F \u043E\u0431\u0449\u0438\u0435 \u0441\u0432\u043E\u0439\u0441\u0442\u0432\u0430 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432. \u0422\u0435\u043E\u0440\u0438\u044F \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432 \u043B\u0435\u0436\u0438\u0442 \u0432 \u043E\u0441\u043D\u043E\u0432\u0435 \u0431\u043E\u043B\u044C\u0448\u0438\u043D\u0441\u0442\u0432\u0430 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u0435\u0441\u043A\u0438\u0445 \u0434\u0438\u0441\u0446\u0438\u043F\u043B\u0438\u043D; \u043E\u043D\u0430 \u043E\u043A\u0430\u0437\u0430\u043B\u0430 \u0433\u043B\u0443\u0431\u043E\u043A\u043E\u0435 \u0432\u043B\u0438\u044F\u043D\u0438\u0435 \u043D\u0430 \u043F\u043E\u043D\u0438\u043C\u0430\u043D\u0438\u0435 \u043F\u0440\u0435\u0434\u043C\u0435\u0442\u0430 \u0441\u0430\u043C\u043E\u0439 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438."@ru , "La th\u00E9orie des ensembles est une branche des math\u00E9matiques, cr\u00E9\u00E9e par le math\u00E9maticien allemand Georg Cantor \u00E0 la fin du XIX si\u00E8cle. La th\u00E9orie des ensembles se donne comme primitives les notions d'ensemble et d'appartenance, \u00E0 partir desquelles elle reconstruit les objets usuels des math\u00E9matiques : fonctions, relations, entiers naturels, relatifs, rationnels, nombres r\u00E9els, complexes..."@fr , "La teoria de conjunts \u00E9s la branca de les matem\u00E0tiques que estudia els conjunts. El primer estudi formal sobre el tema va ser realitzat pel matem\u00E0tic alemany Georg Cantor el segle XIX."@ca , "Teorie mno\u017Ein je matematick\u00E1 teorie, kter\u00E1 se zab\u00FDv\u00E1 mno\u017Einami. Z form\u00E1ln\u00EDho pohledu jsou ve\u0161ker\u00E9 objekty modern\u00ED matematiky mno\u017Einy \u2013 \u010D\u00EDslo je form\u00E1ln\u011B mno\u017Eina, zobrazen\u00ED je form\u00E1ln\u011B mno\u017Eina, stejn\u011B jako v\u0161echny ostatn\u00ED relace."@cs , "Mengdel\u00E6re er den matematiske teorien om mengder, som representerer samlinger av abstrakte objekter. Den omfatter hverdagslige begreper som barn arbeider med i barneskolen, om samlinger av objekter, og elementene i og medlemskap i, slike samlinger. I det meste av moderne matematisk formalisme, er mengdel\u00E6re spr\u00E5ket som brukes for \u00E5 beskrive matematiske objekter."@no , "Teoria mnogo\u015Bci \u2013 dzia\u0142 matematyki a zarazem logiki matematycznej zapocz\u0105tkowany przez niemieckiego matematyka Georga Cantora pod koniec XIX wieku. Pocz\u0105tkowo wzbudza\u0142a wiele kontrowersji, jednak wraz z post\u0119pem matematyki zacz\u0119\u0142a ona pe\u0142ni\u0107 rol\u0119 fundamentu, na kt\u00F3rym opiera si\u0119 wi\u0119kszo\u015B\u0107 matematycznych rozwa\u017Ca\u0144."@pl , "\u96C6\u5408\u8AD6\uFF08\u3057\u3085\u3046\u3054\u3046\u308D\u3093\u3001\u72EC: Mengenlehre\u3001\u82F1: set theory\uFF09\u306F\u3001\u96C6\u5408\u3068\u3088\u3070\u308C\u308B\u6570\u5B66\u7684\u5BFE\u8C61\u3092\u3042\u3064\u304B\u3046\u6570\u5B66\u7406\u8AD6\u3067\u3042\u308B\u3002"@ja , "A halmazelm\u00E9let a matematika egyik alapvet\u0151 tudom\u00E1ny\u00E1ga, mely a halmaz fogalm\u00E1n kereszt\u00FCl a v\u00E9gtelen sok elem\u0171 matematikai \u00F6sszess\u00E9gekkel, illetve a logika matematikai vizsg\u00E1lat\u00E1val foglalkozik. A halmazelm\u00E9let alapvet\u0151 jelent\u0151ss\u00E9g\u00E9t az mutatja, hogy a matematikai \u00E9s logikai fogalmak d\u00F6nt\u0151 t\u00F6bbs\u00E9ge kezelhet\u0151 a halmaz fogalma seg\u00EDts\u00E9g\u00E9vel. Sok\u00E1ig uralkod\u00F3 volt az a n\u00E9zet, hogy a teljes matematika megalapozhat\u00F3 a halmazelm\u00E9let seg\u00EDts\u00E9g\u00E9vel ."@hu , "\u0422\u0435\u043E\u0301\u0440\u0456\u044F \u043C\u043D\u043E\u0436\u0438\u0301\u043D \u2014 \u0440\u043E\u0437\u0434\u0456\u043B \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438, \u0432 \u044F\u043A\u043E\u043C\u0443 \u0432\u0438\u0432\u0447\u0430\u044E\u0442\u044C\u0441\u044F \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u0456 \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0456 \u043C\u043D\u043E\u0436\u0438\u043D. \u0422\u0435\u043E\u0440\u0456\u044F \u043C\u043D\u043E\u0436\u0438\u043D \u043B\u0435\u0436\u0438\u0442\u044C \u0432 \u043E\u0441\u043D\u043E\u0432\u0456 \u0431\u0456\u043B\u044C\u0448\u043E\u0441\u0442\u0456 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0438\u0445 \u0434\u0438\u0441\u0446\u0438\u043F\u043B\u0456\u043D; \u0432\u043E\u043D\u0430 \u0437\u0440\u043E\u0431\u0438\u043B\u0430 \u0433\u043B\u0438\u0431\u043E\u043A\u0438\u0439 \u0432\u043F\u043B\u0438\u0432 \u043D\u0430 \u0440\u043E\u0437\u0443\u043C\u0456\u043D\u043D\u044F \u043F\u0440\u0435\u0434\u043C\u0435\u0442\u0443 \u0441\u0430\u043C\u043E\u0457 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u0438."@uk , "Die Mengenlehre ist ein grundlegendes Teilgebiet der Mathematik. Zahlreiche mathematische Disziplinen werden heute auf der Mengenlehre aufgebaut, darunter Algebra, Analysis, Ma\u00DFtheorie, Stochastik und Topologie."@de , "De verzamelingenleer vormt sinds het begin van de twintigste eeuw een van de grondslagen van de wiskunde. De verzamelingenleer betreft de bestudering en formalisering van het begrip verzameling, en ondersteunt daarmee de axiomatische onderbouwing van andere deelgebieden van de wiskunde."@nl . @prefix skos: . @prefix ns10: . dbpedia:Set_theory skos:subject ns10:Set_theory , ns10:Formal_methods , ns10:Mathematical_logic . @prefix ns11: . dbpedia:Set_theory dbpprop:relatedInstance ns11:portal1 , ns11:portal2 . @prefix ns12: . dbpedia:Set_theory dbpprop:hasPhotoCollection ns12:Set_theory . @prefix dbpedia-owl: . dbpedia:Georg_Cantor dbpedia-owl:knownFor dbpedia:Set_theory . @prefix ns14: . dbpedia:Georg_Cantor ns14:knownFor dbpedia:Set_theory ; dbpprop:knownFor dbpedia:Set_theory . dbpedia:Willard_Van_Orman_Quine dbpprop:mainInterests dbpedia:Set_theory . dbpedia:Ordinary_set_theory dbpprop:redirect dbpedia:Set_theory . dbpedia-owl:knownFor dbpedia:Set_theory ; ns14:knownFor dbpedia:Set_theory ; dbpprop:knownFor dbpedia:Set_theory . dbpedia:Axiomatic_Set_Theory dbpprop:redirect dbpedia:Set_theory . dbpedia:Set_Theory dbpprop:redirect dbpedia:Set_theory . dbpedia:Classical_set_theory dbpprop:redirect dbpedia:Set_theory . dbpedia:Nested_set dbpprop:redirect dbpedia:Set_theory . dbpedia:SetTheory dbpprop:redirect dbpedia:Set_theory . @prefix ns15: . ns15:OldVersion dbpprop:redirect dbpedia:Set_theory . dbpprop:redirect dbpedia:Set_theory . dbpedia:Axiomatic_set_theories dbpprop:redirect dbpedia:Set_theory . dbpedia:Set_theorist dbpprop:redirect dbpedia:Set_theory . dbpedia:Formal_set_theory dbpprop:redirect dbpedia:Set_theory . dbpedia:Set-theoretic dbpprop:redirect dbpedia:Set_theory . dbpedia:Theory_of_sets dbpprop:redirect dbpedia:Set_theory . dbpedia:Transfinite_set_theory dbpprop:redirect dbpedia:Set_theory .