"Na l\u00F3gica matem\u00E1tica e na filosofia, O paradoxo de Skolem \u00E9 uma aparente contradi\u00E7\u00E3o que surge a partir do Teorema L\u00F6wenheim\u2013Skolem. Thoralf Skolem (1922) foi o primeiro a discutir os aspectos aparentemente contradit\u00F3rios do teorema, e descobrir a relatividade das no\u00E7\u00F5es dos conjuntos te\u00F3ricos hoje conhecida como n\u00E3o-absoluto. Embora n\u00E3o seja uma real antinomia como o paradoxo de russel, o resultado normalmente \u00E9 chamado de paradoxo, e foi descrito como \"um estado paradoxal das coisas\" por Skolem (1922: p., 295). O paradoxo de Skolem diz que cada axiomatiza\u00E7\u00E3o cont\u00E1vel da teoria dos conjuntos na l\u00F3gica de primeira ordem, se \u00E9 consistente, tem um modelo que \u00E9 cont\u00E1vel. Isso parece contradit\u00F3rio porque \u00E9 poss\u00EDvel provar, a partir desses mesmos axiomas, uma frase que diz intuitivamente (ou que diz precisamente o modelo padr\u00E3o da teoria) que existem conjuntos que n\u00E3o s\u00E3o cont\u00E1veis. Assim, a aparente contradi\u00E7\u00E3o \u00E9 que um modelo que \u00E9 a pr\u00F3pria contabilidade, e que, portanto, cont\u00E9m apenas conjuntos cont\u00E1veis, satisfaz a primeira frase para que intuitivamente afirma \"h\u00E1 incont\u00E1veis conjuntos\". Uma explica\u00E7\u00E3o matem\u00E1tica do paradoxo, mostrando que n\u00E3o \u00E9 uma contradi\u00E7\u00E3o na matem\u00E1tica, foi apresentada por Skolem (1922). O trabalho de Skolem foi recebido por Ernst Zermelo, que argumentou contra as limita\u00E7\u00F5es da l\u00F3gica de primeira ordem, mas o resultado rapidamente veio a ser aceito pela comunidade matem\u00E1tica. As implica\u00E7\u00F5es filos\u00F3ficas do paradoxo de Skolem foram bastante estudadas. Uma linha de pesquisa questiona se \u00E9 correto afirmar que qualquer senten\u00E7a da l\u00F3gica de primeira ordem de fato afirma \"h\u00E1 incont\u00E1veis conjuntos\". Essa linha de pensamento pode ser estendida para questionar se qualquer conjunto \u00E9 incont\u00E1vel em um sentido absoluto. Mais recentemente, o artigo \"Models and Reality\" de Hilary Putnam e as respostar a ele, levou a um renovado interesse nos aspectos filos\u00F3ficos dos resultados de Skolem."@pt . . . . "1348798"^^ . . . . . . . . "\u65AF\u79D1\u4F26\u6096\u8BBA"@zh . . . . . . "16897"^^ . . . "\u041F\u0430\u0440\u0430\u0434\u043E\u043A\u0441 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u2014 \u043F\u0440\u043E\u0442\u0438\u0432\u043E\u0440\u0435\u0447\u0438\u0432\u043E\u0435 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043D\u0438\u0435, \u043E\u043F\u0438\u0441\u0430\u043D\u043D\u043E\u0435 \u0432\u043F\u0435\u0440\u0432\u044B\u0435 \u043D\u043E\u0440\u0432\u0435\u0436\u0441\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C \u0422\u0443\u0440\u0430\u043B\u044C\u0444\u043E\u043C \u0421\u043A\u0443\u043B\u0435\u043C\u043E\u043C, \u0441\u0432\u044F\u0437\u0430\u043D\u043D\u043E\u0435 \u0441 \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u043D\u0438\u0435\u043C \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u041B\u0451\u0432\u0435\u043D\u0433\u0435\u0439\u043C\u0430 \u2014 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u0434\u043B\u044F \u0430\u043A\u0441\u0438\u043E\u043C\u0430\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0442\u0435\u043E\u0440\u0438\u0438 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432. \u0412 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0420\u0430\u0441\u0441\u0435\u043B\u0430, \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u041A\u0430\u043D\u0442\u043E\u0440\u0430, \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0411\u0443\u0440\u0430\u043B\u0438-\u0424\u043E\u0440\u0442\u0438, \u0433\u0434\u0435 \u043F\u0440\u0438 \u043F\u043E\u043C\u043E\u0449\u0438 \u043B\u043E\u0433\u0438\u0447\u0435\u0441\u043A\u0438 \u0432\u0435\u0440\u043D\u044B\u0445 \u0432\u044B\u0432\u043E\u0434\u043E\u0432 \u043E\u0431\u043D\u0430\u0440\u0443\u0436\u0438\u0432\u0430\u0435\u0442\u0441\u044F \u043F\u0440\u043E\u0442\u0438\u0432\u043E\u0440\u0435\u0447\u0438\u0435, \u00AB\u0437\u0430\u043C\u0430\u0441\u043A\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u043E\u0435\u00BB \u0432 \u0438\u0441\u0445\u043E\u0434\u043D\u044B\u0445 \u043F\u043E\u0441\u044B\u043B\u043A\u0430\u0445, \u00AB\u043F\u0440\u043E\u0442\u0438\u0432\u043E\u0440\u0435\u0447\u0438\u0435\u00BB \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u0432\u043E\u0437\u043D\u0438\u043A\u0430\u0435\u0442 \u043E\u0442 \u043E\u0448\u0438\u0431\u043A\u0438 \u0432 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043D\u0438\u044F\u0445, \u0438 \u0430\u043A\u043A\u0443\u0440\u0430\u0442\u043D\u043E\u0435 \u0440\u0430\u0441\u0441\u043C\u043E\u0442\u0440\u0435\u043D\u0438\u0435 \u0432\u043E\u043F\u0440\u043E\u0441\u0430 \u043F\u043E\u043A\u0430\u0437\u044B\u0432\u0430\u0435\u0442, \u0447\u0442\u043E \u044D\u0442\u043E \u043B\u0438\u0448\u044C \u043C\u043D\u0438\u043C\u044B\u0439 \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441. \u0422\u0435\u043C \u043D\u0435 \u043C\u0435\u043D\u0435\u0435, \u0440\u0430\u0441\u0441\u043C\u043E\u0442\u0440\u0435\u043D\u0438\u0435 \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u0438\u043C\u0435\u0435\u0442 \u0431\u043E\u043B\u044C\u0448\u0443\u044E \u0434\u0438\u0434\u0430\u043A\u0442\u0438\u0447\u0435\u0441\u043A\u0443\u044E \u0446\u0435\u043D\u043D\u043E\u0441\u0442\u044C."@ru . . "\u041F\u0430\u0440\u0430\u0434\u043E\u043A\u0441 \u0421\u043A\u0443\u043B\u0435\u043C\u0430"@ru . "Paradoxa de Skolem"@ca . . . . . . . . . . . . . "Paradoks Skolema \u2013 pozorna sprzeczno\u015B\u0107 dotycz\u0105ca teorii mnogo\u015Bci wynikaj\u0105ca z twierdzenia L\u00F6wenheima-Skolema. Jego autorem jest norweski logik Thoralf Skolem."@pl . . . "La paradoxa de Skolem \u00E9s una paradoxa que apareix a teoria de conjunts i l\u00F2gica com a conseq\u00FC\u00E8ncia paradoxal del . La paradoxa va ser introdu\u00EFda en la discussi\u00F3 del matem\u00E0tic noruec Thoralf Skolem en un article de 1922. D'acord amb el teorema de L\u00F6wenheim-Skolem tota axiomatitzaci\u00F3 d'una teoria matem\u00E0tica utilitzant un llenguatge de primer ordre que tingui un nombre finit de signes distints admet un model numerable. Per exemple, la formalitzaci\u00F3 de la teoria de conjunts usual de Zermelo-Fraenkel afirma l'exist\u00E8ncia de conjunts no numerables. Per\u00F2 existeix un model que satisf\u00E0 els axiomes d'aquesta teoria que \u00E9s ell mateix numerable. La resoluci\u00F3 de la paradoxa requereix distingir adequadament entre els conjunts com objecte descrit per la teoria i els conjunts com a instrument utilitzat per construir un model. La paradoxa a m\u00E9s reflecteix una limitaci\u00F3 de la l\u00F2gica de primer ordre que implica que per una mateixa teoria poden existir models de cardinalitat molt diferents. A m\u00E9s de models numerables, naturalment l'axiomatitzaci\u00F3 de Zermelo-Fraenkel admet models no numerables, que intu\u00EFtivament no resulten paradoxals."@ca . . . "Na l\u00F3gica matem\u00E1tica e na filosofia, O paradoxo de Skolem \u00E9 uma aparente contradi\u00E7\u00E3o que surge a partir do Teorema L\u00F6wenheim\u2013Skolem. Thoralf Skolem (1922) foi o primeiro a discutir os aspectos aparentemente contradit\u00F3rios do teorema, e descobrir a relatividade das no\u00E7\u00F5es dos conjuntos te\u00F3ricos hoje conhecida como n\u00E3o-absoluto. Embora n\u00E3o seja uma real antinomia como o paradoxo de russel, o resultado normalmente \u00E9 chamado de paradoxo, e foi descrito como \"um estado paradoxal das coisas\" por Skolem (1922: p., 295)."@pt . "Paradoks Skolema \u2013 pozorna sprzeczno\u015B\u0107 dotycz\u0105ca teorii mnogo\u015Bci wynikaj\u0105ca z twierdzenia L\u00F6wenheima-Skolema. Jego autorem jest norweski logik Thoralf Skolem."@pl . . "In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward L\u00F6wenheim\u2013Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox and was described as a \"paradoxical state of affairs\" by Skolem (1922: p. 295)."@en . . "Paradoxe de Skolem"@fr . . . . . . . . . . . "In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward L\u00F6wenheim\u2013Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox and was described as a \"paradoxical state of affairs\" by Skolem (1922: p. 295). Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states \"there are uncountable sets\". A mathematical explanation of the paradox, showing that it is not a contradiction in mathematics, was given by Skolem (1922). Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but the result quickly came to be accepted by the mathematical community. The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states \"there are uncountable sets\". This line of thought can be extended to question whether any set is uncountable in an absolute sense. More recently, the paper \"Models and Reality\" by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of Skolem's result."@en . . . . . . "In de wiskundige logica en filosofie, is de paradox van Skolem een schijnbare tegenstrijdigheid die voortvloeit uit de neerwaartse stelling van L\u00F6wenheim-Skolem. Thoralf Skolem was in 1922 de eerste die de schijnbaar tegenstrijdige aspecten van deze stelling besprak en die de relativiteit van de verzamelingtheoretische noties ontdekte die nu bekendstaan als niet-. Hoewel de paradox van Skolem geen werkelijke antinomie is, zoals de Russellparadox, wordt het resultaat meestal een paradox genoemd, en werd hij door Skolem beschreven als een \"paradoxale stand van zaken\"."@nl . "Skolem's paradox"@en . "Paradoxo de Skolem"@pt . "Dragalin"@en . . . . . "A.G."@en . . . . . . "Paradoks Skolema"@pl . . . . . . . . . . . . "1108978273"^^ . . "Paradox van Skolem"@nl . . . . . . "En logique math\u00E9matique et en philosophie analytique, le paradoxe de Skolem est une cons\u00E9quence troublante du th\u00E9or\u00E8me de L\u00F6wenheim-Skolem en th\u00E9orie des ensembles. Il affirme qu'une th\u00E9orie des ensembles, comme ZFC, si elle a un mod\u00E8le, a un mod\u00E8le d\u00E9nombrable, bien que l'on puisse par ailleurs d\u00E9finir une formule qui exprime l'existence d'ensembles non d\u00E9nombrables. C'est un paradoxe au sens premier de ce terme : il va contre le sens commun, mais ce n'est pas une antinomie, une contradiction que l'on pourrait d\u00E9duire dans la th\u00E9orie."@fr . . . . . "Skolem-Paradox"@de . . . . . "\u041F\u0430\u0440\u0430\u0434\u043E\u043A\u0441 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u2014 \u043F\u0440\u043E\u0442\u0438\u0432\u043E\u0440\u0435\u0447\u0438\u0432\u043E\u0435 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043D\u0438\u0435, \u043E\u043F\u0438\u0441\u0430\u043D\u043D\u043E\u0435 \u0432\u043F\u0435\u0440\u0432\u044B\u0435 \u043D\u043E\u0440\u0432\u0435\u0436\u0441\u043A\u0438\u043C \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u043A\u043E\u043C \u0422\u0443\u0440\u0430\u043B\u044C\u0444\u043E\u043C \u0421\u043A\u0443\u043B\u0435\u043C\u043E\u043C, \u0441\u0432\u044F\u0437\u0430\u043D\u043D\u043E\u0435 \u0441 \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u043D\u0438\u0435\u043C \u0442\u0435\u043E\u0440\u0435\u043C\u044B \u041B\u0451\u0432\u0435\u043D\u0433\u0435\u0439\u043C\u0430 \u2014 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u0434\u043B\u044F \u0430\u043A\u0441\u0438\u043E\u043C\u0430\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0442\u0435\u043E\u0440\u0438\u0438 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432. \u0412 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0420\u0430\u0441\u0441\u0435\u043B\u0430, \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u041A\u0430\u043D\u0442\u043E\u0440\u0430, \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0411\u0443\u0440\u0430\u043B\u0438-\u0424\u043E\u0440\u0442\u0438, \u0433\u0434\u0435 \u043F\u0440\u0438 \u043F\u043E\u043C\u043E\u0449\u0438 \u043B\u043E\u0433\u0438\u0447\u0435\u0441\u043A\u0438 \u0432\u0435\u0440\u043D\u044B\u0445 \u0432\u044B\u0432\u043E\u0434\u043E\u0432 \u043E\u0431\u043D\u0430\u0440\u0443\u0436\u0438\u0432\u0430\u0435\u0442\u0441\u044F \u043F\u0440\u043E\u0442\u0438\u0432\u043E\u0440\u0435\u0447\u0438\u0435, \u00AB\u0437\u0430\u043C\u0430\u0441\u043A\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u043E\u0435\u00BB \u0432 \u0438\u0441\u0445\u043E\u0434\u043D\u044B\u0445 \u043F\u043E\u0441\u044B\u043B\u043A\u0430\u0445, \u00AB\u043F\u0440\u043E\u0442\u0438\u0432\u043E\u0440\u0435\u0447\u0438\u0435\u00BB \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u0432\u043E\u0437\u043D\u0438\u043A\u0430\u0435\u0442 \u043E\u0442 \u043E\u0448\u0438\u0431\u043A\u0438 \u0432 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043D\u0438\u044F\u0445, \u0438 \u0430\u043A\u043A\u0443\u0440\u0430\u0442\u043D\u043E\u0435 \u0440\u0430\u0441\u0441\u043C\u043E\u0442\u0440\u0435\u043D\u0438\u0435 \u0432\u043E\u043F\u0440\u043E\u0441\u0430 \u043F\u043E\u043A\u0430\u0437\u044B\u0432\u0430\u0435\u0442, \u0447\u0442\u043E \u044D\u0442\u043E \u043B\u0438\u0448\u044C \u043C\u043D\u0438\u043C\u044B\u0439 \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441. \u0422\u0435\u043C \u043D\u0435 \u043C\u0435\u043D\u0435\u0435, \u0440\u0430\u0441\u0441\u043C\u043E\u0442\u0440\u0435\u043D\u0438\u0435 \u043F\u0430\u0440\u0430\u0434\u043E\u043A\u0441\u0430 \u0421\u043A\u0443\u043B\u0435\u043C\u0430 \u0438\u043C\u0435\u0435\u0442 \u0431\u043E\u043B\u044C\u0448\u0443\u044E \u0434\u0438\u0434\u0430\u043A\u0442\u0438\u0447\u0435\u0441\u043A\u0443\u044E \u0446\u0435\u043D\u043D\u043E\u0441\u0442\u044C."@ru . . "En logique math\u00E9matique et en philosophie analytique, le paradoxe de Skolem est une cons\u00E9quence troublante du th\u00E9or\u00E8me de L\u00F6wenheim-Skolem en th\u00E9orie des ensembles. Il affirme qu'une th\u00E9orie des ensembles, comme ZFC, si elle a un mod\u00E8le, a un mod\u00E8le d\u00E9nombrable, bien que l'on puisse par ailleurs d\u00E9finir une formule qui exprime l'existence d'ensembles non d\u00E9nombrables. C'est un paradoxe au sens premier de ce terme : il va contre le sens commun, mais ce n'est pas une antinomie, une contradiction que l'on pourrait d\u00E9duire dans la th\u00E9orie."@fr . . . "\u5728\u6570\u7406\u903B\u8F91\u4E2D\uFF0C\u7279\u522B\u662F\u96C6\u5408\u8BBA\u4E2D\uFF0CSkolem \u6096\u8BBA\u662F\u5411\u4E0B L\u00F6wenheim-Skolem\u5B9A\u7406\u7684\u76F4\u63A5\u7ED3\u679C\uFF0C\u5B83\u58F0\u79F0\u6240\u6709\u4E00\u9636\u8BED\u8A00\u7684\u53E5\u5B50\u7684\u6A21\u578B\u90FD\u6709\u4E00\u4E2A\u521D\u7B49\u7B49\u4EF7\u7684\u53EF\u6570\u5B50\u6A21\u578B\u3002 \u8FD9\u4E2A\u6096\u8BBA\u89C1\u4E8EZermelo-Fraenkel \u96C6\u5408\u8BBA\u4E2D\u3002\u5EB7\u6258\u5C14\u5728 1874\u5E74\u53D1\u8868\u7684\u66F4\u65E9\u7684\u7ED3\u679C\u662F\uFF0C\u5B58\u5728\u4E0D\u53EF\u6570\u96C6\u5408\u6BD4\u5982\u81EA\u7136\u6570\u7684\u5E42\u96C6\uFF0C\u5B9E\u6570\u7684\u96C6\u5408\uFF0C\u548C\u8457\u540D\u7684\u5EB7\u6258\u723E\u96C6\u3002\u8FD9\u4E9B\u96C6\u5408\u5B58\u5728\u4E8E\u4EFB\u4F55 Zermelo-Fraenkel \u5168\u96C6\u4E2D\uFF0C\u56E0\u4E3A\u5B83\u4EEC\u7684\u5B58\u5728\u53EF\u4ECE\u516C\u7406\u5F97\u51FA\u3002\u4F7F\u7528 L\u00F6wenheim-Skolem \u5B9A\u7406\uFF0C\u6211\u4EEC\u53EF\u4EE5\u5F97\u5230\u53EA\u5305\u542B\u53EF\u6570\u4E2A\u5BF9\u8C61\u7684\u96C6\u5408\u8BBA\u7684\u6A21\u578B\u3002\u4F46\u662F\uFF0C\u5B83\u5FC5\u987B\u5305\u542B\u4E0A\u8FF0\u63D0\u53CA\u5230\u7684\u4E0D\u53EF\u6570\u96C6\u5408\uFF0C\u8FD9\u4F3C\u4E4E\u662F\u4E2A\u77DB\u76FE\u3002\u4F46\u662F\u6B63\u5728\u8BA8\u8BBA\u7684\u8FD9\u4E9B\u96C6\u5408\u662F\u4E0D\u53EF\u6570\u7684\uFF0C\u53EA\u662F\u5728\u6A21\u578B\u5185\u4E0D\u5B58\u5728\u4ECE\u81EA\u7136\u6570\u5230\u8FD9\u4E9B\u96C6\u5408\u7684\u53CC\u5C04(\u6CE8\u610F\u5230\u96D9\u5C04\u51FD\u6578\u4E5F\u662F\u96C6\u5408\uFF0C\u4E5F\u5C31\u662F\u4E00\u7A2E\u7279\u6B8A\u7684\u5173\u7CFB)\u3002\u800C\u6A21\u578B\u5916\u7684\u78BA\u6709\u9019\u6A23\u7684\u53CC\u5C04\u3002"@zh . "La paradoxa de Skolem \u00E9s una paradoxa que apareix a teoria de conjunts i l\u00F2gica com a conseq\u00FC\u00E8ncia paradoxal del . La paradoxa va ser introdu\u00EFda en la discussi\u00F3 del matem\u00E0tic noruec Thoralf Skolem en un article de 1922. D'acord amb el teorema de L\u00F6wenheim-Skolem tota axiomatitzaci\u00F3 d'una teoria matem\u00E0tica utilitzant un llenguatge de primer ordre que tingui un nombre finit de signes distints admet un model numerable. Per exemple, la formalitzaci\u00F3 de la teoria de conjunts usual de Zermelo-Fraenkel afirma l'exist\u00E8ncia de conjunts no numerables. Per\u00F2 existeix un model que satisf\u00E0 els axiomes d'aquesta teoria que \u00E9s ell mateix numerable. La resoluci\u00F3 de la paradoxa requereix distingir adequadament entre els conjunts com objecte descrit per la teoria i els conjunts com a instrument utilitzat per"@ca . . . . . "\u5728\u6570\u7406\u903B\u8F91\u4E2D\uFF0C\u7279\u522B\u662F\u96C6\u5408\u8BBA\u4E2D\uFF0CSkolem \u6096\u8BBA\u662F\u5411\u4E0B L\u00F6wenheim-Skolem\u5B9A\u7406\u7684\u76F4\u63A5\u7ED3\u679C\uFF0C\u5B83\u58F0\u79F0\u6240\u6709\u4E00\u9636\u8BED\u8A00\u7684\u53E5\u5B50\u7684\u6A21\u578B\u90FD\u6709\u4E00\u4E2A\u521D\u7B49\u7B49\u4EF7\u7684\u53EF\u6570\u5B50\u6A21\u578B\u3002 \u8FD9\u4E2A\u6096\u8BBA\u89C1\u4E8EZermelo-Fraenkel \u96C6\u5408\u8BBA\u4E2D\u3002\u5EB7\u6258\u5C14\u5728 1874\u5E74\u53D1\u8868\u7684\u66F4\u65E9\u7684\u7ED3\u679C\u662F\uFF0C\u5B58\u5728\u4E0D\u53EF\u6570\u96C6\u5408\u6BD4\u5982\u81EA\u7136\u6570\u7684\u5E42\u96C6\uFF0C\u5B9E\u6570\u7684\u96C6\u5408\uFF0C\u548C\u8457\u540D\u7684\u5EB7\u6258\u723E\u96C6\u3002\u8FD9\u4E9B\u96C6\u5408\u5B58\u5728\u4E8E\u4EFB\u4F55 Zermelo-Fraenkel \u5168\u96C6\u4E2D\uFF0C\u56E0\u4E3A\u5B83\u4EEC\u7684\u5B58\u5728\u53EF\u4ECE\u516C\u7406\u5F97\u51FA\u3002\u4F7F\u7528 L\u00F6wenheim-Skolem \u5B9A\u7406\uFF0C\u6211\u4EEC\u53EF\u4EE5\u5F97\u5230\u53EA\u5305\u542B\u53EF\u6570\u4E2A\u5BF9\u8C61\u7684\u96C6\u5408\u8BBA\u7684\u6A21\u578B\u3002\u4F46\u662F\uFF0C\u5B83\u5FC5\u987B\u5305\u542B\u4E0A\u8FF0\u63D0\u53CA\u5230\u7684\u4E0D\u53EF\u6570\u96C6\u5408\uFF0C\u8FD9\u4F3C\u4E4E\u662F\u4E2A\u77DB\u76FE\u3002\u4F46\u662F\u6B63\u5728\u8BA8\u8BBA\u7684\u8FD9\u4E9B\u96C6\u5408\u662F\u4E0D\u53EF\u6570\u7684\uFF0C\u53EA\u662F\u5728\u6A21\u578B\u5185\u4E0D\u5B58\u5728\u4ECE\u81EA\u7136\u6570\u5230\u8FD9\u4E9B\u96C6\u5408\u7684\u53CC\u5C04(\u6CE8\u610F\u5230\u96D9\u5C04\u51FD\u6578\u4E5F\u662F\u96C6\u5408\uFF0C\u4E5F\u5C31\u662F\u4E00\u7A2E\u7279\u6B8A\u7684\u5173\u7CFB)\u3002\u800C\u6A21\u578B\u5916\u7684\u78BA\u6709\u9019\u6A23\u7684\u53CC\u5C04\u3002"@zh . . "S/s085750"@en . . "In de wiskundige logica en filosofie, is de paradox van Skolem een schijnbare tegenstrijdigheid die voortvloeit uit de neerwaartse stelling van L\u00F6wenheim-Skolem. Thoralf Skolem was in 1922 de eerste die de schijnbaar tegenstrijdige aspecten van deze stelling besprak en die de relativiteit van de verzamelingtheoretische noties ontdekte die nu bekendstaan als niet-. Hoewel de paradox van Skolem geen werkelijke antinomie is, zoals de Russellparadox, wordt het resultaat meestal een paradox genoemd, en werd hij door Skolem beschreven als een \"paradoxale stand van zaken\". De paradox is dat elke consistente aftelbare eersteordeaxiomatisering van de verzamelingenleer een aftelbaar model heeft. Dit lijkt in tegenspraak met het feit, dat men in de verzamelingenleer kan uitdrukken dat een verzameling overaftelbaar is. Het is dus zo dat een model dat slechts aftelbare verzamelingen bevat een eerste-ordezin vervult, die zegt dat een verzameling overaftelbaar is."@nl . . . . .