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rdfs:label: Aritmética de segunda ordem Arithmétique du second ordre Second-order arithmetic Aritmética de segundo orden
rdfs:comment: Na Lógica matemática, aritmética de segunda ordem é uma coleção de sistemas axiomáticos que formalizam os números naturais e seus subconjuntos. É uma alternativa para a teoria axiomática dos conjuntos como fundamentos, mas não tudo, da matemática. Foi introduzida por David Hilbert e Paul Bernays em seu livro . A axiomatização padrão da aritmética de segunda ordem é denotada Z2. In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted by Z2. En logique mathématique, l'arithmétique du second ordre est une théorie des entiers naturels et des ensembles d'entiers naturels.Elle a été introduite par David Hilbert et Paul Bernays dans leur livre Grundlagen der Mathematik. L'axiomatisation usuelle de l'arithmétique du second ordre est notée Z2. Elle permet en particulier de traiter les nombres réels, qui peuvent être représentés comme des (ensembles infinis) d'entiers, et de développer l'analyse usuelle. Pour cette raison, les logiciens l'appellent également « analyse ». En la lógica matemática, la aritmética de segundo orden es una colección de sistemas axiomáticos que formalizan los números naturales y sus subconjuntos. La aritmética de segundo orden también puede verse como una versión débil de teoría de conjuntos en la que todo elemento es un número natural o un conjunto de números naturales.
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dbo:abstract: Na Lógica matemática, aritmética de segunda ordem é uma coleção de sistemas axiomáticos que formalizam os números naturais e seus subconjuntos. É uma alternativa para a teoria axiomática dos conjuntos como fundamentos, mas não tudo, da matemática. Foi introduzida por David Hilbert e Paul Bernays em seu livro . A axiomatização padrão da aritmética de segunda ordem é denotada Z2. Aritmética de segunda ordem inclui, mas é significativamente mais forte do que, a aritmética de Peano. Ao contrário da aritmética de Peano, aritmética de segunda ordem permite a quantificação sobre conjuntos de números, bem como os próprios números. Como os números reais podem ser representados como conjuntos (infinitos) de números naturais de maneiras bem conhecidas, e dado que a aritmética de segunda ordem permite a quantificação sobre esses conjuntos, é possível formalizar os números reais em aritmética de segunda ordem. Por esta razão, a aritmética de segunda ordem é chamada algumas vezes de "análise". Aritmética de segunda ordem pode também ser vista como uma versão fraca da teoria dos conjuntos, em que cada elemento seja um número natural ou um conjunto de números naturais. Embora seja muito mais fraca do que a teoria axiomática de Zermelo-Fraenkel, a aritmética de segunda ordem pode provar essencialmente todos os resultados de matemática clássica expressível em sua linguagem. Um subsistema de aritmética de segunda ordem é uma teoria na linguagem da aritmética de segunda ordem onde cada axioma é um teorema da aritmética de segunda ordem plena (Z2). Esses subsistemas são essenciais para a , um programa de pesquisa que investiga o quanto a matemática clássica pode ser derivada em certos subsistemas fracos de intensidade variável. A maior parte do núcleo da matemática pode ser formalizada por esses subsistemas fracos, alguns dos quais são definidos a seguir. Matemática reversa também esclarece o âmbito e as modalidades em que a matemática clássica é não-construtiva. En la lógica matemática, la aritmética de segundo orden es una colección de sistemas axiomáticos que formalizan los números naturales y sus subconjuntos. La aritmética de segundo orden también puede verse como una versión débil de teoría de conjuntos en la que todo elemento es un número natural o un conjunto de números naturales. En logique mathématique, l'arithmétique du second ordre est une théorie des entiers naturels et des ensembles d'entiers naturels.Elle a été introduite par David Hilbert et Paul Bernays dans leur livre Grundlagen der Mathematik. L'axiomatisation usuelle de l'arithmétique du second ordre est notée Z2. L'arithmétique de second ordre a pour conséquence les théorèmes de l'arithmétique de Peano (du premier ordre), mais elle est à la fois plus forte et plus expressive que celle-ci. L'arithmétique du second ordre permet la quantification non seulement sur les nombres entiers naturels, comme l'arithmétique de Peano, mais aussi sur les ensembles d'entiers naturels. Elle comprend en particulier une version du schéma d'axiomes de compréhension restreinte aux ensembles d'entiers naturels, qui est exprimable grâce à ces nouvelles quantifications. Le raisonnement par récurrence s'exprime par un seul axiome. Elle permet en particulier de traiter les nombres réels, qui peuvent être représentés comme des (ensembles infinis) d'entiers, et de développer l'analyse usuelle. Pour cette raison, les logiciens l'appellent également « analyse ». L'arithmétique de second ordre peut aussi être considérée comme une version faible de la théorie des ensembles dans laquelle chaque élément est soit un entier naturel, soit un ensemble d'entiers naturels. Bien qu'elle soit plus faible que la théorie des ensembles de Zermelo-Fraenkel, au sens où il existe des énoncés de l'arithmétique du second ordre démontrables en théorie des ensembles, mais pas en arithmétique du second ordre, elle permet de prouver l'essentiel des résultats des mathématiques classiques exprimables dans son langage. L'arithmétique du second ordre, et surtout certains de ses sous-systèmes, construits essentiellement à partir de restrictions des schémas d'axiomes de compréhension et de récurrence, jouent un rôle important pour les mathématiques à rebours. In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called "analysis" (Sieg 2013, p. 291). Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo–Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language. A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive.
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