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In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy. The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical.

Attributes	Values
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rdfs:label	Aritmetika aro (eo) Arithmetical set (en) Арифметическое множество (ru) Conjunto aritmético (pt)
rdfs:comment	En matematika logiko aritmetika aro estas kalkulebla aro kiu povas esti difinita per formulo de aritmetiko de la unua ordo. La aritmetikaj aroj estas orditaj en la aritmetika hierarkio. Funkcio estas nomita kiel aritmetike difinebla se la grafikaĵo de estas aritmetika aro. (eo) Арифметическое множество — множество натуральных чисел , которое может быть определено формулой в языке арифметики первого порядка, то есть если существует такая формула с одной свободной переменной , что . Аналогично, множество кортежей натуральных чисел называется арифметическим, если существует такая формула , что . Также можно говорить об арифметических множествах кортежей натуральных чисел, конечных последовательностей натуральных чисел, формул (при любой их фиксированной гёделевской нумерации) и, вообще, об арифметических множествах любых объектов, кодируемых натуральными числами. (ru) In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy. The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical. (en) Em lógica matemática, um conjunto aritmético é um conjunto de números naturais que pode ser definido por uma fórmula de primeira ordem da aritmética de Peano. Os conjuntos aritméticos são classificados pela hierarquia aritmética. A função é chamada de aritmeticamente definível se o gráfico de é um conjunto aritmético. Um número real é chamado de aritmético se o conjunto de todos os menores números racionais é aritmético. Um número complexo é chamado aritmético se suas partes real e imaginária são ambas aritméticas. (pt)
dcterms:subject	Effective descriptive set theory Mathematical logic hierarchies Computability theory
Wikipage page ID	2342451 (xsd:integer)
Wikipage revision ID	1036518408 (xsd:integer)
Link from a Wikipage to another Wikipage	Peano arithmetic Complement (set theory) Complex number Computable function Countable set Effective descriptive set theory Mathematical logic hierarchies Mathematical logic Function (mathematics) Real and imaginary parts Arithmetical hierarchy Computable number Computable set Dense order Tuple Finitary relation First-order logic Formal language Graph of a function Well-formed formula Halting problem First-order arithmetic Prime number Computability theory Tarski's indefinability theorem Integers Natural number Rational numbers Real number Recursively enumerable set Sequence Set (mathematics) Turing jump Finitary Subset Order-isomorphic Chaitin's constant Ω Gödel number
sameAs	Arithmetical set Arithmetical set Arithmetical set Arithmetical set Arithmetical set Arithmetical set Arithmetical set Arithmetical set
dbp:wikiPageUsesTemplate	dbt:Oclc dbt:Refimprove dbt:Short_description dbt:Number_systems
has abstract	In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy. The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical. A function is called arithmetically definable if the graph of is an arithmetical set. A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number is called arithmetical if its real and imaginary parts are both arithmetical. (en) En matematika logiko aritmetika aro estas kalkulebla aro kiu povas esti difinita per formulo de aritmetiko de la unua ordo. La aritmetikaj aroj estas orditaj en la aritmetika hierarkio. Funkcio estas nomita kiel aritmetike difinebla se la grafikaĵo de estas aritmetika aro. (eo) Em lógica matemática, um conjunto aritmético é um conjunto de números naturais que pode ser definido por uma fórmula de primeira ordem da aritmética de Peano. Os conjuntos aritméticos são classificados pela hierarquia aritmética. A definição pode ser estendida para um conjunto contável A arbitrário (i.e. um conjunto de n-uplas de inteiros, um conjunto de números racionais, um conjunto de fórmulas em alguma linguagem formal, etc.) utilizando números de Gödel para representar elementos do conjunto e declarando um subconjunto de A como sendo aritmético se o conjunto dos correspondentes números de Gödel forem aritméticos. A função é chamada de aritmeticamente definível se o gráfico de é um conjunto aritmético. Um número real é chamado de aritmético se o conjunto de todos os menores números racionais é aritmético. Um número complexo é chamado aritmético se suas partes real e imaginária são ambas aritméticas. (pt) Арифметическое множество — множество натуральных чисел , которое может быть определено формулой в языке арифметики первого порядка, то есть если существует такая формула с одной свободной переменной , что . Аналогично, множество кортежей натуральных чисел называется арифметическим, если существует такая формула , что . Также можно говорить об арифметических множествах кортежей натуральных чисел, конечных последовательностей натуральных чисел, формул (при любой их фиксированной гёделевской нумерации) и, вообще, об арифметических множествах любых объектов, кодируемых натуральными числами. (ru)
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page length (characters) of wiki page	4956 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Arithmetical_set
is Link from a Wikipage to another Wikipage of	Definable set List of mathematical logic topics Constructive set theory Hyperarithmetical theory List of types of sets Turing reduction Outline of logic Arithmetic set Arithmetical numbers Arithmetical real number Arithmetically definable Arithmetically definable function
is Wikipage redirect of	Arithmetic set Arithmetical numbers Arithmetical real number Arithmetically definable Arithmetically definable function
is foaf:primaryTopic of	wikipedia-en:Arithmetical_set

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