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In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets.

Attributes	Values
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rdfs:label	Arithmetical hierarchy (en)
rdfs:comment	In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. (en)
differentFrom	Lévy hierarchy
rdfs:seeAlso	Post's theorem
sameAs	Arithmetical hierarchy Arithmetical hierarchy
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Subject	Complexity classes Computability theory Hierarchy Effective descriptive set theory Mathematical logic hierarchies
thumbnail	wiki-commons:Special:FilePath/Arithmetic_hierarchy.svg?width=300
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prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Arithmetical_hierarchy?oldid=1037077355&ns=0
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Wikipage revision ID	1037077355 (xsd:integer)
Link from a Wikipage to another Wikipage	Union (set theory) Analytical hierarchy Andrzej Mostowski Arithmetical hierarchy Baire space (set theory) Borel hierarchy Borel set Stephen Cole Kleene Tarski–Kuratowski algorithm Tuple Turing degree Turing jump Turing reduction Universal quantification Bounded quantifier Cantor space Cartesian product Complexity classes Computability theory Hierarchy Pairing function Post's theorem Second-order arithmetic Set (mathematics) Hyperarithmetical theory Prenex normal form Primitive recursive function Complement (set theory) Complexity Computability theory Computable function Computable set Computably enumerable Course-of-values recursion E (complexity) Effective Polish space Effective descriptive set theory Hierarchy (mathematics) Oracle machine Effective descriptive set theory Mathematical logic hierarchies Indicator function Existential quantification Halting problem Intersection (set theory) Many-one reduction Peano axioms Lévy hierarchy Mathematical logic Pointclass Polynomial hierarchy Interpretability logic
has abstract	In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Arithmetical_hierarchy
is rdfs:seeAlso of	Post's theorem
is Wikipage redirect of	Kleene-Mostowski hierarchy Arithmetic hierarchy Arithmetic reducibility Arithmetical reducibility Pi-0-1 sentence Pi-0-1 sentences Pi-0-2 sentence Kleene hierarchy Kleene–Mostowski hierarchy AH (complexity)
is Link from a Wikipage to another Wikipage of	Machine that always halts Algorithmically random sequence Analytical hierarchy Andrzej Mostowski Arithmetical hierarchy Arithmetical set

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