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is thus the one with the least n all the other classifications can be determined from it the arithmetical hierarchy of sets of natural numbers edit a set x of natural numbers is defined by a formula φ in the language of peano arithmetic the first order language with symbols 0 for zero s for the successor function for addition for multiplication and for equality if the elements of x are exactly the numbers that satisfy φ that is for all natural numbers n n x n φ n _ displaystyle n in x leftrightarrow mathbb n models varphi underline n where n _ displaystyle underline n is the numeral in the language of arithmetic corresponding to n displaystyle n a set is definable in first order arithmetic if it is defined by some formula in the language of peano arithmetic each set x of natural numbers that is definable in first order arithmetic is assigned classifications of the form σ n 0 displaystyle sigma _ n 0 π n 0 displaystyle pi _ n 0 and δ n 0 displaystyle delta _ n 0 where n displaystyle n is a natural number as follows if x is definable by a σ n 0 displaystyle sigma _ n 0 formula then x is assigned the classification σ n 0 displaystyle sigma _ n 0 if x is definable by a π n 0 displaystyle pi _ n 0 formula then x is assigned the classification π n 0 displaystyle pi _ n 0 if x is both σ n 0 displaystyle sigma _ n 0 and π n 0 displaystyle pi _ n 0 then x displaystyle x is assigned the additional classification δ n 0 displaystyle delta _ n 0 it rarely makes sense to speak of δ n 0 displaystyle delta _ n 0 formulas the first quantifier of a formula is either existential or universal so a δ n 0 displaystyle delta _ n 0 set is not necessarily defined by a δ n 0 displaystyle delta _ n 0 formula in the sense of a formula that is both σ n 0 displaystyle sigma _ n 0 and π n 0 displaystyle pi _ n 0 rather there are both σ n 0 displaystyle sigma _ n 0 and π n 0 displaystyle pi _ n 0 formulas that define the set for example the set of odd natural numbers n displaystyle n is definable by either k n 2 k displaystyle forall k n neq 2 times k or k n 2 k 1 displaystyle exists k n 2 times k 1 a parallel definition is used to define the arithmetical hierarchy on finite cartesian powers of the set of natural numbers instead of formulas with one free variable formulas with k free first order variables are used to define the arithmetical hierarchy on sets of k tuples of natural numbers these are in fact related by the use of a pairing function meaning of the notation edit the following meanings can be attached to the notation for the arithmetical hierarchy on formulas the subscript n displaystyle n in the symbols σ n 0 displaystyle sigma _ n 0 and π n 0 displaystyle pi _ n 0 indicates the number of alternations of blocks of universal and existential first order quantifiers that are used in a formula moreover the outermost block is existential in σ n 0 displaystyle sigma _ n 0 formulas and universal in π n 0 displaystyle pi _ n 0 formulas the superscript 0 displaystyle 0 in the symbols σ n 0 displaystyle sigma _ n 0 π n 0 displaystyle pi _ n 0 and δ n 0 displaystyle delta _ n 0 indicates the type of the objects being quantified over type 0 objects are natural numbers and objects of type i 1 displaystyle i 1 are functions that map the set of objects of type i displaystyle i to the natural numbers quantification over higher type objects such as functions from natural numbers to natural numbers is described by a superscript greater than 0 as in the analytical hierarchy the superscript 0 indicates quantifiers over numbers the superscript 1 would indicate quantification over functions from numbers to numbers type 1 objects the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number and so on examples edit the σ 1 0 displaystyle sigma _ 1 0 sets of numbers are those definable by a formula of the form n 1 n k ψ n 1 n k m displaystyle exists n_ 1 cdots exists n_ k psi n_ 1 ldots n_ k m where ψ displaystyle psi has only bounded quantifiers these are exactly the recursively enumerable sets the set of natural numbers that are indices for turing machines that compute total functions is π 2 0 displaystyle pi _ 2 0 intuitively an index e displaystyle e falls into this set if and only if for every m displaystyle m there is an s displaystyle s such that the turing machine with index e displaystyle e halts on input m displaystyle m after s displaystyle s steps a complete proof would show that the property displayed in quotes in the previous sentence is definable in the language of peano arithmetic by a σ 1 0 displaystyle sigma _ 1 0 formula every σ 1 0 displaystyle sigma _ 1 0 subset of baire space or cantor space is an open set in the usual topology on the space moreover for any such set there is a computable enumeration of gödel numbers of basic open sets whose union is the original set for this reason σ 1 0 displaystyle sigma _ 1 0 sets are sometimes called effectively open similarly every π 1 0 displaystyle pi _ 1 0 set is closed and the π 1 0 displaystyle pi _ 1 0 sets are sometimes called effectively closed every arithmetical subset of cantor space or baire space is a borel set the lightface borel hierarchy extends the arithmetical hierarchy to include additional borel sets for example every π 2 0 displaystyle pi _ 2 0 subset of cantor or baire space is a g δ displaystyle g_ delta set that is a set that equals the intersection of countably many open sets moreover each of these open sets is σ 1 0 displaystyle sigma _ 1 0 and the list of gödel numbers of these open sets has a computable enumeration if ϕ x n m displaystyle phi x n m is a σ 0 0 displaystyle sigma _ 0 0 formula with a free set variable x displaystyle x and free number variables n m displaystyle n m then the π 2 0 displaystyle pi _ 2 0 set x n m ϕ x n m displaystyle x mid forall n exists m phi x n m is the intersection of the σ 1 0 displaystyle sigma _ 1 0 sets of the form x m ϕ x n m displaystyle x mid exists m phi x n m as n displaystyle n ranges over the set of natural numbers the σ 0 0 π 0 0 δ 0 0 displaystyle sigma _ 0 0 pi _ 0 0 delta _ 0 0 formulas can be checked by going over all cases one by one which is possible because all their quantifiers are bounded the time for this is polynomial in their arguments e g polynomial in n displaystyle n for φ n displaystyle varphi n thus their corresponding decision problems are included in e as n displaystyle n is exponential in its number of bits this no longer holds under alternative definitions of σ 0 0 π 0 0 δ 0 0 displaystyle sigma _ 0 0 pi _ 0 0 delta _ 0 0 that allow the use of primitive recursive functions as now the quantifiers may be bounded by any primitive recursive function of the arguments the σ 0 0 π 0 0 δ 0 0 displaystyle sigma _ 0 0 pi _ 0 0 delta _ 0 0 formulas under an alternative definition that allows the use of primitive recursive functions with bounded quantifiers correspond to sets of natural numbers of the form n f n 0 displaystyle n f n 0 for a primitive recursive function f displaystyle f this is because allowing bounded quantifier adds nothing to the definition for a primitive recursive f displaystyle f k n f k 0 displaystyle forall k n f k 0 is the same as f 0 f 1 f n 1 0 displaystyle f 0 f 1 f n 1 0 and k n f k 0 displaystyle exists k n f k 0 is the same as f 0 f 1 f n 1 0 displaystyle f 0 cdot f 1 cdot ldots cdot f n 1 0 with course of values recursion each of these can be defined by a single primitive recursive function relativized arithmetical hierarchies edit just as we can define what it means for a set x to be recursive relative to another set y by allowing the computation defining x to consult y as an oracle we can extend this notion to the whole arithmetic hierarchy and define what it means for x to be σ n 0 displaystyle sigma _ n 0 δ n 0 displaystyle delta _ n 0 or π n 0 displaystyle pi _ n 0 in y denoted respectively σ n 0 y displaystyle sigma _ n 0 y δ n 0 y displaystyle delta _ n 0 y and π n 0 y displaystyle pi _ n 0 y to do so fix a set of natural numbers y and add a predicate for membership of y to the language of peano arithmetic we then say that x is in σ n 0 y displaystyle sigma _ n 0 y if it is defined by a σ n 0 displaystyle sigma _ n 0 formula in this expanded language in other words x is σ n 0 y displaystyle sigma _ n 0 y if it is defined by a σ n 0 displaystyle sigma _ n 0 formula allowed to ask questions about membership of y alternatively one can view the σ n 0 y displaystyle sigma _ n 0 y sets as those sets that can be built starting with sets recursive in y and alternately taking unions and intersections of these sets up to n times for example let y be a set of natural numbers let x be the set of numbers divisible by an element of y then x is defined by the formula ϕ n m t y m m t n displaystyle phi n exists m exists t y m land m times t n so x is in σ 1 0 y displaystyle sigma _ 1 0 y actually it is in δ 0 0 y displaystyle delta _ 0 0 y as well since we could bound both quantifiers by n arithmetic reducibility and degrees edit arithmetical reducibility is an intermediate notion between turing reducibility and hyperarithmetic reducibility a set is arithmetical also arithmetic and arithmetically definable if it is defined by some formula in the language of peano arithmetic equivalently x is arithmetical if x is σ n 0 displaystyle sigma _ n 0 or π n 0 displaystyle pi _ n 0 for some natural number n a set x is arithmetical in a set y denoted x a y displaystyle x leq _ a y if x is definable as some formula in the language of peano arithmetic extended by a predicate for membership of y equivalently x is arithmetical in y if x is in σ n 0 y displaystyle sigma _ n 0 y or π n 0 y displaystyle pi _ n 0 y for some natural number n a synonym for x a y displaystyle x leq _ a y is x is arithmetically reducible to y the relation x a y displaystyle x leq _ a y is reflexive and transitive and thus the relation a displaystyle equiv _ a defined by the rule x a y x a y y a x displaystyle x equiv _ a y iff x leq _ a y land y leq _ a x is an equivalence relation the equivalence classes of this relation are called the arithmetic degrees they are partially ordered under a displaystyle leq _ a the arithmetical hierarchy of subsets of cantor and baire space edit the cantor space denoted 2 ω displaystyle 2 omega is the set of all infinite sequences of 0s and 1s the baire space denoted ω ω displaystyle omega omega or n displaystyle mathcal n is the set of all infinite sequences of natural numbers elements of the cantor space can be identified with sets of natural numbers and elements of the baire space with functions from natural numbers to natural numbers the ordinary axiomatization of second order arithmetic uses a set based language in which the set quantifiers can naturally be viewed as quantifying over cantor space a subset of cantor space is assigned the classification σ n 0 displaystyle sigma _ n 0 if it is definable by a σ n 0 displaystyle sigma _ n 0 formula the set is assigned the classification π n 0 displaystyle pi _ n 0 if it is definable by a π n 0 displaystyle pi _ n 0 formula if the set is both σ n 0 displaystyle sigma _ n 0 and π n 0 displaystyle pi _ n 0 then it is given the additional classification δ n 0 displaystyle delta _ n 0 for example let o 2 ω displaystyle o subseteq 2 omega be the set of all infinite binary strings that aren t all 0 or equivalently the set of all non empty sets of natural numbers as o x 2 ω n x n 1 displaystyle o x in 2 omega exists n x n 1 we see that o displaystyle o is defined by a σ 1 0 displaystyle sigma _ 1 0 formula and hence is a σ 1 0 displaystyle sigma _ 1 0 set while both the elements of the cantor space regarded as sets of natural numbers and subsets of the cantor space are classified in arithmetic hierarchies these are not the same hierarchy in fact the relationship between the two hierarchies is interesting and non trivial for instance the π n 0 displaystyle pi _ n 0 elements of the cantor space are not in general the same as the elements x displaystyle x of the cantor space so that x displaystyle x is a π n 0 displaystyle pi _ n 0 subset of the cantor space however many interesting results relate the two hierarchies there are two ways that a subset of baire space can be classified in the arithmetical hierarchy a subset of baire space has a corresponding subset of cantor space under the map that takes each function from ω displaystyle omega to ω displaystyle omega to the characteristic function of its graph a subset of baire space is given the classification σ n 0 displaystyle sigma _ n 0 π n 0 displaystyle pi _ n 0 or δ n 0 displaystyle delta _ n 0 if and only if the corresponding subset of cantor space has the same classification an equivalent definition of the arithmetical hierarchy on baire space is given by defining the arithmetical hierarchy of formulas using a functional version of second order arithmetic then the arithmetical hierarchy on subsets of cantor space can be defined from the hierarchy on baire space this alternate definition gives exactly the same classifications as the first definition a parallel definition is used to define the arithmetical hierarchy on finite cartesian powers of baire space or cantor space using formulas with several free variables the arithmetical hierarchy can be defined on any effective polish space the definition is particularly simple for cantor space and baire space because they fit with the language of ordinary second order arithmetic we can also define the arithmetic hierarchy of subsets of the cantor and baire spaces relative to some set of natural numbers in fact boldface σ n 0 displaystyle mathbf sigma _ n 0 is just the union of σ n 0 y displaystyle sigma _ n 0 y for all sets of natural numbers y the boldface hierarchy is just the standard hierarchy of borel sets extensions and variations edit it is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol for each primitive recursive function this variation slightly changes the classification of σ 0 0 π 0 0 δ 0 0 displaystyle sigma _ 0 0 pi _ 0 0 delta _ 0 0 since using primitive recursive functions in first order peano arithmetic requires in general an unbounded existential quantifier and thus some sets that are in σ 0 0 displaystyle sigma _ 0 0 by this definition are strictly in σ 1 0 displaystyle sigma _ 1 0 by the definition given in the beginning of this article the class σ 1 0 displaystyle sigma _ 1 0 and thus all higher classes in the hierarchy remain unaffected a more semantic variation of the hierarchy can be defined on all finitary relations on the natural numbers the following definition is used every computable relation is defined to be σ 0 0 π 0 0 δ 0 0 displaystyle sigma _ 0 0 pi _ 0 0 delta _ 0 0 the classifications σ n 0 displaystyle sigma _ n 0 and π n 0 displaystyle pi _ n 0 are defined inductively with the following rules if the relation r ...