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| {{refimprove|date=August 2011}}
| | Alyson is the title individuals use to call me and I think it sounds fairly great when you say it. What I adore doing is soccer but I don't have the time lately. I've always cherished living in Alaska. Since I was eighteen I've been working as a bookkeeper but quickly my spouse and I will start our own business.<br><br>Also visit my weblog - psychic readers ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 mouse click the next site]) |
| In [[mathematical logic]], an '''arithmetical set''' (or '''arithmetic set''') is a [[set (mathematics)|set]] of [[natural number]]s that can be defined by a formula of first-order [[Peano arithmetic]]. The arithmetical sets are classified by the [[arithmetical hierarchy]].
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| The definition can be extended to an arbitrary [[countable set]] ''A'' (e.g. the set of n-[[tuple]]s of [[integers]], the set of [[rational numbers]], the set of formulas in some [[formal language]], etc.) by using [[Gödel number]]s to represent elements of the set and declaring a subset of ''A'' to be arithmetical if the set of corresponding Gödel numbers is arithmetical.
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| A function <math>f:\subseteq \mathbb{N}^k \to \mathbb{N}</math> is called '''arithmetically definable''' if the [[graph of a function|graph]] of <math>f</math> is an arithmetical set.
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| 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.
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| == Formal definition ==
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| A set ''X'' of natural numbers is '''arithmetical''' or '''arithmetically definable''' if there is a formula φ(''n'') in the language of Peano arithmetic such that each number ''n'' is in ''X'' if and only if φ(''n'') holds in the standard model of arithmetic. Similarly, a ''k''-ary relation
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| <math>R(n_1,\ldots,n_k)</math> is arithmetical if there is a formula | |
| <math>\psi(n_1,\ldots,n_k)</math> such that <math>R(n_1,\ldots,n_k) \Leftrightarrow \psi(n_1,\ldots,n_k)</math> holds for all ''k''-tuples <math>(n_1,\ldots,n_k)</math> of natural numbers.
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| A [[finitary]] function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation.
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| A set ''A'' is said to be '''arithmetical in''' a set ''B'' if ''A'' is definable by an arithmetical formula which has ''B'' as a set parameter.
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| == Examples ==
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| * The set of all [[prime number]]s is arithmetical.
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| * Every [[recursively enumerable set]] is arithmetical.
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| * Every [[computable function]] is arithmetically definable.
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| * The set encoding the [[Halting problem]] is arithmetical.
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| * [[Chaitin's constant Ω]] is an arithmetical real number.
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| * [[Tarski's indefinability theorem]] shows that the set of true formulas of first order arithmetic is not arithmetically definable.
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| == Properties ==
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| * The [[complement (set theory)|complement]] of an arithmetical set is an arithmetical set.
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| * The [[Turing jump]] of an arithmetical set is an arithmetical set.
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| * The collection of arithmetical sets is countable, but there is no arithmetically definable sequence that enumerates all arithmetical sets.
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| * The set of real arithmetical numbers is [[denumerable]], [[Dense order|dense]] and [[order-isomorphic]] to the set of rational numbers.
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| == Implicitly arithmetical sets ==
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| Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set. An alternative notion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whether the set itself satisfies some arithmetical property.
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| A set ''Y'' of natural numbers is '''implicitly arithmetical''' or '''implicitly arithmetically definable''' if it is definable with an arithmetical formula that is able to use ''Y'' as a parameter. That is, if there is a formula <math>\theta(Z)</math> in the language of Peano arithmetic with no free number variables and a new set parameter ''Z'' and set membership relation <math>\in</math> such that ''Y'' is the unique set ''Z'' such that <math>\theta(Z)</math> holds.
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| Every arithmetical set is implicitly arithmetical; if ''X'' is arithmetically defined by φ(''n'') then it is implicitly defined by the formula
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| :<math>\forall n [n \in Z \Leftrightarrow \phi(n)]</math>.
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| Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic is implicitly arithmetical but not arithmetical.
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| == See also ==
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| * [[Arithmetical hierarchy]]
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| * [[Computable set]]
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| * [[Computable number]]
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| == Further reading ==
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| *Rogers, H. (1967). ''Theory of recursive functions and effective computability.'' McGraw-Hill. {{ISBN missing|date=September 2013}}
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| {{Number systems}}
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| [[Category:Effective descriptive set theory]]
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| [[Category:Mathematical logic hierarchies]]
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| [[Category:Computability theory]]
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