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| In [[mathematical logic]], '''true arithmetic''' is the set of all true statements about the [[arithmetic]] of [[natural number]]s (Boolos, Burgess, and Jeffrey 2002:295). This is the theory [[Theory (mathematical logic)#Theories associated with a structure|associated]] with the [[Peano axioms#Models|standard model]] of the [[Peano axioms]] in the [[Signature (logic)|language]] of the first-order Peano axioms.
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| == Definition ==
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| The [[Signature (logic)|signature]] of [[Peano arithmetic]] includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the '''language of first-order arithmetic''' are built up from these symbols together with the logical symbols in the usual manner of [[first-order logic]].
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| The [[structure (mathematical logic)|structure]] <math>\mathcal{N}</math> is defined to be a model of Peano arithmetic as follows.
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| * The [[domain of discourse]] is the set <math>\mathbb{N}</math> of natural numbers.
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| * The symbol 0 is interpreted as the number 0.
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| * The function symbols are interpreted as the usual arithmetical operations on <math>\mathbb{N}</math>
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| * The equality and less-than relation symbols are interpreted as the usual equality and order relation on <math>\mathbb{N}</math>
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| This structure is known as the [[nonstandard arithmetic|'''standard model''']] or [[intended interpretation]] of first-order arithmetic.
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| A [[sentence (mathematical logic)|sentence]] in the language of first-order arithmetic is said to be true in <math>\mathcal{N}</math> if it is true in the structure just defined. The notation <math>\mathcal{N} \models \varphi</math> is used to indicate that the sentence φ is true in <math>\mathcal{N}.</math>
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| '''True arithmetic''' is defined to be the set of all sentences in the language of first-order arithmetic that are true in <math>\mathcal{N}</math>, written {{nowrap|1=Th(<math>\mathcal{N}</math>)}}. This set is, equivalently, the (complete) theory of the structure <math>\mathcal{N}</math> (see [[Theory (mathematical logic)#Theories associated with a structure|theories associated with a structure]]).
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| == Arithmetic indefinability ==
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| The central result on true arithmetic is the [[Tarski's indefinability theorem|indefinability theorem]] of [[Alfred Tarski]] (1936). It states that | |
| the set {{nowrap|1=Th(<math>\mathcal{N}</math>)}} is not arithmetically definable. This means that there is no formula <math> \varphi(x)</math> in the language of first-order arithmetic such that, for every sentence θ in this language, | |
| :<math>\mathcal{N} \models \theta \qquad</math> if and only if <math>\mathcal{N} \models \varphi(\underline{\#(\theta)}).</math>
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| Here <math>\underline{\#(\theta)}</math> is the numeral of the canonical [[Gödel number]] of the sentence ''θ''.
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| [[Post's theorem]] is a sharper version of the indefinability theorem that shows a relationship between the definability of {{nowrap|1=Th(<math>\mathcal{N}</math>)}} and the [[Turing degree]]s, using the [[arithmetical hierarchy]]. For each natural number ''n'', let {{nowrap|1=Th<sub>''n''</sub>(<math>\mathcal{N}</math>)}} be the subset of {{nowrap|1=Th(<math>\mathcal{N}</math>)}} consisting of only sentences that are <math>\Sigma^0_n</math> or lower in the arithmetical hierarchy. Post's theorem shows that, for each ''n'', {{nowrap|1=Th<sub>''n''</sub>(<math>\mathcal{N}</math>)}} is arithmetically definable, but only by a formula of complexity higher than <math>\Sigma^0_n</math>. Thus no single formula can define {{nowrap|1=Th(<math>\mathcal{N}</math>)}}, because
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| :<math>\mbox{Th}(\mathcal{N}) = \bigcup_{n \in \mathbb{N}} \mbox{Th}_n(\mathcal{N})</math>
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| but no single formula can define {{nowrap|1=Th<sub>''n''</sub>(<math>\mathcal{N}</math>)}} for arbitrarily large ''n''.
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| == Computability properties ==
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| As discussed above, {{nowrap|1=Th(<math>\mathcal{N}</math>)}} is not arithmetically definable, by Tarski's theorem. A corollary of Post's theorem establishes that the [[Turing degree]] of {{nowrap|1=Th(<math>\mathcal{N}</math>)}} is '''0'''<sup>(ω)</sup>, and so {{nowrap|1=Th(<math>\mathcal{N}</math>)}} is not [[decidable set|decidable]] nor [[recursively enumerable set|recursively enumerable]].
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| {{nowrap|1=Th(<math>\mathcal{N}</math>)}} is closely related to the theory {{nowrap|1=Th(<math>\mathcal{R}</math>)}} of the [[recursively enumerable Turing degree]]s, in the signature of [[partial order]]s (Shore 1999:184). In particular, there are computable functions ''S'' and ''T'' such that:
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| * For each sentence φ in the signature of first order arithmetic, φ is in {{nowrap|1=Th(<math>\mathcal{N}</math>)}} if and only if S(φ) is in {{nowrap|1=Th(<math>\mathcal{R}</math>)}}
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| * For each sentence ψ in the signature of partial orders, ψ is in {{nowrap|1=Th(<math>\mathcal{R}</math>)}} if and only if T(ψ) is in {{nowrap|1=Th(<math>\mathcal{N}</math>)}}.
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| == Model-theoretic properties ==
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| True arithmetic is an [[stable theory|unstable theory]], and so has <math>2^\kappa</math> models for each uncountable cardinal <math>\kappa</math>. As there are continuum many [[type (model theory)|type]]s over the empty set, true arithmetic also has <math>2^{\aleph_0}</math> countable models. Since the theory is [[complete theory|complete]], all of its models are [[elementarily equivalent]].
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| == True theory of second-order arithmetic ==
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| The true theory of second-order arithmetic consists of all the sentences in the language of [[second-order arithmetic]] that are satisfied by the standard model of second-order arithmetic, whose first-order part is the structure <math>\mathcal{N}</math> and whose second-order part consists of every subset of <math>\mathbb{N}</math>.
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| The true theory of first-order arithmetic, {{nowrap|1=Th(<math>\mathcal{N}</math>)}}, is a subset of the true theory of second order arithmetic, and {{nowrap|1=Th(<math>\mathcal{N}</math>)}} is definable in second-order arithmetic. However, the generalization of Post's theorem to the [[analytical hierarchy]] shows that the true theory of second-order arithmetic is not definable by any single formula in second-order arithmetic.
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| Simpson (1977) has shown that the true theory of second-order arithmetic is computably interpretable with the theory of the partial order of all [[Turing degree]]s, in the signature of partial orders, and ''vice versa''.
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| == References ==
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| * {{citation
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| | last1=Boolos
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| | first1=George
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| | last2=Burgess
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| | first2=John P.
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| | last3=Jeffrey
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| | first3=Richard C.
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| | title=Computability and logic
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| | edition=4th
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| | publisher=Cambridge University Press
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| | year=2002
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| | isbn=0-521-00758-5
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| }}.
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| * {{Citation
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| | last1=Bovykin
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| | first1=Andrey
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| | last2=Kaye
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| | first2=Richard
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| | chapter=On order-types of models of arithmetic
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| | year=2001
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| | editor-last=Zhang
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| | editor-first=Yi
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| | title=Logic and algebra
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| | series=Contemporary Mathematics
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| | volume=302
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| | publisher=American Mathematical Society
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| | publication-date=2001
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| | pages=275–285
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| }}.
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| * {{Citation
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| | last1=Shore
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| | first1=Richard
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| | chapter=The recursively enumerable degrees
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| | year=1999
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| | editor-last=Griffor
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| | editor-first=E.R.
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| | title=Handbook of Computability Theory
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| | series=Studies in Logic and the Foundations of Mathematics
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| | volume=140
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| | publisher=North-Holland
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| | publication-date=1999
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| | pages=169–197
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| }}.
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| * {{Citation
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| | last1=Simpson
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| | first1=Stephen G.
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| | title=First-order theory of the degrees of recursive unsolvability
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| | mr=0432435
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| | year=1977
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| | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]]
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| | issn=0003-486X
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| | volume=105
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| | pages=121–139
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| | doi=10.2307/1971028
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| | jstor=1971028
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| | issue=1
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| | publisher=Annals of Mathematics
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| }}
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| * Tarski, Alfred (1936), "Der Wahrheitsbegriff in den formalisierten Sprachen". An English translation "The Concept of Truth in Formalized Languages" appears in Corcoran, J., (ed.), ''Logic, Semantics and Metamathematics'', 1983.
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| [[Category:Model theory]]
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| [[Category:Formal theories of arithmetic]]
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Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Are you considering getting your website redesigned. Storing write-ups in advance would have to be neccessary with the auto blogs. Article Source: Stevens works in Internet and Network Marketing.
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