In mathematical logic, true arithmetic is the set of all true firstorder statements about the arithmetic of natural numbers.^{[1]} This is the theory associated with the standard model of the Peano axioms in the language of the firstorder Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.
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Definition
The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and lessthan relation symbols, and a constant symbol for 0. The (wellformed) formulas of the language of firstorder arithmetic are built up from these symbols together with the logical symbols in the usual manner of firstorder logic.
The structure is defined to be a model of Peano arithmetic as follows.
 The domain of discourse is the set of natural numbers,
 The symbol 0 is interpreted as the number 0,
 The function symbols are interpreted as the usual arithmetical operations on ,
 The equality and lessthan relation symbols are interpreted as the usual equality and order relation on .
This structure is known as the standard model or intended interpretation of firstorder arithmetic.
A sentence in the language of firstorder arithmetic is said to be true in if it is true in the structure just defined. The notation is used to indicate that the sentence is true in
True arithmetic is defined to be the set of all sentences in the language of firstorder arithmetic that are true in , written Th(). This set is, equivalently, the (complete) theory of the structure .^{[2]}
Arithmetic undefinability
The central result on true arithmetic is the undefinability theorem of Alfred Tarski (1936). It states that the set Th() is not arithmetically definable. This means that there is no formula in the language of firstorder arithmetic such that, for every sentence θ in this language,
Here is the numeral of the canonical Gödel number of the sentence θ.
Post's theorem is a sharper version of the undefinability theorem that shows a relationship between the definability of Th() and the Turing degrees, using the arithmetical hierarchy. For each natural number n, let Th_{n}() be the subset of Th() consisting of only sentences that are or lower in the arithmetical hierarchy. Post's theorem shows that, for each n, Th_{n}() is arithmetically definable, but only by a formula of complexity higher than . Thus no single formula can define Th(), because
but no single formula can define Th_{n}() for arbitrarily large n.
Computability properties
As discussed above, Th() is not arithmetically definable, by Tarski's theorem. A corollary of Post's theorem establishes that the Turing degree of Th() is 0^{(ω)}, and so Th() is not decidable nor recursively enumerable.
Th() is closely related to the theory Th() of the recursively enumerable Turing degrees, in the signature of partial orders.^{[3]} In particular, there are computable functions S and T such that:
 For each sentence φ in the signature of firstorder arithmetic, φ is in Th() if and only if S(φ) is in Th().
 For each sentence ψ in the signature of partial orders, ψ is in Th() if and only if T(ψ) is in Th().
Modeltheoretic properties
True arithmetic is an unstable theory, and so has models for each uncountable cardinal . As there are continuum many types over the empty set, true arithmetic also has countable models. Since the theory is complete, all of its models are elementarily equivalent.
True theory of secondorder arithmetic
The true theory of secondorder arithmetic consists of all the sentences in the language of secondorder arithmetic that are satisfied by the standard model of secondorder arithmetic, whose firstorder part is the structure and whose secondorder part consists of every subset of .
The true theory of firstorder arithmetic, Th(), is a subset of the true theory of secondorder arithmetic, and Th() is definable in secondorder arithmetic. However, the generalization of Post's theorem to the analytical hierarchy shows that the true theory of secondorder arithmetic is not definable by any single formula in secondorder arithmetic.
Simpson (1977) has shown that the true theory of secondorder arithmetic is computably interpretable with the theory of the partial order of all Turing degrees, in the signature of partial orders, and vice versa.
Notes
 ^ Boolos, Burgess & Jeffrey 2002, p. 295
 ^ see theories associated with a structure
 ^ Shore 2011, p. 184
References
 Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002), Computability and logic (4th ed.), Cambridge University Press, ISBN 9780521007580.
 Bovykin, Andrey; Kaye, Richard (2001), "On ordertypes of models of arithmetic", in Zhang, Yi (ed.), Logic and algebra, Contemporary Mathematics, vol. 302, American Mathematical Society, pp. 275–285, ISBN 9780821829844.
 Shore, Richard (2011), "The recursively enumerable degrees", in Griffor, E.R. (ed.), Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics, vol. 140, NorthHolland (published 1999), pp. 169–197, ISBN 9780444547019.
 Simpson, Stephen G. (1977), "Firstorder theory of the degrees of recursive unsolvability", Annals of Mathematics, Second Series, 105 (1), Annals of Mathematics: 121–139, doi:10.2307/1971028, ISSN 0003486X, JSTOR 1971028, MR 0432435
 Tarski, Alfred (1936), "Der Wahrheitsbegriff in den formalisierten Sprachen". An English translation "The Concept of Truth in Formalized Languages" appears in Corcoran, J., ed. (1983), Logic, Semantics and Metamathematics: Papers from 1923 to 1938 (2nd ed.), Hackett Publishing Company, Inc., ISBN 9780915144754
External links
 Weisstein, Eric W. "Arithmetic". MathWorld.
 Weisstein, Eric W. "Peano Arithmetic". MathWorld.
 Weisstein, Eric W. "Tarski's Theorem". MathWorld.