In mathematical logic, true arithmetic is the theory Th() of the natural numbers in the language of firstorder Peano arithmetic (Boolos, Burgess, and Jeffrey 2002:295). Tarski's indefinability theorem shows that this theory is not arithmetically definable.
Contents 
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 in this signature are built up in the the usual manner of firstorder logic. The language of firstorder arithmetic consists of all the wellformed formulas in this signature.
The structure is a model of Peano arithmetic defined as follows:
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 φ is true in
True arithmetic is the set Th() of all sentences in the language of firstorder arithmetic that are true in . This set is, equivalently, the (complete) theory of the structure (see theories associated with a structure).
The central result on true arithmetic is the indefinability theorem of Alfred Tarski (1936). It states that the set Th() is not arithmetically definable. This means that there is no "universal formula" φ in the signature of firstorder arithmetic such that, for every sentence θ in this signature,
Here is the numeral of the canonical Gödel number of the sentence θ.
Post's theorem is a sharper version of the indefinability 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.
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 (Shore 1999:184). In particular, there are computable functions S and T such that:
True arithmetic is an unstable theory, and so has 2^{κ} models for each uncountable cardinal κ. Since the theory is complete, all of its models are elementarily equivalent.
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 second order 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.
