The equality and less-than relation symbols are interpreted as the usual equality and order relation on.
This structure is known as the standard model or intended interpretation of first-order arithmetic. A sentence in the language of first-order 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 first-order arithmetic that are true in, written. This set is, equivalently, the theory of the structure.
Arithmetic undefinability
The central result on true arithmetic is the undefinability theorem of Alfred Tarski. It states that the set is not arithmetically definable. This means that there is no formula in the language of first-order 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 and the Turing degrees, using the arithmetical hierarchy. For each natural numbern, let be the subset of consisting of only sentences that are or lower in the arithmetical hierarchy. Post's theorem shows that, for each n, is arithmetically definable, but only by a formula of complexity higher than. Thus no single formula can define, because but no single formula can define for arbitrarily largen.
Computability properties
As discussed above, is not arithmetically definable, by Tarski's theorem. A corollary of Post's theorem establishes that the Turing degree of is 0, and so is not decidable nor recursively enumerable. is closely related to the theory of the recursively enumerable Turing degrees, in the signature of partial orders. In particular, there are computable functions S and T such that:
For each sentence ψ in the signature of partial orders, ψ is in if and only if T is in .
Model-theoretic 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.
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 and whose second-order part consists of every subset of. The true theory of first-order arithmetic,, is a subset of the true theory of second order arithmetic, and 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. has shown that the true theory of second-order arithmetic is computably interpretable with the theory of the partial order of all Turing degrees, in the signature of partial orders, and vice versa.
