In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.
True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.
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.
is defined to be a model of Peano arithmetic as follows.
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
This set is, equivalently, the (complete) theory of the structure
[2] The central result on true arithmetic is the undefinability theorem of Alfred Tarski (1936).
is the numeral of the canonical Gödel number of the sentence θ.
) is arithmetically definable, but only by a formula of complexity higher than
A corollary of Post's theorem establishes that the Turing degree of 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: True arithmetic is an unstable theory, and so has
As there are continuum many types over the empty set, true arithmetic also has
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
The true theory of first-order arithmetic, Th(
), is a subset of the true theory of second-order arithmetic, and Th(
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.
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 degrees, in the signature of partial orders, and vice versa.