# True arithmetic

In mathematical logic, **true arithmetic** is the set of all true first-order 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 first-order Peano axioms.
True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.

## Definition

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

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 less-than relation symbols are interpreted as the usual equality and order relation on .

This structure is known as the

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 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 first-order arithmetic such that, for every sentence *θ* in this language,

Here is the numeral of the canonical

*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

Th() is closely related to the theory Th() of the

^{[3]}

*S*and

*T*such that:

- For each sentence
*φ*in the signature of first-order 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().

## 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

## True theory of second-order arithmetic

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, Th(), is a subset of the true theory of second-order arithmetic, and Th() 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.

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*.

## Notes

**^**Boolos, Burgess & Jeffrey 2002, p. 295**^**see theories associated with a structure**^**Shore 2011, p. 184

## References

- ISBN 978-0-521-00758-0.
- Bovykin, Andrey; Kaye, Richard (2001), "On order-types of models of arithmetic", in Zhang, Yi (ed.),
*Logic and algebra*, Contemporary Mathematics, vol. 302, American Mathematical Society, pp. 275–285, . - .