# Arithmetical set

In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy.

The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.) by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical.

A function $f:\subseteq \mathbb{N}^k \to \mathbb{N}$ is called arithmetically definable if the graph of $f$ is an arithmetical set.

A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number is called arithmetical if its real and imaginary parts are both arithmetical.

## Formal definition

A set X of natural numbers is arithmetical or arithmetically definable if there is a formula φ(n) in the language of Peano arithmetic such that each number n is in X if and only if φ(n) holds in the standard model of arithmetic. Similarly, a k-ary relation $R(n_1,\ldots,n_k)$ is arithmetical if there is a formula $\psi(n_1,\ldots,n_k)$ such that $R(n_1,\ldots,n_k) \Leftrightarrow \psi(n_1,\ldots,n_k)$ holds for all k-tuples $(n_1,\ldots,n_k)$ of natural numbers.

A finitary function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation.

A set A is said to be arithmetical in a set B if A is definable by an arithmetical formula which has B as a set parameter.

## Properties

• The complement of an arithmetical set is an arithmetical set.
• The Turing jump of an arithmetical set is an arithmetical set.
• The collection of arithmetical sets is countable, but there is no arithmetically definable sequence that enumerates all arithmetical sets.
• The set of real arithmetical numbers is countable, dense and order-isomorphic to the set of rational numbers.

## Implicitly arithmetical sets

Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set. An alternative notion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whether the set itself satisfies some arithmetical property.

A set Y of natural numbers is implicitly arithmetical or implicitly arithmetically definable if it is definable with an arithmetical formula that is able to use Y as a parameter. That is, if there is a formula $\theta(Z)$ in the language of Peano arithmetic with no free number variables and a new set parameter Z and set membership relation $\in$ such that Y is the unique set Z such that $\theta(Z)$ holds.

Every arithmetical set is implicitly arithmetical; if X is arithmetically defined by φ(n) then it is implicitly defined by the formula

$\forall n [n \in Z \Leftrightarrow \phi(n)]$.

Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic is implicitly arithmetical but not arithmetical.