# 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 is called **arithmetically definable** if the graph of 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
is arithmetical if there is a formula
such that holds for all *k*-tuples 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.

## Examples

- The set of all prime numbers is arithmetical.
- Every recursively enumerable set is arithmetical.
- Every computable function is arithmetically definable.
- The set encoding the Halting problem is arithmetical.
- Chaitin's constant Ω is an arithmetical real number.
- Tarski's indefinability theorem shows that the set of true formulas of first order arithmetic is not arithmetically definable.

## 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 the sequence of arithmetical sets is not arithmetically definable. Thus, there is no arithmetical formula φ(
*n*,*m*) that is true if and only if*m*is a member of the n-th arithmetical predicates.

- In fact, such a formula would describe a decision problem for all finite Turing jumps, and hence belongs to 0
^{(ω)}, which cannot be formalized in first-order arithmetic, as it does not belong to the first-order arithmetical hierarchy.

- 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 in the language of Peano arithmetic with no free number variables and a new set parameter *Z* and set membership relation such that *Y* is the unique set *Z* such that holds.

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

- .

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

## See also

## Further reading

- Rogers, H. (1967).
*Theory of recursive functions and effective computability.*McGraw-Hill. OCLC 527706