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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 first-order 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 function is called arithmetical if its graph is an arithmetical (k+1)-ary relation.

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



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.

Implicitly arithmetical sets

Each arithmetical set has an arithmetical formula that says whether particular numbers are in the set. An alternative notion of definability allows for a formula that does not say whether particular numbers are in the set but says 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