A subset ${\displaystyle S}$  of the natural numbers is called computable if there exists a total computable function ${\displaystyle f}$  such that:

${\displaystyle f(x)=1}$  if ${\displaystyle x\in S}$ 

${\displaystyle f(x)=0}$  if ${\displaystyle x\notin S}$ .

In other words, the set ${\displaystyle S}$  is computable if and only if the indicator function ${\displaystyle \mathbb {1} _{S}}$  is computable.

• Every recursive language is a computable.
• Every finite or cofinite subset of the natural numbers is computable. Special cases:
  • The empty set is computable.
  • The entire set of natural numbers is computable.
  • Every natural number is computable.^{[note 1]}
• The subset of prime numbers is computable.
• The set of Gödel numbers is computable.^{[note 2]}

• The set of Turing machines that halt is not computable.
• The isomorphism class of two finite simplicial complexes is not computable.^{[clarification needed]}
• The set of busy beaver champions is not computable.
• Hilbert's tenth problem is not computable.

Both A, B are sets in this section.

• If A is computable then the complement of A is computable.
• If A and B are computable then:
  • A ∩ B is computable.
  • A ∪ B is computable.
  • The image of A × B under the Cantor pairing function is computable.

In general, the image of a computable set under a computable function is computably enumerable, but possibly not computable.

• A is computable if and only if A and the complement of A are both computably enumerable(c.e.).
• The preimage of a computable set under a total computable function is computable.
• The image of a computable set under a total computable bijection is computable.

A is computable if and only if it is at level ${\displaystyle \Delta _{1}^{0}}$  of the arithmetical hierarchy.

A is computable if and only if it is either the image (or range) of a nondecreasing total computable function, or the empty set.

• Computably enumerable
• Decidability (logic)
• Recursively enumerable language
• Recursive language
• Recursion

• Cutland, N. Computability. Cambridge University Press, Cambridge-New York, 1980. ISBN 0-521-22384-9; ISBN 0-521-29465-7
• Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1
• Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. ISBN 3-540-15299-7

• Sakharov, Alex. "Recursive Set". MathWorld.