In computability theory two sets A , B {\displaystyle A,B} of natural numbers are computably isomorphic or recursively isomorphic if there exists a total computable and bijective function f : N → N {\displaystyle f\colon \mathbb {N} \to \mathbb {N} } such that the image of f {\displaystyle f} restricted to A ⊆ N {\displaystyle A\subseteq \mathbb {N} } equals B ⊆ N {\displaystyle B\subseteq \mathbb {N} } , i.e. f ( A ) = B {\displaystyle f(A)=B} .

Further, two numberings ν {\displaystyle \nu } and μ {\displaystyle \mu } are called computably isomorphic if there exists a computable bijection f {\displaystyle f} so that ν = μ ∘ f {\displaystyle \nu =\mu \circ f} . Computably isomorphic numberings induce the same notion of computability on a set.