Computable isomorphism

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In computability theory two sets A;B \subseteq \N of natural numbers are computably isomorphic or recursively isomorphic if there exists a total bijective computable function f \colon \N \to \N with f(A) = B. By the theorem of Myhill,[1] the relation of computable isomorphism coincides with the relation of one-one reduction.

Two numberings \nu and \mu are called computably isomorphic if there exists a computable bijection f so that \nu = \mu \circ f

Computably isomorphic numberings induce the same notion of computability on a set.


  1. ^ Theorem 7.VI, Hartley Rogers, Jr., Theory of recursive functions and effective computability