For models that do support primitive recursion, a unique function h(y) satisfying a primitive recursion equation can be shown to exist via external set-theoretic argument.

In the case that \(f(y)\) and \(g(y)\) are primitive recursive, we have remarked that it is possible to show that there exists a unique function \(h(y)\) satisfying (\ref{recex}) by an external set-theoretic argument. But we may also consider the case in which \(g(y)\) is assumed to be computable relative to a model of computation \(\mathbf{M}\) which differs from the partial recursive functions in that it does not natively support recursion as a mode of computation—e.g., the Turing Machine mode