Definition:Enumeration Operator (Recursion Theory)

Definition
Let $\phi \subseteq \N$ be recursively enumerable.

Let $\pi : \N^2 \to \N$ be the Cantor pairing function.
 * Then, $\map \pi {x, y}$ is the pair coding of $\tuple {x, y}$.

Define the mapping $\psi : \powerset \N \to \powerset \N$ as:
 * $\map \psi A = \set {x \in \N : \exists \text { finite } B \subseteq A : \map \pi {x, b} \in \phi}$

where $b \in \N$ is the code number for $B$.

Then $\psi$ is an enumeration operator.