Complete Numbering - Definition

Definition

A numbering of a set is called complete (with respect to an element ) if for every partial computable function there exists a total computable function so that

 \nu \circ h(i) =
\left\{
\begin{matrix}
\nu \circ f(i) &\mbox{if}\ i \in \mathrm{dom}(f), \\
a &\mbox{otherwise}.
\end{matrix}
\right.

The numbering is called precomplete if

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