In computability theory complete numberings are generalizations of Gödel numbering first introduced by A.I.
They are studied because several important results like the Kleene's recursion theorem and Rice's theorem, which were originally proven for the Gödel-numbered set of computable functions, still hold for arbitrary sets with complete numberings.
is called complete (with respect to an element
there exists a total computable function
is called precomplete if the weaker property holds: