Numbering (computability theory)

In computability theory a numbering is the assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some language. A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects.

Common examples of numberings include Gödel numberings in first-order logic and admissible numberings of the set of partial computable functions.

A numbering of a set ${\displaystyle S}$ is a surjective partial function from ${\displaystyle \mathbb {N} }$ to S (Ershov 1999:477). The value of a numbering ν at a number i (if defined) is often written ν_i instead of the usual ${\displaystyle \nu (i)\!}$.

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