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Numbering (computability theory)
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Numbering (computability theory) : ウィキペディア英語版
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.
== Definition and examples ==
A numbering of a set S \! is a surjective partial function from \mathbb 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 \nu(i) \!.
For example, the set of all finite subsets of \mathbb has a numbering γ in which \gamma(\emptyset) = 0 and \gamma(\) = \sum_ 2^ (Ershov 1999:477).
As a second example, a fixed Gödel numbering \varphi_i of the computable partial functions can be used to define a numbering ''W'' of the recursively enumerable sets, by letting by ''W''(''i'') be the domain of φ''i''.

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