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I would like to know if there is a (known and maybe published) way to numbering, in a Gödel style, the functions belonging to every class in the Grzegorczyk Hierarchy and how could it be done.

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I've found it! It 's developed in:

Axt, P., Enumeration and the Grzegorczyk hierarchy, Z. math. Logik und Grundl. Math. 9 (1963) 53–65, MR0144811, Zbl 0112.24602, link

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