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I know for example that the first level of the Grzegorczyk hierarchy contains the functions which enumerate the c.e sets and that it has an interesting relation to the provably total functions in certain axiomatizations of arithmetic due to work of Kriesel, Girard and others. I was wondering if we know of some other interesing connections. For example

Is there a relationship between hierarchy levels and escape or domination properties of the functions enumerated by functions from said levels? I am looking for any connections like this from the literature.

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    $\begingroup$ In this forum, we like precise questions which admit a definitive answer. "Can we say anything about..." tends to annoy us as vague and lacking clarity. $\endgroup$
    – Dave Benson
    Commented Nov 30 at 21:57
  • $\begingroup$ Reformulated the question $\endgroup$
    – H.C Manu
    Commented Dec 1 at 7:21

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