In Chapter 2.2 we have defined how a numbering v of a set S induces a computability theory on the set S. We have shown that the numbering v of a set S is determined uniquely (up to equivalence) by the computability theory induced by it (Lemma 2.2.8, Conclusion 2.2.9). In this chapter we shall study for which effectivity conditions numberings satisfying these conditions exist. As a main result we shall show that for sets which are defined as closures numberings defined via generation trees are natural.
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