The study of uniformly r.e. classes of r.e. sets is a classical topic in recursion theory. In this context the problem to characterize the r.e. classes admitting a recursive enumeration without repetition or a one-one numbering had attracted a great deal of research effort. In 1958, Friedberg [F58], utilizing his insight into the "priority method", proved that the class of all r.e. sets and the class of all partial recursive functions can be enumerated without repetition. Pour-El [P60] rose the question if this could be done for every r.e. class, and answered it negatively [PP65] : Let K be the halting problem (i I
Generalized numberings are an extension of Ershov's notion of numbering, based on partial combinatory algebra (pca) instead of the natural numbers. We study various algebraic properties of generalized numberings, relating properties of the numbering to properties of the pca. As in the lambda calculus, extensionality is a key notion here.
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