On a subrecursive hierarchy and primitive recursive degrees

Abstract

1. Prior to the Herbrand-Godel-Kleene definition of general recursive function, certain classes of functions defined more restrictively by particular recursions had been studied. Subsequently subclasses of the general recursive functions have continued to be of interest, not only because of the simplicity and naturalness of certain types of recursive definition, but because of the insight such classes might provide into recursiveness and effectiveness and because of the need for some measurement of the level or complexity of recursiveness of a function or predicate. Pursuit of this interest led therefore to the devising of various hierarchies of recursive functions. It would seem that such a hierarchy ought to satisfy at least the following conditions: (1) that it be generated on the basis of some general principle, (2) that the order of a class should correspond to the complexity of the functions it contains in the sense that functions which enumerate or majorize a given class should appear in a higher class, (3) that the union of its classes should contain all the recursive functions or, if this fails, should be sufficiently large relative to one's purpose for the hierarchy. The hierarchies of recursive functions which have been studied fall short of fully satisfying these conditions (cf. [4, §l]). In [4] Kleene makes a new attempt at a classification of general recursive functions, by using the notion of relative primitive recursiveness and of the uniform effective enumerability of the functions primitive recursive in an assumed function. A general recursive function hy, and a class C„ of the functions primitive recursive in hy, is associated with each element y of a system O of ordinal notations. If y 2 and to fail at co2. The ^-recursive functions of Peter [6] are located in the hierarchy below the co level. Although it is not yet settled whether all recursive functions are obtained, it is clear that U„eo Cy is a large and interesting class.