Restricted Limits on Natural Functions with Arithmetical Graphs

In this paper we consider the process of defining natural functions by the operation of infinite limit: F (x̄) = limy→∞,y∈A f(x̄, y) (also limes inferior and limes superior are taken into account). But two restrictions are assumed: the given natural function f has a graph belonging to some stage of an arithmetical hierarchy, the index of a limit runs only through a given arithmetical subset A of natural numbers. We investigate the arithmetical class of the graph of the function F , where the respective classes of the graph of f and the set A are known. The corollary for the Turing degrees of F is formulated.