Computability on reals, infinite limits and differential equations

摘要

We study a countable class of real-valued functions inductively defined from a basic set of trivial functions by composition, solving first-order differential equations and the taking of infinite limits. This class is the analytical counterpart of Kleene’s partial recursive functions. By counting the number of nested limits required to define a function, this class can be stratified by a potentially infinite hierarchy—a hierarchy of infinite limits. In the first meaningful level of the hierarchy, we have the extensions of classical primitive recursive functions. In the next level, we find partial recursive functions, and in the following level we find the solution to the halting problem.We use methods from numerical analysis to show that the hierarchy does not collapse, concluding that the taking of infinite limits can always produce new functions from functions in the previous levels of the hierarchy.