Computability on Reals, Infinite Limits and Differential Equations Bruno Loff a,b,∗ , Jos´ e F´ elix Costa a,b , Jerzy Mycka c a Instituto Superior T´ ecnico Technical University of Lisbon Lisbon, Portugal b Centro de Matem´atica e Aplica¸c˜oes Fundamentais do Complexo Interdisciplinar University of Lisbon Lisbon, Portugal c Institute of Mathematics University of Maria Curie-Sklodowska Lublin, Poland Abstract 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. Key words: Real Recursive Functions, Infinite Limits, Differential Equations, Computability on Reals Contents 1 Introduction 2 2 Basic definitions and fundamental results 3 3 The η-hierarchy 8 4 Integration and differentiation in REC(R) 9 5 The universal Ψ function 13 6 The universal Ψn functions 14 7 Conclusions 17 8 Conventions and notational preferences 17 Acknowledgements 18 ∗ Corresponding author. Address is R. Vitorino Nem´ esio, N o 8, 4 o Esq, 1750-307, Lisboa, Portugal. Email addresses: bruno.loff@gmail.com (Bruno Loff), fgc@math.ist.utl.pt (Jos´ e F´ elix Costa), Jerzy.Mycka@umcs.lublin.pl (Jerzy Mycka). Preprint submitted to Elsevier 13 April 2007