Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy

In this paper we show how to construct for every set R of integers in the arithmetical hierarchy a dynamical system \(\mathcal{H}\)with piecewise-constant derivatives (PCD) such that deciding membership in R can be reduced to solving the reachability problem between two rational points for \(\mathcal{H}\). The ability of such simple dynamical systems to “simulate” highly undecidable problems is closely related to Zeno's paradox dealing with the ability to pack infinitely many discrete steps in a bounded interval of time.