E.Asarin, O. Maler, 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 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 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. [Pdf]