Zeno's paradox -- that motion is apparently impossible because infinitely many points must be traversed -- was long disposed of with modern mathematical theory that forms the basis of calculus, including a rigorous analysis of continuity, convergence of infinite series, and related fields of study. Zeno failed to see how an infinite series may have a finite sum.

1) "Mathematicians claim to have done away with Zeno's paradoxes with rigorous analysis of the units of distance and time involved in the problem, and the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. The paradoxes certainly pose no problems in engineering either, as the practical questions as to where and when events such as Achilles passing the Tortoise are satisfactorily handled by unit analysis and calculus. However, some philosophers insist that the deeper metaphysical questions, as raised by Zeno's paradoxes, are not addressed by the calculus. That is, while calculus tells us where and when Achilles will overtake the Tortoise, philosophers do not see how calculus takes anything away from Zeno's reasoning that concludes that this event cannot happen in the first place. Philosophers also point out that Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must be infinite itself, which calculus shows to be incorrect. However, Zeno's problem wasn't with any kind of infinite sum, but rather with an infinite process: how can one ever get from A to B, if an infinite number of events can be identified that need to precede the arrival at B? Philosophers claim that calculus does not resolve that question, and hence a solution to Zeno's paradoxes must be found elsewhere. Physics point out that in the race, after a few dozens of steps, we will have to deal with dimensions where quantum mechanics can’t be disregarded. According to the uncertainty principle those distances are so small that taking a measurement would be pointless, even from a theoretical point of view: uncertainty would be too prominent. Infinite processes remained theoretically troublesome in mathematics until the early 20th century. L. E. J. Brouwer, a Dutch mathematician and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician. However, modern mathematics, with tools such as Kurt Gödel's proof of the logical independence of the axiom of choice and the epsilon-delta version of Weierstrass and Cauchy (or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson), argues rigorous formulation of logic and calculus has resolved theoretical problems involving infinite processes, including Zeno's." Source and further information: http://en.wikipedia.org/wiki/Zeno's_paradoxes#Status_of_the_paradoxes_today 2) "Zeno was right Achilles’s victory in the race does not show that Zeno was wrong, as his aim was to prove that motion does not exist. A similar statement can be found in quantum mechanics. As the Russian physicist Lev D. Landau, who was awarded the Nobel prize in 1962, wrote in the first chapter of his book “Quantum mechanics”, third volume of his most famous “Course of Theoretical Physics”: “In quantum mechanics there is no such concept as the path of a particle. This forms the content of what is called the uncertainty principle, one of the fundamental principles of quantum mechanics, discovered by W. Heisemberg in 1927” Quantum mechanics introduced uncertainty as a principle which is relevant only in the subatomic world; that is due to the small value of Planck’s constant. Its relevance, however, is of general interest, and it could make us think about our conception of the world from a different point of view. We don’t need to think that, in the world outside us, there must be entities that “exist” and are subject to infinitely precise rules. We don’t need to think that motion “exists”; what matters is that we can take measurements and predict their outcomes. In our exam of the race between Achilles and the tortoise, only this unconventional point of view prevents us from coming to a dead end." Source and further information: http://www.riflessioni.it/science/achilles-tortoise-paradox.htm