Zeno of Elea (ca. 490–430 BCE) was the first person in history to show that the concept of infinity is problematic. In Zeno’s Achilles and the Tortoise Paradox, Achilles races to catch a slower runner–for example, a tortoise that is crawling away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones.
Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes. Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Aristotle also distinguished “things infinite in respect of divisibility” (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension (“with respect to their extremities”).
Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Modern calculus achieves the same result, using more rigorous methods. These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.
 Aristotle. Physics 6.9; 6.2, 233a21-31.
 Boyer, Carl (1959). The History of the Calculus and Its Conceptual Development. Dover Publications. p. 295.
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