The Tortoise and Achilles

In this paradox, the tortoise is given a head start in a race against Achilles, and it is argued that because of this no matter how fast Achilles runs, he will never be able to catch up and surpass the tortoise. This isn't true. However, the reason for this isn't that reality proves it wrong (since this paradox takes place outside the real and in the conceptual). The paradox becomes a paradox because most people take velocity as a factor because Achilles is physically faster than the tortoise, while the paradox itself doesn't take speed into consideration. All that matters is distance traveled on a linear path, with the necessity that Achilles and the tortoise must traverse all the same points on that line. Thus, while the tortoise can bypass any point it wants on the line as it goes, Achilles must touch all the points the tortoise has already, making it impossible for Achilles to bypass the tortoise and win, since he must touch every point that the tortoise does. If the tortoise touches all the points on the linear path as he goes, then Achilles must as well, but being bigger than the tortoise, he cannot bypass the tortoise before the tortoise moves as doing so chances him missing a space that the tortoise takes, nor can he stand on the same space as the tortoise.

Taking the above into account, the only way for Achilles to win the race on the linear path is to run in the opposite direction as the tortoise while staying on the same line, until he eventually gets to the space directly in front of the tortoise before the tortoise steps on it.