I can understand discrete space and time and if it’s really how universe works than this paradox is quickly resolved, but there is another solution which my brain can’t understand. How does continuous spacetime work and solve this paradox?
In: Physics
In mathematics this Paradox can be resolved using the idea of limits.
To explain I’ll use a variation of the Paradox where a runner is trying to cross some distance. Before they cross the full distance they need to cross half of the distance. After that they cross half of the remaining distance, and so on. We can write this as 1/2 + 1/4 + 1/8… Ancient mathematicians assumed that since the expression has an infinite number of terms it must be impossible to add them all up.
In modern mathematics we can observe that as you add more and more terms the value of the sum gets closer and closer to 1. It never goes above 1 and it never stops getting closer to 1. In mathematical terms, as the number of terms goes to infinity the limit of the sum is equal to 1. So even though the series has an infinite number of terms the sum of the series is still a finite value.
This same logic applies to the Paradox of Achilles and the hare. Practically we know that Achilles will catch up to the hare after traveling a finite distance in a finite amount of time. The fact that we can describe this process using an infinite number of intermediary steps doesn’t change the final outcome. It simply demonstrates that it is possible to pass through an infinite number of steps in a finite amount of time.
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