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It’s eminently possible that the time it takes to finish each step will still go down: half the original time, a third of the original time, etc, but that the total journey will take an infinite amount of time.
You can check this for yourself by trying to find what the series [½ + ⅓ + ¼ + ⅕ ++ …] sums to. As it turns out, the limit does not exist: this is a diverging series.
Pure mathematics alone cannot provide a satisfactory solution to the paradox, as it isn’t simply about dividing a finite thing up into an infinite number of parts, but rather about the inherently physical concept of a rate
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This is the resolution of the classical “Zeno’s paradox” as commonly stated:
This reasoning is only good enough to show that the total distance you must travel converges to a finite value. It doesn’t tell you anything about how long it takes to reach your destination.
Rate is the amount that one quantity (distance) changes as another quantity (time) changes as well.
To go from her starting point to her destination, Atalanta must first travel half of the total distance.
Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time.
The oldest “solution” to the paradox was done from a purely mathematical perspective. The claim admits that, sure, there might be an infinite number of jumps that you’d need to take, but each new jump gets smaller and smaller than the previous one. Therefore, as long as you could demonstrate that...
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I have always been interested in the concept of time and time travel, as it is always as mind boggling as it is simple, and this article I found useful to introduce paradoxes which arise as part of time travel.
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