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I have known about Zeno's paradox for some time now, but I have never really understood what exactly the paradox is. People always seem to have different explainations.

From wikipedia:

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. __

And we then say that this is a paradox since he should be able to reach the tortoise in finite time? For me it seems like that in the paradox we are slowing down time proportionally. Aren't we then already using the fact that the sum of those "time sequences" make up finite time? I feel like there is some kind of circular logic involved here.

What exactly is the paradox?

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It is not really a paradox. =) –  Pedro Tamaroff May 8 at 22:40

3 Answers 3

It's only a paradox if you assume that the sum of (countable) infinitely many numbers cannot be finite. But modern mathematics has no problem with infinite sums that yield finite results - in the case of Zeno's paradox, the sum in question is $$ \sum_{k=1}^\infty 2^{-k} = 1 \text{.} $$

Not everything that is called a paradox is actually a logical inconsistency. Quite often, things only seem inconsistent because we inadvertedly make an additional assumption, which turns out to be wrong. In the case of Zeno's paradox, that is the assumption that infinite sums cannot yield finite results.

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The paradox is that you need to do infinite "actions" to get to the turtle, therefore you never get to the turtle because humans can't do infinite actions in a finite amount of time.

Of course it's not a true paradox, it's well explained by "you scale time as you scale space therefore you have a finite number". Keep in mind we were in ancient Greece, things like "convergent series" were far from being defined.

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In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow)...

The ancient Greeks didn't have a precise notion of speed. It wasn't until the 16th century that Galileo first measured speed by considering the distance covered and the time it takes.

Assuming constant speeds $S_A$ and $S_T$ (m/s) for Achilles and the Tortoise respectively, we know that, in this example, Achilles would have caught up to the Tortoise in $\frac {100} {S_A-S_T}$ seconds.

With only a vague notion of speed, the ancient Greeks were perplexed by the fact, in that time interval, both racers would have passed through infinitely many points in space, the arrival at each point being an "event". In modern modern mathematics, we have no problem with infinitely many such events occurring in a finite time interval.

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