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So, Zeno assumes that, to go from the mark at $1m$ to the mark at $2m$ we've to do an infinite number of tasks. Like the task of getting to $1.001m$, the task of getting to $1.000005m$,the task of getting to $1.658m$,etc.

So, to perform an infinite number of tasks, one must take an infinite amount of time. I think that's where the loophole is. Well, yes, that would be true only if each of these tasks took a finite time to complete. If each of these tasks took $1s$, then, yes, it'd be true that completing all these tasks would take an infinite number of seconds.

But, the truth is that none of these tasks take a finite time to complete. If we divide the objective of getting from$1m$ to $2m$ into an infinite number of tasks, and let $dx$ be the infinitely small distance, then the infinite number of tasks are:

  1. Going from 1 to 1+dx

  2. Going from 1+dx to 1+2dx...................

$n-1$. Going from 2-2dx to 2-dx

$n$. Going from 2-dx to 2.

Well, none of these tasks take a finite time to complete. The time taken is also infinitely small. In each of these tasks, the distance we've to cover is $dx$ and hence the time taken is the infinitely small, $dt=\frac{dx}{v}$, if our speed is $v$. And, an infinite number of infinitely small times add up to give a finite time required to get from $1m$ to $2m$ like this: $$\int_1^2\frac{dx}{v}=\frac{2-1}{v}=\frac{1}{v}$$ So, I don't see any paradox. Where's the paradox?

I don't understand why people introduce concepts like quantization of space and time to explain Zeno's paradox.

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    One problem with your argument is that there are no "infinitely small" quantities in the standard real numbers. Another problem is that even if there are, there's no proof that there's any such thing in the physical universe. Zeno is not talking about math, he's talking about the world. Modern physics tells us that it's meaningless to talk about intervals of time below the Planck length. You can't reconcile your theory with either physics or standard math. It's true that there are systems of math with infinitesimals, but they won't help your argument. – 2017-02-25
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    "But, the truth is that none of these tasks take a finite time to complete." Every task he talks about takes a finite time to complete because each task is to travel a finite distance, which takes a finite amount of time to do. – 2017-02-25
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    @user4894 The only way to divide going from $1m$ to $2m$ into an infinite number of tasks is to assume that each task is infinitely small. For example, if each task is to cover a finite distance, say 0.01m, and there are an infinite number of such tasks, then the total distance we've to move will be infinite. So, if infinitely small distances don't exist, then there's no way to divide a distance into an infinite number of tasks, and therefore the paradox doesn't exist. – 2017-02-25
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    @ChaiT.Rex If each task is to travel a finite distance, say 0.01m and the number of such tasks is infinite, then the distance we've to travel between 1m and 2m will be infinite. So, if we assume each task to be of a finite length, then there are no infinite no. of tasks and hence there's no paradox. – 2017-02-25
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    In the real numbers there are no infinitely small intervals. Calculus is based on *arbitrarily small* intervals. It's a distinction that makes a big difference. An infinitesimal by definition is a quantity $x$ such that $x < \frac{1}{n}$ for $n = 1, 2, 3, \dots$ Clearly there is no such real number. Secondly, IF the mathematical real numbers are a true model of the physical world, then the mathematical theory of infinite series solves the paradox. However whether the universe is continuous or discrete; and whether it's accurately modeled by the real numbers; are both unknown. – 2017-02-25
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    The distance decreases by half each time in the popular version. For example: 1/2 m, 1/4 m, 1/6 m, 1/16 m. That doesn't sum to infinity. At each step, it's a finite distance, but it's a smaller distance than the previous. Each step takes a finite amount of time. – 2017-02-25
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    You can in fact have infinitely many finite distances adding up to a finite distance. For example, $\dfrac 12 m + \dfrac 14m + + \dfrac 18m + ... = 1m$. None of the distances are infinitesimal, they each have a length. But they get smaller and smaller fast enough so that even infinitely many of them will get you a finite distance. You can also substitute time instead of distance throughout my whole comment. – 2017-02-25
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    @ChaiT.Rex What about the last steps? Do they take finite time? – 2017-02-25
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    There are no last steps. An infinite sequence doesn't end. – 2017-02-25
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    @Ovi 1m is only the limit of that series as the number of tasks go towards infinity. The question is :can we EVER get to 1m in the real world? – 2017-02-25
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    @ChaiT.Rex Do you remember integration as the limit of a sum? We write the last steps of an infinite sequence there. – 2017-02-25
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    There are no last steps to an infinite sequence. Since the steps can be numbered with integers, that's like saying the steps numbered with the highest integers. There are no integers that are the highest. – 2017-02-25
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    That series was in response to you saying that if you add up infinitely many distances, you neccesarily get an infinite distance. $1$ being a limit shows that even if you add up infinitely many terms of that sequence, you get something which is less than or equal to $1$, which is still a finite thing. – 2017-02-25
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    Nobody thinks it's necessary to introduce quantization of space and time to explain Zeno's paradox. It can be explained perfectly well just by noting that $1 + 1/2 + 1/4 + \ldots$ is equal to $2$, a finite number. – 2017-02-25
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    Zeno did not understand that speed was just the ratio of distance and time as every school child knows today. Using the formula $s=\frac {d}{t}$ (from Galileo some two thousand years later), he could have determined precisely when and where the fleet-footed Achilles would overtake the lumbering Tortoise (assuming constant speeds). – 2017-02-26

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Zeno's paradox is fundamentally a philosophical problem. Many philosophy problems can be approached mathematically, but you have to be careful to justify that you're using the right math. In your solution, you're choosing to model time in a certain mathematical way. That needs to be justified. You could also (as you mention) solve the problem by modeling time and space with the integers and just asserting that there's only finite units of both, so the paradox doesn't arise.

However, doing math in these models isn't a complete philosophical argument. You need to argue that time is, in fact, like the real numbers in the relevant way. Otherwise someone can just say "well, in your model your idea works but in the real world it doesn't because the world doesn't look like your model."

There are many ways in which Zeno's paradox can fail. The question is which one is actually the way it can fail.

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    Great answer. So, none of the explanations can be the actual explanation? – 2017-02-25
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    @Dove Mathematics /by itself/ cannot be. If you can present an argument (presumably grounded in physics) as to why your model describes the world, that together with the math is enough. This turns out to be surprisingly hard to do, which is why people often resort to quantized space. It might not be necessary to understand why Zeno's Paradox fails, but it's the easiest way to give a justified explanation for the fact that it does. – 2017-02-25