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Say an object is to travel from point A to point B, a finite distance of 2 meters. Say the object travels at 1m/s.

After 1 meter the speed of the object is halved. After another half of the previous distance the speed is halved again, and so on.

So for each term in the infinite series 1 + 1/2 + 1/4 + 1/8 + ..., the term represents the distance the object travels in meters before the speed is halved.

Now apparently the object does eventually reach point B, but it takes an infinite amount of time to do so.

Yet infinity has no end.

So if the the object does ever reach point B, wouldn't that mark the end of the infinite length of time that it took to reach it and wouldn't that contradict with the definition of infinity?

Yet if we say it never reaches point B, which is to say that there is no point in time that it reaches point B because that point in time would mark an end to infinity, then isn't it consistent with what we mean by infinity?

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This is not the right place for this type of question. Try asking on – Anonymous Sep 18 '13 at 19:48
@Anonymous: I'm not sure it would really fit there, either. It's almost a philosophical question, really. – Cameron Buie Sep 18 '13 at 19:49
The short answer is that if it takes an infinite amount of time to reach point B, then the object never reaches it. Of course, in the real world, no matter how precise your measurement is, there will be some time after which the object is closer to B than your instrument can distinguish. – Javier Sep 18 '13 at 19:49
Assuming that distance is quantized (that is, there is a least possible positive distance, counterintuitive as that seems), then at some point, the object will actually stop before getting to point $B$. Javier's point is also well-taken. – Cameron Buie Sep 18 '13 at 19:52
You are not the first one to think about this: – Daniel R Sep 18 '13 at 19:56
up vote 3 down vote accepted

The key to cracking Zeno's paradox is the realization that an infinite series can have a finite sum. You mention an example of such a series:

$$\sum\limits_{n=0}^\infty \frac{1}{2^n} = 1 +\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 2$$

That yields the 2 meters from your example. You can divide that number infinitely, as Zeno did, but if you add those divisions back up, you still get 2: a finite number. Because it's a finite distance, it can be crossed in finite time.

You still have to cross each partial distance, but each of these distances is also finite (in fact, you can subdivide each of the the same way you divided up the original distance), so they can all be crossed in finite time too. If you move at the same speed through each one, they form the very same sort of infinite series as the distances do. In fact, you don't even have to move at the same speed through each one; it just makes the math easier if you do.

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There is perhaps the bigger question of how close would the object have to be to B to be considered at B as this series seems a lot like the Dichotomy in Zeno's Paradox that would be my suggestion for a place to answer it. For example, how large is the object that is it occupying X space physically?

Zeno's Paradoxes has the Dichotomy example that is pretty much what you are have stated here though you do have to concede the object has to have infinitesimal size or else it will be close after going enough distances.

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Assume the object has length x and the distance between point A and B is (1/1 + 1/2 + 1/4 + ...) - x or that the object is simply a geometric point itself. – RussW Sep 18 '13 at 20:27

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