Something like an inverse Zeno's paradox

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The paradox I am referring to is that which can be resolved by considering the fact that ##\sum_{k=0}^\infty1/2^k=1##. However, before one can travel half of the distance to be travelled, he has to travel half of that half, and half of that half ... Moreover, to say that one can travel by halves implies that he can travel by thirds and fourths ... Therefore, if one can even begin to travel, he has to move ##1/n## of the distance, where ##n\to\infty##, but that is just ##0##, so one cannot start his travelling.
Is there any mistake in the reasoning?
Notice that shifting the problem to travelling the ##1/n,\,n\to\infty## of the distance gets you, once more, to the original problem.
The fact that you can travel ##0## distance doesn't allow you to say that you can go ## 0+ϵ## forward. I am not affirming that you can travel half, a fourth, or a third of a distance, rather what I am saying is that if you are saying that you can travel ##1/n##, then you surely can go ##1/(n+1)##, but can you prove that? You then continue downwards until you reach ##0## which I won't object you can travel, but how can you go upwards? (since I would reason in the same way again)
 
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  • #2
pbuk
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Is there any mistake in the reasoning?
Yes, and Aristotle didn't need infinite sums to see it. If you go forward by 5 metres in 1 second, how far will you go forward in 0.1 seconds? In ε seconds?
 
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PeroK
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The paradox I am referring to is that which can be resolved by considering the fact that ##\sum_{k=0}^\infty1/2^k=1##. However, before one can travel half of the distance to be travelled, he has to travel half of that half, and half of that half ... Moreover, to say that one can travel by halves implies that he can travel by thirds and fourths ... Therefore, if one can even begin to travel, he has to move ##1/n## of the distance, where ##n\to\infty##, but that is just ##0##, so one cannot start his travelling.
Is there any mistake in the reasoning?
Notice that shifting the problem to travelling the ##1/n,\,n\to\infty## of the distance gets you, once more, to the original problem.
The fact that you can travel ##0## distance doesn't allow you to say that you can go ## 0+ϵ## forward. I am not affirming that you can travel half, a fourth, or a third of a distance, rather what I am saying is that if you are saying that you can travel ##1/n##, then you surely can go ##1/(n+1)##, but can you prove that? You then continue downwards until you reach ##0## which I won't object you can travel, but how can you go upwards? (since I would reason in the same way again)
Is yours a mathematical or a physical paradox?
 
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Is yours a mathematical or a physical paradox?
It is physical I guess. I am in some sense saying that if you can eat a cake, then you surely can eat a smaller one, and so on ad infinitum until you reach zero. Now you can a ##0## size cake, but then you cannot prove further than that since if you try to pass on to prove that you can eat an ##\epsilon## size cake, then I would restart the rant again, that you can surely eat a cake of size ##\epsilon/2## and so on.
 
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PeroK
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It is physical I guess. I am in some sense saying that if you can eat a cake, then you surely can eat a smaller one, and so on ad infinitum until you reach zero. Now you can a ##0## size cake, but then you cannot prove further than that since if you try to pass on to prove that you can eat an ##\epsilon## size cake, then I would restart the rant again, that you can surely eat a cake of size ##\epsilon/2## and so on.
If we can assume that the real numbers (and the rational numbers) are valid mathematical sets, then in effect you are saying that neither can be used to model physical time and space? Is this because time and space cannot exist or that time and space can only have a finite or discrete structure?
 
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pbuk
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It is physical I guess. I am in some sense saying that if you can eat a cake, then you surely can eat a smaller one, and so on ad infinitum until you reach zero.
No, you never reach zero, you can only get arbitrarily close.
Now you can [eat] a ##0## size cake
No you can't because it doesn't exist.
, but then you cannot prove further than that since if you try to pass on to prove that you can eat an ##\epsilon## size cake, then I would restart the rant again, that you can surely eat a cake of size ##\epsilon/2## and so on.
If you can eat a whole cake in 1 minute, how long does it take to eat an ## \epsilon ## sized portion?
 
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Thank you both @pbuk and @PeroK for your answers! Please do not think that I am ignoring your comments, I have other things that I am looking forward to do irl right now 😇
 
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PeroK
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If you can eat a whole cake in 1 minute, how long does it take to eat an ## \epsilon ## sized portion?
There may be a limit on how small a piece of cake may be and still be a piece of cake. If you ate one electron, you might be stretching a point to say that you've eaten an electron-sized piece of cake!
 
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If we can assume that the real numbers (and the rational numbers) are valid mathematical sets, then in effect you are saying that neither can be used to model physical time and space? Is this because time and space cannot exist or that time and space can only have a finite or discrete structure?
No, I haven't thought of that.
If you go forward by 5 metres in 1 second, how far will you go forward in 0.1 seconds? In ε seconds?
If you can eat a whole cake in 1 minute, how long does it take to eat an ##\epsilon## sized portion?
Thank you, I see my problem. It is the already made assumption that one can move.
 
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PeroK
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It is the already made assumption that one can move.
You could start with the alternative assumption that motion is impossible and see how far you get.
 
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  • #11
Svein
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Reminds me of an old joke: A male engineer and a male mathematician are led into a room. At one end of the romm is a beautiful woman. They are placed at a starting line 2 meters from the woman, and they are told that the challenge is to embrace the woman. But - they are only allowed to cross half the remaining distance at each step. The mathematician thinks for a second and then leaves in disgust. The engineer starts walking - when asked why, since he is never going to reach the woman, he is still trying. His answer: "Well, after 20 or 30 steps I am close enough for all practical purposes".
 
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Is yours a mathematical or a physical paradox?
To me looks more like a philosophical one :smile:
 

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