# Something like an inverse Zeno's paradox

• B
• archaic
In summary, the conversation discusses a paradox involving traveling distances by halves and the concept of being able to travel by fractions of a distance. The paradox suggests that if one can travel half of a distance, they should be able to travel smaller and smaller fractions of that distance, but since the limit of these fractions is 0, it is impossible to start traveling in the first place. The conversation also touches on the idea of using real and rational numbers to model physical time and space, and whether they are limited by a finite or discrete structure. In conclusion, the paradox highlights the limitations of using mathematical concepts to model physical phenomena.

#### archaic

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 traveling 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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archaic said:
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?

archaic said:
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 traveling 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?

PeroK said:
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.

archaic said:
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?

archaic said:
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.
archaic said:
Now you can [eat] a ##0## size cake
No you can't because it doesn't exist.
archaic said:
, 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?

archaic
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

pbuk said:
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!

Motore
PeroK said:
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.
pbuk said:
If you go forward by 5 metres in 1 second, how far will you go forward in 0.1 seconds? In ε seconds?
pbuk said:
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.

archaic said:
You could start with the alternative assumption that motion is impossible and see how far you get.

Ibix, jbriggs444 and pbuk
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".

archaic
PeroK said:
Is yours a mathematical or a physical paradox?
To me looks more like a philosophical one

## 1. What is an inverse Zeno's paradox?

An inverse Zeno's paradox is a theoretical concept in which the rate of change of a system increases as time progresses, rather than decreasing as in the original Zeno's paradox. This means that the system would approach its final state faster and faster, rather than taking longer and longer to reach it.

## 2. How is an inverse Zeno's paradox different from the original Zeno's paradox?

The original Zeno's paradox states that as time progresses, the rate of change of a system decreases, making it take longer and longer to reach its final state. In an inverse Zeno's paradox, the rate of change increases as time progresses, making the system approach its final state faster and faster.

## 3. Is an inverse Zeno's paradox possible in real life?

While an inverse Zeno's paradox is a theoretical concept, it is not possible to observe it in real life. This is because it goes against the laws of physics and the concept of time, which state that the rate of change of a system should decrease as time progresses.

## 4. How is an inverse Zeno's paradox relevant in scientific research?

An inverse Zeno's paradox is relevant in scientific research as it challenges our understanding of time and the rate of change of systems. It can also be used as a thought experiment to explore the limitations of our current scientific theories and to develop new ones.

## 5. Are there any real-life examples of an inverse Zeno's paradox?

No, there are no known real-life examples of an inverse Zeno's paradox. However, some physicists have proposed potential scenarios in which an inverse Zeno's paradox could occur, such as in the behavior of certain subatomic particles or in the expansion of the universe.

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