Zeno's paradox

[Lord Bennington]

What do we think of Zeno's paradox, which states that you can never catch up to anyone, because by running at twice their speed, you constantly half the distance between you, and nothing /2=0. Personally, I think the whole thing is flawed, because by running twice as fast as someone you don't just half the distance, you go twice as far as they do which allows you to catch up and pass.

[Merlock]

Hm, it's flawed because it assumes some false sense of infinite continuation, where as in actuality the faster runner will eventually run ahead of the slower one, not stay in a constant state of running. In order to constantly stay behind the slower runner, the faster runner would have to stop periodically and wait. The graph actually shows its flaw: the lines intersect and go past that intersection. The stair-like line doesn't for some reason, implying that it nears that intersection point with a constantly decreasing speed, infinitely. But both runners' speeds stay the same. Hence, apparently, the flaw.

[thegnome54]

by running at twice their speed, you constantly half the distance between you, and nothing /2=0. Wait, what? Tell me if my reasoning is wrong here, but this is my first instinct: We have runner A fleeing from runner B. Runner A is going at 1 unit per second, runner B is going at two units per second. If we look at the situation from runner A's point of view, we can pretend that runner A is stationary, while runner B is moving towards him at 1 unit per second. Now, how is running towards something at 1 unit per second "constantly halving the distance between you"?

[Replicon]

It's all about converging and diverging series

[thegnome54]

It's all about converging and diverging series Isn't the series essentially just -5 (behind),-4,-3,-2,-1,0 (you catch up),1,2,3,4,5 (ahead)?

[Cyclic13]

hey gnome, did anyone tell you that the parrot on your shoulder looks like Mr. Hanky the christmas poo?

[Xei]

Of course it's flawed. Are you really so unfit that you've never been able to disprove this experimentally? Basically it's expressing the finite distance (overtaking distance) as an infinite sum of progressively smaller values. Then it makes the false assumption that if you add an infinite amount of positive values, you'll have an infinite answer. This is totally wrong. For example, think of this sum: 0.9 +0.9/10 +0.9/10/10 +0.9/10/10/10 etc. First calculation you have the answer 0.99. Then you have 0.999. Then 0.9999. It's obvious that if you do the sum forever you'll have the answer 1, not infinity.

[TheUncanny]

I think the paradox has been improperly framed by the initial post. The paradox is supposed to be exactly when do you "catch up" to the other person (not if you are able to catch up with the other person). The "when" is represented by the stair-case looking set of lines in the graph that Merlock posted, and is meant to demonstrate the infinite progression of fractions. A similar example is the "unreachable wall". In this example, there is an object and a wall, and the challenge is to find out how many times it would take for the distance between the object and the wall to be cut in half before the object actually reaches the wall. Technically object A will never actually reach the wall because no matter how many (finite) times you divide a value by 2, it will never reach absolute 0. For example, if the object is 1 meter away from the wall, it would go something like this: step 1: .5 meters apart step 2: .25 meters apart step 3: .125 meters apart step 4: etc and so on infinitely. I believe that this example was trying to touch on the same fundamental phenomenon but in a different way. If object A is going 2 units per second, and object B is going only 1 unit per second, they will indeed meet up given a finite amount of time. For example, if object B is 10 units ahead of object A, and B is travling at 1 unit per second and A is travling at 2 units per second, they will meet up in percisely 10 seconds. A: 2 4 6 8 10 12 14 16 18 20 22 B: 11 12 13 14 15 16 17 18 19 20 21 But imagine you have a microscope that could examine these two racing objects to an infinitely small magnification. As the wall theory would suggest, in order to pin-point the ABSOLUTE EXACT moment in which one object met the other object, you would have to look infinitely close between seconds 10 and 9.99 (or 9.999, 9.99999999999, etc.) to no end...(with the wall being analogous to the 10-second mark) I think lol.

[wasup]

Referring to the idea this idea... the distance you have to go gets infinitely smaller, but you always have more to go. But the time it takes to go that distance gets infinitely smaller. Thus, the paradox is solved

[Xei]

Mhm that's how Aristotle did it. You can do it mathematically pretty easily too.

[Replicon]

Isn't the series essentially just -5 (behind),-4,-3,-2,-1,0 (you catch up),1,2,3,4,5 (ahead)? Well, if you define the sequence as "distance between Achilles and the tortoise at leach 'half-way' point", and add it up as an infinite series, you get the total distance traveled by Achilles before he catches up to the tortoise, as a geometric series. Of course, you'd only have to show that the sequence converges to zero (in this case) to prove that the paradox is flawed.

[Universal Mind]

Referring to the idea this idea... the distance you have to go gets infinitely smaller, but you always have more to go. But the time it takes to go that distance gets infinitely smaller. Thus, the paradox is solved I think that is the best explanation I have ever come across. An infinite number of steps are covered, but there are an infinite number of moments for the steps to take place. The fact that distance is infinitely divisible is compensated for by the fact that time is also infinitely divisible. Infinity can be covered by infinity. If that is not the full explanation, the truth lies somewhere in there.

[Xaqaria]

the truth lies Can I quote you on that? (rhetorical, I just did) Ehh. Bad joke. It's late (for me) and there really is nothing meaningful left to contribute to this thread.

[Mysteryhunter]

hey gnome, did anyone tell you that the parrot on your shoulder looks like Mr. Hanky the christmas poo? Were half through Feb, and still with x-mas themes?

[Universal Mind]

Bump.

[Xei]

Answered in the 0.999~ thread to a similar way that I did it in this thread, except formally and using 1/2 instead of 9/10.

[Universal Mind]

We can keep the discussion there, but for those who didn't see it, this was my response... That gives a broad explanation of the phenomenon on the whole, but it does not answer my specific question. We are looking at converging geometric series under a microscope as it applies to motion. There is a last step, and it is when the man reaches the tortoise. Right? Well, that point is also one of the previous tortoise points. Right? The paradox is that the man can never reach the tortoise without reaching the previous tortoise point first, and the tortoise stays in motion. Yet, the man reaches the tortoise, so the tortoise is in a position that is ALSO a previous tortoise point. Apparently the distance between the tortoise and that last previous tortoise point is infinitely small. I know that converging geometric series is the explanation, but I am asking you to look further into the situation. The man always reaches the previous tortoise point before he can reach the tortoise, and the tortoise is always moving away from the last tortoise point. Get it? Tortoise and previous tortoise point end up converging, just like an object falls and hits the ground even though it goes half way to the ground, then half that distance, then half that distance, infinitely (another version of Zeno's Paradox). But I have never come across a detailed enough explanation that satisfactorily answers how any of this happens.

[Never]

I think that is the best explanation I have ever come across. An infinite number of steps are covered, but there are an infinite number of moments for the steps to take place. The fact that distance is infinitely divisible is compensated for by the fact that time is also infinitely divisible. Infinity can be covered by infinity. If that is not the full explanation, the truth lies somewhere in there. I am not grasping how infinity can cover infinity. Can you elaborate on what you mean by this? I have always just though of this as our concept of time being flawed; an attempt to measure something that cannot be exactly measured by virtue of its "infinity" in our measuring system. If we accept that infinity will suffice, then are we not starting over from the beginning?

[Universal Mind]

I am not grasping how infinity can cover infinity. Can you elaborate on what you mean by this? I have always just though of this as our concept of time being flawed; an attempt to measure something that cannot be exactly measured by virtue of its "infinity" in our measuring system. If we accept that infinity will suffice, then are we not starting over from the beginning? I don't think it starts us back at the beginning, but I do think that it is a broad answer that does not fully explain away the details of the paradox. The truth lies somewhere in there, but I can't fully tell you how. I don't know of anybody who can. Xei talked in a similar thread about how a converging geometric series is involved (The sum of the infinite set of fractionally and progressively diminishing parts = 1), and that is an even more specific explanation, but I think it too leaves questions unanswered. The point is that an infinite number of levels are involved in any level of motion, so part of the issue concerns how infinity can ever be reached. Can something travel all the way across an infinite universe? In an eternity, it could, whatever that seemingly impossible process could be. Can an infinite number of steps be covered in motion from point A to point B? In an infinite number of moments, yes. How exactly can either infinity be covered? An infinite number of points exist between A and B, and with limitless time, a stopping of time does not stop the journey. Beyond that, I don't know. (Some people here might be surprised to see me say, "I don't know," but Zeno's Paradox is one thing that has me saying it.)

[wendylove]

. The point is that an infinite number of levels are involved in any level of motion, so part of the issue concerns how infinity can ever be reached. Not, really. The point of the zeno paradox is that something is always in motion, infinitly small motion over a infinitly small distance. Basically, do you agree with the fact of a motion time graph, if you do then you will agree that it has a gradient at a certain point. If you use simple differentiation you will get the acceleration, all you need is this to be positive or negative but decreasing at a certain rate. Can something travel all the way across an infinite universe? Firstly, there is speed to which someone can travel. In the zeno paradox someone is traveling across a infinite amount of points, but not distance. Again, newton mechanics doesn't break down if something travels a meter, as again its just pre-school calculus. How exactly can either infinity be covered? An infinite number of points exist between A and B, and with limitless time, a stopping of time does not stop the journey. Simple calculus. Its funny how this problem was solved ages ago, yet it keeps coming up because people are ignorant of maths, hence that stupid is 0.99999...=1 thread. Beyond that, I don't know. (Some people here might be surprised to see me say, "I don't know," but Zeno's Paradox is one thing that has me saying it.) Anyway, buy a pre-school calculus book and maybe a book mechanics. P.S. 0.99...=1, generally k/9=0.kkkk..... The fact that distance is infinitely divisible is compensated for by the fact that time is also infinitely divisible. Infinity can be covered by infinity. If that is not the full explanation, the truth lies somewhere in there. Distance isn't infinitely divisible, well thats what I heard. Anyway, your argument is not mathematically rigorous. Infinite is a tricky thing and is best left to mathematician and not layman. For example, did you know that the size of set (1,2,3,4,5,6,7,8,9,,) is the same as the set (1/n,1/(n-1),,,,1,1+(1/n),,,2,,,,4,,,,6,,,,7) where n is a natural number. Both of them are equal, yet it kind of defies common sense. However, is the set of all the point between 0 to 1 the same size as both the top sets. The anwser is no, infact its soo big it can't be counted. Lastly, do you think that there is another set between two top sets and the bottom set. Strangely the problem is undecidable at the moment.

[drewmandan]

Is there an "aleph 0.5"? Now that's the question of the ages.

[Universal Mind]

Not, really. The point of the zeno paradox is that something is always in motion, infinitly small motion over a infinitly small distance. Basically, do you agree with the fact of a motion time graph, if you do then you will agree that it has a gradient at a certain point. If you use simple differentiation you will get the acceleration, all you need is this to be positive or negative but decreasing at a certain rate. Isn't that a Jesse Jackson quote? How is "getting the acceleration" an issue? Please stay on topic. Firstly, there is speed to which someone can travel. In the zeno paradox someone is traveling across a infinite amount of points, but not distance. Again, newton mechanics doesn't break down if something travels a meter, as again its just pre-school calculus. Objection. Irrelevant. Simple calculus. Its funny how this problem was solved ages ago, yet it keeps coming up because people are ignorant of maths, hence that stupid is 0.99999...=1 thread. Why do fascinating threads like this attract such insulting posts? The issue is still worthy of discussion. That is why it is still taught at universities. Even if it were not, why act like that? What is the point, other than to troll? Your post does not explain away the paradox, layman. It almost doesn't even address it. You just left a post that would seem really confusing to most people. That's all you did.

[Placebo]

I tend to agree. Wendylove, you realize that all you really said was 'lulz, ur all so dumb. Get da calculus book'. But couched in looking smart. I suspect you do. I suspect you're trolling as well. Unfortunately, you're annoying to prove as a troll. Either actually try to contribute to the thread, or go away

[Xei]

The fact is Zeno just made up some arbitrary step point and claimed it had some sort of bearing in physical reality. Ok, how about this equivalent problem; I want to walk across to the other side of the room. However I conceive of this distance as being divided into infinitely many infinitely small sections. Oh no, how can I ever cross the room if there are infinity sections?? And which section is at the end of the room, considering you can always claim it is the next one after that?? Anybody can see that it just makes no sense.

[wendylove]

How is "getting the acceleration" an issue? Please stay on topic. Okay. Do you know the difference between points and length? Something can have infinite amount of points, but a finite length. Objection. Irrelevant. Learn calculus, its not that hard. Your post does not explain away the paradox, layman. It almost doesn't even address it. You just left a post that would seem really confusing to most people. That's all you did. Actually it does. In calculus you can differentiate stuff, this finds how much a function decreases or increases by at that point. This is improtant as it proves that something can have motion at a infinitely small point, hence why something can travel infinite amount of points at a finite time. Is there an "aleph 0.5"? Now that's the question of the ages. I heard that most research into infinity is too try and prove stuff about large cardinals. Get da calculus book'. But couched in looking smart. I suspect you do. I suspect you're trolling as well. Unfortunately, you're annoying to prove as a troll. The point is this, get a calculus book. Maths isn't a spectator sport to really learn about why the zeno paradox is rubbish you actually need to do some calculus.