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Okay now that you have voted, if you voted yes, then do you also agree that, physically, it is impossible to travle through an infinite amount of points in a finite amount of time? If so, then every time you touch something you are defying your own universe and its physical properties. Because imagine that the end of your finger is a point and, well pick a random point, the spacebar on your keyboard for instance. Now, if you voted yes on the poll then you agree that there is an invinite amount of points now between your finger and your spacebar. If you also agreed that it is impossible to travel an infinite amount of points in any finite amount of time, but yet, there you go. Touching your keyboard, or the wall or your mouse or the light switch. All of it, there is an infinite amount of points between you and everything you can touch that you aren't always touching, but yet we keep on fooling ourselfs. Going through life beliveing in things that aren't real.
If you voted "no" on the poll then please, by all means prove all mathmaticians and physicsists from the last several hundred years wrong. Feel free to explain how there can not be an infinite amount of points between any 2 points in space.
There are an infinite number of points, but they are infintesimally small. It basically takes nothing to travel the distance of one of the points, so traveling the distance of an infinite number of them isn't impossible. It's very obvious that it isn't impossible, because it happens all the time.
Perhaps, but then again it could just be that everything you think you have ever done is nothing more that a computer simulation. Let me give you a real live example. Right now I am 10ft from my door. If I travel exactly half of that distance, I am now 5 ft from the door. If I travel exactly half of this distance again I will be 2.5 ft from the door, so I go half the distance again. Now I am 1.25ft from the door. I am still not at the door so I will go half the distance, now I am 6.25ft from the door. I am still not there so I continue closing in the distance by hal each time until I am touching the door, except that I should never beable to reach it.Originally Posted by Chemboy
I agree. If two points are a distance x points apart then the sum of the distances between all adjacent intervening points must equal x.There are an infinite number of points, but they are infintesimally small. It basically takes nothing to travel the distance of one of the points, so traveling the distance of an infinite number of them isn't impossible. It's very obvious that it isn't impossible, because it happens all the time.
Sorry I voted no because you did not say between any 2 points "in space".Originally Posted by biohazard87
If you had said any two points on a line or some such mathematical object then I would most definitely have said yes.
I will leave the refutation of your Xeno like argument to someone else.
The only precise and correct answer to the question (as it is posted) is: No! It is not always true.
Not if your two points are identical. :D
I think that the points are on the line joining the said points. It is just a mathematical point of view.
Correct me if I am wrong.
Oh? Here's a "mathematical object": Q (this is the rationals). Convince me that it is connected!Originally Posted by mitchellmckain
You can sum an infinite number of values (like the time necessary to travel an infinite set of distances) and get a noninfinite result. The series 1/2 + 1/4 + 1/8 + 1/16... has an infinite number of components, but when you sum them you get a value of 1, a decidedly noninfinite value. That's why even though there are an infinite number of halfdistances between two points, and each halfdistance has a discrete, nonzero value, you can sum the time necessary to travel across all of them and come up with a noninfinite amount of time.Originally Posted by biohazard87
So no, there is no reason to assume that it must take an infinite amount of time to travel through an infinite number of points (or halfdistances, or whatever).
That does not explain how you actually did an infinite number of things in a finite amount of time Scifor. Showing that the sum is finite is not the resolution of the paradox, as it does not show how you arrived at your end point (limits don't handle arrivals, merely approaches)
You don't lose mathematical relationships just by introducing infinity.
eg y = 1 + x, g = x
now as x goes to infinity, g = infinity and y may approx = infinity but it is still exactly 1 greater than g so y  g = 1;
So, as I said earlier, the distance between two points is equal to the sum of the distance between all adjacent intermediate points. Furthermore the time to travel between the two points is equal to the sum of the times to travel between all adjacent intermediate points. The time to travel between each point is finite because time = distance/speed are always related 1/speed which is finite ie the infinities don't outrun each other cos the ratio of distance to time remains constant (or at least finite if acceleration is involved and the speed is bounded by the speed of light).
touch is an illusion,
you do not touch the keyboard. you almost touch the keyboard.
to truly touch the keyboard, you'd have to merge the nucleuses of the finger and your keyboard, which would result in an explosion, and yet, you'd still have not truly touched the keyboard. maybe you'd want to merge not only the nucles, but also the protons into quarks. maybe you'd want to merge the parts of which the quarks consists of, ad infinitum.
the spacing of the nucleus and its electrons should be enough to convince that we never truly touch anything. its more "reacting to our fingers presence"
You can do an infinite number of things in a finite amount of time if each thing takes less time than the previous thing. Suppose I can perform one action in 1/2 second, then a second action in 1/4 second, a third in 1/8 second, etc. How many actions will I have performed after 1 second? Surely you would agree that I have performed an infinite number. In this case each “action” would be traveling half the distance to my destination.Originally Posted by river_rat
He asked how it was possible to travel through an infinite number of halfdistances in a finite time. It seemed that he was making he implicit assumption that if you sum an infinite set of discrete values (such as the time necessary to travel through a distance), you would have to get an infinite result. I was simply pointing out that you can sum a set with infinitely many elements and still get a finite result.Showing that the sum is finite is not the resolution of the paradox, as it does not show how you arrived at your end point (limits don't handle arrivals, merely approaches)
Also, I disagree with your statement "limits don't handle arrivals, merely approaches". Calculating a limit allows you to calculate the exact time (or whatever) of an arrival. The limit is the arrival time. If there is some nonzero difference between the your current time and the limit time, you have not arrived yet. If there is zero difference between your time and the limit time, you have arrived. I hope you aren't going to argue that there is a difference between "approached infinitely close" and "arrived".
And by doing so you would never arrive at your destination for at no point of your journey you could stop and exclaim "I have arrived, it is finished" Supertasks are not as simple as some elementary textbooks would want you to believe, for starters you need transfinite numbers to make sense of them.You can do an infinite number of things in a finite amount of time if each thing takes less time than the previous thing. Suppose I can perform one action in 1/2 second, then a second action in 1/4 second, a third in 1/8 second, etc. How many actions will I have performed after 1 second? Surely you would agree that I have performed an infinite number. In this case each “action” would be traveling half the distance to my destination.
For example the sequence 0.9, 0.99, 0.999, 0.999, ... would represent your task but what is asked for is the sequence 0.9, 0.99, 0.999, ..., 1 which does not make sense unless you extend the naturals a bit.
I am not disputing that fact, merely that this fact (which the ancient Greeks knew from at least before 322BC) magically resolves the paradox.He asked how it was possible to travel through an infinite number of halfdistances in a finite time. It seemed that he was making he implicit assumption that if you sum an infinite set of discrete values (such as the time necessary to travel through a distance), you would have to get an infinite result. I was simply pointing out that you can sum a set with infinitely many elements and still get a finite result.
There is for at no point do you arrive. There is a huge difference between approaching something infinitely close and actually arriving at that value. Lets do a simple example, using a bit of simple math.Also, I disagree with your statement "limits don't handle arrivals, merely approaches". Calculating a limit allows you to calculate the exact time (or whatever) of an arrival. The limit is the arrival time. If there is some nonzero difference between the your current time and the limit time, you have not arrived yet. If there is zero difference between your time and the limit time, you have arrived. I hope you aren't going to argue that there is a difference between "approached infinitely close" and "arrived".
Suppose I offer you a nice function g from the reals to the reals. It is constantly zero unless you are at the point 0 where it is defined to be 1. Now take any sequence which converges to zero and apply our very nice function to it. This new sequence approaches zero, its limit is zero but it never reaches zero. So arrivals and limits are different  you need some sort of continuity to make them the same and how to supply this needed idea of continuity is far from obvious or natural in this setting.
Suppose I want to travel to a destination that is 1 meter away. I can calculate the distance remaining between me and my destination after I cover each halfdistance as 1/n meters, where n = 2, 4, 8, etc. depending on how many halfdistances I've covered. If I approach infinitely close, the distance left between me and my destination will be 1/infinity. So if you agree that 1/infinity equals zero, and if you agree that when the distance between me and my destination is zero then I have arrived at my destination, I don’t see how you could argue that there is any difference between being "infinitely close" and having arrived.
So which of the following do you disagree with:
1. I can calculate the distance between me and my destination as 1/n.
2. When I am infinitely close, n will equal infinity.
3. 1/infinity equals zero.
4. When the distance between me and my destination is zero, I have arrived.
It's the same as 0.999... exactly equaling 1. Since the difference between 0.999... and 1 is infinitely small, they are the same number. Similarly, once I have approached infinitely close to my destination I have arrived at my destination.
Scifor, you are confusing potential and actual infinites here.
The infinity in 0.999.... is a potential infinity, at no point of the definition is anything infinite. The difference is not infinity small, it is zero. You seem to be thinking of infinitesimals here and framing everything in terms of them.
Do you agree that the sequence (1/n) gets "infinity close to zero"? But it is never zero, it never arrives at zero and in so sense is zero an element of the set {1/n: n is natural}. Your idea breaks down at point 2, at no point is n = infinity as your sequence is only defined for natural numbers. To extend it like that takes you beyond the naturals numbers and into transfinite ordinals.
Read up on supertasks and associated bits and pieces  it is fascinating reading.
riverrat, why dont you read up on calculus instead? if what youre saying is true, it would be impossible to calculate the exact velocity or acceleration at a single moment in time.
also i disagree with you when you say that approaching a value is not the same as arriving at it. you learn this when you study limits in calculus. you can prove this with a simple example: take out a calculator and divide 1/9, 2/9, 3/9, and so on. you will see that your answers are infinite repeating decimals of that number (for example 3/9 in decimal form is 0.3 followed by an infinite amount of 3's and 6/9 is 0.6 followed by an infinite amount of 6's). following this pattern 9/9 would be 0.9999 followed by an infinite number of 9's, which means that it is approaching the value of 1. despite this, you can obviously see that 9/9 is also equal to 1, completely disproving what youre saying about limits
but since 0.9999.... = 1 this is no problem. hence substituting the limit of the function x/9 = 0.xxxxx..... is indeed 1 or 0.99999.....
http://www.thescienceforum.com/don%2...illy4125t.php
ok, what if I have two point's where the difference between them is infinitely small?
Ok, will do. Let's us pick 2 points: A and B. Let us say that these two points are 1 meter apart. There cannot be an infinite number of points because there are set perameters which exist, If you take the number of points available as infinite, then that is like saying a 1 meter ruler is the same as a 100 meter ruler. No matter how far down in scale you go there will ALWAYS be a number that can be attributed to the number of points. The problem is we don't have a means of measuring all of themOriginally Posted by biohazard87
I win me thinks...
Huh? Ill ignore the barbed comment about reading calculus and just ask how one example proves something? Now a counterexample is powerful, but a mere example?Originally Posted by ZebraFiesta
Once again inf{1/n : n in N} = 0 but 0 is not an element of {1/n : n in N}. Do you understand that? To include 0 in the set we are constructing requires us to use transfinite induction, not just simple induction. Supertasks live on the transfinite induction level, they are actually infinite and that is the problem.
Finally, what is the definition of 0.9999...? Is anything infinite in that definition? Is anything infinite in the definition of any decimal expansion? Why do we use partial sums? etc etc etc
What does infinitely small mean? Is the difference zero or nonzero?Originally Posted by Nevyn
ignore that one, me being thick. The other one is more relevant
riverrat, did you even read my example?
what i am saying is 9/9 = 0.9999999 (infinite number of 9's following) = 1, therefore an infinitely small value is negligible and = to 0.
The whole basis of calculus is making this assumption and using it to take derivatives/integrals. i was merely pointing that you would understand much better if you were taking a calculus class.
Calculus is used to tell the value of something at an instantaneous point in time. For example, if you set up a position graph as a function of time, any algebra 1 student can find the average velocity of the object during a finite amount of time by taking the slope of the secant line connecting the graph at 2 points. However, it is much harder to calculate the instantaneous velocity, which is the velocity at an exact point in time. Newton did this by setting a limit where the finite amount of time becomes 0, and therefore would give you the instantaneous velocity.
may I also add that Infinity is not a number, it is a concept.
I did read your example ZebraFiesta and once again i do not think you understand the concepts you are throwing around here. What is the definition of 0.9999....? It is not "infinite number of 9's following" as you claimed, it has a proper definition that excludes the use of infinity entirely and for good reason.Originally Posted by ZebraFiesta
Where did you get that calculus rests on infinitely small quantities etc? Calculus rests on limits and there is nothing infinitesimal about those. Now perhaps newton used infinitesimals when he started the calculus but Cauchy fixed up that logical nightmare. To use infinitesimals correctly requires nonstandard analysis and i am certain you are not referring to the hyperreals here!
The infinitesimals have no logical place in standard calculus!
Well thats an entirely different kettle of fish Nevyn as it depends on what infinity you are talking about. For example the infinite ordinals are just as decent a number as any natural number and they are very infinite!Originally Posted by Nevyn
infinite amount of points doesnt mean infinite distance, which is impossible to trevel through in a finite time.Originally Posted by biohazard87
i would like to ask you a question.
do you believe that there are infinite amount of instants between two seconds, or between two inatants?
if your answer is yes, you will be travelling infinite points in infinite instants.
isnt that possible?
Exactly. I agree there isn't an infinate amout of time between 2 seconds but you can divid the time that finite amount of time that does exhist between those two seconds into an infiate amount of sections.Originally Posted by basim
I am not saying that there can be 100 meters in the one meter, but that the one meter can be broken up into an infinate amount of points that add up to the 1 meter. There is a big difference between what I am saying and what you are saying. You are basically saying that there is no unit smaller than a meter and so for an infinate amount of points to exhist between two points set am one meter apart that there has to be someway to add space between them without moving them.Ok, will do. Let's us pick 2 points: A and B. Let us say that these two points are 1 meter apart. There cannot be an infinite number of points because there are set perameters which exist, If you take the number of points available as infinite, then that is like saying a 1 meter ruler is the same as a 100 meter ruler. No matter how far down in scale you go there will ALWAYS be a number that can be attributed to the number of points. The problem is we don't have a means of measuring all of them
Not that I want to get in on this argument, but I just thought I should mention two things.
One is that some theories of the universe (Loop Quantum Gravity for one, IIRC) say that, in fact, there are a finite number of points between any two points. (Also, I think they say that time is discrete too.)
Two is that within any nonzero tolerance of a one meter line, I can fit infinitely many onedimensional meters. That's just because any twodimensional space is infinite when measured with a onedimensional scale.
you have misunderstood what I have said, I am saying that according to you that if a 100m distance and a 1m distance BOTH have an infinate number of point, is this like saying they are Identicle? as too the second part you have also misinterperated that as well, I am saying that no matter how far down the scale you go (even if you count the atoms themselves) you will still end up with a number that is is less than infinity because there are set perametersOriginally Posted by biohazard87
so you came to know scientifically why we can travel like that?Originally Posted by biohazard87
more over what i meant instants is sections of time you said.
No. They are not identical because they amount of space between them is different, but there IS an infinate amount of points in both distances. Thnk og it like this if I draw a line  I can draw any number of perpindicular lines llllllllllllllllllllllllllllllllllllllllllllllllll lll that are still on that line, the number on lines I can draw depends on how far a part they are from each other and how thin or thick they are. The thinner and closer together they are the more lines you can make.Originally Posted by Nevyn
This is true, but going back to the line example, you can define the thickness of the lines to be less that an atom and almost right on top of each other. So because the thickness of the line can always be smaller that what you have and the lines can always be closer together you can infact fit an infiate amount of lines into a finite amount of space. [/quote]Originally Posted by Nevyn
No. They are not identical because they amount of space between them is different, but there IS an infinate amount of points in both distances. Thnk og it like this if I draw a line ____________________ I can draw any number of perpindicular lines llllllllllllllllllllllllllllllllllllllllllllllllll lll that are still on that line, the number on lines I can draw depends on how far a part they are from each other and how thin or thick they are. The thinner and closer together they are the more lines you can make.Originally Posted by Nevyn
This is true, but going back to the line example, you can define the thickness of the lines to be less that an atom and almost right on top of each other. So because the thickness of the line can always be smaller that what you have and the lines can always be closer together you can infact fit an infiate amount of lines into a finite amount of space. [/quote]Originally Posted by Nevyn
I voted no. The reason is simple:
There are no infinities in nature. Infinity is just a concept, a mathematical abstraction. It isn't real.
Biohazard, divide that horizontal line with an infinite number of vertical lines. OK, you spend an hour doing it. That's a lot of lines, but not infinite. Spend a year doing it, that's a hell of a lot of lines, but still not infinite. Spend a zillion years doing it. Still not infinite. It's just not doable. Never is, never will be. There are no infinities in nature.
It's just mind games. Rather like my carpet. The wife said we need to carpet the spare bedroom, and told me to bring home sixteen square metres. So I did. I carried it over my shoulder. It measured... minus four metres in length by minus four metres in width.
This is an assertion desperately in need of proof. Do you have one? While you're at it, also define "real" (or read some Kant, who had some cool things to say on the subject)Originally Posted by Farsight
And the circularity of this argument doesn't bother you? So it takes an infinite amount of time to draw an infinite number of lines? So what? ∞ = ∞, wow!Biohazard, divide that horizontal line with an infinite number of vertical lines. OK, you spend an hour doing it. That's a lot of lines, but not infinite. Spend a year doing it, that's a hell of a lot of lines, but still not infinite. Spend a zillion years doing it. Still not infinite. It's just not doable. Never is, never will be. There are no infinities in nature.
For sure there are "lines" with discontinuities; here's one ......, so what? There are those without, that is, they are infinitely subdivisible. And that is, in the context of line segments, the definition of a continuous line segment
Mind games? So you're against thinking too?It's just mind games.
the distance you travel will cost you a certain time that can e infinitely small,
if I travel 1 meter and I have an infinite number of points, these points will add up to 1meter all together.
For me to travel an infinitely small distance I need an infinitely small time.
so distance is relative to time dilation
so If it takes me 5 seconds to travel 1 meter I can split this 1 meter in an infinite number of points, but I will also have to split the time, so
if I take 5 seconds for 1 meter then it’ll be 2,5 seconds for 0,5 meters and 1,25 for 0.25 (4 points) meters and so on. If you multiply the points times the values (and the points can be an infinite amount) you will always get 1 meter and 5 seconds.
This is something that you learn in 5th grade (multiplying and dividing), so I don't see any Problems for you to understand this.
I also don't see any necessary comments on this topic, so don't post.
Thank you,
miomaz the threadcutter.
I was thinking through this riddle when it occurred to me that when I travel from say my chair to my door, I don't travel in half distances. I travel in finite distances. And while there may be an infinite amount of points between each finite distance I still move that distance. So, I guess that I do not agree with:
Originally Posted by biohazard87
and will the rabit ever catch the turtle?
as someone already said. first of all, no there isnt an infinite points between x and y. it is limited by the quanta.
and two, yes you can go through an infinite number of points in a finite amount of time, as long as the length of the finite time is long enough considering the distance between the first and the last point of your infinite number of points.
this is what i mean by that:
the problem with the paradox is that it talks of never being able to do something, when there is no "never" involved. in breaking the distances down in that way, we are also breaking time into sections so that in a way the length of time we have to travel the said distance is actually not long enough to cover the distance .....
ie we travel 1/2 in 1 minute, 3/4 in 1:30, 7/8 in 1:45, to get to 1 we need 2 minutes, but the way you describe the events by halfing the distance to the next point we will never have it.
I suppose theoretically, if you could travel at the speed of light, and if that would alter your subjective time enough to stop it completely, you could move infinetly far at that speed, and still end up in whatever finite time you felt like slowing down and stopping.
As far as these "points" go, yes you could put an infinite number of points into any distance (unless, as someone posed above, space was discrete & quantifiable which would make infinity space impossible) but you could never travel through an infinite amount of points because you could never fill any measurable distance with enough points before the movement took place.
Either way, the answer to the poll is no, because you are able to "put" infinite points in any distance, but the points aren't there already.
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