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Zeno is a greek philospher who thought motion is impossible. I have to write an essay to refute his argument intelligently (rationally, can't just say stuff like "wow how dumb"). I have an argument in mind but I'd like to see some more insights before I start writing. Here is his argument:
1. Zeno starts by assuming what his opponent says is possible: motion. In particular, the motion of a single body across a finite distance in a finite time.
2. To make things vivid, let’s specify the moving object, and where it is supposed to be moving: Imagine a sprinter, who starts running at one end of a 100 metre straight track, and runs to the other end. Zeno’s opponents (probably including yourself) think that it really is possible that runners can do this. (i.e. They think that it is not just an illusion.) Zeno begins by assuming that his opponents are correct, and then the fun starts.
3. Zeno points out, given that we are assuming that space is continuous, that before the runner can cover the whole 100 metres she has to cover half the distance, i.e. run 50 metres.
4. Zeno then repeats the move just made (point 4) and points out that the move can be repeated an infinite number of times: before the runner can cover 50 metres, she has to run 25, before that she must run 12.5, before that 6.25, etc. Recall that we are assuming that space is continuous, which means that any finite piece of it, such as a sprint track, can be divided into infinitely many parts, which are infinitely small.
5. Zeno then argues that to cover each of these infinitely small parts will take a certain amount of time.
6. But to take a certain amount of time an infinite number of times adds up to an infinite amount of time. So it would take the runner forever to cover 100 metres. But we were assuming that the runner could cover the distance in a finite amount of time, not that she would take forever.
7. We have run into a problem, and Zeno’s conclusion is that motion is impossible, because any finite motion would take forever.
Thoughts?
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On July 24 2005 22:54 Hydrolisko wrote:
Many times when I am driving or even a passenger in a car, I imagine a little elf-like man jumping from tree-top to tree-top right along with my car.
Thoughts?
*fixed
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Sounds like Zeno made a living out of splitting hairs
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wtf i thought this exact shit up myself years ago. Do philosophers still get paid these days?>
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This sounds much like Half Life. The part where u said that, it would take X amount of time for 50m and x amount for 25 and so on. Keeps dividing and cant get a number correct? If not, then i have no idea what you're talking about.
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he confused people into thinking that he actually had something.
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On July 24 2005 23:02 tlstmddn wrote:
wtf i thought this exact shit up myself years ago. Do philosophers still get paid these days?>
He thought of this in 500s BC, about 2500 years ago.
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wow i tought intelligent people would understand the concept of infinite IN infinites.
Ill put it in a veeeery simle way to explain it.
Imaginte an INFINITE row of 1 dollar bills, now:
Imagine an INFINITE row of 100 dollar bills, just under the 1 dolar row, 1 : 1 ratio right?
So in wich of those you have more money?, well you might think, in the $100 bill row right? but how? if you have INIFINITE money on the other one? errr... confusing?
SIMPLE, there are infinites in infinites, i cant remember the whole formula to prove it, can someone post it?
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Well, I don't know. You could just reason that the amount of time required to complete the 100m is a fixed time. Say X. Zeno argues that:
x = x/2 + x/2
x = x/2 + x/4 + x/4
x = x/2 + x/4 + x/8 + x/8
x = x/2 + x/4 + x/8 + x/16 + x/16
and that somehow the more elements you add the longer the time is, but it's always fixed.
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Jus put that he failed to realize that his conclusion isn't so much based on the fact that motion is impossible, but based on a fundamental problem of dealing with infinite numbers
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On July 24 2005 23:22 ReRebanned wrote:
Jus put that he failed to realize that his conclusion isn't so much based on the fact that motion is impossible, but based on a fundamental problem of dealing with infinite numbers
exaaaaactly!, dealing with infinite numbers is way beyond most ppl minds.
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my fist into his face.
there. i just proved his theory wrong.
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Actually, based on the assumptions that Zeno makes, his explanation IS correct and motion would be impossible. The problem however is the assumption that all space is infinitely divideable into smaller parts. At some incredibly small interval, the motion is quantized and can not be further divided.
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On July 24 2005 23:32 RoTaNiMoD wrote:
Actually, based on the assumptions that Zeno makes, his explanation IS correct and motion would be impossible. The problem however is the assumption that all space is infinitely divideable into smaller parts. At some incredibly small interval, the motion is quantized and can not be further divided.
agreed
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On July 24 2005 23:28 ahk-gosu wrote:
my fist into his face.
there. i just proved his theory wrong.
right back at ya
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Use indefinate integration?
He thought this up before invention of calculus.
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On July 24 2005 22:57 imRadu wrote:Show nested quote +On July 24 2005 22:54 Hydrolisko wrote:
Many times when I am driving or even a passenger in a car, I imagine a little elf-like man jumping from tree-top to tree-top right along with my car.
Thoughts? *fixed
Best post ever! I seriously just laughed out loud and woke my puppies up.
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MURICA15980 Posts
On July 24 2005 22:57 imRadu wrote:Show nested quote +On July 24 2005 22:54 Hydrolisko wrote:
Many times when I am driving or even a passenger in a car, I imagine a little elf-like man jumping from tree-top to tree-top right along with my car.
Thoughts? *fixed
WOW~!
I can't stop laughing haha ;D
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Zeno had even better theory.
1. You drop a grain, what do you hear? Nothing.
2. You drop a lot of grains (like pouring them from a bag or sth) and you definitely hear them falling on the floor.
Now, what he says is that if you don't hear one grain falling and when you drop a bag of them it's just many 'one grains' fall then how come you hear it? It's just an imaginary sound ^_^
And to all that despise philosophers: Even if their theories are completly dumb you shouldn't think they are. Philosophers task is NOT to create some superb and rational theory but to force others to think. This way science goes on. (Yeah, 90% of great scientists were philosophers eq. Newton, Einstein, Darwin, La Place, Pithagoras etc.)
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He tricks you into believing that infinite additions gives infinite high numbers. That is not true, however. If the numbers get small enough - fast enough - the sum will go towards a number, never hitting it, but coming very close, and here comes the concept permit limit.
Lets assume he runs 10m/s.
After half the route he will have runned 50 m = 5s
Half from 50m to the finish line = 25m = 2,5s
next: 12,5m = 1,25s
next: 6,25m = 0,625s
next: 3,125m = 0,3125s
next: 1,5625m = 0,15625s
next: 0,78125m =0,078125s
Add these seconds up:
5s + 2,5s + 1,25s + 0,625s + 0,3125s + 0,15625s + 0,078125s = 9,921875s
If we continue:
next: 0,390625m = 0,0390625s
next: 0,1953125m = 0,01953125s
next: 0,09765625m = 0,009765625s
and so on
Add these seconds up:
9,921875s + 0,0390625s + 0,01953125s + 0,009765625s = 9,990234375s
See how it gets closer to 10? If we continue it will go towards 9,99999...s
Here is permit limit useful. Without explaining it in detail the result is:
a -> 10, n ->infinite
A is going towards 10, if n goes towards infinite
a = the whole sum
n = times of additions
You will probably have to rewrite it because my English is quite bad. Hope you got the meaning 
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mrmin123
Korea (South)2971 Posts
Fractals anyone? Chaos theory? Koch curves? Lorenz Attractors? This thing isn't just some philosopher's ramblings anymore (not that all philosopher's ramblings are... ramblings), it's a science.
Ideas like an infinite length in a finite area (or infinitly long coastlines :F) are crazy fun.
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yea, what Kochen said:
An = 50 * 0.5 ^(n-1)
where the sum of the infinite row would be
Sn = A1 / (1-k) = 50 / (1-0.5) = 100
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Pffff, simple to prove Zeno wrong.
The sprinter can run 100m diestance in, lets say, 10 seconds.
Zeno divides the distane by half (infinitely), and you divide the total time by two (also infinitely). This infinite number of distances add up to 100m, and the infinite number of times add up to 10 sec.
There is a part of math which deals with sums of infinite amounts of infinite small "numbers", and it's called calculus. Using it you can easily prove Zeno wrong, and shove his "prove" up his ass.
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On July 25 2005 02:14 qux.afk wrote:
yea, what Kochen said:
An = 50 * 0.5 ^(n-1)
where the sum of the infinite row would be
Sn = A1 / (1-k) = 50 / (1-0.5) = 100
Correct. Kochen was the first to point out the wrong assumption than a sum of infinite many numbers is always infinite. These kind of sums are called convergent sums. Try this.
Edit: Philosphers are jerks. Most of them.
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On July 25 2005 02:21 IcedEarth wrote:
cant divide with zero...
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Sweden33719 Posts
On July 24 2005 22:54 Hydrolisko wrote:
Zeno is a greek philospher who thought motion is impossible. I have to write an essay to refute his argument intelligently (rationally, can't just say stuff like "wow how dumb"). I have an argument in mind but I'd like to see some more insights before I start writing. Here is his argument:
1. Zeno starts by assuming what his opponent says is possible: motion. In particular, the motion of a single body across a finite distance in a finite time.
2. To make things vivid, let’s specify the moving object, and where it is supposed to be moving: Imagine a sprinter, who starts running at one end of a 100 metre straight track, and runs to the other end. Zeno’s opponents (probably including yourself) think that it really is possible that runners can do this. (i.e. They think that it is not just an illusion.) Zeno begins by assuming that his opponents are correct, and then the fun starts.
3. Zeno points out, given that we are assuming that space is continuous, that before the runner can cover the whole 100 metres she has to cover half the distance, i.e. run 50 metres.
4. Zeno then repeats the move just made (point 4) and points out that the move can be repeated an infinite number of times: before the runner can cover 50 metres, she has to run 25, before that she must run 12.5, before that 6.25, etc. Recall that we are assuming that space is continuous, which means that any finite piece of it, such as a sprint track, can be divided into infinitely many parts, which are infinitely small.
5. Zeno then argues that to cover each of these infinitely small parts will take a certain amount of time.
6. But to take a certain amount of time an infinite number of times adds up to an infinite amount of time. So it would take the runner forever to cover 100 metres. But we were assuming that the runner could cover the distance in a finite amount of time, not that she would take forever.
7. We have run into a problem, and Zeno’s conclusion is that motion is impossible, because any finite motion would take forever.
Thoughts?
Eh..
So what if there's an infinite number of distances over the course of a 100 metre track, once you reach the end of it, you have covered them all --
Or am I not getting something ._.?
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On July 25 2005 02:49 ygor wrote:Show nested quote +On July 25 2005 02:14 qux.afk wrote:
yea, what Kochen said:
An = 50 * 0.5 ^(n-1)
where the sum of the infinite row would be
Sn = A1 / (1-k) = 50 / (1-0.5) = 100 Correct. Kochen was the first to point out the wrong assumption than a sum of infinite many numbers is always infinite. These kind of sums are called convergent sums. Try this. Edit: Philosphers are jerks. Most of them.
Not quite, as you see, a provocing philosopher has evedently made us all think twice about a lot of things.
Another of Xeno's paradoxes are the flying arrow:
An arrow flys from A to B
In between these to there is a third point C. In order to occupy space in point C it is requiered to momentarily be at rest in this particular instant of time. However, imagine an infinate number of points like C inbetween A and B. The arrow is then requiered to stay still in every single of these, thus moving at the same time it is at rest.
This is all supposing i remember the paradox correctly, but the point of it all is that the thought of momentary motion is impossible. That motion is allways measured between two places that indeeed can be very close to one and other but not momentarily. Thus in Xenos' opinion making the thought of motion a contradiction to itself. Not that I agree, but you have to realise that he does have a point.
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MURICA15980 Posts
I think you worded it wrong, because I totally disagree and can't even see where the "thinking" part of it is... unless I'm blind at 3:30am :O
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On July 25 2005 03:35 Klogon wrote:
I think you worded it wrong, because I totally disagree and can't even see where the "thinking" part of it is... unless I'm blind at 3:30am :O
Imagine instantaneus motion.
Then try to think outside the box.
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This is the same kind of crap an old greek guy pulled out of his behind about how a rabit could never catch up with a turtle who had a little head start (even tho the rabit is 10 times faster). But as we all know this is just a way of playing with words and math.
Instead of adding up all these infinitely small numbers into an infinite time, what about this?
The runner runs 100 meter in 11 seconds. And now we want to know how long it takes for him to move one of those infinitely small bit forward (1 divided by infinity). The answer is that it takes him no time at all and no matter how long we keep adding these up we see that it takes 0 second to move an infinitely amount of infinitely small distance. So what do we get? The answer is that the runner runs 100 meter in 0 seconds.
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An infinite amount of intervals that are getting infinitly small equals a finite lenght
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On July 25 2005 01:10 Klogon wrote:Show nested quote +On July 24 2005 22:57 imRadu wrote:On July 24 2005 22:54 Hydrolisko wrote:
Many times when I am driving or even a passenger in a car, I imagine a little elf-like man jumping from tree-top to tree-top right along with my car.
Thoughts? *fixed WOW~! I can't stop laughing haha ;D
I do the same thing, only it's a snowboarder usually grinding the fences along the road and jumping and stuff.
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On July 25 2005 04:29 Chaso wrote:
This is the same kind of crap an old greek guy pulled out of his behind about how a rabit could never catch up with a turtle who had a little head start (even tho the rabit is 10 times faster).
lol, it IS the same thing.
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u gotta be kidding
u retard
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why are you all responding to this?
and don't point out that i'm responding too
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On July 25 2005 04:40 WOstick wrote:Show nested quote +On July 25 2005 01:10 Klogon wrote:On July 24 2005 22:57 imRadu wrote:On July 24 2005 22:54 Hydrolisko wrote:
Many times when I am driving or even a passenger in a car, I imagine a little elf-like man jumping from tree-top to tree-top right along with my car.
Thoughts? *fixed WOW~! I can't stop laughing haha ;D I do the same thing, only it's a snowboarder usually grinding the fences along the road and jumping and stuff.
holy shit i do the same thing except its a skateboarder and i imagine him grinding all the road fences. he holds onto a car then lets go to jump up and grind the border and when he dismounts he grabs onto another coming car bahaha
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Belgium6772 Posts
Achilles and the turtle. It wasn't a rabit.
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HnR)hT
United States3468 Posts
calculus came along, end of story k?
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On July 25 2005 05:55 Xeofreestyler wrote:
Achilles and the turtle. It wasn't a rabit.
My bad.
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On July 25 2005 05:55 Xeofreestyler wrote:
Achilles and the turtle. It wasn't a rabit.
Your right but for all the other kids out there it was warners brothers early version of bugs bunny and the turtle.
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Philosophers are wankers. Burn them all!
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Zeno makes a false assumption in his argument. The sum of an infinite number of things is not necessarily infinite.
sum(n=1 to infinity) (1/2^n) = 1. There are a number of proofs of this. If you want one, Ill post one later.
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Bill307
Canada9103 Posts
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sigh...
There are infinite amounts of half points, but the total time it takes to run is finite. The time it takes to reach the next half point will be shorter and shorter. Therefore the runner finish the run.
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There is no such thing as infinity, it is something man made up. Everything is finite, time, the universe, this senten.....
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is awesome32277 Posts
On July 25 2005 19:18 NewbSaibot wrote:
There is no such thing as infinity, it is something man made up. Everything is finite, time, the universe, this senten.....
prove?
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On July 25 2005 19:18 NewbSaibot wrote:
There is no such thing as infinity, it is something man made up. Everything is finite, time, the universe, this senten.....
Human stupidity =]
On July 25 2005 19:24 Hippopotamus wrote:
What about your ego?
that's close to infinite since he will die
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I knew about this concept a long time ago... It's impossible for the runner to do what Zeno says will happen, because eventually, the distance will get so small, the runner won't be able to move that small of a distance.
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Food for thought:
-500 meters is still equal to 500 meters no matter how you split it up. Kochen said it first and its basic math: What you do to one side or part of the equation you must do to the other side or part of the equation. If you divide distance an infinite amount of times, then you must divide time an infinite amount of times.
500M / infinitely = time to run 500M / infinitely
-However, It is also mathematically feasible to prove Zeno correct. Assuming that we are dividing “time” an infinite amount of times, then somewhere along the line there is bound to be a repeating decimal.
3. Zeno points out, given that we are assuming that space is continuous,
Now, here is where it gets tricky. The following formula stats that any that any decimal that repeats -- (and ,thus, is infinite) – will infinitely approach and equal the next highest whole number, thus rounding it up. (Note: the proof was confirmed by my college math teacher.)
http://www.blizzard.com/press/040401.shtml
lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
-You must also look at the perception point of view. You perceive a single grain of rice falling; if you use an amplifier you can perceive the sound that that grain makes. The same goes for a bag of rice.
In this case the question is, “Are we perceiving that the runner is going from point A to point B? Or is he actually moving from point A to point B? Is it possible that both could be true?”
I hope this helps with your essay.
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On July 24 2005 23:19 baal wrote:
wow i tought intelligent people would understand the concept of infinite IN infinites.
Ill put it in a veeeery simle way to explain it.
Imaginte an INFINITE row of 1 dollar bills, now:
Imagine an INFINITE row of 100 dollar bills, just under the 1 dolar row, 1 : 1 ratio right?
So in wich of those you have more money?, well you might think, in the $100 bill row right? but how? if you have INIFINITE money on the other one? errr... confusing?
SIMPLE, there are infinites in infinites, i cant remember the whole formula to prove it, can someone post it?
But the 100 is also infinite, meaning it will go on and on for as long as the $1 line does, right? So, let's just take a small portion of that and end at 1,000. You have either $1,000 or $100,000. I don't see how you could say that the $1 line has the same amount of money as the $100 line.
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[/QUOTE]
But the 100 is also infinite, meaning it will go on and on for as long as the $1 line does, right? So, let's just take a small portion of that and end at 1,000. You have either $1,000 or $100,000. I don't see how you could say that the $1 line has the same amount of money as the $100 line.[/QUOTE]
You foiled it all when you said a small portion of that line. What he was saying was that infinate is infinate. You cant multiply infinity in the same way. infinate times a million still equals infinate.
When speak about a small portion of these lines, they are no longer infinate.
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Korea (South)17174 Posts
On July 26 2005 08:54 SickofLife wrote:Show nested quote +On July 24 2005 23:19 baal wrote:
wow i tought intelligent people would understand the concept of infinite IN infinites.
Ill put it in a veeeery simle way to explain it.
Imaginte an INFINITE row of 1 dollar bills, now:
Imagine an INFINITE row of 100 dollar bills, just under the 1 dolar row, 1 : 1 ratio right?
So in wich of those you have more money?, well you might think, in the $100 bill row right? but how? if you have INIFINITE money on the other one? errr... confusing?
SIMPLE, there are infinites in infinites, i cant remember the whole formula to prove it, can someone post it?
But the 100 is also infinite, meaning it will go on and on for as long as the $1 line does, right? So, let's just take a small portion of that and end at 1,000. You have either $1,000 or $100,000. I don't see how you could say that the $1 line has the same amount of money as the $100 line.
PLEASE TELL ME YOUR JOKING LOLLLLLLLLLLLLL
thats the whole idea behind the concept of infinity ITS NOT SOMETHING THE HUMAN BRAIN CAN COMPREHEND, IT DOESN'T MAKE SENSE
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sickoflife, they both have the same amount of money.
here is a bit of a trickier example.
Define Z+ = the positive integers. So Z+ = {1,2,3,.....}
2*Z+ = {2,4,6,.....}, the even positive integers.
Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size. Thus, there exists a 1 to 1 and onto map between these two sets, which is known as a bijection. Thus, there is an infinite set which is a subset of another infinite set, and can be mapped 1-1 AND onto it.
To move into analysis, take the interval (0,1). (0,1) is NOT countable, there are an infinite number of elements in there. furthermore, any interval is not countable, as long as it has positive measure then it is infinite. However, the example before, 2Z+ IS countable. Although it has an infinite number of elements, it has measure 0.
Infinity is kind of weird, but to me these concepts are intuitive.
But yeah, being on topic, Zeno's proof falls apart when he makes a false assumption. This was often referred to as Zeno's paradox, because SUM_(n=1 to infinity) (.5^n) = 1.
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On July 26 2005 10:21 BigBalls wrote:
But yeah, being on topic, Zeno's proof falls apart when he makes a false assumption. This was often referred to as Zeno's paradox, because SUM_(n=1 to infinity) (.5^n) = 1.
Its lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 ......
Infinite 1 dollar is equal to infinite 100 dollars. And, Zeno is talking about an infinitely small portion of infinity, but
-However, It is also mathematically feasible to prove Zeno correct. Assuming that we are dividing “time” an infinite amount of times, then somewhere along the line there is bound to be a repeating decimal. -------------------------------------------------------------------------------- 3. Zeno points out, given that we are assuming that space is continuous, -------------------------------------------------------------------------------- Now, here is where it gets tricky. The following formula stats that any that any decimal that repeats -- (and, thus, is infinite) – will infinitely approach and equal the next highest whole number, thus rounding it up. (Note: the proof was confirmed by my college math teacher.) http://www.blizzard.com/press/040401.shtmllim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 0.9999... = 1 Thus x = 0.9999... 10x = 9.9999... 10x - x = 9.9999... - 0.9999... 9x = 9 x = 1.
I think the answer is to be shown as:
-2 -1 0 1 2
<--------------------------------------------------------------------------->
(0 < (X / infinity) <_ 1) / infinity
(0 is smaller than (X divided by infinity), (X divided by infinity) is smaller than or equal to 1) divided by infinity.
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On July 26 2005 10:13 Rekrul wrote:Show nested quote +On July 26 2005 08:54 SickofLife wrote:On July 24 2005 23:19 baal wrote:
wow i tought intelligent people would understand the concept of infinite IN infinites.
Ill put it in a veeeery simle way to explain it.
Imaginte an INFINITE row of 1 dollar bills, now:
Imagine an INFINITE row of 100 dollar bills, just under the 1 dolar row, 1 : 1 ratio right?
So in wich of those you have more money?, well you might think, in the $100 bill row right? but how? if you have INIFINITE money on the other one? errr... confusing?
SIMPLE, there are infinites in infinites, i cant remember the whole formula to prove it, can someone post it?
But the 100 is also infinite, meaning it will go on and on for as long as the $1 line does, right? So, let's just take a small portion of that and end at 1,000. You have either $1,000 or $100,000. I don't see how you could say that the $1 line has the same amount of money as the $100 line. PLEASE TELL ME YOUR JOKING LOLLLLLLLLLLLLL thats the whole idea behind the concept of infinity ITS NOT SOMETHING THE HUMAN BRAIN CAN COMPREHEND, IT DOESN'T MAKE SENSE
It does make sense!
take this for example:
lim [n->oo] (100n)/n = ?
do you think it equals 1 because you have infinity in numerator and denominator?
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kryzch thats a completely different question.
The question we are posing would be (lim [n-> oo] 100n) / (lim [n->oo] n), which isnt 1, but instead indeterminate. the answer to your question is clearly 100.
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On July 26 2005 10:21 BigBalls wrote:
(...)
Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size.
Hey, BigBalls, I know you're good at math, but I guess you have screwed something. Can you actually say, that Z+ and 2Z+ are the same size (which means they have the same amount of elements?) and at the same time say that 2Z+ is a subset of Z+ (which means that there are no elements that can be found in 2Z+ and cannot be found in Z+) and that there are some elements that belong to Z+, but not to 2Z+ ???
If there was a finite number of elements that are in Z+ and not in 2Z+ there would be no problem, because both of them have infinite number of elements. But there is an infinite number of odd numbers, so this just looks wrong.
I have never seen proof for what you have said, and my math isn't good enough to prove it right or wrong. Plus it can be infinity which fucks with my brain and doesn't let me to understand this.
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On July 26 2005 12:08 BigBalls wrote:
kryzch thats a completely different question.
The question we are posing would be (lim [n-> oo] 100n) / (lim [n->oo] n), which isnt 1, but instead indeterminate. the answer to your question is clearly 100.
Yup, you're right. But I just wanted to say, that infinity can be comprehended by humans. At least in it's not that vicious incarnation :-)
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so this runner passes an infinite amount infinitely small distances. infinitely small distances are distances of 0, so we get infinity / 0. use L'hopital's rule and it technically could work out, but not necessarily! we need more information! :O
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On July 26 2005 12:20 Krzych wrote:Show nested quote +On July 26 2005 10:21 BigBalls wrote:
(...)
Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size. Hey, BigBalls, I know you're good at math, but I guess you have screwed something. Can you actually say, that Z+ and 2Z+ are the same size (which means they have the same amount of elements?) and at the same time say that 2Z+ is a subset of Z+ (which means that there are no elements that can be found in 2Z+ and cannot be found in Z+) and that there are some elements that belong to Z+, but not to 2Z+ ??? If there was a finite number of elements that are in Z+ and not in 2Z+ there would be no problem, because both of them have infinite number of elements. But there is an infinite number of odd numbers, so this just looks wrong. I have never seen proof for what you have said, and my math isn't good enough to prove it right or wrong. Plus it can be infinity which fucks with my brain and doesn't let me to understand this.
Define a map from 2Z+ to Z+ by x -> x/2. This map is clearly onto, every element in Z+ is mapped to by an element from 2Z+. It is also 1-1. How do we prove 1-1?
Suppose there is an element in Z+ that is mapped to by more than one element. Thus, x in 2Z+ and y in 2Z+ both map to z. Thus, x/2 = z = y/2, which means x=y. thus, the mapping is 1-1.
Since the map is both 1-1 and onto, the sets are the same size, although infinite.
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i guess its wrong to even mention the word size when infinity is around, so saying there is a bijection between sets of different sizes is of enough merit, would you disagree?
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On July 25 2005 18:33 Bill307 wrote:Show nested quote +On July 25 2005 04:38 jtan wrote:An infinite amount of intervals that are getting infinitly small equals a finite lenght Wrong. 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... diverges to infinity, evening though the intervals are getting infinitely small. There are a number of ways to determine whether a series converges to a finite number or diverges to infinity. That's calculus 2 . Btw BigBalls is right. And probably some other people as well, but his answer was particularly terse  .
haha yes I know, and your right, but it's a pretty simple explanation for the problem in question.
Also, btw, Zenos final conclusion was "motion is an illusion" which sounds cool:D
http://mathforum.org/isaac/problems/zeno1.html
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On July 26 2005 12:30 BigBalls wrote:
i guess its wrong to even mention the word size when infinity is around, so saying there is a bijection between sets of different sizes is of enough merit, would you disagree?
Now, that you have cleared that for me I can't disagree :D
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slightly off topic.
im glad im not the only math nerd alive...anyone do math competitions ^^ maybe i've met u guys before
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yeah.
i got a 30 on the putnam this year, which was good enough for 320th out of 3700 or so.
The putnam is the college math competition, very very tough, median score is a 0, probably 80% score under a 10.
I used to do some in high school, got a 115 on the AMC cause i made like 5 stupid mistakes, which ended up costing me a chance at the USAMO cause i only got a 60 on the AIME (the second competition which was much more difficult). oh well, they were fun tests to take, wish i would have concentrated more on the first test so I could had a shot at the USAMO.
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and i find it kinda funny how i say using the word size is bad then use it again in the sentence, ha
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i overslept for putnam b/c i was too drunk the night before lol.
usamo wasn't too bad..i got 1question right...putting me into top 25 percent..i personally think putman practice look so much easier than usamo problems.
god...i havent done serious math since MOSP 3 years ago. keke.
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Well, I think the putnam problems are more challenging, BUT, americans are generally awful at geometry, and that's all the USAMO is, so it's probably a bit more intimidating, and I guess harder overall because of the lack of algebra, calculus and discrete math problems.
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i'll make sure not to drink before this year's putnam...my strength is geometry and inequalities.....any of those on last year's putnam?
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yeah, those are always the questions i avoid.
im an algebra/discrete/logic guy, not much of a geometry person
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i think inequalities are the easiest problems ever...just takes some time....it always reduces to AMGM or cauchy.
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On July 26 2005 10:26 Tontow wrote:Its lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 ...... Infinite 1 dollar is equal to infinite 100 dollars. And, Zeno is talking about an infinitely small portion of infinity, but Show nested quote +-However, It is also mathematically feasible to prove Zeno correct. Assuming that we are dividing “time” an infinite amount of times, then somewhere along the line there is bound to be a repeating decimal. -------------------------------------------------------------------------------- 3. Zeno points out, given that we are assuming that space is continuous, -------------------------------------------------------------------------------- Now, here is where it gets tricky. The following formula stats that any that any decimal that repeats -- (and, thus, is infinite) – will infinitely approach and equal the next highest whole number, thus rounding it up. (Note: the proof was confirmed by my college math teacher.) http://www.blizzard.com/press/040401.shtmllim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 0.9999... = 1 Thus x = 0.9999... 10x = 9.9999... 10x - x = 9.9999... - 0.9999... 9x = 9 x = 1. I think the answer is to be shown as: -2 -1 0 1 2 <---------------------------------------------------------------------------> (0 < (X / infinity) <_ 1) / infinity (0 is smaller than (X divided by infinity), (X divided by infinity) is smaller than or equal to 1) divided by infinity.
On July 26 2005 12:08 BigBalls wrote:
kryzch thats a completely different question.
The question we are posing would be (lim [n-> oo] 100n) / (lim [n->oo] n), which isnt 1, but instead indeterminate. the answer to your question is clearly 100.
Could you do a proof of that?
I’m not shore 100 is correct Because the way you wrote it:
-The limit of N is infinity. (N = infinity)
- 100N = Infinity
-N/N = Infinity
And remember that you stated that:
On July 26 2005 12:30 BigBalls wrote:Show nested quote +On July 26 2005 12:20 Krzych wrote:On July 26 2005 10:21 BigBalls wrote:
(...)
Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size. Hey, BigBalls, I know you're good at math, but I guess you have screwed something. Can you actually say, that Z+ and 2Z+ are the same size (which means they have the same amount of elements?) and at the same time say that 2Z+ is a subset of Z+ (which means that there are no elements that can be found in 2Z+ and cannot be found in Z+) and that there are some elements that belong to Z+, but not to 2Z+ ??? If there was a finite number of elements that are in Z+ and not in 2Z+ there would be no problem, because both of them have infinite number of elements. But there is an infinite number of odd numbers, so this just looks wrong. I have never seen proof for what you have said, and my math isn't good enough to prove it right or wrong. Plus it can be infinity which fucks with my brain and doesn't let me to understand this. Define a map from 2Z+ to Z+ by x -> x/2. This map is clearly onto, every element in Z+ is mapped to by an element from 2Z+. It is also 1-1. How do we prove 1-1? Suppose there is an element in Z+ that is mapped to by more than one element. Thus, x in 2Z+ and y in 2Z+ both map to z. Thus, x/2 = z = y/2, which means x=y. thus, the mapping is 1-1. Since the map is both 1-1 and onto, the sets are the same size, although infinite.
Here is how I arrived at the answer of (0 < (X / infinity) <_ 1) / infinity :
I take into account the theorem (It is important to keep this in mind):
lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
X/infinity
X can’t equal 100 because it is impossible to get a repeating decimal that is greater than 1.
X/infinity will eventually have a repeating decimal. And thanks to the theorem, ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”, we will eventually run into a paradox.
-It is impossible for X/infinity to equal 0 and so I use “ 0 < “.
-It is impossible for X to equal anything greater than 1 since we are constantly dividing. However, X can equal 1 thanks to the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1” and so I use “<_ 1”
And so I end up with:
(0 is smaller than (X divided by infinity), (X divided by infinity) is smaller than or equal to 1) divided by infinity.
To summarize and simplify for everyone:
(1). Start with any given number.
(2). Continue to divide that number until you end up with a repeating decimal; I can guarantee that the repeating decimal will not be greater than 1.
(3). Given the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”. That repeating decimal is = 1
(4). The given number we have now is 1. And your back at step (1).
Thus: (0 < (X / infinity) <_ 1) / infinity 
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what are you even arguing against me???
his problem was lim(n -> infinity) (100n/n), which is 100
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o snaps.. so many replies haha thanks yall for putting in your insights... well today i took my solution to my professor in class.. and he said that this paradox cannot solved by mathematics, instead you have to take it from a philosophical perspective. This is what I said to him in class:
"As you break down the 100 meters into many intervals, up to an infinite amount, the distance of each interval then becomes smaller and smaller; which then means that the distance of each interval begins to approach zero, not infinity.. which means that each interval is a finite amount, thus it can be covered."
He then said to me that I make an excellent point, and he thought that a few years ago it would be the correct solution (not sure what that meant). However, he said that this problem is more about time than distance, the correct way to look at this is to look at the time it takes to cover each interval. Each interval is assigned a temporal value, and the total time is infinity, thats what matters. He then told me that math is not the way to solve this paradox, as advanced mathematics didn't exist in 500 BC, when this was first supposedly disproven. Well gotta think aabout it more then. Peace.
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it's the same as
lim (x->1) (x^2-2x+1)/(x-1)
you do the inside operation first, then apply the limit.
if you are taking infinity/infinity, then that is indeterminate, it's value doesnt exist
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can someone explain the achilles and the tortoise paraddox? Too lazy to google it~_~ if you can, also post how it is proven wrong.
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tortoise leaves point a time 0.
achilles leaves point a time 10.
although achilles moves faster than the tortoise, he will never catch him, because while achilles moves closer to the tortoise, it moves a little further forward.
http://mathworld.wolfram.com/ZenosParadoxes.html
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hydro what course are u taking?
Philosophy?
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I think you should have just handed in a blank piece of paper, and when he asks you what it was, just say you are unable to argue with zeno because motion does not exist, hence you couldnt write your essay.
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Hydrolisko, I told you how to approach it.
Let's assume the runner covers a 100m track in 10 seconds.
Let Zeno divide the distance infinitely (so he gets infinitely small distances) and you divide the time needed to run the distance (so you'll get infinitely small time). Whatever operation Zeno does with distance you apply that to time.
Now he tries to tell you that running that partial (infinitely small) distance takes a certain amount of time (which you keep a track of), and running all of them (which sums to 100 meters) will take infinity. Now you should ask him why the partial distances sum up to a nice finite number (100m) and the partial times (which were divided the exactly same way) would sum up to infinity? They should, in the very same way, sum up to 10 seconds. And that's all.
It just sounds stupid without using math though.
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On July 26 2005 18:41 Krzych wrote:
They should, in the very same way, sum up to 10 seconds. And that's all.
Isnt that the paradox? That they should, but they dont... I dont think youre understanding the argument or youre making assumptions which they dont make. Like saying that we can time how long it takes, hes saying we cant time it. Hes arguing for monoism, that everything is one and that there is no change. We arent moving 100m just as 10 seconds didnt take place, there is no change. Now argue it without all the assumptions you used that are what the argument is about.
If something weighs one gram, you split that into an infinite amount of parts each part weighs something (it cant just not exist anymore) and anything x infinity = infinity. So therefore 1g weight the same as the universe. In 10 seconds, you can break that down infinite times and each time will be a certani amount of time, that amount of time x infinity = infinity, thus 10 seconds and 68982389423940 billion years are the same amount of time ... infinite. You dont run 100m in 10 seconds, hes saying motion and time dont exist, there is no change.
Your post is just so full of assumptions that you assume we will infer. Its kind of written like that last sentence.
Just say that motion isnt infinitely constant or we are unable to understand infinite.
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No NO nONONONONON
Just argue that the sum of an infinite number of things IS NOT NECESSARILY INFINITE. PROVIDE AN EXAMPLE. HIS ARGUMENT DOES NOT HOLD. DONE
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On July 26 2005 19:09 BigBalls wrote:
No NO nONONONONON
Just argue that the sum of an infinite number of things IS NOT NECESSARILY INFINITE. PROVIDE AN EXAMPLE. HIS ARGUMENT DOES NOT HOLD. DONE
Then provide your example. Of course he is going to argue that the sum of infinite number of things isnt infinite otherwise we would be monoists.
He is using the old definition of infinity, the one that basically the whole world believes in still to this day. That infinity is endless. How can something that is endless have an end?
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Sum (n=1 to infinity) (1/2)^n = 1
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which happens to be the exact thing zeno is hypothesizing is infinite
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the proof i remember of it.....
Define S_n = 1/2 + 1/4 + 1/8 + ... + (1/2)^n. Thus, S_n is the sum of the first n terms.
2*S_n = 1 + 1/2 + 1/4 + ... + (1/2)^n-1.
S_n = 2*S_n-S_n = 1-(1/2)^n-1.
Let n approach infinity.
lim (n -> infinity) S_n = lim (n -> infinity) 1-(1/2)^n-1 = 1-0 = 1. Done.
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I thought they proved that theory wrong, and they now use...
Sum (n=1 to limit) (1/2)^n = 1
Does 1/3 = .33333.......infinite ?
Can mathematics provide a number for 1/3?
Im too high right now to begin thinking about it. Disregard this.
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On July 26 2005 19:20 Triton wrote:
I thought they proved that theory wrong, and they now use...
Sum (n=1 to limit) (1/2)^n = 1
Does 1/3 = .33333.......infinite ?
Can mathematics provide a number for 1/3?
Im too high right now to begin thinking about it. Disregard this.
1/3 is approximately equal to .3 repeating (infinitely) and is considered an inaccurate calculation. Math has a number for 1/3 already. That would be 1/3
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HnR)hT
United States3468 Posts
No, .3 repeating is exactly the same thing as 1/3.
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Just a question (I dont pretend to be amazing at math, i havent been in school for a number of years so any knowledge has long since left me).
In 1000000000 years will we be using the same system of mathematics? I dont believe we will be, and I believe that our system that we are using now (using words like approximately) will no longer exist. The reason our system is wrong is the same reason we are unable to fully disprove Zeno.
And going with Hnr)HT -- if .33.... infinite = 1 then what is .9999... infinite / 3. Is there a difference between 1 and .999... infinite then? Im fairly certain that all this infinite isnt actually used anymore, and they use limits instead... So instead of disproving infinite (other than saying I am moving right now which I am perfectly okay with) we just conveniently choose to use something else instead.
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What? I just completely disproved Zeno. He made a false assumption in his argument, thus the conclusion he makes is worthless.
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On July 26 2005 19:41 BigBalls wrote:
What? I just completely disproved Zeno. He made a false assumption in his argument, thus the conclusion he makes is worthless.
Thomson's lamp
From Wikipedia, the free encyclopedia. ( http://en.wikipedia.org/wiki/Thomson's_lamp )
Thomson's lamp is a puzzle that is a variation on Zeno's paradoxes. It was devised by philosopher James F. Thomson, who also coined the term supertask.
Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of all these progressively smaller times is exactly two minutes.
Questions
* If after two minutes the switch is no longer toggled, will the switch (and lamp) be in the on-state or the off-state?
* Would it make any difference if the lamp had started out being on, instead of off?
Contrast with Zeno's Paradoxes
Two notable features of contrast between Thomson's Lamp and Zeno's Paradoxes is that in the case of the lamp the focus is on two discrete positions and there is a pause between them. Several proposed solutions to Zeno's Paradoxes fail if there is a pause before each movement in the series.
-------------------
It says proposed solutions, nothing has been "proven" or "disproven" least of all by you.
I dont believe that 1 = (1/2)^infinite now calculate it for me to prove it to me... go ahead, you have til infinity to finish.
edit: times up
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No offense, but i'm a 4th year math major who finished his math degree 2 years ago and im not going to argue with a stoner about something i already proved.
but thomson's lamp paradox is a nonsensical question. It's the equivalent of asking if the last integer is even or odd, which doesnt make any sense.
just for your own curiousity...
http://mathworld.wolfram.com/ZenosParadoxes.html
The resolution of the paradox awaited calculus and the proof that infinite geometric series such as can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances.
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http://philsci-archive.pitt.edu/archive/00001197/02/Zeno_s_Paradoxes_-_A_Timely_Solution.pdf
I believe they place your theory (even quote the site I think) in "3. Their Historical Proposed Solutions" which they say are wrong.
And in that link, it will also give the answer to the original posters question that his teacher is looking for.
To return to Zenos paradoxes, the solution to all of the mentioned paradoxes then,9 is that there isnt
an instant in time underlying the bodys motion (if there were, it couldn’t be in motion), and as its
position is constantly changing no matter how small the time interval, and as such, is at no time
determined, it simply doesn’t have a determined position. In the case of the Arrow paradox, there isnt
an instant in time underlying the arrows motion at which its volume would occupy just one block of
space, and as its position is constantly changing in respect to time as a result, the arrow is never static
and motionless. The paradoxes of Achilles and the Tortoise and the Dichotomy are also resolved
through this realisation: when the apparently moving bodys associated position and time values are
fractionally dissected in the paradoxes, an infinite regression can then be mathematically induced, and
resultantly, the idea of motion and physical continuity shown to yield contradiction, as such values are
not representative of times at which a body is in that specific precise position, but rather, at which it is
passing through them. The bodys relative position is constantly changing in respect to time, so it is
never in that position at any time. Indeed, and again, it is the very fact that there isnt a static instant in
time underlying the motion of a body, and that is doesnt have a determined position at any time while
in motion, that allows it to be in motion in the first instance.
In case you dont want to click the link, this is the theory it states is wrong, seems the same as yours.
The paradoxes of Achilles and the Tortoise and the Dichotomy are often thought to be solved
through calculus and the summation of an infinite series of progressively small time intervals and
distances, so that the time taken for Achilles to reach his goal (overtake the Tortoise), or to traverse the
said distance in the Dichotomy, is in fact, finite. The faulty logic in Zeno’s argument is often seen to be
the assumption that the sum of an infinite number of numbers is always infinite, when in fact, an
infinite sum, for instance, 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +...., can be mathematically shown to be
equal to a finite number, or in this case, equal to 2.
This type of series is known as a geometric series. A geometric series is a series that begins with one
term and then each successive term is found by multiplying the previous term by some fixed amount,
say x. For the above series, x is equal to 1/2. Infinite geometric series are known to converge (sum to a
finite number) when the multiplicative factor x is less than one. Both the distance to be traversed and
the time taken to do so can be expressed as an infinite geometric series with x less than one. So, the
body in apparent motion traverses an infinite number of "distance intervals" before reaching the said
goal, but because the "distance intervals" are decreasing geometrically, the total distance that it
traverses before reaching that point is not infinite. Similarly, it takes an infinite number of time
intervals for the body to reach its said goal, but the sum of these time intervals is a finite amount of
time.
So, for the above example, with an initial distance of say 10 m, we have,
t = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + .
+ 1 / 2 n Difference = 10 / 2 n m
Now we want to take the limit as n goes to infinity to find out when the distance between the body in
apparent motion and its said goal is zero. If we define
S n = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + .
+ 1 / 2 n
then, divide by 2 and subtract the two expressions:
S n - 1/2 S n = 1 - 1 / 2 n+1
or equivalently, solve for S n:
S n = 2 ( 1 - 1 / 2 n+1)
So that now S n is a simple sequence, for which we know how to take limits. From the last expression it
is clear that:
lim S n = 2
as n approaches infinity.
Therefore, Zeno’s infinitely many subdivisions of any distance to be traversed can be mathematically
reassembled to give the desired finite answer.
edit: definitely a lot of differing views on his paper, obviously dont take it as fact.
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On July 26 2005 17:22 Tontow wrote:Show nested quote +On July 26 2005 10:26 Tontow wrote:Its lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 ...... Infinite 1 dollar is equal to infinite 100 dollars. And, Zeno is talking about an infinitely small portion of infinity, but -However, It is also mathematically feasible to prove Zeno correct. Assuming that we are dividing “time” an infinite amount of times, then somewhere along the line there is bound to be a repeating decimal. -------------------------------------------------------------------------------- 3. Zeno points out, given that we are assuming that space is continuous, -------------------------------------------------------------------------------- Now, here is where it gets tricky. The following formula stats that any that any decimal that repeats -- (and, thus, is infinite) – will infinitely approach and equal the next highest whole number, thus rounding it up. (Note: the proof was confirmed by my college math teacher.) http://www.blizzard.com/press/040401.shtmllim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 0.9999... = 1 Thus x = 0.9999... 10x = 9.9999... 10x - x = 9.9999... - 0.9999... 9x = 9 x = 1. I think the answer is to be shown as: -2 -1 0 1 2 <---------------------------------------------------------------------------> (0 < (X / infinity) <_ 1) / infinity (0 is smaller than (X divided by infinity), (X divided by infinity) is smaller than or equal to 1) divided by infinity. Show nested quote +On July 26 2005 12:08 BigBalls wrote:
kryzch thats a completely different question.
The question we are posing would be (lim [n-> oo] 100n) / (lim [n->oo] n), which isnt 1, but instead indeterminate. the answer to your question is clearly 100. Could you do a proof of that? I’m not shore 100 is correct Because the way you wrote it: -The limit of N is infinity. (N = infinity) - 100N = Infinity -N/N = Infinity And remember that you stated that: Show nested quote +On July 26 2005 12:30 BigBalls wrote:On July 26 2005 12:20 Krzych wrote:On July 26 2005 10:21 BigBalls wrote:
(...)
Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size. Hey, BigBalls, I know you're good at math, but I guess you have screwed something. Can you actually say, that Z+ and 2Z+ are the same size (which means they have the same amount of elements?) and at the same time say that 2Z+ is a subset of Z+ (which means that there are no elements that can be found in 2Z+ and cannot be found in Z+) and that there are some elements that belong to Z+, but not to 2Z+ ??? If there was a finite number of elements that are in Z+ and not in 2Z+ there would be no problem, because both of them have infinite number of elements. But there is an infinite number of odd numbers, so this just looks wrong. I have never seen proof for what you have said, and my math isn't good enough to prove it right or wrong. Plus it can be infinity which fucks with my brain and doesn't let me to understand this. Define a map from 2Z+ to Z+ by x -> x/2. This map is clearly onto, every element in Z+ is mapped to by an element from 2Z+. It is also 1-1. How do we prove 1-1? Suppose there is an element in Z+ that is mapped to by more than one element. Thus, x in 2Z+ and y in 2Z+ both map to z. Thus, x/2 = z = y/2, which means x=y. thus, the mapping is 1-1. Since the map is both 1-1 and onto, the sets are the same size, although infinite. Here is how I arrived at the answer of (0 < (X / infinity) <_ 1) / infinity : I take into account the theorem (It is important to keep this in mind): Show nested quote +
lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1
Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
X/infinity X can’t equal 100 because it is impossible to get a repeating decimal that is greater than 1. X/infinity will eventually have a repeating decimal. And thanks to the theorem, ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”, we will eventually run into a paradox. -It is impossible for X/infinity to equal 0 and so I use “ 0 < “. -It is impossible for X to equal anything greater than 1 since we are constantly dividing. However, X can equal 1 thanks to the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1” and so I use “<_ 1” And so I end up with: Show nested quote +
(0 is smaller than (X divided by infinity), (X divided by infinity) is smaller than or equal to 1) divided by infinity.
To summarize and simplify for everyone: (1). Start with any given number. (2). Continue to divide that number until you end up with a repeating decimal; I can guarantee that the repeating decimal will not be greater than 1. (3). Given the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”. That repeating decimal is = 1 (4). The given number we have now is 1. And your back at step (1). Thus: (0 < (X / infinity) <_ 1) / infinity
On July 26 2005 17:33 BigBalls wrote:
what are you even arguing against me???
his problem was lim(n -> infinity) (100n/n), which is 100
I think I'm arguing the formula used to represent Zeno's paradox. Or at least try to show another way to mathmatically represent it.
To clearify: the given number that you start out with should be eather time or distance.
To summarize and simplify for everyone:
(1). Start with any given number.
(2). Continue to divide that number until you end up with a repeating decimal; I can guarantee that the repeating decimal will not be greater than 1.
(3). Given the theorem ”lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1”. That repeating decimal is = 1
(4). The given number we have now is 1. And your back at step (1).
Thus: (0 < (X / infinity) <_ 1) / infinity Where X is eather time or distance.
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On July 26 2005 19:20 Triton wrote:
Can mathematics provide a number for 1/3?
Decimal(1/3) = Trinary(0.1)
Trinaries own
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On July 26 2005 10:21 BigBalls wrote:
sickoflife, they both have the same amount of money.
here is a bit of a trickier example.
Define Z+ = the positive integers. So Z+ = {1,2,3,.....}
2*Z+ = {2,4,6,.....}, the even positive integers.
Now, 2*Z+ is a SUBSET of Z+. This means that every element in 2Z+ is an element of Z+. It is a proper subset, meaning there are elements in Z+ that are not in 2Z+. Both of these sets are the same size. Thus, there exists a 1 to 1 and onto map between these two sets, which is known as a bijection. Thus, there is an infinite set which is a subset of another infinite set, and can be mapped 1-1 AND onto it.
To move into analysis, take the interval (0,1). (0,1) is NOT countable, there are an infinite number of elements in there. furthermore, any interval is not countable, as long as it has positive measure then it is infinite. However, the example before, 2Z+ IS countable. Although it has an infinite number of elements, it has measure 0.
Yes and No, i cant remember who proved that infinites in infinites theory, but when he solved it he said "i see it, but i cant believe it".
Its the same, math is way beyond our minds, math will solve something our brains cant, for example, think it visually, a never ending line of 1 dollar bills and a never ending line of 100 dollars bill, the 1:100 ratio is there, so infinite becomes really an utopic 90° turned "8", nothing more.
There are only 2 theories, infinite and finite universe, thing we will never know, and any numbers that have the infinite number are absolutely uselss besides having sophistic arguments
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btw the only proof of infinity is human stupidity , man that thing its really infinite :D
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Thats pretty cool, never really thought about that.
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On July 26 2005 19:28 Echo wrote:Show nested quote +On July 26 2005 19:20 Triton wrote:
I thought they proved that theory wrong, and they now use...
Sum (n=1 to limit) (1/2)^n = 1
Does 1/3 = .33333.......infinite ?
Can mathematics provide a number for 1/3?
Im too high right now to begin thinking about it. Disregard this. 1/3 is approximately equal to .3 repeating (infinitely) and is considered an inaccurate calculation. Math has a number for 1/3 already. That would be 1/3
lol yeah, its like saying
"I wonder when will math put a number for 4, those silly white coat morons"
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On July 24 2005 22:54 Hydrolisko wrote:
Zeno is a greek philospher who thought motion is impossible. I have to write an essay to refute his argument intelligently (rationally, can't just say stuff like "wow how dumb"). I have an argument in mind but I'd like to see some more insights before I start writing. Here is his argument:
1. Zeno starts by assuming what his opponent says is possible: motion. In particular, the motion of a single body across a finite distance in a finite time.
2. To make things vivid, let’s specify the moving object, and where it is supposed to be moving: Imagine a sprinter, who starts running at one end of a 100 metre straight track, and runs to the other end. Zeno’s opponents (probably including yourself) think that it really is possible that runners can do this. (i.e. They think that it is not just an illusion.) Zeno begins by assuming that his opponents are correct, and then the fun starts.
3. Zeno points out, given that we are assuming that space is continuous, that before the runner can cover the whole 100 metres she has to cover half the distance, i.e. run 50 metres.
4. Zeno then repeats the move just made (point 4) and points out that the move can be repeated an infinite number of times: before the runner can cover 50 metres, she has to run 25, before that she must run 12.5, before that 6.25, etc. Recall that we are assuming that space is continuous, which means that any finite piece of it, such as a sprint track, can be divided into infinitely many parts, which are infinitely small.
5. Zeno then argues that to cover each of these infinitely small parts will take a certain amount of time.
6. But to take a certain amount of time an infinite number of times adds up to an infinite amount of time. So it would take the runner forever to cover 100 metres. But we were assuming that the runner could cover the distance in a finite amount of time, not that she would take forever.
7. We have run into a problem, and Zeno’s conclusion is that motion is impossible, because any finite motion would take forever.
Thoughts?
According to dictionary.com, motion can be defined as:
The act or process of changing position or place.
A meaningful or expressive change in the position of the body or a part of the body; a gesture.
Active operation: set the plan in motion.
The ability or power to move: lost motion in his arm.
The manner in which the body moves, as in walking.
A prompting from within; an impulse or inclination: resigned of her own motion.
We know that for each movement we make, we are traveling through time and space.
So lets say a sprinter has to run 100meters in 10seconds. And then apply Zeno's argument.
Then, he concludes that the sprinter isn't moving at all just by saying he must run through a series of infinite parts?...!!!
I MEAN WTF!!! HAHAHAHHAHA
Dude the sprinter WILL HAVE TO MOVE in order to cover the 100meters in 10seconds.
Meaning that even if he is moving THROUGH those series of infinite parts, he is still moving.
Which means that there is movement, and movement=motion. And in the end, he WILL cover the 100meters.
The point here in his conclusion is about that motion is impossible, and not infinity is impossible.
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is .9 repeating exact same thing as 1?
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On July 26 2005 23:19 MPXMX wrote:
is .9 repeating exact same thing as 1?
i could be wrong, but i think the whole .99999 (extc.) = 1 thing was made just to show flaws in our math system when dealing with infinite numbers? So the short answer is no.
Plz correct me if im wrong.
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On July 26 2005 23:19 MPXMX wrote:
is .9 repeating exact same thing as 1?
No, it isn't.
lim n->oo 0,9...9 = 1, but 0,9...9 < 1.
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it got sence,
but im not sure do i "got it"
.
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HnR)hT
United States3468 Posts
On July 27 2005 01:20 geod wrote:No, it isn't. lim n->oo 0,9...9 = 1, but 0,9...9 < 1.
He said .9 *repeating*, which IS lim .9, .99, etc.
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HnR)hT
United States3468 Posts
On July 26 2005 22:18 baal wrote:
Yes and No, i cant remember who proved that infinites in infinites theory
That would be Cantor. These things are called transfinite numbers; the "number" of natural numbers is an infinity called aleph-0, the "number" of real numbers is a much "bigger" infinity called continuum. Cantor's theorem roughly says that for any infinity there is always a bigger infinity, ad infinitum.
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stop this rabbit and tortoise thing already.
these stupid philosophers probably skipped math lessons all the time. It's about limits, and we all know that decreasing the time intervals makes no sense whatsoever.
something that annoys me more than the philosophers are the people who dig these 'proofs' and will post them on forums.
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The Rabbit and Tortoise paradox:
Zeno and other philosophers argued that motion was impossible in this way:
The rabbit (A) and the tortoise (B) was going to run 100 m to settle who was the fastest. To be fair the tortoise was given a 10 m. headstart
The rabbit runs 10m/s.
The tortoise runs 1m/s.
...
The race starts: A moves 10m (1 sec), B moves 1m (1sec). A moves 1m (0,1sec), B moves 0,1m (0,1sec)... Continue this an infinite number of times.
The total time will be 1 + 0,1 + 0,01 + 0,001 + ... = 1.111 ... sec., which is less than 2 sec. Again the philosophers tricks us into believing that infinite time intervals becomes an infinite big number. But instead the infinite time intervals becomes smaller and smaller.
A catches B after 1,1111...sec. = 10/9 sec.
Math way of typing it:
a1 = 1, a2 = 1.1, a3 = 1.11, ... , a(n) = 1.111..1 (n 1-numbers).
a(n) → 10/9 for n → ∞ .
a(n) is converging towards the permit limit 10/9, for n going towards ∞
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HnR)hT
United States3468 Posts
On July 27 2005 05:46 aseq wrote:
stop this rabbit and tortoise thing already.
these stupid philosophers probably skipped math lessons all the time. It's about limits, and we all know that decreasing the time intervals makes no sense whatsoever.
lol. Zeno lived BCE and limits were invented during the 19th century by mathematicians like Cauchy and Weierstrass. Philosophers are usually very well-informed about contemporary math and science.
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ive taken enough analysis to shun those names and kill a small horse
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HnR)hT
United States3468 Posts
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Belgium9947 Posts
On July 26 2005 00:33 Tontow wrote:Food for thought: -500 meters is still equal to 500 meters no matter how you split it up. Kochen said it first and its basic math: What you do to one side or part of the equation you must do to the other side or part of the equation. If you divide distance an infinite amount of times, then you must divide time an infinite amount of times. 500M / infinitely = time to run 500M / infinitely -However, It is also mathematically feasible to prove Zeno correct. Assuming that we are dividing “time” an infinite amount of times, then somewhere along the line there is bound to be a repeating decimal. Now, here is where it gets tricky. The following formula stats that any that any decimal that repeats -- (and ,thus, is infinite) – will infinitely approach and equal the next highest whole number, thus rounding it up. (Note: the proof was confirmed by my college math teacher.) http://www.blizzard.com/press/040401.shtmllim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 0.9999... = 1 Thus x = 0.9999... 10x = 9.9999... 10x - x = 9.9999... - 0.9999... 9x = 9 x = 1. -You must also look at the perception point of view. You perceive a single grain of rice falling; if you use an amplifier you can perceive the sound that that grain makes. The same goes for a bag of rice. In this case the question is, “Are we perceiving that the runner is going from point A to point B? Or is he actually moving from point A to point B? Is it possible that both could be true?” I hope this helps with your essay.
my math teacher of first grade in high school proved that shit wrong a couple of years ago when I gave her that
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On July 27 2005 11:55 imRadu wrote:
ahahahha
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On July 26 2005 21:46 0x64 wrote:Show nested quote +On July 26 2005 19:20 Triton wrote:
Can mathematics provide a number for 1/3?
Decimal(1/3) = Trinary(0.1) Trinaries own
I remember I once heard at a lecture, that computer based on trinary elements would be better than binary based one in the respect of the number of elements needed to build one (logic gates, transistors, etc.). We're using binary computers only because they're easier to build.
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On July 27 2005 04:49 HnR)hT wrote:Show nested quote +On July 26 2005 22:18 baal wrote:
Yes and No, i cant remember who proved that infinites in infinites theory That would be Cantor. These things are called transfinite numbers; the "number" of natural numbers is an infinity called aleph-0, the "number" of real numbers is a much "bigger" infinity called continuum. Cantor's theorem roughly says that for any infinity there is always a bigger infinity, ad infinitum.
Cantor did so much research into infinity, that eventually he developed mental illness. And he was not the only one.
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On July 27 2005 10:29 BigBalls wrote:
ive taken enough analysis to shun those names and kill a small horse
No matter how clearly you prove that an infinite sum of a geometrical series doesn't have to equal infinity, there are people who refuse to stop saying that you can't prove Zeno wrong. :-/
C'mon people, stop skipping math classes and start thinking!
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