Are you guys familiar with this philosophical riddle of Zeno's paradox?

Here's a snippet:

Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

Now I've heard solutions to this over and over, but am never satisfied with them. Could someone explain this to me? In idiot terms?

-Xenu

You cover half a distance, in half the time. Another quarter of distance, in a quarter of the time. Another eighth of the distance in an eighth of the time, and so on. Your total time for crossing the room is the sum of all those times, and your total distance covered is the sum of all those distances.

Is that idiotic enough? :D

ok
You keep halving so the number you are taking away gets smaller but is never 0

example

100-50
50-25
25-12.5
ect

it will never be 0

this is the half life rule for radioactivity

The solution is setting the end-point of the journey infinitely small piece further than where you want to go.
in that case you'll get there with no problem at all...

Overdoze and Asguard,

I'm sorry if I wasn't clear. I know how the problem works. What I wanted to know was how the problem can be explained to be wrong mathematically. In real life it's obvious that the arrow get there, but logically when it's worked out the way it is, it doesn't work.

Merlijn,

I think your solution has a logical hole. When you change your endpoint you have changed your endpoint, you are now shooting for a new goal, which you can still no longer reach. Whether or not this passes through the old goal is arbitrary.

For example, if I wanted to walk to a point right in front of a wall, I don't adjust my aim to walk into the wall.

If you can explain your solution mathematically, maybe it'll make more sense.

It doesn't work because the entire analogy is false to begin with. It uses division by two as an anology of motion in a way which simply never happens. Yes, if in maths you continually divide by two you get half the previous number. Big deal. It has nothing whatsoever to do with actual travelling/motion. Why? Because you will not halve the length of your strides as you move. Your steps will stay around one yard, for example. When the remaining distance is less than one yard/step, the next step takes you to the end. Basically the entire analogy from the start is false. It compares division by two with regular decrements.

EDIT: Or, mathematically, it is comparing:

1)
Start at 10, and divide by 2 at each step: 5, 2.5...

2)
Start at 10, and use regular decrements representing steps: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0!

The paradox is trying to say that 2 can not happen because of 1. Which is totally ridiculous. It is a comparison of two entirely difefrent things.

Okay, let's have it again...
Since the aiming point is never overt, we can only percieve the end-point of the motion. What was the aiming point need not necessarily be the aiming point!
So I want to go from A to B, but my cerebellum calculates thet I have to aim for C.
A............................................B..C

In fact this is not the true solution, I know. But i always found that is the most insightful.
I think the actual solution has to do with a second-order differential equation. But it's rather stricky.

Adam, the length of strides do not matter at al, because originally the story was about an arrow flying to a mark.

The stride wa san example. The same applies for arrows in flight. In any given unit of time during the flight it will cover a certain distance. That distance per time unit does not halve with each successsive time unit.

Oh I forgot...
In my A...................B..C, the distance between B and C is zero, still it will in fact solve Zeno's paradox.

Adam, I know... I was annoying you. sorry. So you think the paradox is there because of our choice of reference. You're right. However, that does not solve the paradox. Also it's a fun exercise, don't you agree?

what i learned in science is you touch at infinity

adam: this DOSE have a practical purpose, its used for half lives of radiactive isotopes

I know division by two serves a purpose. But the fact remains it is still only division by two. It's really not that interesting. Zeno got his name remembered for using a false preposition which basically doesn't work.

EDIT: Fixed a BOLD marker.

Alright, let me restructure the problem so that your guys' previous solutions and answers wouldn't work.

****"Xenu's Paradox"****

The first concept to understand should be simple.

The whole is equal to the sum of its parts

This isn't true in things such as say... Gestalt Psychology, but for mathematics it seems to be a logical truth.

For example, I live in city A and am driving to city C, in between is city B. The distance between A and C is equal to the sum of distances of A and B plus B and C. This is a no brainer, I know.

Now, let's take a line segment. It's 2 cm (feet, inches, or whatever) long.

Represented on a number line:

0--------1---------2

Mathematically this line segment can be divided into an infinite number of line segments. How can this be so? Think about it. For instance, I could have a start of a segment at the real number 1.99998 and have an end point at 1.99999, making the segment .00001 cm long. But between these two point are real numbers such as 1.99999999999999...repeating off into infinity, or 0.44444444444 repeating, or what have you - logically making it possible to divide the line segment into an infinite number of parts.

Now the major question is, how big are each of these line segments? I'd have to guess that each line segment would have to be infinitely small, represented by:

1/Infinity

So given the rule above, The whole is equal to the sum of its parts, the line segments mathematic length is equal to:

Infinity * 1/Infinity or simplified:

Infinity/Infinity which is equal to:

1? - seems initially intuitive, 0? - infinity divided by anything = 0, but mathematically I think it is equal to: Undefined (or that big fat E you get on your calculator at times).

Even if you didn't say that each segment was infinitely small, anything divided by infinity equals 0.

It seems that no matter how you work it, you don't get 2 cm.

****************************

Alright guys, please show me the answer to this, and definitely show me any logical flaws. This should be fun. I'm a little befuddled myself.

-Xenu

Xenu,
Your post reminds me of an early book by Paul Davies. Beyond the Edge of Infinity or something it was called.
When you seperate all the infintate points of a line segment and paste them together, you can have a new line segment of any length (e.g. l=0).

This isn't true in things such as say... Gestalt Psychology, but for mathematics it seems to be a logical truth.

Gestalt psychology is ancient. Hahaha. Better talk about Dynamical Systems Approach of psychology. This also has some Gestalt like elements, but it is based upon the mathematical study of complex systems.. which do have emergent properties!
So, in mathematics it is sometimes true that the whole is more than the sum of the parts!

But the logical flaw is that infintity cannot be treated as a number. So you cannot do mathematical operations on them.

Gestalt psychology is ancient. Hahaha.

So is Freud, and they still don't stop talking about that bastard ;)

Better talk about Dynamical Systems Approach of psychology. This also has some Gestalt like elements, but it is based upon the mathematical study of complex systems.. which do have emergent properties!

Heard of it, but not much about it. Could you post some info on this, sounds interesting.

But the logical flaw is that infintity cannot be treated as a number. So you cannot do mathematical operations on them.

I guess that this is kind of my point too. However, calculus deals with infinities and limits (approaching infininity). I guess I don't know enough about calculus to make any other judgements about there systems. Maybe someone else could enlighten me?

Xenu,

The problem is far easier than you make it sound. Basically, assuming you travel with constant speed, if you divide distance travelled into smaller chunks you will cover each chunk in appropriately less time. The "paradox" says you will never cover that distance, but that's not true. Since each chunk of the distance corresponds to an appropriately small piece of time, adding all the chunks of distance and all the time gives you a finite time for covering the entire distance.

With respect to infinities, sure you can do math with them. But you have to be careful not to fudge the math. There are different "degrees" of infinity. For example, there are infinitely many distinct integer numbers, and there are infinitely many distinct real numbers, but there are more real numbers than integers. So in effect, you have one infinity greater than another infinity (which is bad language, but that's what you have.) If you want to be more PC, you'd say that the cardinality of the set of real numbers is greater than the cardinality of the set of integers even though both sets are infinite in size. So you can indeed compare infinities based on their cardinality.

If you divide a distance into infinitely many chunks, this set of chunks will have a certain cardinality. In one-to-one correspondence to it will be the infinity of time intervals, each corresponding to its own chunk of distance. So as long as you keep track of that one-to-one correspondence you will be fine. You can split the distance into an infinity of segments, and you can put it back together again to get the same distance. For a slightly different distance the set of segments of the same length will have a slightly different cardinality, so adding it back together will give you back that slightly different distance.

True. In Zeno's day they didn't have the concept of limits. If you calculate the limit of that paradox, you get a finite answer. Also, there is a humbling, practical answer to that. You can walk across a room, duh.

Poor Zeno, he wasted a lifetime on some useless thing that a high schooler could calculate in five seconds today. Case closed.

In Zeno's day they didn't have the concept of limits. If you calculate the limit of that paradox, you get a finite answer.

Could you (or anyone else) show me how to calculate either/both Zeno's paradox or Xenu's paradox?

You can walk across a room, duh.

Well of course you can, and Zeno wouldn't deny it.

Poor Zeno, he wasted a lifetime on some useless thing that a high schooler could calculate in five seconds today. Case closed.

You consider it a waste of time because you don't see the importance of it, the way he did. Also, the case is not closed for me.

-Xenu

I'm not satisfied with you answer to Zeno's paradox. I'll break apart your argument here.

Basically, assuming you travel with constant speed, if you divide distance travelled into smaller chunks you will cover each chunk in appropriately less time.

You have changed the nature of the problem. The way it's set up, the problem has nothing to do with time. The problem that I posted here only has to do with distance.

Since each chunk of the distance corresponds to an appropriately small piece of time, adding all the chunks of distance and all the time gives you a finite time for covering the entire distance.

This sentence is confusing. I am only going to discuss distance here. What I think you are getting at is this: You have a distance. All of the segments, that make up that distance add up to that distance, no matter how many of them there are. Duh, simple.

Yes it's easy to see a solution when you view it post hoc, after the fact. But you have changed the nature of the problem again. What your reasoning relies on is that the distance has already been traveled, and then going back and showing how. The actual Zeno's paradox is showing the movement in process. The goal is never reached in the first place, so you can't start your math from there. Does this make sense?

*************************

Now I want to discuss what you have to say about infinities and degrees in respects to Xenu's paradox.

If you divide a distance into infinitely many chunks, this set of chunks will have a certain cardinality. In one-to-one correspondence to it will be the infinity of time intervals, each corresponding to its own chunk of distance. So as long as you keep track of that one-to-one correspondence you will be fine. You can split the distance into an infinity of segments, and you can put it back together again to get the same distance. For a slightly different distance the set of segments of the same length will have a slightly different cardinality, so adding it back together will give you back that slightly different distance.

Again, this can only be done post hoc (after the fact). In order for this to work you have to start out saying that the segment is two cm long, chop it into the infinite segments and then attach a degree to it. What if I started with just the infinite segments. Add up all their distances and what do you get? You won't get 2 cm.

"Wait a minute" you might say. When I give you the infinite segments, I also have to give you a degree with it. For instance a degree of "2 cm". This doesn't make sense either. This implies the answer is the problem. It's like asking what is the color of Chester's white horse? It's a cop out.

-Xenu

The problem lies on the difference between Zeno's desire and his action. From his mind, he wants to across a room, but in reality, his action dosen't. He just tries to get as close as possible to the other side of the room, but never reach it. Duh!

It's obvious that this Zeno guy is not doing what he wanted to do. How do you expect he gets what he wanted?

Try to look at it in another way. If Zeno is only one micron away from his goal position, can you tell from just standing next to him and say he has not cross the room yet? ;)

It is pointless. So what if there are an infinite number of half-spaces? You don't even need limits.

If you walk at constant speed, then the time it takes for you to cross the half distance is half that of the previously covered distance. And half of that is again half that time. And so on. It is utter stupidity to think of it that way. In constant speed problems...time required is total distance divided by speed.

If you don't walk at constant speed and you keep getting slower, then you might not reach it. Duh. If you stop midway and you stay there forever, then you don't reach it.

Zeno wasted his time. He doesn't even bother to make any real assumptions, such as how the walker's speed changes with respect to time, and he just twists words so that people will get confused. It apparently worked, because he confused himself. It is such a waste of time. Case closed for me, it is idiocy to discuss this further. Everyone I advise you to leave and go spend your time on something constructive.

Last bit of prooof that Zeno's ideas are dead. No one ever joins his sect of philosophy. It is dead and gone.

A few of you are posting things like "so what". "Zeno can't get across the room, big deal." "This is a waste of time." etc.

I think there is a very important lesson from this. Infinity is something that we demonstrate with reason. For example, we can imagine that the number 1.9999999... repeating can go on forever. It's a logical possibility. But when infinity is inserted into basic math problems like distance problems, they break down. Problems such as Zeno's or Xenu's paradox show how reason can be used to screw itself.

I'm not saying that reason is "bad" or "should be abandoned". Reason is a tool of mankind used to make "understanding" out of an infinite universe. It's a model of the universe.

-Xenu

Originally posted by Xenu
Problems such as Zeno's or Xenu's paradox show how reason can be used to screw itself.Reason doesn't screw itself. As I stated earlier, it's Zeno screwing himself. He knows his action won't get him the wanted result in reality, but he feels it should? What an idiot!!! :rolleyes:

Pure reason is useless and it is idiocy. All reason and theoretical knowledge came from empirical knowledge. Read Kant's Critique of Pure Reason, for heaven's sake.

The solution is simple. Infinity does not exist in reality. It exists only as an abstract concept in math. You cannot keep travelling half the distance, because soon the minimum distance you can travel will be greater than half the previous distance because space and time are quantized.
One way to show that you cannot keep dividing in this way is this:

Proof
x = 0.9999....
10x = 9.9999....
10x-x = 9.9999.... - 0.9999....
9x = 9
x = 1
Therefore 0.9999.... = 1

Infinity is not a number, it is a concept. It is self contradictory and therefore doesn't exist.

Alpha,

You cannot keep travelling half the distance, because soon the minimum distance you can travel will be greater than half the previous distance because space and time are quantized

Ummm, what? You better explain yourself here, because right now I feel you're talking out your ass. If this truly makes sense, please share, then I'll be the idiot. :D

Proof
x = 0.9999....
10x = 9.9999....
10x-x = 9.9999.... - 0.9999....
9x = 9
x = 1
Therefore 0.9999.... = 1

This is a nice little trick, but that's all it is, a number trick. Firstly, it doesn't prove anything about infinity (as applied to math). Secondly, it doesn't disprove the paradox.

Let's say for instance, Zeno's paradox was in 1/4 's instead of 1/2 's. Your new infinite number would be .3333....(repeating). Now let's put this into the formula....

x = 0.3333....
10x = 3.3333....
10x - x = 3.3333.... - 0.3333....
9x = 3
x = 0.3333....
Therefore 0.3333.... = 0.3333....

Infinity does not exist in reality.

It all depends on what you define "reality" as.

Originally posted by Zero
Pure reason is useless and it is idiocy. All reason and theoretical knowledge came from empirical knowledge. Read Kant's Critique of Pure Reason, for heaven's sake.

Is this post directed towards me? If so, you're arguing basically the same thing I am, but I'm showing how reason is "useless and idiocy", at least when it's applied to infinity. I think we're on the same side.

Also, what happened to:

Case closed for me, it is idiocy to discuss this further. ??? :p

And no, I will not read Kant. ;)

-Xenu

Originally posted by daktaklakpak
Reason doesn't screw itself. As I stated earlier, it's Zeno screwing himself. He knows his action won't get him the wanted result in reality, but he feels it should? What an idiot!!! :rolleyes:

Yes, he knows his "action" (the paradox) won't get him realistic results. No he doesn't feels it should. He is showing how reason fails in determining reality. Reason is a model. Models have faults. This is one of reason's faults - Infinity.

If you are still against Zeno, make a rational case against what he says, rather than criticize him and dish out your subjective opinion.

-Xenu

Originally posted by Xenu
Alpha,

Ummm, what? You better explain yourself here, because right now I feel you're talking out your ass. If this truly makes sense, please share, then I'll be the idiot. :DInfinity is a paradox. It is self contradictory. Imagine a point on a flat plane. Now there are an infinite number of directions from that point. (I'm starting with the assumption that infinity exists in order to find a contradiction and therefore prove it doesn't exist.) If you imagine 2 lines radiating from the dot and extending to infinity, you can draw another line in between them, and another between them and so on. You can always fit more, so there's an infinity of directions. Remember the lines have no thickness, so they don't overlap. And you can always go further away from the originating point until the lines diverge enough that you can fit yet more lines in there. More directions.
Now imagine a point in 3d space. There's an infinite number of directions from that point. Yet there are clearly more directions than on the plane, because all those directions are included and multiplied by infinity in three dimnesions! So Infinity is more than itself? It doesn't make sense. There's your contradiction.

Also, the minimum length is the Plank length, and the minimum time is the Plank time. You can't travel a smaller distance than the Plank length, so eventually one half of the previous distance would be less than the Plank distance, and you wouldn't be able to move forward by such a small amount. So the argument is invalidated because you can't keep travelling half (or a quarter) the distance.
This is a nice little trick, but that's all it is, a number trick. Firstly, it doesn't prove anything about infinity (as applied to math). Secondly, it doesn't disprove the paradox.

Let's say for instance, Zeno's paradox was in 1/4 's instead of 1/2 's. Your new infinite number would be .3333....(repeating). Now let's put this into the formula....

x = 0.3333....
10x = 3.3333....
10x - x = 3.3333.... - 0.3333....
9x = 3
x = 0.3333....
Therefore 0.3333.... = 0.3333....No, it's not a "nice little trick." Yours was a nice little trick. How do you get from 9x = 3 to x = 0.3333....? In my equation you divide both sides of the equation by nine so they stay equal. In yours, you divided one side by nine and the other by 3 or something. That doesn't work. It's not allowed in math. Go here (http://www.drmath.com/dr.math/faq/faq.0.9999.html) for a more detailed explanation.
It all depends on what you define "reality" as.Well, something that exists without paradox for one.

No, it's not a "nice little trick." Yours was a nice little trick. How do you get from 9x = 3 to x = 0.3333....? In my equation you divide both sides of the equation by nine so they stay equal. In yours, you divided one side by nine and the other by 3 or something.

9x = 3

now divide the left side by 9, you get x.

now divide the right side by 9

3 divided by 9 = 0.3333....(repeating)

Pull out your calculator. It might amaze you. ;)

Like I said, your first problem doesn't disprove anything to me, it's just a mathematical trick that only applies to the 1/2 distance infinity and not the 1/4 distance infinity.

Also, the minimum length is the Plank length, and the minimum time is the Plank time. You can't travel a smaller distance than the Plank length, so eventually one half of the previous distance would be less than the Plank distance, and you wouldn't be able to move forward by such a small amount. So the argument is invalidated because you can't keep travelling half (or a quarter) the distance.

This doesn't disprove anything either. Firstly, I'll state this again, the Zeno's paradox that I posted has nothing to do with time. Secondly, why does there need to be a minimum distance, a Plank length? What calls for a minimum length?

Originally posted by Xenu
Like I said, your first problem doesn't disprove anything to me, it's just a mathematical trick that only applies to the 1/2 distance infinity and not the 1/4 distance infinity.Did you even check the link I posted? I think you should. 0.9999.... = 1
The proof is there. And yes it does apply to 1/4 distance, or any other division for that matter, because you cannot keep travelling a smaller and smaller distance. Read some physics books!
This doesn't disprove anything either. Firstly, I'll state this again, the Zeno's paradox that I posted has nothing to do with time. Secondly, why does there need to be a minimum distance, a Plank length? What calls for a minimum length?I didn't say it had anything to do with time.
Honestly, does it matter why right now? There is, just accept it. Do you really need it explained to you?

Originally posted by Xenu
Yes, he knows his "action" (the paradox) won't get him realistic results. No he doesn't feels it should. He is showing how reason fails in determining reality. Reason is a model. Models have faults. This is one of reason's faults - Infinity.First he wants to reach point B from point A.
Second he tries to apporach point B from point A.
Third he starts to complain that he will never reach point B by the method stated above and claims a paradox has occurs.

From the list above, all I see is a failed logic. In fact, I don't see any paradox, unless he assumes reaching a goal is same as approaching a goal but never reaches it.

Originally posted by daktaklakpak
First he wants to reach point B from point A.
Second he tries to apporach point B from point A.
Third he starts to complain that he will never reach point B by the method stated above and claims a paradox has occurs.

From the list above, all I see is a failed logic. In fact, I don't see any paradox, unless he assumes reaching a goal is same as approaching a goal but never reaches it.

no, no, no,

1) Something (it doesn't have to be a person!) approaches point B from Point A.
2) The manner in which the thing moves is half the distance that it moves before, and then half that, and then half that, (this isn't necessarily in step increments, thinking this way however makes it easier to comprehend)
3) The paradox is, when something moves in this manner, it can move forever and ever and never reach its goal. He isn't trying to prove that trying to reach a goal in this manner is the same as actually reaching a goal (this is what Alpha is trying to say with his .9999... equation) - that's not the paradox.

He is showing that reasoning in this way, although logical, doesn't work. The thing moves forever and never reaches its goal. When it comes to infinity, reasoning becomes faulty.

Alpha,

Did you even check the link I posted? I think you should. 0.9999.... = 1
The proof is there. And yes it does apply to 1/4 distance, or any other division for that matter

Yes I did. That equation may or may not be proof for the 1/2 decreasing distance problem, but it doesn't prove anything about a problem that decreases by 1/4 ths. Otherwise you have to be willing to accept that 0.3333... is equal to 1.

Alpha, do you believe that 0.3333... is equal to 1? Are you this irrational? Do I have to show you exactly how I got this again?

you cannot keep travelling a smaller and smaller distance. Read some physics books!

why not, it's mathematically possible, numbers can be decreased forever That's what the whole 0.9999.... (or 0.3333....) thing is about isn't it? If you are saying that the numbers can't decrease smaller and smaller, then you have just contradicted yourself, because that's exactly what your whole 0.9999... = 1 argument relies on.

Honestly, does it matter why right now? There is, just accept it. Do you really need it explained to you?

It matters because you are trying to say that you can rationally dissprove Zeno's paradox, but you can't. You make things up like "you cannot keep travelling a smaller and smaller distance" and try to prove it with the "Plank length". Then when I confront you on what this "minimum Plank length" is you say "Honestly, does it matter why right now? There [it] is, just accept it." Which makes me think you really don't know what the Plank length is and you have no idea what you are talking about.

It's a shame that these time zones exist. Otherwise I would have replied earlier.

First of all: in reality there is no such thing as infinity, in any case not that it affects our lives. However, in mathematics it does exist. Are we not allowed to talk math in the math dept. now?!\
So maybe IRL we have probems at quantum level, but in mathematics, we can ignore them.

Secondly: 0,999... <> 1 , even though you think to have proven otherwise.
And 3 divided by 9 does not equal 0,333...
3/9 = 1/3. something completely different! 0,333.... and 0,999... are elements of the array of real numbers (R) but not of the array of all quotients (Q). 1 and 1/3 are in both Q and R. The decimal notation is not to be taken lightly.Whenever possible, use the "normal" (pre-Huygens) notation.

Thirdly: Alpha, It is NOT your place to lecture others! Your posts show a lack of knowledge (or of understanding, in the more negative scenatio) of several concepts.
Infinity cannot be used as an operator in the way numbers can be used. True, there are diffent levels of infinity (called: aleph-0 aleph-1,...)
What we're discussing here is all aleph-null stuff.
The fun thing of infintity is that the total number of elements of an infinityely large set, equals the total number of elements of two of such sets (or of an infinite amount of infinitely large sets). It's mind-boglint. But don't worry, just accept it.
It's just like general relativity: very confusing and contra-intuitive (there is another thread regarding this).

Virtus Rationis. ('the power of ratio')
Merlijn

Cmon, Kant is not dumb. Everyone has read Kant. You should catch up. Critique of Pure Reason. After that, Phenomenology of Spirit by Hegel is also useful.

Yes I did. That equation may or may not be proof for the 1/2 decreasing distance problem, but it doesn't prove anything about a problem that decreases by 1/4 ths. Otherwise you have to be willing to accept that 0.3333... is equal to 1.Man, it doesn't matter if you say 1/4 the distance! Where the hell are you getting the idea that it makes a difference when you make the fraction smaller?
Alpha, do you believe that 0.3333... is equal to 1? Are you this irrational? Do I have to show you exactly how I got this again?Irrational?! Of course I don't believe that. And I can review your post anytime thank you.
Let's review your equation.
x = 0.3333...
10x = 3.3333...
10x - x = 3.3333... - 0.3333...
9x = 3
Stop. Here you divide both sides by nine and end up with a repeating decimal. But if you divide both sides by 3 you get:
3x = 1, which is the same as x = 1/3 which converts to the same repeating decimal, but it's inaccurate. It doesn't convert properly. And this is shown when you put 0.9999... into the equation instead.
why not, it's mathematically possible, numbers can be decreased forever That's what the whole 0.9999.... (or 0.3333....) thing is about isn't it? If you are saying that the numbers can't decrease smaller and smaller, then you have just contradicted yourself, because that's exactly what your whole 0.9999... = 1 argument relies on.I'm starting to wonder if you're capable of rational thought! The equation doesn't rely on the fact that numbers can't "decrease smaller and smaller", it disproves it!!!
It matters because you are trying to say that you can rationally dissprove Zeno's paradox, but you can't. You make things up like "you cannot keep travelling a smaller and smaller distance" and try to prove it with the "Plank length". Then when I confront you on what this "minimum Plank length" is you say "Honestly, does it matter why right now? There [it] is, just accept it." Which makes me think you really don't know what the Plank length is and you have no idea what you are talking about.Yes, I am saying you can disprove the "paradox" rationally. I'm not "making things up." I said I didn't want to go into it because I didn't have the time, and it could make this whole thing more complicated than it needs to be. The plank length is 1.6160505*10^-35 meters. I did not make it up.
Secondly: 0,999... <> 1 , even though you think to have proven otherwise.Oh, really? Prove otherwise! The proof is there. Ask a math teacher!
And 3 divided by 9 does not equal 0,333...
3/9 = 1/3. something completely different! 0,333.... and 0,999... are elements of the array of real numbers (R) but not of the array of all quotients (Q). 1 and 1/3 are in both Q and R. The decimal notation is not to be taken lightly.Whenever possible, use the "normal" (pre-Huygens) notation.That's essentially what I was saying.
Thirdly: Alpha, It is NOT your place to lecture others! Your posts show a lack of knowledge (or of understanding, in the more negative scenatio) of several concepts.Excuse me?! This is a forum for discussion. This is more of a place for "lectures" than others. And who are you to say who can and cannot "lecture" others. Freedom of speech. I can say what I want. If I think I know the solution to a problem I have a right to say my piece. It is NOT your place to tell others that they can't speak their mind.
OK, trying to calm down before I continue...
Infinity cannot be used as an operator in the way numbers can be used. True, there are diffent levels of infinity (called: aleph-0 aleph-1,...)Exactly.
What we're discussing here is all aleph-null stuff.
The fun thing of infintity is that the total number of elements of an infinityely large set, equals the total number of elements of two of such sets (or of an infinite amount of infinitely large sets). It's mind-boglint. But don't worry, just accept it.Sorry, I can't just accept things. I have an insatiable curiosity. I'm always wondering "Why?" But I'll let it go for now. Especially since everyone who's come close to the solution so far has gone insane. (Godel for example).

Alright, let's look at this problem from another perspective. You travel half the distance to your destination. How did you get there? You must have travelled half the distance from your starting point to where you are now, and then half the distance again, and so on. Yet you are clearly there. If Zeno's paradox were true, then you would be unable to move at all, because if you want to move to any point, you must first move half the distance, no matter how close the point is. In this way the paradox is solved through rational thinking.

Alpha,

Stop. Here you divide both sides by nine and end up with a repeating decimal. But if you divide both sides by 3 you get:
3x = 1, which is the same as x = 1/3 which converts to the same repeating decimal, but it's inaccurate. It doesn't convert properly. And this is shown when you put 0.9999... into the equation instead.

OK I agree with both you and Merlijn, when the problem is worked out you should get 1/3 and not 0.3333... But still does 1/3 = 1? In a 1/4 decreasing Zeno's paradox does the thing that moves get to 1/3 and then stops, Alpha? Does the thing get instantly teleported to the goal once it reaches 1/3 of the distance? Does it sit down and contemplate Kant and Hegel? ;)

Merlijn,
0,999... <> 1 , even though you think to have proven otherwise.

Here is the best rationality that 0.9999... = 1, taken from the math site that Alpha posted earlier:

.9999... is equal to 1 because no matter how small a difference
between .9999... and 1 you ask for, I can write enough 9s to get
within that difference. So suppose you want it within .00001 of 1.
I can write .99999. This can go on and on.

It at first kind of makes sense. But by the same rational, I could describe how similar I am to a toad infinitely (because language is infinite) and no matter how many similarities you require as a minimum, I can keep coming up with more similarities. So I must be a human and a toad right? Alright, my reasoning's a little shakey. :D

Alpha, since you failed to describe what the Planck length is, I looked it up. Here's a definition that I found:

Definition: The length scale at which a classical description of gravity ceases to be valid and quantum mechanics must be taken into account. The value of the Planck length is of the order of 10-35 m (twenty orders of magnitude smaller than the size of a proton, 10-15 m).

I found this at: http://physics.about.com/library/dict/bldefplancklength.htm

So what I make of it is that below the Planck length, matter is no longer affected by gravity and then follows by quantum mechanics. So what. The definition itself implies that lengths below the Planck length are possible. Different Quantum particles are smaller are they not. So why can't a length be reduced smaller than the Planck length?

But like Merlijn said, we're talking about physics with this, Zeno's paradox is strict mathematics. Math is almost pure reason (very little science, or observation). Physics is more science (based on experience. My point all along is that reason alone isn't enough. It's a good model, but it's got holes.

Alright, let's look at this problem from another perspective. You travel half the distance to your destination. How did you get there? You must have travelled half the distance from your starting point to where you are now, and then half the distance again, and so on. Yet you are clearly there. If Zeno's paradox were true, then you would be unable to move at all, because if you want to move to any point, you must first move half the distance, no matter how close the point is. In this way the paradox is solved through rational thinking.

You've changed the nature of the problem. The paradox isn't whether or not the thing reaches it's goal. The paradox is that something can move in such a manner forever (1/2's 1/4'th, whatever) and never reach your goal. You might think that it never reaches it's goal because it keeps decreasing speed infinitely. But that doesn't matter, even if the object was in constant acceleration, it would still never reach the goal. Speed is irrelevent.

Yes the manner of movement is ridiculous and not in real life, but it is rationally possible. The paradox doesn't exist in real life. It doesn't in physics either. It exists in reason. Reason can be faulty. Plato was wrong! (alright this is a little strong ;) )

Zero, I've read parts of Kant and Hegel before when I was a philosophy student, and at this stage in my I don't need anymore. Among my philosophy peers, Hegel was a 5 letter dirty word. If I said it, they'd turn around and give me dirty looks. :D

Merlijn, I want to thank you for your post. You have indirectly solved something for me. I think it is what I needed to solve by posting this thread in the first place. Thanks again.

Oh, really? Prove otherwise! The proof is there. Ask a math teacher!

I am a math teacher!

Excuse me?! This is a forum for discussion. This is more of a place for "lectures" than others. And who are you to say who can and cannot "lecture" others. Freedom of speech. I can say what I want. If I think I know the solution to a problem I have a right to say my piece. It is NOT your place to tell others that they can't speak their mind.
OK, trying to calm down before I continue...
I was merely referring to your own behaviour here. You yourself told us to take physics classes, and the like. Whereas you yourself do not show complete control of the subject yourself.
Bu never mind. it's true.... it's an open forum.

AGAIN: 0,999...<> 1 because 0,999... is an element of the array of real numbers (R) but not of the array of all quotients (Q). 1 is both in R and Q. There is a small difference. it's infinately small, but it's there.

Alright, let's look at this problem from another perspective. You travel half the distance to your destination. How did you get there? You must have travelled half the distance from your starting point to where you are now, and then half the distance again, and so on. Yet you are clearly there. If Zeno's paradox were true, then you would be unable to move at all, because if you want to move to any point, you must first move half the distance, no matter how close the point is. In this way the paradox is solved through rational thinking.
Exactly.
Still, I do not know the formal reason why it's wrong... I will look it up, or try to find out myself (when not succesfull).

In case you were wondering: the aim for an infinately small distance away from the point you want to go to, was not completetly serious. :)
I was just curious how you would reply.

keep up the good work

I am happy to have helped you Xenu.
May I know what it was? :)

Originally posted by Xenu
3) The paradox is, when something moves in this manner, it can move forever and ever and never reach its goal.

He is showing that reasoning in this way, although logical, doesn't work. The thing moves forever and never reaches its goal. When it comes to infinity, reasoning becomes faulty.Err, how is that a paradox? What's wrong with the idea of infinitely close but never equal? A simple graph of y=1/x is all you need to see. If you still think it's a paradox, try limit in math. It's the right tool to solve your paradox, and it sounds pretty logical.

OK I agree with both you and Merlijn, when the problem is worked out you should get 1/3 and not 0.3333... But still does 1/3 = 1?Huh? No. Where the hell are you getting that from?
In a 1/4 decreasing Zeno's paradox does the thing that moves get to 1/3 and then stops, Alpha? Does the thing get instantly teleported to the goal once it reaches 1/3 of the distance? Does it sit down and contemplate Kant and Hegel?I'm sorry, but what the hell are you on? Please stop being sarcastic and say it in a way that makes sense.
It at first kind of makes sense. But by the same rational, I could describe how similar I am to a toad infinitely (because language is infinite) and no matter how many similarities you require as a minimum, I can keep coming up with more similarities. So I must be a human and a toad right? Alright, my reasoning's a little shakey.Yet another reason why infinity doesn't exist.
Alpha, since you failed to describe what the Planck length is, I looked it up.How's that?
So what I make of it is that below the Planck length, matter is no longer affected by gravity and then follows by quantum mechanics. So what. The definition itself implies that lengths below the Planck length are possible. Different Quantum particles are smaller are they not. So why can't a length be reduced smaller than the Planck length?There are no particles smaller than the Planck length (guess I was spelling it wrong). It is the smallest distance that can exist. Anything smaller has no meaning. According to the Heisenberg uncertainty principle there is a certain degree of uncertainty that cannot be overcome at the quantum level. The level of the Planck length. If you cannot measure anything smaller, and have no way to detect it, then what's the point of speculating about something that effectively doesn't exist?
But like Merlijn said, we're talking about physics with this, Zeno's paradox is strict mathematics. Math is almost pure reason (very little science, or observation). Physics is more science (based on experience. My point all along is that reason alone isn't enough. It's a good model, but it's got holes.But reason is enough! I just showed that with my last post!

You've changed the nature of the problem. The paradox isn't whether or not the thing reaches it's goal. The paradox is that something can move in such a manner forever (1/2's 1/4'th, whatever) and never reach your goal. You might think that it never reaches it's goal because it keeps decreasing speed infinitely. But that doesn't matter, even if the object was in constant acceleration, it would still never reach the goal. Speed is irrelevent.But there is no paradox because NOTHING can keep moving half the distance to it's destination, then half the remaining distance, etc. It cannot be done!
Yes the manner of movement is ridiculous and not in real life, but it is rationally possible. The paradox doesn't exist in real life. It doesn't in physics either. It exists in reason. Reason can be faulty. Plato was wrong! (alright this is a little strong )How is it rationally possible when it has been rationally proven that it cannot happen? Maybe reason can be faulty (I don't know, though I don't believe so), but it's not demonstrated by this "paradox."
I am a math teacher!LOL! Sorry. Then prove it wrong.
I was merely referring to your own behaviour here. You yourself told us to take physics classes, and the like. Whereas you yourself do not show complete control of the subject yourself.
Bu never mind. it's true.... it's an open forum.I didn't mean to take classes on the subject, I just said to read some books on the subject. How can anyone have complete control over the subject? Obviously you think my arguments flawed. Show me where.
AGAIN: 0,999...<> 1 because 0,999... is an element of the array of real numbers (R) but not of the array of all quotients (Q). 1 is both in R and Q. There is a small difference. it's infinately small, but it's there.Please read the info at the site I posted. There is a good explanation in there. Specifically about the distance between 0.9999.... and 1.
In case you were wondering: the aim for an infinately small distance away from the point you want to go to, was not completetly serious.
I was just curious how you would reply.

keep up the good work.Er, thanks.

I have yet another argument (or another way of putting it). If you travel to 0.9999.... m away from your destination, you're already at your destination as shown by the fact that 0.9999.... is equal to one. Until someone proves otherwise....

I would advise all interested in this subject to read the following link, which is the best I've found on the net in regards to infinity and transfinite numbers; it directly addresses zeno's pair o' docks. It was a little clearer than the explanations in Rudy Rucker's 'Infinity and the Mind,' which is a good book on the subject, and how our brains try to contemplate such heady stuff. A math professor and writer, Rucker's work is fun and mind-bending, and he writes some of the best science fiction I've ever read.

http://www.techno.net/pcl/archive/gg_fiction/achill1.htm

John Le Coq

Xenu...

Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

This sounds awfully familiar... :rolleyes::D

From "Where is the limit?????" (http://www.sciforums.com/t8231/s/showthread.php?s=&threadid=8231&perpage=20&pagenumber=3) posted by... me...:
If you don't stop using your mind, you'll never get to any point. You can divide things in the infinitesimal, going to atoms, subatomic particles, quarks...
You can "multiply" things in millions and millions of times, going to solar system, galaxy, universe...

But... whatever...
Get a calculator and press the number 1.
Then divide it by 10. Again. Again. Again. Again. Again...
Well... when you get to a number like that:
0,000000000000000000000000000000000000000000000000 001
You might get tired and stop doing it...
Press the number 1 again.
Now, multiply it by 10. Again. Again. Again. Again. Again...
You will probably get tired when you get to somthing like:
10000000000000000000000000000000000000000000000000 00

Well... see... you can divide something infinite times and you will never really get to a point. You can also multiply and you won't get to any point either...

So...
Where is the limit??

Thanks Xenu... :)

"Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room."

Isn't this the graph y = x/1 ??? I'm actually curios, not meant to be sarcastic.


Nelson; Xenu sounds nothing like you. You can't grasp infinite. He can.

Isn't this the graph y = x/1 ???
No actually it isn't. The equation that it applicable is very dependant of what variables you have in the equation.

Here are the most obvious:

If x is the time, and y the distance traveled then (with v is the velocity)
y=v.x
The graph is a straight line.

If x is time and y is the 'stepsize' you'll get (again v is the velocity)
y=v
this is a (constant) horizontal line. You may notice: it's the difference equation of the first equation.

If x is the distance travelled and y is the 'stepsize', you'll get almost the same thing as the previous equation. Again no strange effects of infinity.

now... if x is some measure of the number of steps and y is stepsize the equation will be something like
y= v / (x+1) (x >= 0)
Now this one is interesting... or, maybe not. Because how would one define the steps taken? is it possible to just replace a continuous variable by a discontinuous one?
(what's the proper english word?)

I doubt it.

Merlijn

This is a rather confused thread.

I'll respond to <b>Merlijn</b> first, since he seems to think he knows what he's talking about. :)

<i>First of all: in reality there is no such thing as infinity, in any case not that it affects our lives.</i>

That's debateable.

<i>Secondly: 0,999... <> 1 , even though you think to have proven otherwise.</i>

Oops! Actually 0.999.... is exactly equal to 1. The proof given before is perfectly valid. All you need is an infinite number of 9s after the decimal point.

<i>And 3 divided by 9 does not equal 0,333...</i>

Yes it does. And yes, it's also equal to 1/3.

<i>0,333.... and 0,999... are elements of the array of real numbers (R) but not of the array of all quotients (Q).</i>

Wrong. 0.333.... is a rational number. Just because you express something as a decimal doesn't make it irrational. Nor does the fact that the decimal happens to repeat. In fact, <b>every</b> repeating decimal is a rational number (can be written as a fraction).

A number like 1/3 is still 1/3 whether it is written as a fraction, as a decimal, in binary, or hexadecimal or in base 17. It doesn't become irrational when you change the way you write it.

<i>Thirdly: Alpha, It is NOT your place to lecture others! Your posts show a lack of knowledge (or of understanding, in the more negative scenatio) of several concepts.</i>

Ironic, isn't it? (My turn to lecture.) :)

Back to Zeno's paradox.

The relevant insight which resolves the apparent paradox is that it is possible for an infinite sum to add to a finite value. For example:

1/2 + 1/4 + 1/8 + 1/16 + ... = 1

This is an example of a <b>geometric series</b>. If you have a series of terms of the form

r, r², r³, ...

where r is less than 1, and you want to find the sum of the first n of them, it can be shown that the formula is:

Sum = [(1 - rⁿ⁺¹) / (1 - r)] - 1

In the above example, r = 1/2. Let's try adding the first 3 terms.

1/2 + 1/4 + 1/8 = 7/8

Let's also try the formula:

Sum = [1 - (1/2)⁴) / (1 - (1/2)] - 1 = 7/8

So what about the full sum:

1/2 + 1/4 + 1/8 + 1/16 + ... = ?

To get that we look at the sum formula. Remembering that r is less that 1, if we make n very big (infinite, in fact) then rⁿ⁺¹ is a very very small number (actually zero if n is infinity). So, the sum formula becomes, when we take an infinite number of terms:

Sum = [1 / (1-r)] - 1

In the above case we get

Sum = [1 / (1/2)] - 1 = 1

So the infinite series sums to 1.

First..from a post elsewhere here is the heap of paradoxes...there are more than one Zeno paradox for those who are curious:
The paradoxes of Zeno (http://plato.stanford.edu/entries/paradox-zeno/#3)

A link for the enjoyment of all.

Secondly..some snippets from previously posts regarding the halving of distances:

The continual summing of half the remaining distance will always be in the form of (n-1)/n.
(n-1)/n [in the Zeno paradox] as it stands remains a mathematical paradox. Mathematically, the summing of the continuing halving of remaining distance will never reach one...never. From the high school math teacher to triple PhD brainiac in math, the fact that the mathematical formula as stated will never end cannot be contradicted. The adding limits, to this geometric series is an adjunct constraint in order to 'stop the madness' so to speak. If you change the parameters of a paradox, then you are not dealing with the original paradox. Setting a limit side-steps the mathematical paradox by giving it a specific 'sandbox' to play in. In this way the paradox is limited to real world scenarios. The taming of a geometric series is an important step in the mathematical realm, but to say it solves the paradox is like saying a lion, in a cage would never eat humans, because we have put limits on it. The cage...the limits put on the lion is what stops the devouring.
A mathematical formula may not correspond to reality. If it does not correspond to reality, then the formula needs to be tested for its application to reality [if this is the point of the formula.] If the limited formula serves a purpose, by dropping the remainders, then so be it; the modified formula becomes a useful tool. These thoughts are directly related to the need to test 'elegant formulas' for validity in physics.

Read up on the original paradoxes. Zeno was no moron in his development of these paradoxes. There is an aspect of warning what happens when you use mathematical forulas to prove reality. These paradoxes crop up anew in many fields; The development of film and television, for example.

I loves my Zeno...
137

Tyler,

Nelson; Xenu sounds nothing like you. You can't grasp infinite. He can.

We are talking about exactly the same thing...!:bugeye:

James R,
I seriously doubt that

SUM(i=0 -> infinity) [2^-i] = 2

you're right: I think I made a mistake earlier... I mean that the sum is part of the rational numbers Q (thanks for the translation james!), but not of Z. (since the sum of quotients must be element of Q). I admid. Probably half my brain was asleep.

Still: 0,999... <>1

Indeed, the notation of a number does not affect it's meaning.
However, (now I am going to give a less serious argument, just for the fun of it) since the universe is of limited size and information is a physical entity, we can never fully write 1/3 as 0,333....

Actually, 0,333... <> 1/3 because 0,999... <>1
(if I am wrong about this, my whole arumentation flaws)

So what it boils down to is: is 0,999.... element of Z (or: 0,999... =1) ?

"Remembering that r is less that 1, if we make n very big (infinite, in fact) then rn+1 is a very very small number (actually zero if n is infinity). "
here you're wrong.
let W stand for infinitely large.
then: W* 1/W = n, whereas W*0 = 0.
If you take 0 grams of salt infinately often, you end up with no salt!
Hence 0,999... <> 1
Ergo: me right, you wrong! ;)

You haven't lectured me yet.

THIS IS FUN....
(even if you'll disprove my argument)

Edited to paste a minus in the equation. like I never make typing errors... eeeh, and replaced a Q by a Z... that's what you get from confusing me.

Halfway writing my reply StarTrek started... so I didn't see 137's reply.

"Mathematically, the summing of the continuing halving of remaining distance will never reach one...never. " hee...that's my line! :D
I don't think I have encountered you before, so welcome 137. (anything to do with Turk 137, or was that a different number?)

Le Coq,

Thanks for the link, I read through it last night, kind of got lost, am going to have to read through it a few more times. It talks about a different paradox than what I posted, but it still might be relevant. There are a number of them like 137 posted. It did seem to solve the paradox why achilles couldn't beat the tortoise, but it didn't seem to address why both of them would run forever, never reaching their goal. Like I said I'll have to read through it several more times.

Alpha,

Huh? No. Where the hell are you getting that from?
and
I'm sorry, but what the hell are you on? Please stop being sarcastic and say it in a way that makes sense.

If I sound sarcastic, that's because I'm frustrated with you. Since you have no idea what I'm talking about when I say "Does 1/3=1?", I feel that you either have not been reading my posts, just reacting to them, or not attempting to digest anything that I say, you're only goal is to be "right" and not trying to come to any real understanding of what's going on. These are my observations.

So...."Does 1/3 = 1?"

Obviously not. My point is you tried to disprove Zeno's paradox with the notion that 0.9999....=1 (remember your little math trick?). Let's say this is correct (for the time being). Well if you change the paradox a bit, instead of moving by 1/2's you move by 1/4's. When you do this your new distance is 0.3333.... (considering 1 is the end point). Now by that same math trick that you posted, this is turned into 1/3. I really don't care if it is 0.3333... or 1/3, either way they don't equal 1, which is where the end point is. So does the object stop at 1/3? This shows that your math trick doesn't adequately solve Zeno's paradox, just one of its "manifestations".

If you still don't understand, reread all of my posts and attempt to see things through my view. If you still don't understand what I'm saying here, I give up on you.

-Xenu

p.s. going to post the Planck stuff in a different thread, this is off topic, like merlijn said, we're talking math not physics.

Xenu, if it's any conselation: I understood the 1/3 thing. :D

137,

Just wanted to add. Yes!!! Finally someone who understands me!!! ( you too Merlijn, don't want to leave you out ;) ). Yes, when things such as limits, you change the nature of the problem, turning infinity into the finite (look at the article that Le Coq posted too), when you do a geometric series, you work backwards, starting at the end, as if the goal had already been reached. Again, changing the nature of the problem.

Also, the whole 0.9999.... = 1 thing to me shows me another thing how reason alone is flawed....er limited ;) How can something be two things at once, how can something be infinite and finite at the same time? I can see it when I approach the outside world, but only without reason (or with reason combined with non-reason). Reason who's number 1 maxim is that "things can't be contradictory" is contradictory itself in the terms of mathematics.

Merlijn: Here is one citation of the 137. "One hundred thirty-seven is the value of a number called the fine-structure constant. This constant, 137, is the way physicists describe the probability that an electron will emit or absorb a photon. Because this is the basic physical mechanism of electricity and magnetism, the fine-structure constant has its own symbol, the Greek letter a, “alpha.”" type 137 in Google and see all the fun range of 137s.
I dunno what Turk 137 is what is it?

I think if we had a common focus on what constitutes A] a mathematical paradox, B] the calculus of limits, C] an acknowledgement of the difference between mathematical formulas [and the strange world of pure numbers] and D] the real world where mathematical chaos is reigned in for useful purposes, then these psuedo-conflicts would not exist.

My only other posts are related to the thread 'Does Light have mass?'

Glad to check in when I can...Each day I can only spend half as much time as the day before...since a mathematical calculation of this time can be just as infinite...I gues I will be coming back here forever!

:p 137:(

:D

Xenu,

Also, the whole 0.9999.... = 1 thing to me shows me another thing how reason alone is flawed....er limited

Wow! You got the the same conclusion I did! :)
You can see it in my thread "Where is the limit?????", in the thrid page... (which by "coincidence" begins with the discussion about infinite...) :bugeye:


137,

Glad to see you are also aware about that... :eek:
Btw... I'm also coming back here forever... you see... I told people I wouldn't be here anyomre today, but I'm still here...!!! :p:D:D

...I may be here forever then...who knows...:p:eek:

Xenu, Seeker,

Also, the whole 0.9999.... = 1 thing to me shows me another thing how reason alone is flawed....er limited

"Remembering that r is less that 1, if we make n very big (infinite, in fact) then rn+1 is a very very small number (actually zero if n is infinity). "
here you're wrong.
let W stand for infinitely large.
then: W* 1/W = n, whereas W*0 = 0.
If you take 0 grams of salt infinately often, you end up with no salt!
Hence 0,999... <> 1


Didn't I just show with pure reason that 0,999...=1 is wrong?

Glad to check in when I can...Each day I can only spend half as much time as the day before...since a mathematical calculation of this time can be just as infinite...I gues I will be coming back here forever!

LOL! It seems that each day I spend twice as much time as the day before, well it looks like I'm here forever too!

Xenu, if it's any conselation: I understood the 1/3 thing.

Good, I was beginning to think that I was either a) the only rational person on this thread or b) going completely insane.

"HOW MANY FINGERS AM I HOLDING UP ...." *holds up 4 fingers*

"Uh, er .....5" :p

Didn't I just show with pure reason that 0,999...=1 is wrong?

I'm not following your guys' argument, I tried to reread it several times, but not sure. Could you explain it in a more simplistic way, idiot terms? You're probably correct in disproving .9999....=1, but at the same time Alpha's math trick is at least mathematically true. I guess I don't know exactly what this says about reason. I think for myself I need to define, reason, logic, and mathematics, because for me it's all sort of blending and distorting some of my conclusions.

Also on a side note, I've heard that Infinity divided by Infinity (W* 1/W = n) is undefined in mathematics, is this true?

that's true...
if a/b = c if a=b*c.
simple... 8/2 = 4 because 2*4=8;
now 3/0 = ....? ...*0 =3??? no can't do that, but
0/0 = 431 because 431*0=0 ! but also 0/0=15 or so.
and W/W = 127 becaquse 127*W=W.

Now is the question: W/0=0 or W/0=2 (or any other positive integer)?

If W/0<>0 then I am right, otherwise JamesR is right and I am wrong.

is that more clear to you?

0,999...<> 1 because 0,999... is an element of the array of real numbers (R) but not of the array of all quotients (Q). 1 is both in R and Q. There is a small difference. it's infinately small, but it's there.Since it's "infinitely small," it's not there!
"First of all: in reality there is no such thing as infinity, in any case not that it affects our lives."

That's debateable.Why's that?
"And 3 divided by 9 does not equal 0,333..."

Yes it does. And yes, it's also equal to 1/3No actually it doesn't. The decimal system is inaccurate. At least, it's not as accurate as fractions. 0.3333.... is technically not exactly the same as 1/3.
Wrong. 0.333.... is a rational number. Just because you express something as a decimal doesn't make it irrational. Nor does the fact that the decimal happens to repeat. In fact, every repeating decimal is a rational number (can be written as a fraction).

A number like 1/3 is still 1/3 whether it is written as a fraction, as a decimal, in binary, or hexadecimal or in base 17. It doesn't become irrational when you change the way you write it.Since the decimal system is not as accurate, it does make a difference when you convert in some cases, as I've shown.
To get that we look at the sum formula. Remembering that r is less that 1, if we make n very big (infinite, in fact) then rn+1 is a very very small number (actually zero if n is infinity). So, the sum formula becomes, when we take an infinite number of terms:

Sum = [1 / (1-r)] - 1

In the above case we get

Sum = [1 / (1/2)] - 1 = 1

So the infinite series sums to 1Thank you!
Mathematically, the summing of the continuing halving of remaining distance will never reach one...never. From the high school math teacher to triple PhD brainiac in math, the fact that the mathematical formula as stated will never end cannot be contradicted.Not so, as it has already been shown.
Actually, 0,333... <> 1/3 because 0,999... <>1
(if I am wrong about this, my whole arumentation flaws)Well, 0.3333... isn't really equal to 1/3 because you can never put enough threes, but at the limit of infinity (which is a contradiction in terms) it would be exactly equal. Just like 0.9999... would be equal to 1.
So...."Does 1/3 = 1?"

Obviously not. My point is you tried to disprove Zeno's paradox with the notion that 0.9999....=1 (remember your little math trick?). Let's say this is correct (for the time being). Well if you change the paradox a bit, instead of moving by 1/2's you move by 1/4's. When you do this your new distance is 0.3333....How do you figure? 0.3333.... is one third, not one quarter. And that's not the limit you approach.
(considering 1 is the end point). Now by that same math trick that you posted, this is turned into 1/3. I really don't care if it is 0.3333... or 1/3, either way they don't equal 1, which is where the end point is. So does the object stop at 1/3?No, seeing as you're travelling by 1/4 the limit is not 1/3.
This shows that your math trick doesn't adequately solve Zeno's paradox, just one of its "manifestations". Explain.
p.s. going to post the Planck stuff in a different thread, this is off topic, like merlijn said, we're talking math not physics.We're talking math and physics aren't we? They both apply. I've used both math and physics to disprove Zeno's paradox, and even just pure logic.
Xenu, if it's any conselation: I understood the 1/3 thing.Then perhaps you could explain it?
the whole 0.9999.... = 1 thing to me shows me another thing how reason alone is flawed....er limited How can something be two things at once, how can something be infinite and finite at the same time? I can see it when I approach the outside world, but only without reason (or with reason combined with non-reason). Reason who's number 1 maxim is that "things can't be contradictory" is contradictory itself in the terms of mathematics.Reason is not flawed, just some people's implementation of it. As you can see, the flaw in the "0.9999...=1" thing was also pointed out by reason. Seeming flaws in reason can be pointed out by further reason. Since no-one is perfect, sometimes it takes someone else to come along and reason it out better.
I'm not following your guys' argument, I tried to reread it several times, but not sure. Could you explain it in a more simplistic way, idiot terms? You're probably correct in disproving .9999....=1OK, if you have 0.3333.... and you keep trying to add threes, you'll never reach exactly 1/3. But at the limit of infinity it is equal. But since infinity has no limit, it will never be exactly equal, only almost. That's a flaw in the decimal numbering system.
at the same time Alpha's math trick is at least mathematically true. I guess I don't know exactly what this says about reason. I think for myself I need to define, reason, logic, and mathematics, because for me it's all sort of blending and distorting some of my conclusions.It really isn't a math "trick." It is a valid proof that illustrates the fact that if you could have an infinit number of nines after the decimal (at the limit of no limit) it would be equal to one. All this can be shown by reason. Does this help?
Also on a side note, I've heard that Infinity divided by Infinity (W* 1/W = n) is undefined in mathematics, is this true?that's true...
if a/b = c if a=b*c.
simple... 8/2 = 4 because 2*4=8;
now 3/0 = ....? ...*0 =3??? no can't do that, but
0/0 = 431 because 431*0=0 ! but also 0/0=15 or so.
and W/W = 127 becaquse 127*W=WXeno, if you want a clearer version of that, read Mathematical Fallacies and Paradoxes (http://www.amazon.com/exec/obidos/ASIN/0486296644/ref=ase_ericstreasuretroA/002-3412380-6316059). It'll state what he was trying to say in a clearer way, and there's lots of other stuff in there too. I read it yesterday. Well, most of it.
I would say that you can't divide infinity by infinity because it's not a number, it's more of a process than anything. You could only divide by it if you used the result of the "process," but since it has no limit, the process never ends, so you'd never get to divide by it.

<b>Merlijn</b>:

<i>I seriously doubt that

SUM(i=0 -> infinity) [2^-i] = 2</i>

Where did that come from? Seems to be irrelevant to the current argument.

<i>Still: 0,999... <>1</i>

It has been shown that it is, above. Do you have a disproof, or just a gut feeling?

<i>...since the universe is of limited size and information is a physical entity, we can never fully write 1/3 as 0,333... </i>

You just wrote it! It must be understood, of course, that the ellipsis "..." means the 3s go on forever. You're not the only person here who has failed to grasp that point. When I write 0.333..., I mean, literally, an infinite string of 3s after the decimal point. That is very different to the number 0.3333...3, where the "..." stands for a finite number of 3s.

<i>So what it boils down to is: is 0,999.... element of Z (or: 0,999... =1) ?</i>

Yes. 0.999... = 1, exactly. If you think otherwise, please provide a disproof.

I said: <i>Remembering that r is less that 1, if we make n very big (infinite, in fact) then rn+1 is a very very small number (actually zero if n is infinity).</i>

This was in the context of summing a geometric series. In reply, you say:

<i>here you're wrong.
let W stand for infinitely large.
then: W* 1/W = n, whereas W*0 = 0.</i>

My statement has nothing to do with transfinite arithmetic, so I don't see how your statement is relevant.


<b>alpha</b>:

<i>Since it's "infinitely small," it's not there!</i>

You have to be careful of statements like that. An infinitessimal still exists.

<i>No actually it doesn't. The decimal system is inaccurate. At least, it's not as accurate as fractions. 0.3333.... is technically not exactly the same as 1/3.</i>

See my comments above concerning the ellipsis (...). That makes 0.333... precisely equal to 1/3. It is an accepted convention for writing infinite decimal expansions.

<i>Since the decimal system is not as accurate, it does make a difference when you convert in some cases, as I've shown.</i>

No. The decimal system is as accurate as any other system of writing real numbers. In any base there will be some numbers with infinite expansions. Write 1/2 in base 3 and you get an infinitely long expansion. Write it in base 10 and you get 0.5. Nothing is lost in the conversion from one base to another, if you do it properly.

Thanks for the link, I read through it last night, kind of got lost, am going to have to read through it a few more times. It talks about a different paradox than what I posted, but it still might be relevant. There are a number of them like 137 posted. It did seem to solve the paradox why achilles couldn't beat the tortoise, but it didn't seem to address why both of them would run forever, never reaching their goal. Like I said I'll have to read through it several more times.

I would say it is very relevant, since you posted Zeno's first "dichotomy" paradox of motion, and the Achilles' Paradox is essentially the same paradox, as stated in 137's link, which is comparing two bodies in motion (with no end to the motion) vice one body with reference to a fixed point (end of motion).

Keep reading the Achilles' link I sent. It may take a week to "get", and this doesn't mean you're dumb. I'm no math whiz, but after first reading and rereading it a year or so ago, along with Rucker's book, I got little "eureka" moments of clarity that, unfortunately, fade with distance from the material. I have to go back when the mood strikes me and recoat those neurons from time to time.

Gothard writes at the end of the article's first page in question, "The next installment of my article will show that Cantor's theory of the transfinite cardinal number c, the Aleph of the continuum, provides a genuine solution to Zeno's paradox." I can't verify this, and maybe the other matheaux here can comment, but it is the closest thing that I know of that can answer your quest(ion), as it satisfied mine some time ago.

John Le Coq

JamesR

SUM(i=0 -> infinity) [2^-i] = 2

Where did that come from? Seems to be irrelevant to the current argument.
Well... it is precisely what this is all about. since Zeno's paradox is about the equation D= 0,5*SUM(i=0 -> infinity) [2^-i] = 1 ?
I just left out the scalair.

Remembering that r is less that 1, if we make n very big (infinite, in fact) then rn+1 is a very very small number (actually zero if n is infinity).
here you're wrong.
let W stand for infinitely large.
then: W* 1/W = n, whereas W*0 = 0.

This again is totally relevant!
An infinite large amount of zero's still add up to zero. Thus, infinately small is larger than 0.
In the x>0, the > will resemble a = but is will never be.
AND HEEEE LOOK WHAT WE'VE GOT HERE:
You have to be careful of statements like that. An infinitessimal still exists.
Since when are you on our side now? :D
That is exactly my point! Strange enough, the very next thing you say is the direct opposite of the infinitessimal you brought up.... get your story straight :)

STILL: you're wrong about the decimal system. the conversion of an n-base system to an m-base system is no problem. HOWEVER, bringing in the dot (or comma) notation is a totally different conversion. And its inherently inaccurate. Remember your infinitissimal!

I think it is you time to disprove that. I think I have done my share of giving evidence for now.

Merlijn, I think I'm beginning to follow your argument but not entirely. Let me show you my thought processes to your post and you can show me where I'm wrong...


if a/b = c if a=b*c.
simple... 8/2 = 4 because 2*4=8;

ok, simple, understood


now 3/0 = ....? ...*0 =3??? no can't do that, but

yeah you can't do that, can't divide anything by zero

0/0 = 431 because 431*0=0 ! but also 0/0=15 or so.

still can't do that, can't divide something by zero.

and W/W = 127 becaquse 127*W=W.

yes, can't do this either, can't divide infinity by infinity, get the fat E (error)

Now is the question: W/0=0 or W/0=2 (or any other positive integer)?

yes, neither is correct, anything divided by zero gets the fat E (unless infinity is a special case, which doesn't seem rational)

If W/0<>0 then I am right, otherwise JamesR is right and I am wrong.

Ok, W/0<>0, I agree. It doesn't equal anything but the fat E to me. I would say W*0 = 0. So now what I need now is the logic that makes this disprove that 0.9999...<>1. Let me know if I got any of the above wrong.

Thanks for taking the time to clarify some of this.

Alpha,

How do you figure? 0.3333.... is one third, not one quarter. And that's not the limit you approach.

Multiply it out. If something 1/4's its distance, and then 1/4's that and then 1/4's that , you get .3333....

Use a calculator
O is your beginning, 1 is your end point

1 * 1/4 = 0.25
0.25 * 1/4 = 0.0625
0.0625 * 1/4 = 0.015625
0.15625 * 1/4 = 0.00390625
etc. keep doing this until your calculator can't divide anymore

Now add up all your answers, this is the distance traveled

0.25 + 0.0625 + 0.015625 + 0.00390625 + etc. = .3333.... or some might say 1/3

This object will either be infinitely approaching 1/3 or make it to 1/3. In either case, it doesn't come close to the end point, which is 1; it'll only make it, no matter how much or how fast it moves, at best, about 1/3 of the way there.

The point is, it doesn't matter whether or not 0.9999...=1, it doesn't solve Zeno's paradox when something moves in 1/4's rather than 1/2's.

-Xenu

Xeno, if you want a clearer version of that, read Mathematical Fallacies and Paradoxes. It'll state what he was trying to say in a clearer way, and there's lots of other stuff in there too. I read it yesterday. Well, most of it.

Thanks for the recommendation for the book Alpha, I appreciate it. :)

No problem. You might find it interesting.
Multiply it out. If something 1/4's its distance, and then 1/4's that and then 1/4's that , you get .3333....OK, my bad.
The point is, it doesn't matter whether or not 0.9999...=1, it doesn't solve Zeno's paradox when something moves in 1/4's rather than 1/2's.

-XenuActually it does. The same argument I used before still applies. Eventually the minimum distance you can travel will be more than one quarter the distance (or whatever fraction you choose). So it doesn't matter how small the fraction is. Eventually you will have to break the limit.
And guys, once again:
0.3333... isn't really equal to 1/3 because you can never put enough threes, but at the limit of infinity (which is a contradiction in terms) it would be exactly equal. Just like 0.9999... would be equal to 1.

from 137: Mathematically, the summing of the continuing halving of remaining distance will never reach one...never. From the high school math teacher to triple PhD brainiac in math, the fact that the mathematical formula as stated will never end cannot be contradicted.
--------------------------------------------------------------------------------

From Alpha: Not so, as it has already been shown.

So you cannot comprehend the difference between a mathematical infinite geometric series and the adjunct tool of the calculus of limits? The sum halvseys will always and forever be in the form of (n-1)/n. This cannot ever reach one, mathematically. Putting limits on what infinitesimal quantity, and below, is worth throwing away is a separate issue from the proposed mathematical paradox. A human mind developed a separate tool and artificially ASSIGNS a limit; without this artificially assigned limit, the series marches on in a spiral of smallness with no end. David Berlinkski in a Tour of Calculus [fantastic book by the way] has some good illustration of Zeno's paradox. To paraphrase, the solution of limits does not satisfy the Paradox, but it allows the unmanageable to be managed. Where Zeno's paradox intersects the real world, of course there is a limit where our separation between the wall and our self becomes negligible.
A human inability to comprehend a never ending decrease, does not change the fact that it never ends. Why the inability for some folks? Because a decrease of distance between physical objects HAS a limit. It is counter-intuitive to our experience. BUT, in the playground of mathematics, there are formulas which do not have real world correspondents. Why do you think the Pythagorians romped in the world of mathematics as their mystical world and source of the infinite divine mystery.
Is pi really 3.14? Have you found the end of pi? Given, this is a different order of mathematical usage, but the hunt goes on and on. We put a limit on pi for practical usage, but the same mathematical brainiacs who make a living studying the endless non-repetitive nature of pi's trailing decimal numbers would laugh you out of the room if you 'proved' by limits that pi has been proven to = 3.14. Alpha, you have been omega-ed.

137

I have some questions... it seems that our matemathics are not very good... :o

0.25 + 0.0625 + 0.015625 + 0.00390625 + etc. = .3333.... or some might say 1/3

This object will either be infinitely approaching 1/3 or make it to 1/3. In either case, it doesn't come close to the end point, which is 1; it'll only make it, no matter how much or how fast it moves, at best, about 1/3 of the way there.

Well... and if we do the same thing for 1/4? And 1/5? And 1/6? And 1/2485928460976? And 1/W???? :confused:

Will it ever get to the same paradox??

I mean... if we divide a number, any number, "infinite" times, or multiply a number "infinite" times, will we get to a limit?

Is there a limit?????:bugeye: :bugeye: :eek:

Truthseeker,

You could use all of those fractions ( not entirely sure about the 1/W though) and they would go on forever. Just like 137 said right above the post of yours, a limit is something created to make infinite numbers into finite concepts, so that we can deal with them in mathematical equations. Limits are basically "close enoughs". So, no, there are no true limits, mathematically speaking.

This however is pure mathematics which doesn't necessarily apply to the "real"world. Alpha keeps spouting about Planck lengths being the minimum possible length (which still doesn't apply to a mathematical problem, I might add), which may or may not be true, so maybe in "real" life there is a limit. I would personally say that infinities exist everywhere, but you can't rationally debate about them because rationality can't handle infinities too well, as shown by this whole debate over Zeno's paradox.

Xenu,

Rationally, we are unable to discern about things without using limitations. See how we categorized and ordered all living beings of this planet. When you see them in their natural habitat, are they ordered? No. We created an order just to make it easier for us to observe and to understand. However, it limits our reality to our own little perspective.

Besides that, mathematics are the bases of all our physics, chemistry, and even to buy something in the market we use mathematics. If mathematics don't aply to the "real" world, why do we use them to determine things in the "real" world!?!?:bugeye:

If we can't explain infinity in a rational manner, this is an evidence that our rationalism limits itself and can't handle with the task of discovering what is(are) the basic(s) Truth(s) of the universe.

Do you realize how many tortroises the Ancient Greeks went through before the gave up on the half then half then half (or hthth) premise?
You fire an arrow at a tortoise. The arrow not only has to cover half the remaining distance for eternity but also has to make up the additional distance the tortoise has moved.

Rationally, we are unable to discern about things without using limitations.

All things have limitations which we are able to observe.

See how we categorized and ordered all living beings of this planet. When you see them in their natural habitat, are they ordered? No.

What about an ant colony ? There exists one of most ordered of natural habitats known to man.

We created an order just to make it easier for us to observe and to understand. However, it limits our reality to our own little perspective.

We do not observe the order, we observe the reality. Our perspective is that of reality, nothing more, nothing less.

Besides that, mathematics are the bases of all our physics, chemistry, and even to buy something in the market we use mathematics. If mathematics don't aply to the "real" world, why do we use them to determine things in the "real" world!?!?

Mathematics does apply to the real world. Our universe follows mathematics. Mathematics was derived from our knowledge of the universe and our surroundings. Mathematics is the one true universal language that all may speak and understand.

If we can't explain infinity in a rational manner, this is an evidence that our rationalism limits itself and can't handle with the task of discovering what is(are) the basic(s) Truth(s) of the universe.

The concept of infinity can be explained quite easily and in a rational manner. Perhaps it is your inability to understand the explanations that is causing you to question the concept.

From a basic mathematical stance:

let d[p] be distance in meters from start point
let d[y] be distance in meters from start point + 1 meter
let t be time in elapsed seconds
let x be d[y]-d[y] at t+ >plank time
(t(d[y] - d[p])) / 2 = x

No matter how many times you increase s, you never reach a point where x = 0
Why?

Rationally, we are unable to discern about things without using limitations. See how we categorized and ordered all living beings of this planet. When you see them in their natural habitat, are they ordered? No. We created an order just to make it easier for us to observe and to understand. However, it limits our reality to our own little perspective.

I agree with this.

Besides that, mathematics are the bases of all our physics, chemistry, and even to buy something in the market we use mathematics. If mathematics don't aply to the "real" world, why do we use them to determine things in the "real" world!?!?

If we can't explain infinity in a rational manner, this is an evidence that our rationalism limits itself and can't handle with the task of discovering what is(are) the basic(s) Truth(s) of the universe.

I see where you are going with this, but I don't entirely agree. Reason, Mathematics, etc. may not describe things completely, but they are one of the best tools we have in describing the universe. We didn't evolve it for no reason. I like to describe it as a model. Reason is a good model, but all models don't describe what they are describing completely. I believe in a balance between Reason and the things that we can't describe with reason. But by all means, Reason should not be eradicated. This is all my viewpoint.

(Q),

We do not observe the order, we observe the reality. Our perspective is that of reality, nothing more, nothing less.

This is a direct perception standpoint. I will be starting a rather lengthy set of threads in the near future, and part of this will be showing how this isn't necessarily true.

The concept of infinity can be explained quite easily and in a rational manner.

Is this so? I would like to be enlightened.

Merlijn:

<i>SUM(i=0 -> infinity) [2^-i] = 2</i>

I'm sorry. I agree with this. I didn't read it carefully enough. I didn't notice that i was the index you were using, and thought it must have been i = sqrt(-1). As written above, the statement is true.

<i>An infinite large amount of zero's still add up to zero.</i>

I agree.

<i>Thus, infinately small is larger than 0.</i>

Yes. I agree.

<i>STILL: you're wrong about the decimal system. the conversion of an n-base system to an m-base system is no problem. HOWEVER, bringing in the dot (or comma) notation is a totally different conversion. And its inherently inaccurate. Remember your infinitissimal!

I think it is you time to disprove that. I think I have done my share of giving evidence for now.</i>

This is easily settled. I say that I can write an exact decimal expansion of any rational number. You presumably dispute that by saying decimal is "inherently inaccurate". So, all you need to do is to provide an example of a rational number for which I cannot write an exact decimal.

Remember that I've already explained my use of the ellipsis, so for example, 0.333... is an exact decimal expansion of 1/3.

So? Your counter-example?

Thanks James R, you just defeated yourself:

An infinite large amount of zero's still add up to zero.
I agree.
Thus, infinately small is larger than 0.
Yes. I agree.

Thus we can conclude:
0,999... < 1

0,999... / 3 = 0,333... AND 1 / 3 = 1/3

Therefor we can conclude 1/3 - 0,333... > 0

The overall conclusion is that Zeno's paradox still stands.
This is not a real surprise is it? After thousands of years, suddenly a group of sciforums users solve the paradox, and even in a very simple manner.

And now for Phase Two.
Why don't we differntiate the functions first, and then re-examine the problem?
You start ;)
I'm going to take a shower (one of my favorite things in life)

PS... I was not very certain which notation to use.... Why are the math symbols not standard?
my opteions were


SUM (2^-i) = 2 (leaving out the index, I decided I did not want confusion about i.... the one you ran into)
SUM(2^0 ... 2^- infinity) = 2 (here I found the - and the infinity a bit unclear, so I opted for: )
SUM(i=0 -> infinity) [2^-i]=2 (which I thought to be closest to the original mathematical notation)

Sorry to have caused confusion.

Xenu,

I see where you are going with this, but I don't entirely agree. Reason, Mathematics, etc. may not describe things completely, but they are one of the best tools we have in describing the universe. We didn't evolve it for no reason. I like to describe it as a model. Reason is a good model, but all models don't describe what they are describing completely. I believe in a balance between Reason and the things that we can't describe with reason. But by all means, Reason should not be eradicated. This is all my viewpoint.

I agree with that. However we should try to find a better model... a model that transcends the limitations of reason.

Originally posted by 137
Is pi really 3.14? Have you found the end of pi? Given, this is a different order of mathematical usage, but the hunt goes on and on. We put a limit on pi for practical usage, but the same mathematical brainiacs who make a living studying the endless non-repetitive nature of pi's trailing decimal numbers would laugh you out of the room if you 'proved' by limits that pi has been proven to = 3.14. Alpha, you have been omega-ed.LOL! That's not what I'm saying at all! I can see you don't understand. Perhaps you should re-read the thread.
Thus we can conclude:
0,999... < 1

0,999... / 3 = 0,333... AND 1 / 3 = 1/3

Therefor we can conclude 1/3 - 0,333... > 0

The overall conclusion is that Zeno's paradox still stands.Question, why do you keep using a comma for a decimal point?
0.3333.... is equal to 1/3 only at the limit. Since you could never reach the limit, you would never get it to equal 1/3. Sure, we can type a representation of it at the limit (with a bar or elipses), but then you're defining the limit of infinity which is a contradiction. And if you accept a limit of infinity, where it does equal 1/3, then at the same limit for 0.9999... it equals 1. (!)

OK, lemme try this again, arguing from a math standpoint. If you travel towards your destination you have to travel half the distance. (Or whatever fraction, lets say half so we're not altering the parameters). Well, if you have to travel half the distance first, and you think of that as your new destination, then you must first travel half the distance to that halfway point, but then with that as your new destination, you must first travel half the distance to there, and so on, and so on.... Essentially, you can't move! Because any distance away from where you are now that you can state, (no matter how close) will always be further than half that distance!

I use the comma, because I am European. But if you prefer (like my computer and calculator) I can use the decimal point.

let me get this straight... your last story leads me to conclude the same again:

If the distance to travel equals 1
and any distance you have travelled is already too far away, thus you'll never be able to get from 1 to 0.999...
(I think you're going to disagree)
ERGO: 1 <> 0.999...

:)

Originally posted by Merlijn
I use the comma, because I am European. But if you prefer (like my computer and calculator) I can use the decimal point.Whatever you prefer.
If the distance to travel equals 1
and any distance you have travelled is already too far away...Huh?
thus you'll never be able to get from 1 to 0.999...
(I think you're going to disagree)
ERGO: 1 <> 0.999...How did you guess? ;)
Could you please reword that?

Alpha, my cousin (http://www.sciforums.com/showthread.php?s=&postid=127814#post127814) is this leading anywhere?

BTW what's whith the difference between calcuator and computer?
I mean... calculating is not that different than computing, is it?

Merlijn:

<i>Thanks James R, you just defeated yourself</i>

Um, no.

<i>Thus we can conclude:
0,999... < 1</i>

No. 0.999... and 1 are two different notations for exactly the same number. There is no infinitessimal difference between 0.999... and 1. The ellipsis means that there is an infinity of 9s in 0.999..., and that makes all the difference. If the number of 9s was finite, I would agree with your argument, but it isn't.

<i>And now for Phase Two.
Why don't we differntiate the functions first, and then re-examine the problem?</i>

Sorry, you've lost me again. Differentiate which functions?

<i>PS... I was not very certain which notation to use.... Why are the math symbols not standard?</i>

Which maths symbols?

<i>SUM(i=0 -> infinity) [2^-i]=2 (which I thought to be closest to the original mathematical notation)</i>

This isn't your problem at all. It's mine. What you wrote is clear enough. I would have written

Sum (i=0 to infinity) 1/2ⁱ = 2

but its the same thing. As I said, I just mistook your i for the imaginary number rather than the index. My mistake.

LOL! That's not what I'm saying at all! I can see you don't understand. Perhaps you should re-read the thread.

Alpha...as an example of calculating a never-ending number...possibly, never repeating number, is valid. If you prefer to disclude it, then focus on my comments that limitations on a formula change the original paradox.

cheers,
137

Here are my thoughts on the subject 1<>0.99999...

I am concvinced that 1 = 0.9999... from the definition of reals (please look at http://www.math.vanderbilt.edu/~schectex/courses/thereals)

It seems that reals are defined as "Dedekind complete ordered fields". This definition uniquely defines reals up to isomorphism. Dedekind completeness is the principle that alowes us to take limits. There are several formulations for this property. One of which is that of Cauchy sequence. Every real can be denoted by a Cauchy sequence (this is a sequence of rational numbers for which the differences between the numbers become eventually arbitrarily small). 0.9999.. is a Cauchy-sequence as follows:

0, 0.9, 0.99, 0.999, 0.9999, 0.99999, ....

The other number we are interested in is 1. An associated Cauchy sequence is:

1, 1, 1, 1, 1, 1, 1, ....

Now, these two sequences are equal since the difference between the two can be made arbitrarily small. These two sequences are in the same equivalence class and are thus only different representations of one and the same number.

Sorry.

People that want 1<>0.9999 should consider another set built on top of the reals in which infinitesimals are contained. Again, look at the same website as mentioned before for details. In this set there are also the reciprocals of infinitesimals. Of couse in this set you loose the possibility to take limits (all limits?).


Greetings,


Han.

There are an infinite number of ways to set limits and define an infinite series into a manageable and solidified form.
The mathematical formula of the endless halved distances still remains unfettered in the realm of pure mathematics.
<1 will never equal 1 unless it is defined as equal to 1. When this type of defined limit is imposed, it only exists for the duration of its usefulness to the 'solver' of the inequality. Once the temporary solution is shelved, the secret world of the infinite smallness continues in its darkly mysterious and endless pathway.

If you change the parameters of the orignally stated formula, you are not solving the unsolvable.
If we set up a rule that says, without using calculus and defining limits, AND using a continuous halving formula of a distance A to B, will there ever be an un-rounded sum of all halves which equals the numerical distance between A & B?

The answer based on these original paradox parameters must be no.

Originally posted by 137
<1 will never equal 1 unless it is defined as equal to 1.That's the thing! As it's defined, it IS equal to one.
If you change the parameters of the orignally stated formula, you are not solving the unsolvable.The reason we are changing the parameters is because it was shown that there is no paradox as it was originally defined. We changed the original parameters ever so slightly to create a slightly altered paradox, which was also shown to be a fallacy. Because there can never be a true paradox! It only seems that way due to faulty logic and assumptions.

"That's the thing! As it's defined, it IS equal to one."
We are powqerful being, but not that powerful.

I have done a bit more reading on the subject of reals. There is a different presentation of "reals" (in fact it is an augmentation of the reals) including infinities. This follows better the intuition we have about reals and allows us to calculate with infinities and infinitesimals. This branch of mathematics was invented (discovered for Platonists) by Abraham Robinson in the sixties. Leibniz also calculated with infinitesimals, but never got it entirely right. Robinson did. Oh yes, the name for this analysis is "non-standard analysis (http://members.tripod.com/PhilipApps/nonstandard.html).

There is also another kind of analysis which deals with limits in another way: Unified Analysis. I havent read much about it, but it seems very strange to me (for instance the limit of all the natural numbers = sqrt(2*pi) ).


Greetings,


Han.

Originally posted by Merlijn
We are powqerful being, but not that powerful. What are you talking about?

What I meant of course was
We are powerful beings, but not that powerful.

With that I mean that I am a Platonists, meaning I believe that mathematical truths (or those in other sciences for that matter) are independant of the definitions we choose. It is possible we do not know them, and can only approximate them. I stronly believe that we do not just make up mathematics, but that we try to describe, to the best of our abilities, an objectvely true world. That's why Han wrote that a "branch of mathematics was invented (discovered for Platonists)".

In fact, I do not see the use of doing sciences when you are not persuing objectively (not culturally, individually, or whatever) dependant truths!

Han, you're a a smart and sensible person; how come you're not a Platonist.

Ah, but it's only discovered if it's true! If a flaw is found then it must have been "invented."

Originally posted by Merlijn
Han, you're a a smart and sensible person; how come you're not a Platonist. [/B]

You just gave the answer ;)

Seriously, the reason that I am not a Platonist is examplified by the discussion about 1<>0.99999... If you want the numbers to be equal then you choose to use the standard analysis, if you don't like this you just design another kind of analysis, as Robinson did.


Greetings,


Han.

Alpha,

The fact that the mathematical setup is self-referenced to equal a totality does not erase the mathematical paradox.
The reason we are changing the parameters is because it was shown that there is no paradox as it was originally defined. We changed the original parameters ever so slightly to create a slightly altered paradox, which was also shown to be a fallacy. Because there can never be a true paradox! It only seems that way due to faulty logic and assumptions.To slightly alter a paradox is still alterating a paradox. Can someone be slightly dead? Who ever said there can never be a true paradox? If you study your logic 101 then you will find plenty of logical paradoxes....
The mathematics of a line and numerical progressions are full of paradoxes. In Mathematics there are paradoxes...how can you deny this?

Because I've never seen a paradox in my life!
Every paradox I've come across has either been a false paradox (a fallacy, it can be explained). There's always something wrong with the initial assumptions or something. A paradox only exists in abstract thought, not in reality. A paradox by definition can't exist. It just doesn't bloody well make sense!
All those logical paradoxes can be explained.

Just because paradoxes don't make sense doesn't mean they can't exist.

(sry I gave up reading further at a quarter of the discussion, maybe someone else said this already)

I see it this way: By constantly dividing the remaining distance in 2, you also divide the remaining TIME in two. At the first step (halfway) you make, you set a maximum timespan:

(time for step 1) times 2 minus (infinitely small amount of time).

The reason why you will never end up where you want, is that the time it'll take to reach your destination (when using a constant speed, which you do as you divide time AND distance each step) will be equal to (time for first step) times 2.

problem solved :P :bugeye: :confused: :D

Originally posted by Xenu
Just because paradoxes don't make sense doesn't mean they can't exist. Yes it does.

All of you are assuming that you can divide distance into smaller fragments indefinately.

What if everything is quantumized, including distance and time? Wouldn't this defeat Zeno's paradox since this math would not "fit" reality?

Tom

Originally posted by Prosoothus
All of you are assuming that you can divide distance into smaller fragments indefinately.

What if everything is quantumized, including distance and time? Wouldn't this defeat Zeno's paradox since this math would not "fit" reality?

Tom

well then we just might have to jump the final quark distance. Anyway, whether it's possible or not, my explanation worked either way :confused:

Originally posted by Prosoothus
All of you are assuming that you can divide distance into smaller fragments indefinately.

What if everything is quantumized, including distance and time? Wouldn't this defeat Zeno's paradox since this math would not "fit" reality?
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