xeno's paradox

[sage]

you have heard this before perhaps. its about a runner trying to run d metres. he covers d/2 in t1 second, then half of the distance that is left in t2 seconds, then half of the rest in t3 seconds and so on.as there is always a finite distance left, according to the paradox he can never cover d metres. so how does he do it?

[Hurkyl]

This infinite sequence of actions can be accomplished in finite time, so he does them all and then keeps going.

[AndersHermansson]

We had this over at sciforums recently. The short answer is that a sum of infinite series can be finite, which is where it might seem confusing. So that if you add an infinite amount of lengths the total length can still be finite. So the original question simply assumes it is not so.

[quartodeciman]

"there is always a finite distance left"

really means

"there is for any time before d/v (, with v being the speed of the runner) a finite distance left".

[sage]

yes this infinite sequence converges. but the point is if we go on adding the successive elements of the sequence one by one (as must be done here) we never reach the end of the sequence precisely because it is infinite. as we cannot reach the end of the sequence we cannot cover this finite distance in the calculated finite time. consider the finite time interval between n-th second and n+1-th second. first half a second passes by, then another 1/4-th, then another 1/8-th and so on. another infinite sequence converging at the limit, but that limit can never be attained. that is the problem.

[Hurkyl]

But why should one think that sequence of events cover the entire range of motion? Try this transfinite sequence:

Cover half the distance.
Cover half of what's left.
Cover half of what's left.
...
(countably finite repetitions)
...
Arrive at the destination.


Each step in the sequence picks up right there you left off if you perform all previous steps, includes the "Zeno sequence", and continues on afterwards to arrive at the destination.

[drnihili]

You have to be quite clear on what the question is. If you take Zeno to merely be asking how an infinite sequence can occupy a finite space, then calculus indeed answers the question. However, if you taking him to be asking the question of how one can complete and infinite sequence one member at a time, then calculus not only doesn't answer the puzzle but is entirely irrelevant to it. I think the latter question is the better way to understand the point of the paradox.

There are a host of related paradoxes which highlight the central issues. SOmetimes it helps to look at them instead of just the runner paradox.

[drnihili]

Originally posted by Hurkyl
But why should one think that sequence of events cover the entire range of motion? Try this transfinite sequence:

Cover half the distance.
Cover half of what's left.
Cover half of what's left.
...
(countably finite repetitions)
...
Arrive at the destination.


Each step in the sequence picks up right there you left off if you perform all previous steps, includes the "Zeno sequence", and continues on afterwards to arrive at the destination.

Ah, but this sequence can't be right. It presumes that after you've completed all the half distances you still have to do something further to arrive. If your sequence were correct, it would be possible to travel all the distances and yet still fail to arrive. But arriving cannot amount to traversing a distance or you give up the continuity of the reals. So on your account two runners could travers precisely the same distance and yet one of them would run d meters and the other wouldn't.

[Hurkyl]

It presumes that after you've completed all the half distances you still have to do something further to arrive.

Covering all of the half distances means covering the interval [0, d). If I run 1 meter per second, I cover all the half distances over the time interval [0, d).

You actually have to get to time d to have arrived at distance d. Zeno's paradox is a paradox because it presumes that you can't continue beyond the infinite sequence of covering half distances.

By continuity, any possible continuation of motion would have to include being at distance d at time d.

[drnihili]

Originally posted by Hurkyl
Covering all of the half distances means covering the interval [0, d). If I run 1 meter per second, I cover all the half distances over the time interval [0, d).

You actually have to get to time d to have arrived at distance d. Zeno's paradox is a paradox because it presumes that you can't continue beyond the infinite sequence of covering half distances.

By continuity, any possible continuation of motion would have to include being at distance d at time d.

The problem is that the open and closed intervals have the same distance. Closing the interval does not add any distance. Continuity comes in because the LUB of the two intervals is the same. If the runner really has completed all of the open intervals, he must have arrived at d.

Suppose otherwise, i.e that the runner has completed [0, d) but has not yet arrived at d. Call the runner's position r. r must be between the open interval and d. But this contradicts the fact that d is the least upper bound of the interval. So if r<d, then r must be in the open interval. But if r is in the open interval, then the runner has not yet completed the interval. This is because for every point in the interval there are infinitely many other points beyond it that are still in the interval. So r cannot be in the interval. thus the earliest point which can be r is d.

And the paradox isn't that you can't continue beyond the open interval, it's that you can't complete the interval at all.

[Hurkyl]

I'm aware the lengths of [0, d) and [0, d] are the same.


Anyways, a paradox is typically a contradiction that arises from an unfounded assumption. They usually get cleared up once you try to do everything rigorously.

So tell me, as precisely as possible, what you think the problem is.

[drnihili]

Well, I don't think I agree with your view of what a paradox is, but we'll leave the general theory of paradox for another thread.

The paradox in this case is that the runner, Achilles, must accomplish an infinite sequence of tasks. We know that he can complete them, we can even calculate precisely by when he will have completed them. The problem is in explaining how he completes them.

Achilles starts out with an infinite number of tasks to do. By the description of the problem, he must complete them one at a time. After he has accomplished his first task, there are an infinite number of tasks left. After he completes his second taks, there are an infinite number of tasks left. In fact after each task that he completes, there's always an infinite number left. As he moves down his list of tasks, he never gets any closer to the end of it. He always has just as many left to do as he started out with. As long as he is still working on the list, he has infinitely many left. The first point at which he has fewer than infinitely many tasks left is when he is all done, and at that point he has zero. He never decreases his list, he just suddenly finds that it is already done. So how is it that he manages to get to the end?

Geometry can predict the point at which Achilles will be done. Calculus can explain how it is that all the decreasing segments have a finite sum. But neither of them explains how it is that Achilles counts through the list, one task at a time - how he manages to complete an endless sequence.

[Hurkyl]

You still haven't answered the big question; why should an infinite sequence of tasks be impossible?


In particular (if I'm predicting your response correctly), why should every task in a sequence of tasks have a previous and a next task? (except, of course, for the first and last task, should they exist)

[drnihili]

Originally posted by Hurkyl
You still haven't answered the big question; why should an infinite sequence of tasks be impossible?


In particular (if I'm predicting your response correctly), why should every task in a sequence of tasks have a previous and a next task? (except, of course, for the first and last task, should they exist)

Because there's a function that given any task in the sequence returns the next task, and another function that returns the previous. If you take an ordering that lack that property it gets even more difficult. But Zeno's ordering does have the property.

[Hurkyl]

But why should an infinite series of tasks be impossible?


The resposne I was anticipating was something equivalent to saying that in my sequence of tasks, there is no task previous to "arrive at d". (it is eqiuvalent to say that there is no last task in Zeno's sequence)

[drnihili]

That response doesn't quite get it right. I've tried to explain it a couple times, but I'll have another go at it.

If Achilles accomplishes an infinite series of tasks, there must be some action of his which counts as completing all the tasks. But none of the tasks can be that action as each of the tasks leaves an infinite number remaining. So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action.

If there were such an action, then it would be theoretically possible for Achilles to accomplish each of the tasks and yet still fail to complete all of them. This is absurd. Hence there can be no such action.

[Hurkyl]

If Achilles accomplishes an infinite series of tasks, there must be some action of his which counts as completing all the tasks.

For the problem at hand, there must be some task which counts as the completion of all (previous) tasks, though this isn't always the case. But the question is why must that task be one of the infinite series of tasks?

Continuity (and completeness) guarantees that there must be a unique limiting event, but it does not guarantee that the unique limiting event must be one of the members of the infintie sequence.

In particular, the limiting task is the "arrive at destination" step I listed.

[drnihili]

Originally posted by Hurkyl
For the problem at hand, there must be some task which counts as the completion of all (previous) tasks, though this isn't always the case. But the question is why must that task be one of the infinite series of tasks?

Continuity (and completeness) guarantees that there must be a unique limiting event, but it does not guarantee that the unique limiting event must be one of the members of the infintie sequence.

In particular, the limiting task is the "arrive at destination" step I listed.

Obviously it can't be one of the listed tasks. But your proposal is no solution. What exactly does one do to arrive at the destination and when does one do it? Do you really mean to imply that one might complete each of the tasks and still not arrive at the destination?

[Hurkyl]

But your proposal is no solution. What exactly does one do to arrive at the destination and when does one do it?

One traverses the position interval [0,d) over the time interval [0, d). That is sufficient to be at position d at time d. (I'm assuming the traversal is in the manner being discussed)


Do you really mean to imply that one might complete each of the tasks and still not arrive at the destination?

I mean to imply that one does not reach the destination during the time interval in which one is performing Zeno's tasks. In this case, the time interval [0, d). One arrives at the destination at time d, after all of Zeno's tasks have been completed.

[drnihili]

Originally posted by Hurkyl
One traverses the position interval [0,d) over the time interval [0, d). That is sufficient to be at position d at time d. (I'm assuming the traversal is in the manner being discussed)

Here you've essentially said that completing all the tasks is sufficient for arrival. But you haven't said how that is accomplished. I agree that it's sufficient, that's not the issue. The issue is saying how it is done.




I mean to imply that one does not reach the destination during the time interval in which one is performing Zeno's tasks. In this case, the time interval [0, d). One arrives at the destination at time d, after all of Zeno's tasks have been completed.

This can't be right. One doesn't first complete the tasks and then arrive. If that were the case then there would have to be a moment in between finishing the tasks and arriving. (given infinite divisibility.) But that would contradict what you said above about completing the tasks being sufficient for arriving. Arriving can't be separate from completing all the tasks. It can't occur after completing them, nor can it occur before completing them. It has to occur simultaneously with completing them. But this still leaves the problem of saying what it means to complete and endless sequence.

[Hurkyl]

The issue is saying how it is done.

You do it by crossing the entire path between you and the destination. What's wrong with that?


If that were the case then there would have to be a moment in between finishing the tasks and arriving. (given infinite divisibility.)

Why must there be a moment between finishing the tasks and arriving? There is zero time between finishing the tasks and arriving at the destination; no matter how you infinitely divide zero, all of the pieces must be zero.

[drnihili]

Originally posted by Hurkyl
You do it by crossing the entire path between you and the destination. What's wrong with that?

That's just begging the question.


Why must there be a moment between finishing the tasks and arriving? There is zero time between finishing the tasks and arriving at the destination; no matter how you infinitely divide zero, all of the pieces must be zero.

If there is zero time between the two events, then they are simultaneous. You stated one was after the other.

[Hurkyl]

If there is zero time between the two events, then they are simultaneous. You stated one was after the other.

But we're not talking about the time between two individual events, are we?

[drnihili]

Originally posted by Hurkyl
But we're not talking about the time between two individual events, are we?

If we are not, then there must be just one event. In that case, please say what that event is, and what specific action of Achilles' brings it to pass.

Also, if it is just one event, then I'm puzzled why you said it occurred after itself.

[Hurkyl]

We are talking about the time between a single event and an infinite sequence of events. There is a big difference there, and this time may be zero, even if the single event occurs strictly later than every event in the infinite sequence.

[drnihili]

Originally posted by Hurkyl
We are talking about the time between a single event and an infinite sequence of events. There is a big difference there, and this time may be zero, even if the single event occurs strictly later than every event in the infinite sequence.

Again, without begging the question this time, what is this single event, and what action does Achilles take to bring it about?

[Hurkyl]

The single event is arriving at the destination. He does this by covering all the ground between his starting point and this destination.

[drnihili]

Originally posted by Hurkyl
The single event is arriving at the destination. He does this by covering all the ground between his starting point and this destination.

You'll recall I asked for a non-question begging answer.

[Hurkyl]

I don't see what's wrong with this task.

But instead of debating the merits of this description, I'll ask how this is any different from any of the other tasks in Zeno's sequence? How is "Go from here to there" any different from "Go from here to half way between here and there"?

[NeutronStar]

Originally posted by sage
you have heard this before perhaps. its about a runner trying to run d metres. he covers d/2 in t1 second, then half of the distance that is left in t2 seconds, then half of the rest in t3 seconds and so on.as there is always a finite distance left, according to the paradox he can never cover d metres. so how does he do it?

Sage, I haven't read every detail of this thread because it appears to have deteriorated into a mathematics argument that can never be fully justified in a comprehensible intuitive way. The inevitable conclusion of any such mathematical arguments can only be had by accepting the abstract axioms of mathematics and forfeiting an intuitive comprehension of any physical explanation of quantity.

For a physical explanation I would like to offer the following:

In physics we have discovered that the nature of the universe is indeed quantized. For this reason it is not physically possible to continue to divide up time and distance in half indefinitely. There comes a point when we reach a length of distance that has no half distance. In other words, it makes no sense to talk about space between these points. Therefore Zeno's paradox is not a paradox at all.

Zeno's paradox would only be a paradox if we lived in a purely continuous universe. But we don’t. We live in a quantized universe. Therefore there is no paradox.

It may very well be impossible to move if the universe is indeed continuous.

[ahrkron]

The resolution of the paradox lies within well understood math (convergence of series). The universe may well have continuous quantities, and they would still be able to change. What we need to understand is how our description of such changes has to be used.

[Hurkyl]

The gross mischaracterization of mathematics aside...

Zeno's paradox is only a paradox when one makes unfettered attempts at extrapolating the properties of finite sets to those of infinite sets without any attempt at proof. (though at this point, it's merely boiled down to drnihili complaining that I don't have acceptable semantics for describing motion)


As for the physical theory, only bound systems have been shown to be quantized. Free particles are not quantized. And no aspect of the geometry of space-time has yet been shown to be quantized. Even in theoretical physics, no theory implies that distance is quantized (though LQG implies area and volume are quantized).

[NeutronStar]

Hurkyl wrote:
As for the physical theory, only bound systems have been shown to be quantized. Free particles are not quantized. And no aspect of the geometry of space-time has yet been shown to be quantized. Even in theoretical physics, no theory implies that distance is quantized (though LQG implies area and volume are quantized).


It would seem to me that Zeno being a macro object would be made up of bound particles. He could hardly be thought of as an unbound system. The same would go for his start and finish lines that would necessarily be locations on a macro scale. (After all, if he isn't moving relative to some other macro object can he even be said to be moving at all?)

As far as distance being quantized goes, it follows from Planck's constant of energy. After all, if energy is quantized then so must be time. And of course if time is quantized then so must be distance. I'm sure that I've seen references to the Planck length of quantized distance. In fact I believe that Brian Green refers to the Planck distance in his book on string theory called "The Elegant Universe". I think it was on the order of ten to the -33 centimeters or something like that. A distance that below which it is absurd to talk about space as a dimension.

In fact, I quite sure that he also referred to the concept of the unit of Planck time in that book too. A time duration below which time no longer holds meaning.

It would seem to me that these concepts would need to apply to free particles as well as bound ones. But maybe not, since free particles are really nothing more than free waves of probability while bound ones are restricted to standing waves of probability. [:D]

Alright, so I have no idea what I'm talking about. Big deal. It still makes more sense to me than the abstract mathematical converging series. (See my next post in response to the capabilities of calculus)

[Hurkyl]

It still makes more sense to me than the abstract mathematical converging series.

One does not need to consider infinite series to describe motion through a connected space.

[NeutronStar]

Ahrkron wrote

The resolution of the paradox lies within well understood math (convergence of series). The universe may well have continuous quantities, and they would still be able to change. What we need to understand is how our description of such changes has to be used.



"well understood math"?

I remember studying the series convergences in calculus II. In fact, I just had a refresher course in calculus II this last spring. We did many series convergence problems.

I disagreed with those conclusions. I mean from a static or absolute point of view.

In other words, all of calculus is based on the idea of the limit. This is the foundation of calculus. Everything in calculus rests upon the definition of the limit.

Well, if you learn that definition well, you can clearly see that mathematicians use it incorrectly all the time. The definition for the limit of f(x) at c clearly states:


For every epsilon greater than zero there exists a delta greater than zero such that,
If the distance between x and c exists and is less than delta.
Then the distance between f(x) and L exists and is less than epsilon,


Yet mathematicians never fail to claim that calculus can prove something about when epsilon equals zero

That's totally invalid. The definition of the limit is dependent on the fact that epsilon is greater than zero. As soon as epsilon actually becomes zero the formalism of calculus is no longer applicable. Yet mathematicians continually claim that calculus proves that something like a converging series actually converges when epsilon equals zero.

It does not prove this, nor can it be used to prove this.

All it can possibly prove is that as the series converges it gets close to some number. Period amen. To claim that it actually converges is to totally ignore the definition of the limit upon which calculus rests.

All of calculus rest on the definition of the limit. And anyone who truly understands that definition should clearly undestand that calculus cannot make any statements whatsoever about what might happen should epsilon actually become zero. Such statements are outside of the scope of calculus.

Edited to correct the order of the if-then statement in the definition of the limit. Sorry about that.

[ahrkron]

Originally posted by NeutronStar
We did many series convergence problems.

I disagreed with those conclusions. I mean from a static or absolute point of view.


Mathematicians use the same reasoning capabilities as you and I. Theorems are proven using such logic. "Points of view" (static, absolute or otherwise) need not enter in the discussion.

Well, if you learn that definition well, you can clearly see that mathematicians use it incorrectly all the time.

Either that, or you misunderstood the use they make of it.

Do you seriously sustain that all mathematicians since the formalization of calculus have used a definition incorrectly? I find that extremely hard to believe.

But it is not a matter of faith. I have gone over the resoning myself, as do quite a few high schools students (and college students and grad students and profesional mathematicians) every year, and find the to be quite clean.


The definition for the limit of f(x) at c clearly states: ...

Yet mathematicians never fail to claim that calculus can prove something about when epsilon equals zero


There's no such "claim". The definition of limit allows you to consider situations in which:
1. the case epsilon=0 is not relevant for the discussion at hand,
2. the value at epsilon=0 is not defined
3. the value at epsilon=0 is different from the limit.
4. such value is defined and is equal to the limit.

When studying the behavior of a series, it is possible to use these tools. No dogmas or errors are involved; rather, further concepts are founded upon these studies.


calculus cannot make any statements whatsoever about what might happen should epsilon actually become zero. Such statements are outside of the scope of calculus. [/B]

No, they're not. Look up the definition of continuity. It is one of the basic concepts in calculus and it is concerned with the case epsilon=0. It is not an invalid extrapolation, but a case that has to be considered due to its enormously frequent occurrence on the formalism.

[NeutronStar]

Oops! I do owe you an apology. I have made a terrible error in my last post.

I was thinking in terms of the calculus limit, but I was just typing the word calculus alone.

The reason that I did this is because our discussion is focusing on the convergence of a series, and so for all intents and purposes we are talking solely about a one-sided limit. We also must take note that we have no functional values for the value c in question (which would be the sum of the series). We are using a one-sided limit to come up with this value.

This is the context in which our discussion resides.

So if I may, I would like to rephrase my last comment by adding the following words in red.

The calculus limit cannot make any statements whatsoever about what might happen should epsilon actually become zero. Such statements are outside of the scope of the calculus limit.

Now, in response ahrkron gave the following comment.

Originally posted by ahrkron

No, they're not. Look up the definition of continuity. It is one of the basic concepts in calculus and it is concerned with the case epsilon=0. It is not an invalid extrapolation, but a case that has to be considered due to its enormously frequent occurrence on the formalism.


I am well aware of the definition (and limitations of) continuity.

If you carefully read all of the requirements for the definition of continuity you will first notice that we must have two limits. In the case of Zeno's paradox we have only one limit. (Also, in the case of a convergent series we have only one limit).

This in an of itself forbids any mention of continuity in this problem. The definition of continuity cannot be applied to a single-sided. This is a direct violation of its very own definition.

Secondly, even if we had two limits (one from the right, and one from the left), and they both existed and were the same value, we still can't conclude that a function is continuous unless we know the functional value at the point in question.

Why? Because in the very definition of continuity, it clearly states that continuity can only be said to exist if the functional value agrees with the values of the two limits. Therefore, it is not possible to conclude continuity without knowing the functional value at the point in question. This is clearly a restriction of the definition of continuity.

So even if we did have two limits here, we still could not conclude continuity even if the two limits were the same. Because we don’t' have a functional value for the point in question. The definition of continuity requires that we know this value!

So again, I apologize for my last post. The things that I said about calculus in general I meant to say about the calculus limit.

It is true that we can make statements about when epsilon equals zero (if and only if) we know the functional value at the point in question.

However I must also point out here that this knowledge of the value of the point when epsilon equals zero does not come from calculus, it comes from the functional value! In other words, if we have a function and we know the value of the point in question we don’t' need calculus to find it! Not to mention the fact that calculus is completely useless to prove the existence of such a point if we don't know the functional value in the first place. The best calculus can do is say that the function has a limiting value as it approaches this point. Calculus in and of itself cannot make any direct statements about the existence of such values at the point where epsilon equals zero (without referring to the functional value at that point).

What we use can use calculus for, is to say that the entire function is continuous at a point. (Assuming of course that we know the functional value at that point.)

So at any rate, the concept of continuity cannot even be applied to Zeno's paradox (or the convergence of a series). Because in both of these cases we have only one limit, and we do not know the function value at the point in question. We would need to have both of these conditions present before we could even begin to apply the concept of continuity.

Continuity has no application in this problem. Pure and simple.

[Hurkyl]

I'll go off on this tangent to reply to your statements about calculus (though we are deviating quite a bit from Zeno's paradox):

If you read carefully the definition of a continuity, it says that f is continuous at a iff

f(x) -> f(a) as x -> a

(this means the limit of f(x) as x approaches a is f(a); this notation is much cleaner for text-based communication)

The definition of continuity says nothing about one-sided limits.


If you recall, the business about proving f(a) is equal to both the left-sided and the right-sided limits was based on a theorem about limits when the domain is the real numbers. In particular:

f(x) -> L as x -> a
iff
f(x) -> L as x -> a+ and f(x) -> L as x -> a-

This theorem rests heavily on the fact that for any real number a, we can seperate the real numbers into a left hand side and a right hand side. More generally, this theorem works for any interior point of an ordered space. However, when our domain does not permit such seperation, we can't even define one-sided limits, let alone prove this theorem.



Now, recall the definition of an infinite series;

Letting S(n) = sum over x = 1 .. n of f(x):

S(n) -> sum over x = 1 .. &infin; of f(x) as n -> &infin;

iow the infinite sum is equal to the limit of the sequence partial sums.

If we extend the domain of S(n) to the extended natural numbers by defining

S(&infin;) = sum over x = 1 .. &infin; of f(x)

then S(n) is a continuous function of n at &infin;! The definition of the infinite series is precisely the definition of continuity:

S(n) -> S(&infin;) as n -> &infin;


Also, it is worth recalling that the &epsilon;-&delta; definition of limits is only used when the domain and range of the limit are finite real numbers. You may recall doing &epsilon;-M or N-&delta; or even N-M limit proofs in your calc classes... alternatively one may use a topological definition of limits that is equally applicable to both the finite and infinite cases.


Secondly, even if we had two limits (one from the right, and one from the left), and they both existed and were the same value, we still can't conclude that a function is continuous unless we know the functional value at the point in question.

Quite often, we're told from the outset that the function in qusetion is continuous. In such cases, finding the limit of the f(x) as x approaches a yields the value of f(a).


and we do not know the function value at the point in question.

But we do; we know the limit of the sequence of positions is the destination point.


Also, one should note that the term "continuous" when applied to topological spaces is not the same term as "continuous" when applied to functions.

[NeutronStar]

(P.S. for the sake of correctness, I should point out you have the if statement in your definition of continuity backwards)


Yes, I did type that in backwards. Thanks for pointing that out. [:D]

Some of my math books print definitions as "whenever" statements, and some of them print them as "if-then" statements.

I prefer to use the "if-then" style. Unfortunately I was looking at a book that uses the "whenever" style when I typed the information into my post. During the translation to the "if-then" style I forgot to swap things around.

Sorry for the confusion. I've corrected that post for future readers.

[drnihili]

Originally posted by Hurkyl
I don't see what's wrong with this task.

But instead of debating the merits of this description, I'll ask how this is any different from any of the other tasks in Zeno's sequence? How is "Go from here to there" any different from "Go from here to half way between here and there"?

The problem with the task is that it is comprised simply of all the other tasks. So you have a sequence of discrete tasks, and a further task which is doing all of them.

Since the original problem is to say how it is that Achilles can complete the entire series, saying that he does is by completing all the tasks is no help - it just begs the question.

The "all tasks" task is different in that it requires no action on Achilles' part that is not already required by some other task on the list. It can be dropped from the list of required task with absolutely no effect on Achilles. In this sense it is an empty task.

Since the task is empty, and since it's proposal merely begs the question, it cannot constitute an answer to the paradox.

Achilles must complete an infinite series of tasks. We know that completing them only requires a finite period of time. But we don't know how he can complete them at all.

[Hurkyl]

But we also don't have a logical reason why he can't complete them.

[drnihili]

Originally posted by Hurkyl
But we also don't have a logical reason why he can't complete them.

Yes, we do. I've given it above, but here it is again. He can't complete them because there is nothing he could do which would count as completing them.

The list of tasks is unbounded (in the sense of not containing it's own bound.) So there is no task in the list the doing of which would count as completing the list. But neither is their any task not on the list the doing of which would count as completing the list. If there were, then Achilles could do each of the tasks on his list and still not arrive at d - which is absurd. Thus there is no task either on or off the list the doing of which would count as completing the list. Hence there is nothing Achilles can do which would count as completing the list, since if there were it would be describable as a task. Thus Achilles cannot complete the list.

[Hurkyl]

If there were, then Achilles could do each of the tasks on his list and still not arrive at d

During the time interval when Achilles is completing the tasks, [0, d), he does not arrive at position d.

But any single point in time at which it can be said "Achilles finished the tasks", he has (had) arrived at d, precisely at time d.


Hence there is nothing Achilles can do which would count as completing the list, since if there were it would be describable as a task.

The task is to complete the list. (actually, I've heard this type of task called a supertask since it's a task to complete tasks)



Anyways, the individual tasks in Zeno's sequence are of the form "Go from point a to point b". If these are acceptable tasks, then so must "go from 0 to d". And in the process of performing this task, every task in Zeno's sequence is completed.

[drnihili]

Originally posted by Hurkyl
During the time interval when Achilles is completing the tasks, [0, d), he does not arrive at position d.

But any single point in time at which it can be said "Achilles finished the tasks", he has (had) arrived at d, precisely at time d.


This is true, but not especially relevant to the question at hand


The task is to complete the list. (actually, I've heard this type of task called a supertask since it's a task to complete tasks)

Anyways, the individual tasks in Zeno's sequence are of the form "Go from point a to point b". If these are acceptable tasks, then so must "go from 0 to d". And in the process of performing this task, every task in Zeno's sequence is completed.

Every task requires doing something. The problem with your supertask isn't that it requires going from point a to point b. The problem is that it either doesn't require anything at all, or begs the question.

I can, for example, specify a task that requires that Achilles complete his first two tasks. But the task is empty since it doesn't require Achilles to do anything that he's not already required to do by his other tasks. We can also add the task that Achilles should exist while he runs. There are an infinite number of empty tasks that can be added to the list. But these aren't real tasks since they don't require anything more of achilles than is already required by the tasks on the list. These sorts of "tasks" dont really specify something that must be done, they aren't additions to the list.

Furthermore, even if I were to grant that your supertask was nonempty, it doesn't help answer the original question. I ask "How can Achilles complete all the tasks on his list?" You answer "by ccompleting the super-task." I ask "And what must Achilles do to complete the super task?" You answer "He must complete all the tasks on his list." We've just come full circle. I want to know how he completes all the tasks, you answer that he does it by completing all of them. Perhaps we can grant that your answer is true, but it is entirely vacuous.

[NeutronStar]

Hurkyl wrote:

But any single point in time at which it can be said "Achilles finished the tasks", he has (had) arrived at d, precisely at time d.




Tsk, tsk, Hurkyl. All you are saying here is that it is obvious that we can move so it therefore it must also be obvious that we have completed all of the tasks.

Zeno was well aware that we can move. His whole point is that there is no logical explanation for it. So the argument that we can obviously move, and therefore we must be able to move, is actually quite silly don't you think? [s(]

Zeno wants an explanation of how it can be done, and so do we.

As drnihili has suggested, the task of actually finishing is unimportant, it's the question of how the infinite many tasks in between are completed in a finite way. What constitutes the completion of these tasks, save for the trivial response that we can obviously move.

As I've already posted my answer is to conclude there simply aren't an infinite number of tasks. The concept of distance cannot be broken up continuously and indefinitely. The whole paradox suggests to me that the universe (including motion) must necessarily be quantized.

Mathematical tricks of limits and or convergent series just don't satisfy my quest for an explanation. Those definition are mathematical abstractions and kind of miss the point of physical reality. Just because we say that a mathematical series can converge doesn't mean that it actually can. We are just satisfied that as the additions become smaller and small they don't add anything significant because they are continually decreasing in the amount that they add. So we ignore them. They are never really *completed*. And calculus doesn't claim that they ever do get completed. It only shows that in the limit, the additions tend toward zero so we can ignore them. But we can never have claim to actually have stopped the process! If we did we would have an finite number of additions and not an infinite sum.

This problem actually reminds me a lot about the number of points in a finite line.

Anyone who accepts that a finite line can contain an infinite number of points should have absolutely no problem with Zeno's paradox. Or, maybe better said, if Zeno's paradox bothers a person, then the idea of an infinite number of points in a line should also bother them.

After all, to say that you have an infinite number of points in a finite line is to also accept the idea that a finite line contains an infinite number of finite distances. Anyone who can accept that should have no problem imagining an infinite number of distances being completed in a finite amount of time. Just imagine the line as being a timeline. [:D]

Personally I don't accept the idea that a finite line can contain an infinite number of points, and so Zeno's paradox holds interest for me. But I solve it in the same way that I solve the finite line problem. Real physical distance cannot be divided up into infinitely many parts, and therefore it does not require an infinite number of tasks to move. No paradox here. [;)]

Now I go back to work fellas! These forum boards can be addictive!

[Hurkyl]

This is true, but not especially relevant to the question at hand

I bring it up because I get the impression that you are confusing the properties of individual events with the properties of infinite sets of events.


Every task requires doing something...

Then why are you saying there must be a task that counts as completing the list? Such a task is as "empty" as my supertask.

[NeutronStar]

drnihili wrote:

I ask "How can Achilles complete all the tasks on his list?"



Be careful drnihili. If you are willing to accept that Achilles has completed the very first task then you are done. He moved.

All that is needed now is to move the finish line up to where he moved to by completing his first task and the race is over. Motion is possible and there is no paradox.

But Achilles can not even complete his first task, because in order to do this he must first complete the task of moving halfway toward that first destination.

It works in both directions. Not only can Achilles never finish the race, he can't even start it!

He cannot move at all.

Why? Because the assumption is that space can be divided up into an infinite many parts. So in order to move to any point we must first move to the point halfway between that point and where we are, etc, etc, etc. We can't even make our first move, or complete our first task, because to do so would assume that we had already completed an infinite number of tasks just to get to that first halfway point!

At this point I can't help but bring in Sherlock Holmes.

We have a mystery. The mystery of how motion can be achieved in an infinitely divisible space. After investigating all of the evidence, and removing all of the impossibilities only one explanation remains.

And in the spirit of Sherlock Holmes whatever remains must be the truth.

"Space must be finitely divisible my dear Watson. For this is the only suspect that we cannot eliminate."

[Hurkyl]

Tsk, tsk, Hurkyl. All you are saying here is that it is obvious that we can move so it therefore it must also be obvious that we have completed all of the tasks.

...

Zeno wants an explanation of how it can be done, and so do we.

You're not asking for an explanation of how motion can be done, you're asking how to reconsile apparent paradoxes in the ability to describe motion in terms of an infinite sequence of tasks. My response is that the paradoxes only arise from incorrectly extrapolating properties of the finite to the infinite.


As to why we can move at all is somewhat more fundamental than any questions about infinite sequences.


Mathematical tricks of limits and or convergent series just don't satisfy my quest for an explanation.

As I mentioned before, mathematical "tricks" of limits and convergence (iow ordinary tools for working with infinite sequences) only arise when you insist on describing things with an infinite sequence.

If you don't like dealing with infinite sequences... then don't go seeking them and then complain you don't like dealing with them! [:)] Incidentally, differential and integral calculus can be introduced without any use of limits and infinite sequences.


Edit: I was wondering when someone was going to get around to Zeno's other paradoxes. [:)]

[drnihili]

Originally posted by Hurkyl
I bring it up because I get the impression that you are confusing the properties of individual events with the properties of infinite sets of events.

Fine, lets try this. Specifically what properties do you think I am confusing?


Then why are you saying there must be a task that counts as completing the list? Such a task is as "empty" as my supertask. [/B]

That is the point. There cannot be any task that completes the list. But absent such a task, there is no reason to think that the list can be completed and indeed there is every reason to think that it cannot.

[drnihili]

Originally posted by NeutronStar
Be careful drnihili. If you are willing to accept that Achilles has completed the very first task then you are done. He moved.



Yes I know. However this version grants the completion of each task and shows that even with that assumption Achilles cannot get to d.

I actually prefer the form in which he cannot start. My favorite rendition is the following.

Imagine that Achilles wants to proceed from point A to point B, but that an infinite number of Gods have vowed to prevent him. God 1 vows that if Achilles makes it 1/2 way, he will kill him with a Lightning bolt. God 2 vows that if Achilles makes it even as far as 1/4 of the way he will kill him with a lightning bolt. God 3 vows that if he makes it 1/8 of the way .... In general, God N vows that if Achilles makes it 1/2^Nth of the way, he will be killed with a lightning bolt before proceeding further.

It turns out, that Achilles can't start across the road. For suppose that he travels some distance. In that case he must have already traveled 1/N^2 of the distance for some N. But that means that God N+1 would have already killed him. In turn that means that he wouldn't have made it 1/N^2 of the distance. Since the assumption that Achilles travels some distance across the road yields a contradiction, it can't be right. So Achilles doesn't travel any portion of the distance.

Now here's the clincher. Since Achilles doesn't travel any portion of the distance, none of the God's have to fulfill their vow. Their vows alone make it logically impossible for him to cross the road. But it seems absurd that Achilles can be prevented from crossing the road without any of the gods having to carry out their vow. WHat, after all, is stopping him?

[Hurkyl]

Fine, lets try this. Specifically what properties do you think I am confusing?

If a point X is not in a finite set S, the distance between S and X must be greater than zero (in nonpathological topological spaces). However, such a guarantee is not possible when S is infinite.

In particular, it is certainly possible for an event E to occur after every element of an infintie set S of events, yet there be no time elapsed between S and E.


Ordered finite sets have a well-defined first and last element. That is not always true of infinite sets.


Achilles covers the distance [0, d] in time [0, d]. If we consider everything but the endpoint, Achilles covers the distance [0, d) in time [0, d). The finishing time of a task is simply the first instant of time at which it can be said the task was completed. This time does not necessarily occur during the time interval in which the task was performed. Finishing times are what's called a "least upper bound" or a "supremum". The finishing time of each individual task occurs in [0, d). The finishing time of the entire sequence of tasks is d.

But the supremum of a finite set is an element of that set.


(all three of these comments are three different wordings of the same fact about infinite sets; unlike finite sets, infinite sets are not always topologically closed)


I actually prefer the form in which he cannot start. My favorite rendition is the following.

I was expecting the arrow paradox, but this one is nifty too.

Again the question is the same, what logical reason do we have to think this is or is not a valid scenario?

At least with Zeno's paradox, the fact that we can move from start to finish and that science allows infinite divisibility allowed us to prove that the scenario was a valid one. (You just failed to show that the consequences were impossible)

With this paradox, you've shown the consequences are impossible (actually, this part is still debatable, but irrelevant to my argument), but you've failed to prove the scenario is a valid one.

[drnihili]

Originally posted by Hurkyl

With this paradox, you've shown the consequences are impossible (actually, this part is still debatable, but irrelevant to my argument), but you've failed to prove the scenario is a valid one.

Valid in what sense?

[Hurkyl]

In the sense of it being anything more than an imaginative story.


I've gotten the impression throughout the thread that you aren't just presenting food for thought; you actually think that the paradoxes represent an actual inconsistency in the modern view of things. (If I'm wrong on this point, I apologize for dragging this thread out for so long)

In order for it to be an actual inconsistency, you have to both justify each step in the construction, and to prove that the consequences are contradictory with science.

With Zeno's paradox, we can rigorously justify the construction of the sequence, using the fact motion is possible and infinite divisibility. (I've taken the liberty of implicitly giving the tasks a precise meaning; to go from point A to point B means that at some time Achilles was at point A and at some later time Achilles was at point B, and at the time Achilles arrived at B, and any future time, it is correct to say the task has been completed)

We can't do that with the vows; at least I don't know of a way to rigorously prove it. Induction only tells that for any n, it is possible for there to be n gods with vows... it does not tell us the infinite case is possible.

[drnihili]

Originally posted by Hurkyl
In the sense of it being anything more than an imaginative story.


I've gotten the impression throughout the thread that you aren't just presenting food for thought; you actually think that the paradoxes represent an actual inconsistency in the modern view of things. (If I'm wrong on this point, I apologize for dragging this thread out for so long)

I think there are still unresolved problems with our formalizations of infinity. Those problems are not always inconsistencies in the formalizations themselves but are problems in the relationship between the formal models and the reality which they are taken to be models of.


In order for it to be an actual inconsistency, you have to both justify each step in the construction, and to prove that the consequences are contradictory with science.

With Zeno's paradox, we can rigorously justify the construction of the sequence, using the fact motion is possible and infinite divisibility. (I've taken the liberty of implicitly giving the tasks a precise meaning; to go from point A to point B means that at some time Achilles was at point A and at some later time Achilles was at point B, and at the time Achilles arrived at B, and any future time, it is correct to say the task has been completed)

We can't do that with the vows; at least I don't know of a way to rigorously prove it. Induction only tells that for any n, it is possible for there to be n gods with vows... it does not tell us the infinite case is possible.

It's actually a lot easier to validate the Gods example than the race example.

Let each vow be represented as a material conditional of the form "If Achilles reaches a point 1/2^n alive, then he will be instantly killed by God n." The conditionals are built by recursion on the counting numbers. Now let N be the conditional built at nth step of the recursion. If N is false, then each M>N must also be false. However, there is no contradiction in all of the Ns being true.

Either I don't understand your point about induction or you're confused. It is enough that we have a god for each n. We only need an omega sequence, not an omega+1 sequence.

[NeutronStar]

Hurkyl wrote:

If you don't like dealing with infinite sequences... then don't go seeking them and then complain you don't like dealing with them!



Hey! Wait a minute!

I came into this thread to offer a physical explanation of Zeno's paradox. This is a physics forum is it not?

My comments on the failure of pure abstract mathematical formalism are merely an observation. (not a complaint)

After having just completed four calculus refresher courses last fall and this spring I feel that I can safely say the following:

Calculus' answer to Zeno is simply this:

We are not concerned with whether or not the tasks can be completed. All we are concerned with is showing that the tasks will get closer and closer to the goal, and that they will continue to do so at an every increasing rate, and that this behavior will not change not matter how arbitrarily close we get.

Prove these things for convergence and you will get an A on a calculus exam!

That's it. That is all that calculus can say about the matter. Calculus cannot be used to claim that an infinite number of tasks have ever been completed. That is not the purpose of calculus, nor is it the formalism of it.

So I am not complaining when I say that calculus cannot answer Zeno's question. I am stating this as a fact with absolute confidence based on my understanding of the rules, and definitions of the calculus used in this problem. And believe me, I've had long discussions with my mathematics professors about these very concepts. I am confident that they would agree with what I have said above in bold blue letters.

So in the spirit of physics let me offer the following

There mere fact that the tasks are getting smaller as they continue on indefinitely is not good enough. There must be a 'physical' explanation as to how they can be completed.

I am satisfy that an infinite number of tasks can never be completed. No explanation possible.

So my answer is that one of our assumptions are wrong. And the assumption that I take to be wrong is that space can be divided up into infinitely many parts. Take away that assumption and the paradox goes away.

Space can only be divided up into a finitely small amount. This does not imply an absolute space by the way! It simply implies that whatever we mean by space, that concept is going to necessarily be quantized.

I think that QM, and even to a much greater extent, String Theory, both indicate that this is most likely the case anyhow. So unless we know differently why do we keep assuming that space can be divided up indefinitely?

Who told us that we could do that?

[ahrkron]

Hurkyl is right on the money when he says that

If you don't like dealing with infinite sequences... then don't go seeking them and then complain you don't like dealing with them!

Zeno's paradox hides a self-inconsistency in the treatment of infinities that is neatly cleaned up by calculus.

The basic structure of the paradox is:

1. Think of a trajectory (i.e., our name for something that necessarily has a non-zero extension in two quantities: space and time).
2. Divide the space extension infinitely (i.e., "assume we can infinitely divide the distance to be traversed").
3. Think of the times needed for each section defined in step 2.
4. The total time should be the sum of all times we found on step 3.
5. But summing those times is impossible, since there is an infinity of them.

There is a blatant contradiction between the assumption hidden in step 5 (an infinite sum cannot add up to a finite value), and the explicit division put by hand into step 2 (lets divide this distance into an equivalent infinite number of subintervals)!!

The resolution of this has nothing to do with the physics of motion. It is just a matter of self-consistency.

What calculus has to say about this is that, doing things carefully, continuous magnitudes and infinite sums (as those assumed to exist in the formulation of the paradox) can be solidly dealt with.

[drnihili]

Originally posted by ahrkron
Hurkyl is right on the money when he says that



Zeno's paradox hides a self-inconsistency in the treatment of infinities that is neatly cleaned up by calculus.

The basic structure of the paradox is:

1. Think of a trajectory (i.e., our name for something that necessarily has a non-zero extension in two quantities: space and time).
2. Divide the space extension infinitely (i.e., "assume we can infinitely divide the distance to be traversed").
3. Think of the times needed for each section defined in step 2.
4. The total time should be the sum of all times we found on step 3.
5. But summing those times is impossible, since there is an infinity of them.

There is a blatant contradiction between the assumption hidden in step 5 (an infinite sum cannot add up to a finite value), and the explicit division put by hand into step 2 (lets divide this distance into an equivalent infinite number of subintervals)!!

The resolution of this has nothing to do with the physics of motion. It is just a matter of self-consistency.

What calculus has to say about this is that, doing things carefully, continuous magnitudes and infinite sums (as those assumed to exist in the formulation of the paradox) can be solidly dealt with.

If that were the problem, you would be correct that calculus solves it. But since you don't understand the problem, it's not surprising that your answer is irrelevant to it. Yes, calculus is a wonderful mathematical theory. Bully for it. It just doesn't have anything to do with the problem at hand.

The problem is not finding a finite sum of an infinite series. The problem is saying what it would be to complete an infinite series one step at a time. Calculus doesn't even begin to adress this issue, let alone provide an answer for it.

[NeutronStar]

ahrkron wrote:

5. But summing those times is impossible, since there is an infinity of them.

…(snip)…

What calculus has to say about this is that, doing things carefully, continuous magnitudes and infinite sums (as those assumed to exist in the formulation of the paradox) can be solidly dealt with.



I agree that calculus solidly deals with these concepts from a purely mathematical point of view. I love calculus!

I'm not out to put down calculus.

But I also understand the definitions and the methods of how calculus deals with these problems. And I would like to state some 'facts' regarding this issue.


Fact 1. There is nothing in the formalism of calculus that says that an infinite number of tasks can be completed.


If you can find a calculus book that states otherwise I would be glad to read it. But to my knowledge calculus does not make that claim anywhere in its formalism.


Fact 2. Calculus handles the problem of infinite sums by basically 'cornering' the problem


I'll grant you that this statement is purely off-the-cuff, but I feel that it genuinely describes the situation.

Proving that a series converges in calculus is all about proving the behavior of the quantities that are being added together. That they have certain limit properties. And if all of these properties are satisfied then we can ignore the fact that the sums can never be completed. We then say with confidence that the sum converges to a particular value, and for all intents and purposes equals that value.

This is not at all the same as proving why Achilles can complete an infinite number of tasks.

In fact, all calculus is really saying is that if Achilles can manage to complete all his tasks he will definitely end up at the finish line because this is where he is converging to.

The same thing goes for an infinite sum. If you could possibly complete it then you most certainly will end up with the precise value that calculus has predicted because this is where the sum is going to converge.

I mean, calculus is perfectly correct in both cases. It calculates the correct point where Achilles will end up, and it calculates the correct sums for infinite series. But it does NOT explain how an infinite number of task can be completed. Neither does it claim to have completed the infinite additions. It simply gives what the results would be if they could be completely.

I think calculus is great! But it does what it does. And it doesn't explain how an infinite number of tasks can be completed. Nope, it doesn't do that, nor does it claim to do that anywhere in the formalism that I am aware of. It just corners the convergence.

But it does deal with that solidly, I do agree with that! [:)]

[Hurkyl]

Fact 3:

Nobody here (but me) has made any attempt at making a rigorous notion of what a "task" is, and what it means to "complete" a task.

[HallsofIvy]

I would agree that "There is nothing in the formalism of calculus that says that an infinite number of tasks can be completed."

But you will have to agree that calculus then gives ways of arriving at the same result AS IF an infinite number of tasks were completed!

[NeutronStar]

I'm just going to agree with what drnihili wrote way back at the beginning of this thread:


drnihili wrote:

You have to be quite clear on what the question is. If you take Zeno to merely be asking how an infinite sequence can occupy a finite space, then calculus indeed answers the question. However, if you taking him to be asking the question of how one can complete and infinite sequence one member at a time, then calculus not only doesn't answer the puzzle but is entirely irrelevant to it. I think the latter question is the better way to understand the point of the paradox.



I firmly believe that if Zeno were here today, and was totally versed in calculus he would agree with what drnihili said here.

And I honestly believe that anyone who fully understands both calculus, and Zeno's paradox would also agree.

I can only suggest that those who do not agree either misunderstand calculus, or have misunderstood what Zeno is asking.

Calculus convergence does not answer Zeno's questions in my opinion. I'll just leave it at that. I've already given more than my fair share of explanations of why I believe that calculus doesn't answer Zeno's question. And I've even given my solution to the paradox by suggesting that space and time must necessarily be finitely divisible.

There's where I stand on the topic. [:)]

I do appreciate all contributions to the conversation regardless of the viewpoints. Exchanging differences of opinions is always good. I'm not really interested in discussions on the topic of proving that an infinite number of tasks can be finitely completed. I don’t believe that such a proof can be constructed without destroying the meaning of infinity as I understand the concept. As I say, my solution is to simply accept that our universe is finitely divisible in time and space.

Although, I also have a second possible solution: There is no such thing as time or space, and therefore it's actually absurd to talk about quantifying or dividing them. This is probably the real solution to the paradox. [;)]

[NeutronStar]

Originally posted by HallsofIvy
I would agree that "There is nothing in the formalism of calculus that says that an infinite number of tasks can be completed."

But you will have to agree that calculus then gives ways of arriving at the same result AS IF an infinite number of tasks were completed!

I don't have to agree to anything. [:D]

But I do agree.

I believe that I actually said this same thing in one of my posts.

[Hurkyl]

The "solution" to Zeno's paradox (like almost every other paradox) is to state precisely what's going on.

As I tried to hint, the way it's been brought up here (as a question about tasks) is not well-formulated; e.g. what is a task and what does it mean to complete one, and why should one think a sequence of tasks should or should not be completable?

If one cannot precisely answer those questions, this "paradox", like most others, serves to demonstrate the pitfalls of intuitive logic.

If one can precisely answer (and justify) those questions, then the "paradox" becomes a full-fledged contradiction and something has to give. (But, of course, it's awfully arrogant to think your just-invented theory of tasks is pristine and millenia of efforts at refining mathematics have gone astray; it's far more likely that a slight modification to your theory of tasks would make it compatable with mathematics)

[drnihili]

Originally posted by Hurkyl
Fact 3:

Nobody here (but me) has made any attempt at making a rigorous notion of what a "task" is, and what it means to "complete" a task.

Say What?!?!?!

I suggest you reconsider what "rigorous" means. It doesn't mean just dressing something up with math-speak. Math-speak is neither necessary nor sufficient for rigor.

Thus far your contributions have mainly been to say that other people haven't yet satisfied your desire for proof. You've offered next to nothing by way of positive argument, instead insisting that it's up to the other side to convince you that there's a problem. Each time you've asked, you've gotten a reply which you have for the most part failed to address. Instead you've picked out pieces, tried to guess what other people might say, and then responded to that.

While I suspect your contributions are sincere and well intentioned, they fall well short of rigor. And to suggest that you are the only one who has even attempted rigor is utterly laughable.

[drnihili]

Originally posted by Hurkyl
The "solution" to Zeno's paradox (like almost every other paradox) is to state precisely what's going on.

As I tried to hint, the way it's been brought up here (as a question about tasks) is not well-formulated; e.g. what is a task and what does it mean to complete one, and why should one think a sequence of tasks should or should not be completable?

If one cannot precisely answer those questions, this "paradox", like most others, serves to demonstrate the pitfalls of intuitive logic.

If one can precisely answer (and justify) those questions, then the "paradox" becomes a full-fledged contradiction and something has to give. (But, of course, it's awfully arrogant to think your just-invented theory of tasks is pristine and millenia of efforts at refining mathematics have gone astray; it's far more likely that a slight modification to your theory of tasks would make it compatable with mathematics)

Tell me, how much effort have you given to the formal study of paradox? You seem quick to put all paradoxes in one easy box as having to do with lack of rigor. Do you know anything about these things? Have you, for example, studied fixed point theories? Paraconsistency? Revision theory? or how about advanced logic in any form whatsoever? Do you have any idea what kinds of paradoxes there are? Can you distinguish those that trade on omega-inconsistency versus negation-inconsistency? Do you know any of the history of the subject you are pontificating on? Or perhaps do you think because you are well versed in some part of mathematics that you therefore are qualified to sit in jugement over all of it, and also all other areas that are related to mathematics?

Do not presume that others are ignorant just because you disagree with them. It may just be that you are the one in the dark.

[Hurkyl]

I suggest you reconsider what "rigorous" means. It doesn't mean just dressing something up with math-speak.

It would entail, at the very least, a set of premises (ideally ones upon which we can both agree) from which you argue a point.

As is, you have some intuitive notion of what it means to complete a task that seems to have no relationship to my comprehension of the term task other than sharing the same name.

This phenomenon, to varying degrees, is, in my experience, very common in threads such as this where one side (or both) refuses to make any attempts at making their argument precise. Experience has shown the only practical recourse in such situations is to reduce the problem to its fundamentals (such as discuss the precise meaning of "task" and "completion"). When I've been in debates where the other side goes along with this, progress gets made. When the other side steadfastly refuses to participate, well, I get practice expressing my thoughts clearly, but have no expectation of the argument going anywhere.


you've picked out pieces, tried to guess what other people might say, and then responded to that.

Bingo! This is why I emphasize the need to precisely state one's arguments. Because if you don't fill in the details, I have to, and since I don't think the same as you, you probably won't like my choices. [;)]


You've offered next to nothing by way of positive argument, instead insisting that it's up to the other side to convince you that there's a problem.

For what do you want me to argue positively?

The whole point of the thread is whether or not Zeno's paradox is or is not an actual flaw in the view point of modern science, is it not? (I assume so because I've offered you several opportunities to say otherwise which you haven't taken)

Then, please tell, why I should be expected to do anything but try and ascertain why Zeno's paradox is perceived to be a flaw, and then attempt show flaws in the reasoning leading to the perception?


Tell me, how much effort have you given to the formal study of paradox?

I haven't had the opportunity to take a class on paradoxes.


You seem quick to put all paradoxes in one easy box as having to do with lack of rigor. Do you know anything about these things?

I have plenty of experience with "paradox" as it appears in a typical forum setting, and I do know the role paradoxes (real ones) played in the formulation of modern (mathematical) logic and set theory and the usefulness of paradoxes to force one to think clearly and carefully about concepts.


Or perhaps do you think because you are well versed in some part of mathematics that you therefore are qualified to sit in jugement over all of it, and also all other areas that are related to mathematics?

Nope. I only argue as if I know something when I think I know it.


Do not presume that others are ignorant just because you disagree with them. It may just be that you are the one in the dark.

Yep it may. But when I'm pretty sure I'm not in the dark, I continue operating under that hypothesis unless I see a light that proves otherwise.

[Hurkyl]

It's actually a lot easier to validate the Gods example than the race example.

Let each vow be represented as a material conditional of the form "If Achilles reaches a point 1/2^n alive, then he will be instantly killed by God n." The conditionals are built by recursion on the counting numbers. Now let N be the conditional built at nth step of the recursion. If N is false, then each M>N must also be false. However, there is no contradiction in all of the Ns being true.

Either I don't understand your point about induction or you're confused. It is enough that we have a god for each n. We only need an omega sequence, not an omega+1 sequence.


The point I tried to make is why is it reasonable to think we can have a god for each n? (I was granting, for the sake of argument, that we can have at least one god, and for any number of gods we can add one more)

[drnihili]

Originally posted by Hurkyl
It would entail, at the very least, a set of premises (ideally ones upon which we can both agree) from which you argue a point.

As is, you have some intuitive notion of what it means to complete a task that seems to have no relationship to my comprehension of the term task other than sharing the same name.

This phenomenon, to varying degrees, is, in my experience, very common in threads such as this where one side (or both) refuses to make any attempts at making their argument precise. Experience has shown the only practical recourse in such situations is to reduce the problem to its fundamentals (such as discuss the precise meaning of "task" and "completion"). When I've been in debates where the other side goes along with this, progress gets made. When the other side steadfastly refuses to participate, well, I get practice expressing my thoughts clearly, but have no expectation of the argument going anywhere.

In this case you are the one refusing to participate as you routinely ignore what others have said and instead imply that they are ignorant and that the problems they raise have simple solutions.




Bingo! This is why I emphasize the need to precisely state one's arguments. Because if you don't fill in the details, I have to, and since I don't think the same as you, you probably won't like my choices. [;)]

This is a prime example. You extracted a bit and utterly ignored the primary point. All the while pretending that somehow your unwillingness to deal with the argument presented is a failing on the other side.


For what do you want me to argue positively?


I've already said this. Try reading.


The whole point of the thread is whether or not Zeno's paradox is or is not an actual flaw in the view point of modern science, is it not? (I assume so because I've offered you several opportunities to say otherwise which you haven't taken)


Ok, let me hold your hand through this reading business as it's apparently a new skill for you. Go back to the first post. You'll notice it ends with a question mark. That's your clue that the poster is asking a question. Now, let's read the question. "How does he do it?" There. The thread is asking how Achilles manages to cover distance d given infinite divisibility. It's not about whether there's a flaw in the view point of modern science. Modern science doesn't have a view point on this question. In fact several people have pointed this out. We've blatantly said "Calculus is irrelevant". Notice we said "irrelevant", not "wrong". There's a reason for that, and it's not just that we like bigger words.


Then, please tell, why I should be expected to do anything but try and ascertain why Zeno's paradox is perceived to be a flaw, and then attempt show flaws in the reasoning leading to the perception?


Well the obvious answer would be because that's how communication and conversation go. A less obvious answer might be because you need to justify the claims you keep making.


I haven't had the opportunity to take a class on paradoxes.

I have plenty of experience with "paradox" as it appears in a typical forum setting, and I do know the role paradoxes (real ones) played in the formulation of modern (mathematical) logic and set theory and the usefulness of paradoxes to force one to think clearly and carefully about concepts.


So from your experience with paradoxes in forums you're prepared to pronounce "Anyways, a paradox is typically a contradiction that arises from an unfounded assumption. They usually get cleared up once you try to do everything rigorously." and "The "solution" to Zeno's paradox (like almost every other paradox) is to state precisely what's going on." That's an amazing amount of hubris to ditch an entire field of study solely on the basis of what you've seen in forums. And this is what we're supposed to take as the example of rigor to which we should all aspire?


Nope. I only argue as if I know something when I think I know it.


Yep it may. But when I'm pretty sure I'm not in the dark, I continue operating under that hypothesis unless I see a light that proves otherwise.

One last time, try reading what others post, and then try figuring out why they posted it. It can be very enlightening. After that, try actually studying a subject before dismissing it out of hand.

[drnihili]

Originally posted by Hurkyl
The point I tried to make is why is it reasonable to think we can have a god for each n? (I was granting, for the sake of argument, that we can have at least one god, and for any number of gods we can add one more)

Obviously the example is meant to be fantastical. But remember, we're not creating a sequence of gods, we're taking them as a given whole. There is no problem, finitism aside, with the exitence of infinite collections. The problem is in the stepwise construction of such. Unless you don't like recursion, I can't see why you would object to the gods.

[Hurkyl]

In this case you are the one refusing to participate as you routinely ignore what others have said and instead imply that they are ignorant and that the problems they raise have simple solutions.

That is because the "problems raised" bear an extraordinarily striking resemblence to the mistakes people tend to make when first learning about noncompact sets.


This is a prime example. You extracted a bit and utterly ignored the primary point. All the while pretending that somehow your unwillingness to deal with the argument presented is a failing on the other side.

I've already given a go at your game, and determined it was a fruitless endeavor unless I could discover what you mean by "task" and "completing a task". It's not a coincidence that my game implies focusing on the meaning of these phrases. When will you have a go at mine?


I've already said this. Try reading.

I'm sorry; I'll stop browsing the forums through osmosis.


Modern science doesn't have a view point on this question.

Infinite divisibility of paths is required by modern science. Since you appear to be attempting to cast a shadow of doubt on the notion of infinite divisibility, modern science most certainly factors into this discussion.


A less obvious answer might be because you need to justify the claims you keep making.

The text of any claims I have made to this point are irrelevant (at least most of them are, anyways); the goal is to try and determine what you mean by "task", et cetera.


That's an amazing amount of hubris to ditch an entire field of study solely on the basis of what you've seen in forums.

If asked, I'd certainly admit that my use of "paradox" is how it relates to the way it typically appears in debates, and to my experience in seeing it used (and using it myself) to guide the formulation of theories and ideas. Who said anything about ditching an entire field of study?


One last time, try reading what others post, and then try figuring out why they posted it. It can be very enlightening.

I have, and I'm quite certain that any sort of progress revolves around you conveying to me what you mean by phrases like "task", "completion a task", and "how one completes a task".


Unless you don't like recursion, I can't see why you would object to the gods.

The problem is that recursion guarantees only unbounded sets of gods, not infinite sets of gods.

Since the conclusion is absurd, one must turn to validate the hypotheses. Aside from the fact we're dealing with mythological creatures, the fact we require an infinite set of gods stands out (to me) as the most likely candidate for a problem, so I naturally ask myself "Why would I think it's reasonable to consider this?"

The other (physical) sticky point for this particular paradox is that death is assumed to be instantaneous... and while death can be short, science gives us every reason to presume it cannot be truly instantaneous. The paradox dissolves if there is a minimum time interval between life and death for Achilles, even if we permit an infinite number of gods; although there is no "first" god to strike Achilles, we can clearly indicate an infinite collection of them which participated in his death.

This is why I ask the question "Why would we think this is valid?"; if one can answer this question, it is quite frequently clear what the likely problems are. If one can't answer the question, then there really isn't a reason to think it's valid now, is there? [;)]

[NeutronStar]

Hurkyl wrote:

The whole point of the thread is whether or not Zeno's paradox is or is not an actual flaw in the view point of modern science, is it not?



I've given up on trying to read everything that you guys are discussing, but this comment caught my attention.

I certainly didn't view this thread to be about any flaws in science. I am well aware that science cannot answer every question. Does someone think it can?

Zeno's question is almost philosophical in nature. Is it even important to science right now? I can imagine a time when it may actually become important. However for now I simply see it as interesting. And having value in the sense that it does spark food for thought. I see it as a worthy question to consider, but not to cry about if we can't answer it. That doesn't imply any flaw in science. Sheesh! There are a lot of questions science can't answer!



Hurkyl wrote:

The "solution" to Zeno's paradox (like almost every other paradox) is to state precisely what's going on.

As I tried to hint, the way it's been brought up here (as a question about tasks) is not well-formulated; e.g. what is a task and what does it mean to complete one, and why should one think a sequence of tasks should or should not be completable?



Seems to me that it's pretty straight-forward in this particular problem

What is a task?

To move 1/2 of the distance to finish line.

What does it mean to complete one?

You are 1/2 of the distance closer to the finish line than you were before.

Why should one think that this sequence of tasks cannot be completed?

Because these tasks are never ending. No matter how close you get to the finish line you can always move 1/2 of the distance closer.

You had suggested a "super task" a while back that would constitute reaching the final point. But who said that you could make up that task? That task isn't part of the problem. Your task is to step only 1/2 of the distance to the finish line with each step, and then explain how you can actually reach the finish line by completing only these tasks. Making up a new task to step directly to the finish line is cheating.

Your first reaction was to use calculus. But as has been pointed out, calculus will only guarantee that you will indeed reach the finish line if you manage to complete all of your tasks. Calculus in no way makes any claim that you should be able to actually complete your infinite many tasks. It only says that if you do manage to complete them you will end up at the finish line. But we already know that so we don’t even need calculus at all.

What we do need is a rigorous proof that an infinite number of tasks can actually be completed. That is impossible! This is why it's a paradox. Achilles can actually reach the finish line in the real world, but logically it isn't possible!

Even if you managed to prove that an infinite number of tasks could be completed, then all you would have proven is that infinity is finite. You aren't going to be able to prove that with rigor.

So going back to your original worry that this means that science is somehow flawed why not consider some alternatives.

1. Space and time cannot be divided up infinitely because they are quantized.

or,

2. Space and time are not absolute properties of the universe and so we can't even talk about dividing them up at all because the don't even exist!

I actually like both of these solutions to the paradox.

QM and String theory suggest solution number 1

SR and GR suggest solution number 2

It's interesting food for thought. Trying to prove that infinity can be finite seems to be futile to me. I'd much rather pick one of the other 2 possible solutions that I've offered.

Perhaps someone else has other possible solutions? [:)]

[Hurkyl]

What is a task?

To move 1/2 of the distance to finish line.

What does it mean to complete one?

You are 1/2 of the distance closer to the finish line than you were before.

If this is all one means by completing task, we should have been done on the first page. As I mentioned, one can clearly demonstrate that if Achilles follows the trajectory x(t) = t for t in [0, d], each task is completed.

Why should one think that this sequence of tasks cannot be completed?

Because these tasks are never ending. No matter how close you get to the finish line you can always move 1/2 of the distance closer.

But why does that imply they cannot be completed?


Given the meanings I attach to the words involved, that trajectory is rigorous proof that the tasks can be completed. (you just need to show it is smooth and to prove that the set of points {d - d/2n | n > 0} lie in [0, d]) That is sufficient to prove that geometry and infinite divisibility is not an obstacle to completion.

However you obviously don't accept this, which is why I've spent so much effort trying to get you guys to say what you mean by the words involved.


What we do need is a rigorous proof that an infinite number of tasks can actually be completed.

May I presume that if Achilles has completed the n-th task, he can complete the (n+1)-th task? The proof then follows by the standard calculus argument; it is then elementary to prove that each of the points in {d - d/2n | n >= 0} lie on Achilles's path, and continuity proves that d lies on it.


Alternatively we may use transfinite induction; if we add a final task "arrive at d" to Zeno's sequence, and agree that this final task can be completed if the entirety of Zeno's sequence can be completed, then transfinite induction guarantees that the "arrive at d" task is completed.


Even if you managed to prove that an infinite number of tasks could be completed, then all you would have proven is that infinity is finite. You aren't going to be able to prove that with rigor.

In the theory I am working, it is proven with rigor. Which is why I spend such an effort trying to figure out in what theory y'all are working.


So going back to your original worry that this means that science is somehow flawed why not consider some alternatives.

1. Space and time cannot be divided up infinitely because they are quantized.

I do consider alternatives; science is flawed, as demonstrated at the very least by the incompatability between the standard model and general relativity. I would be happy to see position quantized though at the moment there is no indication that will occur.

But my point is that Zeno's paradox is not sufficient to prove that an alternative must be considered.



And there is one comment I can't believe I missed because I was all but specifically requesting it!

I think there are still unresolved problems with our formalizations of infinity. Those problems are not always inconsistencies in the formalizations themselves but are problems in the relationship between the formal models and the reality which they are taken to be models of.

but I don't have time to respond to it at the moment, so I'll get it when I'm back from work... I'm quoting it so I'll remember it later.

[NeutronStar]

Hurkyl wrote:

May I presume that if Achilles has completed the n-th task, he can complete the (n+1)-th task? The proof then follows by the standard calculus argument; it is then elementary to prove that each of the points in {d - d/2n | n >= 0} lie on Achilles's path, and continuity proves that d lies on it.


Alternatively we may use transfinite induction; if we add a final task "arrive at d" to Zeno's sequence, and agree that this final task can be completed if the entirety of Zeno's sequence can be completed, then transfinite induction guarantees that the "arrive at d" task is completed.



With all due respect Hurkyl you are looking at the problem backwards. You are assuming that Achilles can obviously finish the race and reach the finish line. You therefore feel that can use that information as part of your proof.

Zeno's paradox is purely a thought problem. He was well aware that motion is possible, he could move himself! The whole point to Zeno's question is how can we explain motion without already assuming that it can be achieved.

Your first proof above just says what we have already allowed. That if Achilles can complete all of his tasks he will end up at d. So how does that prove that he has completed all his tasks? Simply because we know that he can actually reach the finish line in reality? That is not a permitted assumption! Secondly, even it we did permit that assumption that still doesn't prove that Achilles completed an infinite number of tasks, he may very well have cheated and moved though a space that is finitely divisible and have only completed a finite number of tasks. In other words, he may performed your supertask at the end where he ran out of half-way points and was forced to step completely to the last point because space is ultimately finitely divisible.

So your first so-called proof would prove nothing about completing an infinite number of tasks even if you were permitted to use the fact that Achilles actually finishes the race.

Your second proof is almost identical only in this case you are using induction. But you are still assuming two things. You are assuming that space can be infinitely divided, and you are assuming that Achilles can indeed reach the finish line. That second assumption is not permitted.

In Zeno's problem you are only permitted to use the first assumption and then logically prove that Achilles can finish the race. The mere fact that Achilles can finish the race in reality doesn't prove that space can be divided up infinitely. And this is really all that Zeno is saying with his paradox. He is saying, "Show me how Achilles can logically complete and infinite number of tasks without using the assumption that he can do it!"

In other words, prove that space is infinitely dividable without assuming it!

But every so-called proof that you have shown thus far contains both the assumption that space can be infinitely divisible and that Achilles can reach the finish line!

Therefore, I can only conclude that you don't fully appreciate Zeno's concern.

[NeutronStar]

Hurkyl,

I do want to agree with your proofs. Seriously.

If we allow both of your assumptions:

1. Space is infinitely divisible.
2. Achilles can obviously reach the finish line.

Then your induction proof would be valid. You would have indeed proven that infinity is finite!

And in reality we all know that assumption number 2 makes a whole lot of sense because we do this sort of thing everyday. (this is what I would call self-evident).

But we don't have any real reason to assume that number 1 is true. We have absolutely no reason to believe that space can be divided up infinitely. But do we have any reason to believe that can't be?

This is really where Zeno is at. Imagine this very wise old man asking you to PROVE you assumption number 1. To do so you cannot start by assuming it. You can't just assume that space is infinitely divisible, Zeno wants proof! And this is really what his paradox is all about. He is asking, "If space is infinitely divisible than how is it possible that we can move?".

I surrender to Zeno and openly admit that I cannot prove that space is infinitely divisible. Therefore Zeno has forced me to consider other alternatives.

This is the essence of Zeno's paradox.

It's a paradox because if anyone can prove it they will have proven that infinity is finite! It will always be a paradox.

It cannot be solved. Zeno offers it as food for thought. Any so-called solutions (like the ones that I have offered) are not really solutions at all. They are merely an acceptance that Zeno is right, and that we must reject the idea that space and/or time can be infinitely divisible.

I bow to Zeno's genius. He is my hero of the ancient Greeks. [:)]

Although, some of those other Greek dudes were pretty cool too! [;)]

[Hurkyl]

My goal is merely to observe that infinite divisibility can be internally consistent. Expecting to give a logical proof would be misguided; logical proofs are always relative to the hypotheses... and if the exceptional success science has had using models incorporating infinite divisibility isn't sufficient evidence that it is a practical idea, then I don't think it's worth spending additional effort on that angle.


I surrender to Zeno and openly admit that I cannot prove that space is infinitely divisible. Therefore Zeno has forced me to consider other alternatives.

Semantic correction; Zeno has forced you to consider that infinite divisibility is not the only option.

[NeutronStar]

Hurkly worte:

My goal is merely to observe that infinite divisibility can be internally consistent.



"merely observe"?

What happen to all the logical rigor that you claimed to be interested in?

That's Zeno's whole point! What do you mean "you merely observe"? What kind of a proof do you call that? Where's the logic behind your observation? What exactly is it that you have observed?

If your observation makes logical sense you should be able to state it in an unambiguous logical way with rigor.

It sounds to me like you are just saying, "My goal is merely to believe that infinite divisibility can be internally consistent.

And if this is what you choose to believe that's fine. But at least own up to the fact that you merely believe it and that you can't prove it at all!

If you could prove this with rigor, I'm sure that the entire world would love to see your proof!

You would win a Nobel Prize for such a proof to be sure!

[Hurkyl]

"merely observe"?

What happen to all the logical rigor that you claimed to be interested in?

Ala Godel, it's futile to use a logical theory to prove itself consistent. The best we can do is to seek potential contradictions and analyze them for flaws. Or, of course, we may pass the buck and appeal to a different theory within which we can prove the original theory consistent. (e.g. a relative consistency proof using set theory, but since I know you have issues with ZF, that's not really an option for this discussion)

[NeutronStar]

Originally posted by Hurkyl Ala Godel, it's futile to use a logical theory to prove itself consistent. So in other words, when you are up against the wall and you can't make your case just cite Gödel's inconsistency proof.

I see. [:D]

I'll have to remember that one. [;)]

The best we can do is to seek potential contradictions and analyze them for flaws. Or, of course, we may pass the buck and appeal to a different theory within which we can prove the original theory consistent. (e.g. a relative consistency proof using set theory, but since I know you have issues with ZF, that's not really an option for this discussion)
Actually you do have a valid point here. I do have issues with ZF. So you're right that it isn't an option for this discussion.



From way back in this thread,...
drnihili worte:
I think there are still unresolved problems with our formalizations of infinity. Those problems are not always inconsistencies in the formalizations themselves but are problems in the relationship between the formal models and the reality which they are taken to be models of.
I never did get a chance to formally state that I agree with this. Even though it should be obvious. [:D]




Other than this I suppose I'm done on this thread. For real this time. [:)]

By they way, I would like to say that I did gain many ideas and food for thought from this discussion. I believe that is it always fruitful when great minds come together no matter how much they disagree. I am quite sure that everyone here has a very good understanding of what they understand. [8)]

We simply understand things differently. [:D]

Thanks to everyone who has contributed.

[Joy Division]

I've always been amused with Zeno's argument because even today people love to bring it up to argue various points. It's funny that an argument that Zeno used to show that motion and change were impossible is still brought up today.

Really though the logic in this argument can be very subtle and one has to tread carefully before saying what it really means. (I just hope I'll heed my own advice.)

Let's look at the perscribed motion of Achilles. Each of his steps covers half the distance of his last step and his first step covers half the overall distance. Let's look at what we've assumed here.

First we've assumed that motion is possible. So he can take his steps in the first place. Second we've assumed that space is infinitely divisible (unless we add some extra content to how he moves.) We get that because if space is not infinitly divisible then we get problems that we don't want when on one of his steps Achilles can no longer cross the minimum distance in our space and we get a contradiction.
It's not one we want though because all that contradiction proves is that unless we say what happens when to Achilles' motion when he reaches that point then the motion we've perscribed is not possible in a finitely divisible space.

Ok so if we want to consider that space may be finitely divisible then we have to either say achilles moves like we tell him to and stops when he can no longer make a step over the minium distance or that he stops taking smaller steps and continues covering the smallest distance each time. Notice that each one of these will end his movement in a finite number of steps, but in the latter he will actually reach his goal.

Ok the big thing to notice now is that Zeno's argument has not shed any light on whether or not space is infinitly divisible. In fact it's very consistant in all 3 cases we've considered. Neither does this shed any light on what would happen if you did move as Achilles does, we just really can't say.

Now let's consider the case when space is infinitly divisible. This one is the one in which he makes an infinite number of steps. Now our feelings and what is "intuitively obvious" isn't going to cut it anymore. We're dealing with infinities so mathematical formalism becomes key. The only sane thing to do is to use calculus. Really it is, we've got an ideal situation when we consider this argument and reality is what doesn't have any buisiness poking it head in here. Even if Calculus were completely wrong at describing the real world ,it is perfectly suited to the task of describing Achilles' motion.

Ok so what's the problem here. He takes an infinite number of steps!! That must be crazy. Wrong because the series that describes his movement converges to a finite value there is no problem here. Really what's so wrong, the set of all his steps may be infinite but the distances he covers with each step, the time it takes for each step and even the work he does for each step approach zero.(That is in a limiting sense). So the amount of effort however you want to define it for each task ( or step as we've defined the tasks earlier) approaches zero too. That means because our long haired achean is an ideal runner he finishes those tasks like nobody's business.

Now this argument gets interesting when you actually ask, what if we tried to model Achilles' movement on a computer? It would take an infinite time for our digital Achilles to finish. That's the interesting part, because computers can't use the axiom of choice while our calculus running ideal Achilles can. What this means is that for the digital Achilles the two definitions of continuity (the one based on functions the other on sequences) aren't equivalent. You need the axiom of choice to prove they are.
That's where at least for me the interesting mathematics comes out from this argument.

[Ether]

The main point of contention here seems to be whether or not Achilles can complete an infinite number of tasks in a finite period of time. Hurkyl says he can. NeutronStar and drnihili say he cannot, and insist that Hurkyl give an explanation of how he can. This insistence, however, is based on the assumption that the completion of infinitely many tasks in finite time is (self-evidently) undoable.

My understanding of Hurkyl is that while the series of ever-shrinking segments Achilles covers converges to d, the time interval to complete each task converges to zero and the total time converges to some finite value. Calculus is thus telling us that the thing is indeed doable. Hurkyl's response as to how the thing is done, that Achilles merely runs from 0 to d, is thus sufficient, because performing an infinite number of tasks in finite time is no obstacle at all (in a sense it happens "automatically") and its undoability is not self-evident to everyone. Asking how it is done is like my asking how I will cross the river in front of me when in fact there is no river there.

Sorry if I misrepresented any of your points, Hurkyl.

[Hurkyl]

I think you have the semantics of the consistency argument right, but that point is not the one upon which neutron star and drnihili are focusing. To summarize the facts of the thread in two sentences:


One cannot logically prove an infinite divisibility model correctly describes the physical universe.

Zeno's paradox does not logically prove infinite divisibility models cannot correctly describe the physical universe.


neutron star and drnihili are focusing on the first fact, the rest have focused on the second fact.

[NeutronStar]

Hurkyl wrote:
One cannot logically prove an infinite divisibility model correctly describes the physical universe.

Zeno's paradox does not logically prove infinite divisibility models cannot correctly describe the physical universe.


neutron star and drnihili are focusing on the first fact, the rest have focused on the second fact.


Yes, I would agree that this is a good assessment of the conversation.

I just had some new thoughts about the infinite divisibility of space.

A while back Hurkyl mentioned that QM only states the bound particles are quantized, and it does not restrict free particles to this behavior.

As an important append to that I would like to add the following: (and please correct me if I am wrong, which I know you all will of course. [:D])

QM does not precisely describe the behavior of free particles. In other words, it only describes them as a probability function. It does not assign to them any specific vector in space.

And as an additional prerequisite piece of information I would just like to remind everyone that the idea of an absolute space died with classical mechanics, and with Relativity came the idea that space, and time are merely relative notions.

With that I'd like to possibly spark a seminar on the discussion of the practicality of even thinking of an infinitely divisible space from a physical perspective.

I will begin the discussion with the following piece of information:

I just started studying classical analytical mechanics as a refresher course. (I've been trying to refresh this tired old brain lately. [;)]) Just in case anyone is interested the book is Analytical Mechanics by Fowles & Cassiday ISBN 0-03-022317-2.

In any case, the book starts off with some rather interesting historical information about the definitions of particular physical quantities. These definitions are really all we have to go on as physicists. Length (which is really the same thing as distance) was originally quantified as a unit by the length of a particular physical object. By that physical definition the notion of length (and therefore distance) would necessarily be dependent on the bound state of matter. Thus the very notion of distance would indeed be subject to the quantized nature of QM.

As time progressed we moved on to redefine length as the distance that light travels over the duration of a specific time. At first this may sound like we have freed the definition of distance from the grasp of QM. However, further analysis reveals otherwise.

We have defined time as the oscillations between particular states of the cesium-133 atom, a quantum object to be sure. Our concept of time is based on this quantum jumping little fellow, so these quantum ticks of time are necessarily pushed onto our idea of distance as defined by the distance that light travels during these ticks. In other words, we can only time the light in an intermittent fashion (with each jump of the quantum atomic clock).

So by our very definitions of space and time we have no choice but to own up to the fact that space and time is indeed finitely quantified as we have defined them to exist. The finite nature of it all stems from the quantum jumpiness of our universe. It is an unavoidable part of the universe that we live in.

Now to speak abstractly about some concept of absolute space being infinitely divisible it purely a fabrication of our minds. We have no reason to believe that such an infinitely divisible absolute space exists. In fact, Albert Einstein would be giving us all the tsk, tsk if he knew we were still attempting to hang onto such an ideal.

So with this in mind let me quote Hurkyl once again:

Hurkyl wrote:
One cannot logically prove an infinite divisibility model correctly describes the physical universe.

Zeno's paradox does not logically prove infinite divisibility models cannot correctly describe the physical universe.


neutron star and drnihili are focusing on the first fact, the rest have focused on the second fact.


Based on everything that I've just said here I would have to side with neutron star and drnihili. [:D]

Hey! I'm allowed to take my own side on this!

Of course whether or not Zeno has proven that infinite divisibility models cannot correctly describe the physical universe is certainly arguable.

Whether Zeno's Paradox proves it or not, it sure looks like he hit the nail on the head to me.

I yield the floor to whomever wishes to jump in next. [:)]

[Joy Division]

While it is interesting that quantum mechanics shows us that we can't observe the universe in an infinitly divisible way (as far as I know), whether or not that means that space itself is actually discrete is an interesting question.

I don't have an answer. But the idea that space may be discrete does not invalidate the fact that calculus is still internally consitant.

I know some people think that any math not related to the real world is uninteresting (or worse nonexistant [s(] ). I just don't share their opinions. This is why there is in fact a separate study of mathematics and physics.

Of course calculus alone can't describe our universe at all times.

[Ether]

Originally posted by NeutronStar
Now to speak abstractly about some concept of absolute space being infinitely divisible it purely a fabrication of our minds.

How does an assumption of infinite divisibilty imply that space is absolute?

The mathematics of both Special and General Relativity is a continuum math. I'm not sure Einstein wouldn't take you to task.

[NeutronStar]

Ether
The mathematics of both Special and General Relativity is a continuum math. I'm not sure Einstein wouldn't take you to task.

I wish Einstein were alive today. I would most certainly like to ask him how he justifies quantifying space when he himself claims that there is no such thing to quantify?

Please understand that I am in no way suggesting that Einstein would be stumped by this question. I am simply saying that I would genuinely like to hear his response to it.

Einstein's genius is unquestionable. However, his tendency to stubbornly hang onto preconceived notions was also historically reordered as being notorious. He totally could not accept the idea the quantum mechanics predicts that god plays dice with the universe. He was a determinist through and through. So while he was willing to reject the idea of an absolute space, he was not prepared to reject the idea of an absolutely determined universe. Very ironic I think.

Einstein's General Relativity describes gravity as warped space-time. That doesn't amount to a very comprehensible idea coming from someone who has denounced the very idea of an absolute space-time that can be warped.

However, we do know that QM and GR are not compatible. So something's got to give. Will it be QM, or GR? OR some combination of the two? We don't yet know. All we do know is that QM hasn't failed yet. And while the relativity of space and time have been experimentally verified their infinite divisibility has not.

Einstein was not even a very good mathematician actually. He relied on other mathematicians to formulate his ideas. Minkowski was a big player in this among others. In fact it was actually Minkowski who first suggested that time should be a dimension. Einstein actually opposed that idea in the very beginning.

Einstein's mathematical descriptions are necessarily continuous because he is using a continuous mathematics. He had no choice. I have reason to believe that mathematical formalism is actually incorrect in this regard. (See the thread entitled "What Makes Mathematics?")

If we correct the basis of mathematical formalism, then Einstein's GR would automatically become discrete because all of mathematics would become discrete. [:)]

Who knows? That alone may automatically make it compatible with QM! I'm not far enough into these topics to really say.

We may actually have a complete theory of everything and just don't know it! [8)]

(Disclaimer) This particular post was an extreme instantaneous ramble. I just happened to be passing by my computer and noticed the response on this thread, so I sat down and typed in my thoughts. [:D]

[Hurkyl]

QM does not precisely describe the behavior of free particles. In other words, it only describes them as a probability function. It does not assign to them any specific vector in space.

The classical notion of particles is incorrect. The "particles" of quantum mechanics are particular states of a field that is approximately like a classical particle. (the very nature of how this approximation works yields the uncertainty principle)


By that physical definition the notion of length (and therefore distance) would necessarily be dependent on the bound state of matter. Thus the very notion of distance would indeed be subject to the quantized nature of QM.

Recall that it is the generally the energy of the state that is quantized in a bound state; not position. (e.g. the modern electron cloud model of the atom has replaced the Bohr model)


We have no reason to believe that such an infinitely divisible absolute space exists.
That doesn't amount to a very comprehensible idea coming from someone who has denounced the very idea of an absolute space-time that can be warped.

It's unclear how the notion of "relative" and "absolute" measurement bears any relation to these ideas.


All we do know is that QM hasn't failed yet.

IIRC, QM fails spectacularly when we push it too far out of its flat space assumption.



If we correct the basis of mathematical formalism, then Einstein's GR would automatically become discrete because all of mathematics would become discrete.

GR depends crutially on continuity; it simply does not work if you use a discrete space-time. People have studied such models, but no acceptable model has been discovered.

[drnihili]

Originally posted by Hurkyl
May I presume that if Achilles has completed the n-th task, he can complete the (n+1)-th task? The proof then follows by the standard calculus argument; it is then elementary to prove that each of the points in {d - d/2n | n >= 0} lie on Achilles's path, and continuity proves that d lies on it.

Alternatively we may use transfinite induction; if we add a final task "arrive at d" to Zeno's sequence, and agree that this final task can be completed if the entirety of Zeno's sequence can be completed, then transfinite induction guarantees that the "arrive at d" task is completed.


Let's start with the case of finite induction above. Yes, if Achilles can complete task n, then he can complete task n+1. I grant you that there is no task such that Achilles cannot complete it. However, it does not follow that Achilles can complete the list.

Your argument assumes that if each task on a list is completable, then the list as a whole is completable. This is an example the fallacy of composition. You need to explicitly show that the completability of each of the tasks implies the completability of the collection. It's easy, and perhaps instructive, to come up with examples where this sort of reasoning leads us astray.

If the nth task preceds infinitely many tasks, then the n+1th task also precedes infinitely many tasks. Thus each task precedes infinitely many tasks. However, it does not follow that the list as a whole precedes infinitely many tasks. "Precedes infinitely many tasks" is not an composable property.

Not all numeric properties are composable. For example, for any natural number n, if there is there is a natural number bigger than n, then there is a natural number bigger than n+1. However, there is no natural number bigger than all natural numbers.

Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."
Now for any n, the initial segment whose last member is S_n is consistent. However, the entire seequence of sentences is inconsistent. (This, btw, is Yablo's paradox. Perhaps it's inclusion here is a hint that we may really be dealing with omega-inconsistency. I'll have a think on that.)

What your proof above really does is to show that if Achilles completes the list, he will have completed each task and arrived at d. But that has already been granted. What has yet to be shown is that Achilles can complete the list as a whole.

Transfinite induction won't help either as you would need to justify the limit step in order to not merely beg the question.

[I should be clear that NeutronStar and I are grinding entirely different axes. I have no special problem with continuous space. It does have some oddities, but as near as I can tell, discrete space does not fare any better.

I do not know how to resolve the paradox. Thus I am not arguing for any particular resolution of it.]

[NeutronStar]

Hurkyl wrote:
Recall that it is the generally the energy of the state that is quantized in a bound state; not position. (e.g. the modern electron cloud model of the atom has replaced the Bohr model)


Actually that is totally irrelevant. When using an atomic clock we are only concerned with the oscillations between energy states. We are not tracking any imagined position that an electron might be in an imaginary electron cloud.

To think of the oscillations of a cesium-133 atom as a sine wave would be a totally incorrect picture. QM tells us that an atom is either in one energy state, or it is in the other. There are no partial states in between. This is the famous quantum leaping. This is what makes quantum mechanics so mysterious to us. We simply can't comprehend how this can be so.

So if we were to graphically draw the frequency of a cesium-133 atom as it changes states that graphic would necessarily have to be a square wave with zero rise and fall times. (Not a sine wave that we would be tempted to intuitively imagine it to be). QM tells us that this must be so. Therefore our clock is necessarily quantized. And since our definition of distance is based on time then it too must be quantized.

Now that I have had time to think about this a little bit I now realize that Einstein would be forced to accept this.

When Einstein started to postulate relativity he said the following:

Distance (or space) is what we measure with a measuring rod

Time is what we measure with a clock

He was a very practical physicist. He didn't want to deal with any erroneous made-up definitions, he wanted to conceptualize the problem at hand.

Based on these conceptions for space and time we have no choice but to conclude that both space and time are indeed quantized by these definitions.

Therefore any claim that that space can indeed be infinitely divisible is not supported by this model. To make such a claim would require a new definition for both space and time.

Whether anyone else is buying this or not doesn't really concern me. I'm selling it to myself here big time. [:D]

I'm glad that I came to this forum. This is very useful for me to talk these things out. [:)]

Hurkyl wrote:
GR depends crutially on continuity; it simply does not work if you use a discrete space-time. People have studied such models, but no acceptable model has been discovered.


If what you say here is true then GR might ultimately be incorrect.

Gee, I've been studying QM because I thought it was weird. But if GR demands a continuous space then it has its own sort of interesting weirdness.

However, when you say that it depends on continuity exactly what does that mean? I mean, if space is quantized and you move between the only points available isn't that a form of continuity? I mean, there's nowhere else to move to!

We might just be misinterpreting the meaning of infinity as it has to do with continuity there. In fact, I'm willing to bet that this is all there is to that. [;)]

That just goes back to what I keep saying about the problems associated with how we have incorrectly defined the concept of One (and therefore infinity) in mathematics.

[drnihili]

Originally posted by NeutronStar
[argument that our measurement of time must be discrete rather than continuous]

Based on these conceptions for space and time we have no choice but to conclude that both space and time are indeed quantized by these definitions.

Therefore any claim that that space can indeed be infinitely divisible is not supported by this model. To make such a claim would require a new definition for both space and time.



You can't get rid of infinite divisibility so easily. It may be the case that any way of dividing time yields discrete units. However, all that is needed for infinite divisibility is that given any way of dividing time into discrete units, there is another way which divides it into a greater number of discrete units. What you need to show is that this process has an end. For example, you could show that there is no way to divide time more finely than by the motions of a cesium atom.

Of course that example is impossible to prove since we can divide time more finely by using two out of synch cesium atoms. So if time is quantized, it smallest unit can be no bigger than the period of oscilation of a cesium atom divided by the number of out of synch cesium atoms within a light cone. If this were the case, it would have the interesting side effect that size of a quanta was relative to one's location in spacetime.

[Hurkyl]

So if we were to graphically draw the frequency of a cesium-133 atom as it changes states that graphic would necessarily have to be a square wave with zero rise and fall times. (Not a sine wave that we would be tempted to intuitively imagine it to be). QM tells us that this must be so. Therefore our clock is necessarily quantized

First off, only the energy is quantized. The times at which QM allows the atom to change states is not.

Secondly, even if the times at which the atom can change states were quantized, that provides no reason to suppose that the quantization would have the same grain across all possible clocks.

Thirdly, even if there were reason to suppose that the quantization was the same grain across all possible clocks, that provides no reason to think that the allowed transitions would be synchronized across all possible clocks... in fact, because of relativity of simultaneity, such a possibility is forbidden by special relativity.


In summary, this doesn't even provide a mildly compelling reason to think time comes in discrete steps.


This is very useful for me to talk these things out.

I agree with the sentiment; even when arguing against the most hopeless crackpot (I'm not referring to you, incidentally), I find the practice of expressing ideas and concepts useful. (and I gain a greater precision of thought and expression every time)


However, when you say that it depends on continuity exactly what does that mean?

The mathematics of GR use the idea of a "4 dimensional differentiable manifold", which has as part of its definition that sufficiently small open sets on a manifold can be mapped to R4 with an invertible continuous function. This lets us lift the infintie divisibility of R to that of a differentiable manifold.


At a more fundamental level, the topology of "discrete" spacesis radically different from those of "continuous" spaces. The EEP is an axiom about local properties which "discrete" sets don't really have; as you shrink your perspective on a discrete set, it looks different.

(From here on, by a "discrete" topological space I mean a space such that neighborhoods contain finitely many points)

The fact that the neighborhoods of a discrete space have only finitely many points allows you to prove, for each pointX , one of the two following things must be true:

(a) X is topologically "well-seperated" from the rest of the space; the only sets which the point is near are those that it is a part of. It has a neighborhood consisting of only itself.

(b) the point is not seperable from some other point. There is some point Y distinct from such that whenever X is part of an open set, Y is also part of that set.

[Hurkyl]

Your argument assumes that if each task on a list is completable, then the list as a whole is completable.

I did misspeak; I was saying "can complete" when I was thinking "completes", which makes a huge difference.

[drnihili]

Originally posted by Hurkyl
I did misspeak; I was saying "can complete" when I was thinking "completes", which makes a huge difference.

Yes it does. Now you've brought tense into play. As a result your assumption is either incorrect or begs the question depending on where you set the present moment.

[Hurkyl]

"can" connotes something vastly different from temporal concerns. (at least depending on your presumptions!)

But that's a little misleading. The current hypothesis with "can complete" gets formalized to:

P(n) := the set of the first n tasks can be completed. (whatever "can be completed" means)

while the "completes" version gets formalized to

P(n) := Achilles completes the n-th task (meaning he is at the specified destination at some point in time)


The present moment has nothing to do with it.

[drnihili]

Originally posted by Hurkyl
"can" connotes something vastly different from temporal concerns. (at least depending on your presumptions!)

But that's a little misleading. The current hypothesis with "can complete" gets formalized to:

P(n) := the set of the first n tasks can be completed. (whatever "can be completed" means)

while the "completes" version gets formalized to

P(n) := Achilles completes the n-th task (meaning he is at the specified destination at some point in time)


The present moment has nothing to do with it.

"completes" introduces temporal concerns. "can" is just an alethic modality.

The present moment has a lot to do with it. If you take "completes" to be present tense, then your assumption that if Achilles completes task n, then he completes task n+1 is simply wrong. If you are using "completes" atemporally or with a specious present, then the assumption begs the question as it tacitly assumes the completion of the list just as the use of a future tense would.

In any case you still would not have addressed the issue of composition.

[Hurkyl]

And we go back to why I was so insistent on getting precise definitions of terms. The english language is not only horribly imprecise, but highly subjective.

[drnihili]

Unfortunately you're stuck with the messiness of natural language any time you try to make mathematics connect to the real world. Take something as simple as 1+1=2. Is it true? Of course it is mathematically speaking. But once you try to tell me that if I put one apple in a bag and then another that I will have two apples and you give as your reason the fact that 1+1=2, then you've gone and botched it up.

Since the original quesiton deals with the physical rather than the mathematical world, we have to show that the math correctly depicts the real world. Doing that invariably requires some messines. In particular, our original problem is inherently temporal. The question isn't whether the sequence of ever decreasing segments has a finite sum. The question is whether it can be completed when done one task at a time.

So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.

[NeutronStar]

drnihili wrote:
Not all numeric properties are composable. For example, for any natural number n, if there is there is a natural number bigger than n, then there is a natural number bigger than n+1. However, there is no natural number bigger than all natural numbers.


Actually if you are happy with this condition then you should have no problem accepting that even an imaginary space can only be divided up into finite many distances.

My whole basis for the argument that a finite line can only be said to contain an infinite number of points is precisely the same logic used here. Just because the set of natural numbers itself is infinite doesn't force that property of infinity onto any of its elements. In other words, no natural number itself is infinite.

Imagine starting with a line defined only by its two endpoints. (That line can represent a distance or space) We can now place a point between the two endpoints. We now have 3 points. We can place points halfway between those existing points. We now have 5 points. Toss in more points between those, we have 9 points. Do it again we get a line with 17 points and so on.

We are building a set of the finite elements, {2, 3, 5, 9, 17,…}

There is no end to how many times we can do this. In other words we can add more elements to this set endlessly by simply placing more points between existing points. However, it is quite clear (to me anyway) that just because there is no end to the number of times we can do this, it does not follow that we can make the line contain an infinite number of points. On the contrary this example shows that no matter how hard we try we can never force the property of infinity onto any of the elements within the set. So while the number of points that we can add to a line appears to be unbounded, it must also always retain the property of being finite.

I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

drnihili wrote:
Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."
Now for any n, the initial segment whose last member is S_n is consistent. However, the entire seequence of sentences is inconsistent. (This, btw, is Yablo's paradox. Perhaps it's inclusion here is a hint that we may really be dealing with omega-inconsistency. I'll have a think on that.)


I don’t see the problem of a finite number of points in a finite line, or the finite divisibility of a finite distance, being a problem of self-reference.

However you might have a point concerning the completion of an infinite number of tasks. If so, all this would do is move you from Zeno's Paradox into Yablo's Paradox which may very well be a similar situation. (i.e. Something that isn't solvable: a Paradox)

If that’s the case, then you will have just added yet another way of saying that it isn't possible to infinitely divide up a finite space. [:)]

[NeutronStar]

Hurkyl wrote:
First off, only the energy is quantized. The times at which QM allows the atom to change states is not.

Alright, I confess I jumped the gun on that one. [:(]

I'm starting to just post my first thoughts now. That's not good on a forum board. [:D]

This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I'm still convinced that they are, and I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

I do like that way of thinking, and this does imply that time and space are related to material objects rather than to some imaginary idea of an absolute space.

So while I confess to being a complete idiot here, I still claim to be an idiot who's on the right track! [6)]

And this was still good food for thought too! [:)]

[drnihili]

Originally posted by NeutronStar
Actually if you are happy with this condition then you should have no problem accepting that even an imaginary space can only be divided up into finite many distances.

That depends on what you mean by "be divided". If you mean that any activity of dividing yields only finitely many divisions, then I agree. If you mean that there is some inconsistency in supposing that the line is constituted by infinitely many points, then I disagree.


I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.


Simlarly here. I'm with you until you switch to set theory. Set theory is not about tasks or actions. When one talks of constructing the hierarchy of sets, there is no pretense that one actually builds the entire hierarchy. Rather one is setting out rules which define the hierarchy. Those definitions cover an infinite number of cases. However no one can actually carry out the construction for an infinite number of cases.



I don’t see the problem of a finite number of points in a finite line, or the finite divisibility of a finite distance, being a problem of self-reference.

Yablo's paradox does not contain any self reference, so I'm not sure where you're headed with this one.

[NeutronStar]

[I]drnihili wrote:[/]
Simlarly here. I'm with you until you switch to set theory. Set theory is not about tasks or actions. When one talks of constructing the hierarchy of sets, there is no pretense that one actually builds the entire hierarchy. Rather one is setting out rules which define the hierarchy. Those definitions cover an infinite number of cases. However no one can actually carry out the construction for an infinite number of cases.

It was in fact the idea of the rules that I was referring to with the set theory example.

There is no rule (or axiom) in set theory that states that the quantitative property of a set must necessarily be transferred onto any of it's elements. There mere fact that we can continue to constructs elements without bound does not in itself imply that any of the elements within the set must necessarily become infinite.

All I'm saying here is that there is no rule in set theory that allows us to do that, or make that conclusion in any way. If we jump to any such conclusion then we are doing that intuitively on our own, and we really have no logical reason to do it.

On the contrary the rules of addition tell us just the opposite. We have started out with a finite number of point (the two end points). We then begin to add points half way between these two end points, etc. Well each time we do this we are adding a finite number of points to a finite number of points, and we have an axiom somewhere that says that the result of any such operations on sets must also produce a finite set. So from the operation of addition we have proven that all individual elements in our infinite set must indeed be finite.

It seems to me that putting all of this together set theory has indeed shown that any finite line must contain a finite number of points. Even though there is no bound on the finite number that we may choose to express this quantity.

The Points Salemans:

There is a points salesman who has cards that each contain a finite number of points on them. He has an infinite number of cards. Before you begin the race you much purchase a card to describe the number of points in your line. Then you may begin your race. No matter which card you chose from his infinite inventory you will end up with a card that contains a finite number of points. Then when you run the race, you will only need to complete a finite number of tasks.

Zeno's Paradox:

Zeno comes along and says, "Hey wait! Try doing this instead,…". And then he tells you to try stepping half way to the next point each time instead of using the points on a card from the set that defines the possible number of points in a line.

All that Zeno has done was to force you to use the last card in an infinite set of cards. But that is impossible because an infinite set has no last card. Zeno has set you up to fail. He has transferred the infinite property of the main set onto the elements.

How did he do that? It should be obvious. Imagine that you begin with a line that contains only two points. To step half way to the end point you must trade in your card with the 2 on it and request a card with a 3 on it. (in other words, you must dynamically redefine your line and move to the next card in the set). But after you make that step you again run out of points and must trade in your 3 card for a 5 card. Only then can you make the next step. This will continue forever because you are not actually completing the number of points of any particular card, instead what you are being forced to do is to continually choose a new card from the infinite deck with each step. In other words, you are being forced to redefine your line with every step you take!''

To me Zeno's Paradox simply proves that we can't construct an axiom that would allow us to force the quantitative property of a set onto its elements, because to do so would cause a logical contradiction. [:)]

Anyhow, this is how I think of the problem from the point of view of set theory. I'm sure that there are other ways to think of it as well.

[I]drnihili wrote:[/]
Yablo's paradox does not contain any self reference, so I'm not sure where you're headed with this one.

I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!

[Hurkyl]

This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I would too like to see a discrete model of space-time (marginally different than "quantized", but I think you mean discrete anyways)... but I don't go promoting the idea because there's no proof.


I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

Right. In fact, in my casual reading of both GR and QM, the issue of what constitutes a "rod" and "clock" is one of the more interesting and still active research topics. One interesting fact is that, according to QM, any physical clock has a finite probability of giving the same reading twice!


So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.

I am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)


I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

Logic does not tell us this is an impossible task; logic merely tells us that induction is not sufficient to prove it possible.


Incidentally, the question about lines is not whether we can place a set of points on a line, but a question about the set of points that are on the line. So while induction cannot (by itself) prove that we can place infinite points on a line, it can prove that there are an infinite number of points on a line.

(given the reponse written while I wrote this, I think this warrants proof, so I'm writing it now)

[Hurkyl]

(the context is Euclidean geometry)

Theorem: Any line has an infinite number of points on it.

Proof:

Let L be any line.
Choose P on L.


Lemma: For n > 1, there is a set of n points on L, Sn, such that there is a point Qn in Sn such that for any other point R in Sn, R*Qn*P. (That is, Qn is between R and P)

Base case:
An incidence axiom guarantees that any line has at least two distinct points. One of those points has to be different than P; call it A.

A betweenness axiom guarantees a point Q such that A*Q*P. Define S2 := {A, Q}. This satisfies the statement of the lemma for n = 2.

Inductive step:
Let Sn satisfy the statement of the lemma. The same betweenness axiom guarantees the existance of a point Qn+1 such that Qn*Qn+1*P.

For every point R in Sn different from Qn, it is true that R*Qn*P. We also have Qn*Qn+1*P, thus we can conclude R*Qn+1*P. (I don't know wehre my text is, I can't remember if this is an axiom or a theorem) This allows us to conclude that R is distinct from Qn+1

Define Sn+1 := Sn U {Qn+1}. This set satisfies the statement f the lemma.

Thus, the lemma has been proven for all natural numbers n > 1.


Let T be the union of all of the Sn for natural numbers n > 1

Every point in T lies on L (because it is an element of one of the Sn's)

|T| >= |S| = n for all natural numbers n > 1, so T cannot be finite.

Therefore T is an infinite set of points on L, and this concludes the theorem.

[drnihili]

Originally posted by NeutronStar
I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!

Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

But whatever, this isn't a thread about self-reference. I was merely pointing out that there are lots of cases where we can find things such that if they are true of n, they are true of n+1 and yet they are not true of the natural numbers taken as a set.

[drnihili]

Originally posted by Hurkyl
II am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)



That's fine. In that case you have to agree that calculus (nor any branch of mathematics) cannot by itself resolve Zeno's paradox. Any resolution would have to consist not only of a set of axioms and cosequences but also an interpretation of the axioms in terms of the real world and also an argument that the axioms correctly model the world. The first of these tasks might, perhaps, be mathematical, but the second clearly isn't.

So from your revised formalist perspective, you still owe an argument that you have an accurate model of what's going on. At this stage it's entirely unclear how such an argument might go.

[Hurkyl]

Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely seperate question.

[drnihili]

Originally posted by Hurkyl
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely seperate question.

Ah, no answer. Not surprising I suppose.

[NeutronStar]

drnihili wrote:
Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

I was referring to logical self-reference not just any self-reference. Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

Now if the sentence were to be rewritten as ""The first sentence of this post begins with the letter 'S' only if this statement is false." Then we would have a logically self-referenced situation. It's true that the self-reference would be with respect to the entire post, and not merely with respect to the sentence itself. But it is still a form of logical self-reference none the less.

drnihili wrote:
Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."

In this situation each element of the proposed set is making a demand that a certain truth value exist with respect to other specific elements within the same set to which this element belongs. I clearly see the logical self-reference here with respect to the set as a whole. I don't understand how you can seriously deny this. [?]

[drnihili]

I have no idea what you mean by "logical self-reference". However I can assure you that it's pretty uncontroversial that there's no self reference there. Try looking up the original article (by Yablo in Analysis. The whole point of the paradox is to avoid self reference. If you can find a critic who has claimed that Yablo failed in that, I'd be interested.

I grant you that there is a certain oddity to the sentences, but the oddity has absolutely nothing to do with self reference. You also said


Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.


which strikes me as exceedingly odd. Of course the sentence's truth depends on the condition stated. How could it be otherwise?

**Edit**
Are you prehaps worried because the sentence refer to the truth or falsity of other sentences rather than making direct claims?

[NeutronStar]

drnhilili wrote:

NeutronStar wrote:
Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.


which strikes me as exceedingly odd. Of course the sentence's truth depends on the condition stated. How could it be otherwise?

My point is that your example here would not be an example of logical self-reference. It would merely be an independent form of self-reference.

The only time that statements get involved in logical self-reference is when they start making statements about the state of their own truth value, or the truth value of other statements that are directly related to them (like other members of the same set that they themselves belong to).

If expert logicians believe that Yablo's Paradox is not a form of logical self-reference then I have lost complete faith in expert logicians. I can see the logical self-reference clear as day. Not as a self-reference of any particular element onto itself, but as a self-reference relative to the set as a whole.

All of the elements in the set make up the entire set. Therefore if any specific element within the set demands that the truth value of a statement contained in any other element in the same set is dependent on its position relative to the original specific element,… Well, it's clear to me that any set composed on such a rule is definitely referencing its own composition to decide a particular truth state about its own composition.

If that's not logical self-referencing then I don't know what is. [:)]

[HallsofIvy]

drnhili wrote:
quote:
--------------------------------------------------------------------------------
Originally posted by Hurkyl
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely seperate question.
--------------------------------------------------------------------------------



Ah, no answer. Not surprising I suppose.

You mean- no answer that you understood.

[drnihili]

Originally posted by HallsofIvy
You mean- no answer that you understood.

No, I mean no answer. I understood what was said. If you'll take the time to read my post just prior to the one you quoted from Hurkyl, you'll see that I point out that a certain kind of explanation is required based upon Hurkyl stated position. Instead of providing such an explanation Hurkyl merely retreated to his "psuedoparadox" refrain. Then he goes on to say that the question of modeling is a separate one, which is of course absurd given his earlier post about the demarcation.

So, he provided no answer to my post. He has yet to provide any justification for his position that Zeno's is a pseudoparadox. His demarcation post does finally admit that calculus is not sufficient to resolve it. But faced with that admission he simply retreats to a domatic position and avoids the question at hand.

Please understand, I am not saying that calculus is somehow flawed or wrong. It's just not the right sort of theory. Neither is arithmetic, but that doesn't mean arithmetic is wrong.

But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it.

[Hurkyl]

If you'll take the time to read my post just prior to the one you quoted from Hurkyl, you'll see that I point out that a certain kind of explanation is required based upon Hurkyl stated position.

I'd like to see, in your own words, what you think my position is.

[drnihili]

Originally posted by Hurkyl
I'd like to see, in your own words, what you think my position is.

Your position in this regard is that science and mathematics have different purposes. Mathematics is supposed to tell us what follows from what. Science is supposed to tell us what axioms correctly model the real world (or some relevant portion thereof). Further, you believe that science and mathematics have some overlap, at least in practice, but probably also in theory.

So, how close is that?

[Hurkyl]

Well, yes.

But I was asking about my position with regard to Zeno's paradox; the position to which you make an accusation like

"But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it."

[drnihili]

Originally posted by Hurkyl
Well, yes.

But I was asking about my position with regard to Zeno's paradox; the position to which you make an accusation like

"But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it."

Ah, since you quoted a place where I was talking about your position on science and math, I assumed that was the postion you were asking about.

I'm on my way out the door now, so I can't say much more than that you think there is no contradiction involved. When I get back I'll try to give a fuller account.

[drnihili]

Ok, as I understand it, your position is roughly the following.

1. The consistency of calculus demonstrates the consistency of an inifinite divisibility model.

2. Induction is sufficient to show that all of the ever decreasing segments lie on Achilles' path from 0 to d.

3. Since d is the limit of the series of segments, completing the series is sufficent for reaching d.

(I agree with you on 1, 2, and 3 by the way.)

4. The above points suffice to show that Achilles can reach d by covering each of the ever decreasing segments in order. (At least barring a fully rigorous derviation to the contrary.)

(Point 4 is where I see the disagreement)

This is all from memory of posts, so I may be mistaken at one or two spots, but that's where I see your position.

Oh, you also seem to have a general background belief that all paradox is the result of a lack of clarity.

[Hurkyl]

4. The above points suffice to show that Achilles can reach d by covering each of the ever decreasing segments in order. (At least barring a fully rigorous derviation to the contrary.)

You sure you stated this right? If Achilles does cover all of the ever decreasing segments (in order), then we just have point 3.

(actually, this is contingent on the assumption that paths are continuous, and that there aren't any silly tricks like Achilles ceasing to exist at the very instant he would be at d)


Oh, you also seem to have a general background belief that all paradox is the result of a lack of clarity.

I think this warrants further explanation.

Mathematicians have known about the various paradoxes for a while, and to the best of my knowledge they have sufficiently tweaked the formalism so that all known paradoxes are either impossible to state (such as the liar's paradox) or have been reduced to pseudoparadoxes (such as Russel's paradox).

The prototypical example of "lack of clarity" is that of the Twin paradox; the applied formulas require that the reference frame be inertial, but this condition is overlooked and the formula is applied in a noninertial frame. A number of paradoxes arise in this manner merely by making logical mistakes.

[drnihili]

Originally posted by Hurkyl
You sure you stated this right? If Achilles does cover all of the ever decreasing segments (in order), then we just have point 3.

(actually, this is contingent on the assumption that paths are continuous, and that there aren't any silly tricks like Achilles ceasing to exist at the very instant he would be at d)




Yes, I stated it correctly, but there is sufficient ambiguiity in the English to warrant further explanation. A lot depends on emphasis that isn't recoverable from text.

I think you hold that 1-3 demonstrate that the series is completable, that by running one segment at a time Achilles can run all of the segments.

I think we're all in agreement that if Achilles completes the series, then he will have arrived at d. The question is whether Achilles can complete the series. I think you hold that 1-3 provide sufficient argument that he can.

[drnihili]

Originally posted by Hurkyl
I think this warrants further explanation.

Mathematicians have known about the various paradoxes for a while, and to the best of my knowledge they have sufficiently tweaked the formalism so that all known paradoxes are either impossible to state (such as the liar's paradox) or have been reduced to pseudoparadoxes (such as Russel's paradox).

The prototypical example of "lack of clarity" is that of the Twin paradox; the applied formulas require that the reference frame be inertial, but this condition is overlooked and the formula is applied in a noninertial frame. A number of paradoxes arise in this manner merely by making logical mistakes.

All I can say is that the best of your knowledge isn't very good on this point. The Liar paradox continues to be an active area of research and there is no generally accepted solution to it. Tarski's hierarchical approach and Kripke's fixed point theories are perhaps the most commonly cited, but not far behind are defalationsist and various context dependent approaches including at least one based on set thoery with an anti foundational axiom. Not all solutions make the paradox impossible to state, revision theory and paraconsistency are good examples of theories that aim to preserve the paradox while removing it's sting.

There are wide variety of ways of tweaking the formalism to avoid the repurcussions of the Liar in formal systems. Even so, it remains a very open question which, if any, of them correctly models the Liar in natural languages. It would be a gross misrepresentation to say that Liar paradox has been shown to be the result of a lack of clarity. Three are some theories that take that stance, but by no means all of them, nor even the most important of them.

If by Russell's paradox, you mean the Barber paradox, then I could perhaps agree. But that was never really meant as a full blown paradox anyway. It was merely an example of the problems with unrestricted comprehension. Set theory was modified to have only restricted comprehension and thus avoided inconsistency. However the root problem remains, and set theory's response just isn't very illuminating. It is by no means clear that restricted comprehension is ultimately the right answer. Though admittedly it does pretty well for most things. I suspect that Neutron Star would have similar misgiving about resorting to restricted comprehension as he does about resorting to the empty set.

In general, initial mathematical responses to paradox are primarily frantic attempts to avoid inconsistency. They generally succeed in doing so, but it doesn't follow from that that the paradox has been laid to rest or shown to be just the result of foggy thinking.

[Hurkyl]

The question is whether Achilles can complete the series. I think you hold that 1-3 provide sufficient argument that he can.

Ok, this is where you have me wrong.

When we went over Zeno's paradoxes in my Greek philosophy class, they were presented as being an actual contradiction in Greek times; geometery permitted space to be infinitely divisible, but Greeks held infinite sequences to be impossible, and Zeno could derive from that a contradiction... we know Achilles can get from here to there, but we know Achilles cannot get from here to there because of the infinite sequence of tasks in the way.

Most of the time, when the paradox arises on forums, the poster tends to be of that same persuasion, so I typically start off by addressing the typical reasons why one might think an infinite sequence of tasks is impossible... thus the discussion about calculus and infinite series.


You, however, are asking a completely different question. The root of your question is why should one accept a given model of reality. Dressing it up in the guise of Zeno's paradox only serves to obfuscate the issue. And bundling your question up with Zeno's paradox connotes that there's some aspect about the paradox that is relevant to your question, causing me to respond attempting to show that your question + Zeno's paradox is no deeper than your question by itself. (I did comment on this earlier, but it seems to have been missed)

IOW, if you're serious about pressing the "Why should we use model X to describe reality", you really should ask it in its own context, at which point I make the boring statement that mathematics doesn't care about your question, science gives only empirical evidence that model X is practical, and that there has not yet been given any good reason to accept any other particular model. Certainly not logical proof of anything, but I think you're savvy enough to realize you're asking an unanswerable question. (though I do think it was bad form to complain that the unanswerable question was, in fact, not answered)


All I can say is that the best of your knowledge isn't very good on this point. The Liar paradox continues to be an active area of research and there is no generally accepted solution to it.

I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)


However the root problem remains, and set theory's response just isn't very illuminating.

But it is practical. As with logic, the idea of ZFC was to take a minimalist approach that permits us to do everything we like to do, but not have enough power to derive the contradictions. I don't see anything wrong with that, and unless you really want to work with pathologically large sets, nothing is lacking in the solution either.


In summary, one can answer all the paradoxes. It might not be the best answer, but it is an answer, and it is probably contained in any better answer to boot.

[NeutronStar]

On Restricted Comprehension

drnihili wrote:
I suspect that Neutron Star would have similar misgiving about resorting to restricted comprehension as he does about resorting to the empty set.

Unfortunately I'm not familiar with the idea behind restricted comprehension so I can't say.

On Russel's "set of all possible sets" paradox

As far as Russel's Paradox is concerned I was actually thinking that Hurkyl was referring to Russel's set of all possible sets. I do see this as a legitimate paradox, but I also see it as being caused by the flaws associated with the empty set. Repair set theory and Russel's Paradox is not longer possible.

On Russel's "Barber" Paradox

"I shave all those men, and only those men, who do not shave themselves."

As I see it, the paradox of which set the barber himself belongs to is based on the assumption that the above quote is true. My simple solution is to say that the barber's statement is simply incorrect, it can't be true. Where's the paradox? The barber is simply mistaken. That's all.

On The Twin Brothers Paradox

The twin brothers paradox is only a paradox if we insist on keeping absolute time. Since we have accepted the concept of time dilation there is no paradox here. It is well understood, and has just kept its original title as a paradox. No modern physicist sees it as a real paradox.

We may still have questions concerning the actual physics of how time dilation is accomplished, but in general we have accepted that this is the case.

On Zeno's Paradox

I'm afraid that I must take a different stance than Hurkyl on this one. I do see Zeno's paradox as a valid logical paradox. I don't accept the methods of calculus as a solution because, as I understand them they do not profess to solve the problem of completing an infinite number of tasks. They simply give the results of what would happen should they somehow be completed.

In some ways, I might consider mathematical induction in this case. However, as I pointed out many posts ago, assuming the Achilles can make the first step is really cheating, because if we turn the problem around Achilles can never even start the race. In order to start he would first need to step half-way to the first point that he is trying to step to! Ouch! Actually I see this as a mirror-type self reference. (trying to touch a mirror at a point other then were your finger is reflecting on it!) That's not a paradox, its just an impossibility. Unless you silver the mirror on the back side and use very thick glass! [;)]

Like the mirror reflections I see Zeno's Paradox as a valid reflection of reality. I take Zeno's Paradox quite seriously. And just like the Twin Brothers Paradox, which is only a paradox is you are unwilling to let go of absolute time, I see Zeno's Paradox as only being a paradox if you are unwilling to give up the idea of space and time being continuous. Give that up and Zeno's Paradox becomes nothing more than another strange property of reality. It's no longer a paradox if you accept that space and time must be finitely divisible.

Finally, to Hurkyl I would like to say the following:

Hurkyl wrote:
I would too like to see a discrete model of space-time (marginally different than "quantized", but I think you mean discrete anyways)... but I don't go promoting the idea because there's no proof.

However, you appear to be going around promoting the idea that space must be infinitely divisible. Yet, you don't seem to have any real proof for that either!

It's true that I can't prove that space can only be finitely divided. But I can't prove that it can be infinitely divided either! So I promote what I believe to be the most likely case based on other things that I know.

As I have mentioned, I have reason to believe that a finite line can be logically said to only contain a finite number of points. This supports my case logically (I don't claim that it is a proof). However, I haven't seen anything that I would consider to be a stronger proof to the contrary.

I have reason to believe that our conception (and experience) of time and space is based solely on bound quantum states.

I have reason to believe that bound quantum states are quantized.

I have reason to believe that any idea of absolute space must be abandoned.

In short, I have more reasons to believe that space and time must be finitely divisible than I have to believe otherwise. So while I can't prove which case is true, I promote the idea that space and time are quantized and finitely divisible. It seems the more reasonable thing to do based on my current understanding of things.

[Hurkyl]

Unfortunately I'm not familiar with the idea behind restricted comprehension so I can't say.

It's the axiom of subsets thing I mentioned earlier. Unrestricted comprehension means that it is rigorously justifiable to construct the set S such that:

P(x) := x in x
S := {x | ~P(x) }

And then, it is in the language of logic to allow one to ask if S is in S. Both true and false lead to contradictions, thus the problem.


The restricted comprehension axiom (axiom of subsets) only gives us the ability to say

S := {x in A | P(x) }

so with this axiom one cannot construct the problematic set. Now, the axioms of ZF don't say you cannot construct that set; instead the paradox becomes proof you cannot construct the set in ZF (Assume you can, derive contradiction, thus you can't), and thus we say that ~P(x) is a proper class; we can only refer to this class with logical propositions, we cannot make a set contining all of its elements and analyze the whole thing with set theory.


I see Zeno's Paradox as only being a paradox if you are unwilling to give up the idea of space and time being continuous.

I only see it as a paradox if one insists on holding the notion of continuous time and space along with the notion that infinite sequences of tasks are impossible. Really, discrete space is not free from pseudoparadoxes; we can again go with a modified Zeno for this one: if distinct units of time are seperated from each other, rather than being connected in a continuous fashion, then at each instant of time no motion can occur; there is no difference between a stationary arrow and an arrow in flight, so motion is impossible!

(incidentally, one reason science thus far rejects discrete spaces is that every discrete space model proposed predicts blurring of distant objects that simply isn't observed)


Incidentally, QM and GR are why I argue that infinite divisibility does seem to be the right answer; both derive their results by presuming space and time are continuous, and thus far, no discrete model has duplicated their success... and certainly by no means has the continuous model been disproved.

[drnihili]

Originally posted by Hurkyl
I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)

I'm guessing that you're restricting mathematical logic to a sort of standard first order logic as one might see it in a beginning logic course? If so I grant you that the Liar isn't stateable in that language, but so what? Remove the words "true" and "false" from English and it's not stateable in English either. If your only point is that for an paradox we can restrict the expressive power of our language so that the paradox is not stateable, then so what? That's guaranteed by the construction of a null language.

The fact is that mathematics does not typically restrict itself to that sort of logic. But even if it did, there's nothing here about a lack of clarity or precision. There are perfectly clear mathematical treatments of the Liar which don't make it unstatateable. That you choose to dismiss the majority of the field of logic is irrelevant here. Fortunately mathematics is not generally so cavalier.


But it is practical. As with logic, the idea of ZFC was to take a minimalist approach that permits us to do everything we like to do, but not have enough power to derive the contradictions. I don't see anything wrong with that, and unless you really want to work with pathologically large sets, nothing is lacking in the solution either.

Certainly one can choose to work in such a system. Of course one thus gives up a lot of interesting mathematics. However, choosing to work in a restricted system does not amount to solving the paradox.


In summary, one can answer all the paradoxes. It might not be the best answer, but it is an answer, and it is probably contained in any better answer to boot.

Again, merely restricting the expressive power of a language does not amount to answering a paradox. If anything it amounts to a refusal to answer it. As for being contained in any better answer, that's simply false as even a cursory glance and the literature would show.

[drnihili]

Post deleted by the moderator.

[drnihili]

Originally posted by Hurkyl
Ok, this is where you have me wrong.

When we went over Zeno's paradoxes in my Greek philosophy class, they were presented as being an actual contradiction in Greek times; geometery permitted space to be infinitely divisible, but Greeks held infinite sequences to be impossible, and Zeno could derive from that a contradiction... we know Achilles can get from here to there, but we know Achilles cannot get from here to there because of the infinite sequence of tasks in the way.

[...]

You, however, are asking a completely different question. The root of your question is why should one accept a given model of reality. Dressing it up in the guise of Zeno's paradox only serves to obfuscate the issue. And bundling your question up with Zeno's paradox connotes that there's some aspect about the paradox that is relevant to your question, causing me to respond attempting to show that your question + Zeno's paradox is no deeper than your question by itself. (I did comment on this earlier, but it seems to have been missed)

IOW, if you're serious about pressing the "Why should we use model X to describe reality", you really should ask it in its own context, at which point I make the boring statement that mathematics doesn't care about your question, science gives only empirical evidence that model X is practical, and that there has not yet been given any good reason to accept any other particular model. Certainly not logical proof of anything, but I think you're savvy enough to realize you're asking an unanswerable question. (though I do think it was bad form to complain that the unanswerable question was, in fact, not answered)

I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)

Let me just put it this way. As a person with a PhD focusing on Logic who has been empolyed teaching the same for several years, and as a person who has been doing active research on paradoxes for over a decade and who has been focusing on aspects of Zeno's paradox for about 5 years now, I can assure you that your apparent grasp of the fundamentals here is somewhat less that complete. I would suggest that before you go around trying to correct others, that you first acquaint yourself with the related literature.

It's all fine to have a discussion that is based solely on the arguments at hand. I actually prefer that. But when you start making general claims about a field, and when you take it upon yourself to correct others views about the state of that field, it's best to have some acquaintance with the field that goes beyond a brief exposure as part of one class.




Most of the time, when the paradox arises on forums, the poster tends to be of that same persuasion, so I typically start off by addressing the typical reasons why one might think an infinite sequence of tasks is impossible... thus the discussion about calculus and infinite series.

May I suggest that in the future you simply read what others post and attempt to understand them, rather than just adopting a pro forma approach based on the assumption that you think you already know what they are going to say. I think you'll find that it makes discussions more productive for all concerned. You may even learn a bit in the process.

As for the current discussion, I had a try at stating your position, how about you see if you can state mine?

[Hurkyl]

I'm so astonished by your arrogance that at this time I cannot respond to the bulk of your last couple posts without personal attack.


However, choosing to work in a restricted system does not amount to solving the paradox.

It's no longer a contradiction, paradox solved. The restricted system is sufficient to do any mathematics I have ever used and read.

If this does not count as a solution, then maybe you should explain what you mean by "solution".


May I suggest that in the future you simply read what others post and attempt to understand them,

To be frank, you do not have the gift to say a couple sentences and have everyone who hears magically understand precisely what you mean. You might recall that I have spent quite an effort to get you to explain better what you mean (which you have, for the most part, steadfastly refused to do).

Maybe you should heed your own advice. For example, did you notice that you went off on your lecture to explain to me the composition fallacy after I had already brought it up in response to the gods with vows paradox? (though I didn't refer to it by that name)


As for the current discussion, I had a try at stating your position, how about you see if you can state mine?

Aside from the assertion I don't have a clue about what I speak, I honestly don't know what your position is or what point you're trying to make. My most recent best guess was that you were trying to bring up the issue of how we know which model describes reality, but you haven't responded to any of my comments attempting to draw a discussion out of you on the topic.

[drnihili]

Originally posted by Hurkyl
I'm so astonished by your arrogance that at this time I cannot respond to the bulk of your last couple posts without personal attack.




It's no longer a contradiction, paradox solved. The restricted system is sufficient to do any mathematics I have ever used and read.

If this does not count as a solution, then maybe you should explain what you mean by "solution".


Perhaps you should instead set out what mathematics you have used and read and what your basis is for saying that the logic is sufficent. I've given you references to the literature and provided the basis for my own claims to know what the field contains. Thus far all you've done is stated that you once took a philosophy course that mentioned Zeno. If that is your only basis for the sweeping claims you make, then the arrogance that should astonish you is your own.




To be frank, you do not have the gift to say a couple sentences and have everyone who hears magically understand precisely what you mean. You might recall that I have spent quite an effort to get you to explain better what you mean (which you have, for the most part, steadfastly refused to do).

Maybe you should heed your own advice. For example, did you notice that you went off on your lecture to explain to me the composition fallacy after I had already brought it up in response to the gods with vows paradox? (though I didn't refer to it by that name)


Whether what you brought up was the fallacy of composition or just a worry about induction in a particular case is not decipherable from your posts. In any case, whether you previously brought it up is irrlelevant to whether you committed it. When I see someone blatantly committing a fallacy, I think it's appropriate to point it out. If you were aware that you were committing it, then perhaps you owe an expanation of why you did.

And no, you've spent little to no effort looking for explanations. By your own admission (repeated now) you reacted to a thread based upon what you had seen in similar past threads. As for your critique of my writing style, I'm entirely uninterested. When you become an editor, let me know.



Aside from the assertion I don't have a clue about what I speak, I honestly don't know what your position is or what point you're trying to make. My most recent best guess was that you were trying to bring up the issue of how we know which model describes reality, but you haven't responded to any of my comments attempting to draw a discussion out of you on the topic.

Then again I recommend the art of reading. Careful reading that pays attention to what is said. And when you really feel like you don't have a clue about what another person is saying, that's not generally a good time to start telling them that they've got the position wrong.

Now, lest we devolve into simply expressing disgust with each other, let me say again what the issue is, from my point of view of course.

The question is how does Achilles complete an infinite sequence one element at a time, given that the sequence is unbounded.

Caluculus can tell us how a finite segment can be composed of an infite sequence of ever smaller finite segments, but it says nothing on the issue of how one can proceed through the sequence from start to finish. You can't simply appeal to induction in this case, since induction equally shows that the end can never be reached, that there will always remain an infinite set of segments to traverse no matter how many have already been traversed.

So, in short how does one reach the end of a sequence that has no end, but only an upper bound? Not, what is the upper bound, but how can one reach it one element at a time?

[Hurkyl]

Now, lest we devolve into simply expressing disgust with each other

Before moving on, I'd like to point out that it's extremely bad form to get in your last insults in the same breath you suggest we stop insulting each other.



The question is how does Achilles complete an infinite sequence one element at a time, given that the sequence is unbounded.

Caluculus can tell us how a finite segment can be composed of an infite sequence of ever smaller finite segments, but it says nothing on the issue of how one can proceed through the sequence from start to finish. You can't simply appeal to induction in this case, since induction equally shows that the end can never be reached, that there will always remain an infinite set of segments to traverse no matter how many have already been traversed.

So, in short how does one reach the end of a sequence that has no end, but only an upper bound? Not, what is the upper bound, but how can one reach it one element at a time?

I will restate what was my primary question on this. What precisely do you mean by "How can one ....?" An attempt at precise definition or some explicit properties would be nice, but barring that, picking some thing and answering "How can one do this thing?" just to give an example may be helpful.

[ahrkron]

Originally posted by Hurkyl
Before moving on, I'd like to point out that it's extremely bad form to get in your last insults in the same breath you suggest we stop insulting each other.

I completely agree.


Stick to arguments. Credentials have no weight in any discussion, and showing them tends to deteriorate discussions.

Instead of saying "I've studied so and so for so many years", it would be much better for all if you just share some of your allegedly solid understanding. Some may like your arguments, some may not, but such evaluation should come from the arguments themselves, regardless of the time anyone has spent on them.

[drnihili]

Originally posted by ahrkron
I completely agree.


Stick to arguments. Credentials have no weight in any discussion, and showing them tends to deteriorate discussions.

Instead of saying "I've studied so and so for so many years", it would be much better for all if you just share some of your allegedly solid understanding. Some may like your arguments, some may not, but such evaluation should come from the arguments themselves, regardless of the time anyone has spent on them.

Credentials have weight when the issue turns to the question of what the accepted view in a field is. Ideally, this thread should have stayed on the topic of Zeno's paradox and whether it is resolvable. However, it instead strayed to the topic of what the accepted view in a field is. For example, my claims that the Liar paradox is still widely considered to be unresolved requires some substantiation. That substantiation can only be supplied by reference to literature or some other statement of authority as the question is not answerable by reason alone. Likewise, Hurkyl's claims regarding the status of paradoxes, the proper interpretation of Zeno, and the status of mathematical logic cannot be resolved by pure reason. They require some documentation. When documentation is not supplied, credentials are the only thing saving the discussion from brute assertion.

I supplied my credentials only when you specifically brought up the idea of doing so via PM. I do find it passing odd that you should now dismiss them.

So let's put credentials behind us if you wish. But let us also put behind us the sorts of claims that can only be resolved by appeal to them. Or at the very least let's supply some sort of reference to the literature when we make sweeping claims about the status of a field.

[drnihili]

Originally posted by Hurkyl
Before moving on, I'd like to point out that it's extremely bad form to get in your last insults in the same breath you suggest we stop insulting each other.

qed


I will restate what was my primary question on this. What precisely do you mean by "How can one ....?" An attempt at precise definition or some explicit properties would be nice, but barring that, picking some thing and answering "How can one do this thing?" just to give an example may be helpful.

And I'll repeat my answer from before.


If Achilles accomplishes an infinite series of tasks, there must be some action of his which counts as completing all the tasks. But none of the tasks can be that action as each of the tasks leaves an infinite number remaining. So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action.

If there were such an action, then it would be theoretically possible for Achilles to accomplish each of the tasks and yet still fail to complete all of them. This is absurd. Hence there can be no such action.


I obviously can't give you an example of completing an infinite sequence as the possibility of doing so is what is at issue. In a finite case examples are easy to come by. Suppose that Achilles' task is to count all the integers form 1 to 100,000 in sequence. There is an action he does which counts as finishing the sequence - the act of counting the number 100,000. By doing that act he finishes the sequence.

So, either tell me what action of Achilles' counts as finishing the sequence or explain why no such action is needed.

[Hurkyl]

A few questions (to try and flesh out the meaning of things):


If, by some means, it could be proven that Achilles could run a straight line from point A to point B, would that be sufficient to prove that Achilles can complete the infinite sequence of tasks:

{ Go from A to the 1/2 point between A and B,
Go from the 1/2 point to the 3/4 point,
Go from the 3/4 point to the 7/8 point,
... }


If so, would the above be an answer to the question "How can Achilles complete this infinite sequence of tasks?"


If Achilles counted:
1 - 2 - 3 - 4 - 6 - 7 - 8 - ... - 99,998 - 99,999 - 100,000
I presume that this would not count as finishing the task of counting from 1 to 100,000. Can you flesh out the steps in your proof with some rigor? Or if not, would it be correct to say that the completion of the overall task was accomplished, not because it can be said Achilles completed a final task, but because it can be said every task in the list was accomplished?


And in light of the previous observation, would it be correct to say that the sequence of tasks does not need to have a final task in order for Achilles to complete the sequence?


And finally, is the inquiry of the form:

Can Achilles complete the infinite sequence of tasks required to get from point A to point B?

or of the form

Does there exist points A and B such that Achilles can complete the infinite sequence of tasks required to get from point A to point B?

[drnihili]

Originally posted by Hurkyl
A few questions (to try and flesh out the meaning of things):


If, by some means, it could be proven that Achilles could run a straight line from point A to point B, would that be sufficient to prove that Achilles can complete the infinite sequence of tasks:

{ Go from A to the 1/2 point between A and B,
Go from the 1/2 point to the 3/4 point,
Go from the 3/4 point to the 7/8 point,
... }


If so, would the above be an answer to the question "How can Achilles complete this infinite sequence of tasks?"


No, there's no question about whether Achilles can run from A to B. That is physically demonstrable. What is not demostrable is whether his run can be understood as the completion of an infinite sequence of shorter runs. Infinite divisiblity gives some reason to think that it can, but does not give an answer to the question of what counts as completing the series.


If Achilles counted:
1 - 2 - 3 - 4 - 6 - 7 - 8 - ... - 99,998 - 99,999 - 100,000
I presume that this would not count as finishing the task of counting from 1 to 100,000. Can you flesh out the steps in your proof with some rigor? Or if not, would it be correct to say that the completion of the overall task was accomplished, not because it can be said Achilles completed a final task, but because it can be said every task in the list was accomplished?


I'm not entirely sure what you're asking here. Are you asking whether the entire sequence may be taken as a single task? If so, then the answer is that given the 100,000 tasks already on the list there can be no further task which counts as completing them all. The reasons for this are given in my previous post. Given this your proposal in the last sentence is a non-starter.


And in light of the previous observation, would it be correct to say that the sequence of tasks does not need to have a final task in order for Achilles to complete the sequence?


I don't know which sequence you are referring to here. But I am not presupposing that a sequence must have a final member in order to be completed. It is of course easier to understand how such a list sequence might be completed. Understanding what completion of an unending sequence amounts to is part of the current problem though. Appeals to supertasks are fruitless for reasons already stated.


And finally, is the inquiry of the form:

Can Achilles complete the infinite sequence of tasks required to get from point A to point B?

or of the form

Does there exist points A and B such that Achilles can complete the infinite sequence of tasks required to get from point A to point B?

Of those two the former is better, though I prefer the inquiry in the form "How can Achilles complete the infinite sequence of tasks required from point A to B" as it emphasizes that more is required than a simple yes or no response.

[Hurkyl]

Infinite divisiblity gives some reason to think that it can, but does not give an answer to the question of what counts as completing the series.

Are you merely asking for a definition of what counts as completing an infinite sequence of tasks?


I'm not entirely sure what you're asking here.

Mainly I'm asking for you to present the gritty details of a proof.

Your post appears to equate "the completion of a finite sequence of tasks" with "the performance of the final task".

My example was to demonstrate that you're leaving something fairly important implicit.


Is "performing the final task" synonymous with "completing a sequence of tasks"?

Is "all of the tasks in the sequence have been completed" sufficient to conclude "the sequence of tasks has been completed"?


If so, then the answer is that given the 100,000 tasks already on the list there can be no further task which counts as completing them all.

If I understand you correctly, you are justifying this with "So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action."

I don't see how that follows from the description of the problem. Unless you've been leaving it implicit this whole time, the problem does not state "Achilles does not perform any action you can imagine other than those that complete a single task in this sequence... not even actions that can be described as being composed of the aforementinoed allowed actions".

And it's unclear precisely what you mean by "action" anyways. Do actions occur over extended periods of time? Can actions occur instantaneously? Must the completion of a task (or sequence of tasks) be describable by an action? What about the converse?



Of those two the former is better, though I prefer the inquiry in the form "How can Achilles complete the infinite sequence of tasks required from point A to B" as it emphasizes that more is required than a simple yes or no response.

Is the quoted statement synonymous with:

"Can Achilles complete the infinite sequence of tasks required from point A to B? Support your answer with proof."

[drnihili]

Originally posted by Hurkyl
Are you merely asking for a definition of what counts as completing an infinite sequence of tasks?

No, I am not merely asking for a definition. A definition might be part of an answer, but it certainly wouldn't be an answer all by itself. At the very least it would require also demonstrating that Achilles' run could satisfy the definition. It would probably also require at least a prima facie argument that the definition was apt.


Mainly I'm asking for you to present the gritty details of a proof.

Your post appears to equate "the completion of a finite sequence of tasks" with "the performance of the final task".


No, my post explicitly denies that equation.


My example was to demonstrate that you're leaving something fairly important implicit.


If you believe I'm leaving something implicit, then come right out and spell it out. Provide, as you put it, the "gritty details" of a proof on the matter. At the very least provide a clear statement of what you take to be implicit and some reason for thinking that it is implicit based upone what I've said.


Is "performing the final task" synonymous with "completing a sequence of tasks"?


As stated in my previous post, I am not assuming that a sequence needs a final element to be completable. Hence I am not even holding that the two phrases are coextensive, let alone synonymous.

Is "all of the tasks in the sequence have been completed" sufficient to conclude "the sequence of tasks has been completed"?

Quantifiers in English are notorious for introducing ambiguities, but I'll provisionally say yes given something like a substitutional interpretation of them.


If I understand you correctly, you are justifying this with "So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action."


No, if you'll look at the context in which that quote occurs you'll see it's part of a proof by cases. The first case deals with one of the tasks counting as completing the list. Because that case does not hold when the sequence is infinite, I then take the case in the list is completed by some action which is not one of the tasks. However in the finite sequence scenario, the first case does hold.

The problem with a super task, as we have seen before, is that it is empty. As a result it has no explanatory value.

I don't see how that follows from the description of the problem. Unless you've been leaving it implicit this whole time, the problem does not state "Achilles does not perform any action you can imagine other than those that complete a single task in this sequence... not even actions that can be described as being composed of the aforementinoed allowed actions".


No, that's not implicit. Achilles can do lots of things. If you believe he does something beyond the tasks in the list, then please state what it is. Then we can see whether the task could be usefully used to resolve the issue.


And it's unclear precisely what you mean by "action" anyways. Do actions occur over extended periods of time? Can actions occur instantaneously? Must the completion of a task (or sequence of tasks) be describable by an action? What about the converse?


I am willing to countenance both instantaneous and extended actions. As for what an action is, I think the intuitive notion is sufficient for the nonce. If there is some specific ambiguity or vagueness you're worried about, then say what it is and I'll attemtp to clarify it. But at this stage your response is sounding more like a blanket call to define all terms. This of course cannot be done, so pace specific worries about particular words, I'm not going to attempt it.


Is the quoted statement synonymous with:

"Can Achilles complete the infinite sequence of tasks required from point A to B? Support your answer with proof."

No, though I would probably accept "Can Achilles complete the infinite sequence of tasks required from point A to B? Support your answer with an explanation."

[NeutronStar]

I'd like to make a couple of comments on the topic. However, before I do I feel a need to touch on the topic of credentials, simply because it had been brought up previously. I personally have no formal degree at all. However, it would be a gross mistake to take this to mean that I have no formal education. I have actually spent more than enough time in college classrooms to earn a PhD had that been my goal. And much of that time was spent as instructor rather than student so no credits were accumulated during those years.

Furthermore, just recently I have returned to college to basically start over from scratch at an undergraduate level. Why? Because I am more interesting in gaining a solid understanding of the fundamental concepts than I am about gaining credentials that supposedly show that I know what I am talking about when in fact I don't.

Albert Einstein has clearly shown us that a sincere interest to understand a problem, and a genuine dedication to focus your mind on the fundaments is more productive than merely going through the motions to earn credentials. When Einstein wrote his papers on the photoelectric effect, and on Relativity many PhD's did not want to believe him. They were taught to believe in absolute time and space and they did not want to go against what they thought they knew. They were wrong. Einstein was right. So much for credentials.

Now for the problem at hand I can't help but offer the following observation.

There is necessarily a distinction between infinite divisibility and unbounded divisibility. Unfortunately I can’t point to a formal distinction between the two because I'm not even sure whether mathematical formalism has made such a distinction. But if they haven't, they most certainly should.

A while back I posted a comment about "The Points Salesman". He has an infinite deck of cards (an infinite set) and each card (each element of the set) represents a finite number of points. By allowing the runner to chose a card (any card) prior to the race we have a distance that is finitely divisible but it also has unbounded divisibility. There is no bound on how fine we can divide it up. Yet, there is a bound on how we can divide it up - we must divide it finitely.

However, Zeno comes along and rejects the Points Salesman approach. Instead he demand that you continually redefine your line while you run the race. How does he doe this? By demanding that you step to the half-way point to the finish line with each step that you take. This forces the infinite property of the deck (set) onto the cards (elements). But in reality none of the cards (elements) actually has the property of being infinite.

So from a mathematical point of view I can clearly see what Zeno is doing. He is forcing the property of the set onto its elements. Yet in mathematical formalism we have no rule or axiom that allows us to do this!

Zeno is forcing infinite divisibility onto a case of finite but unbounded divisibility.

This is my purely logical resolution to the paradox. It resolves the paradox with respect to our understanding of mathematical formalism. Even though our mathematical formalism is not very rigorous in such matters!

However, I am much more interested in the reality of Zeno paradox. Achilles was a real person in the real universe running a real race to a real finish line. And the cheering from the crowd only serves to emphasize that fact that he can indeed complete the race.

So my question is this: As physicists how can we explain the finite unbounded divisibility of the material universe. And is it really unbound? As far as I'm concerned it is obvious that it is not infinitely divisible. For if it were, we would need to conclude that an infinite number of task can be finitely completed. That flies in the face of our very meaning of infinity. So for this very reason we must reject it. We simply cannot accept the idea that infinity is finite. That is a contradiction of concepts. So we must come up with a concept that has no contradiction. And if we simply accept that space cannot be infinitely divided we have solved the paradox. Thus, this is my solution.

I actually think that Zeno was quite the genius to figure out how to create an illusion of forcing the property of a set onto its elements. But there is really nothing in mathematical formalism that permits us to do this. Even using Zeno's very method to build another set all we end up with is yet another infinite set containing an infinity of finite elements. Zeno's method does not actually transfer the property, it merely give the illusion of property transfer.

[HallsofIvy]

Albert Einstein has clearly shown us that a sincere interest to understand a problem, and a genuine dedication to focus your mind on the fundaments is more productive than merely going through the motions to earn credentials. When Einstein wrote his papers on the photoelectric effect, and on Relativity many PhD's did not want to believe him. They were taught to believe in absolute time and space and they did not want to go against what they thought they knew. They were wrong. Einstein was right. So much for credentials.

Yes, Einstein's papers on relativity (not so much the photo-electric effect) were controversial and resulted in a lot of argument- as they should.

One of the reasons that Einstein's theories were given that consideration was that he HAD the "credentials"- a Ph.D from a prestigious German University and several prior publications.

[sage]

ok this is too long for me to read. i have skimmed through some of the arguments though. here is my question. suppose there is before me an infinite number of apples in an infinite basket. my task is to pick up the apples one by one and throw each into a second infinite basket. my task will be regarded as complete when the first basket is
completely empty. now when will i finish my task if (1)the time i need to pick up an apple and throw it in the second basket remains constant throughout the duration of my task? and (2)the time taken decreases by half for each successive apple i choose?
i have given here two conditions. the second case seems to me to be somewhat equivalent to the xeno's problem. the question is whether it is possible to complete the said task in a finite time and if so how is that physically possible?

[drnihili]

Originally posted by sage
ok this is too long for me to read. i have skimmed through some of the arguments though. here is my question. suppose there is before me an infinite number of apples in an infinite basket. my task is to pick up the apples one by one and throw each into a second infinite basket. my task will be regarded as complete when the first basket is
completely empty. now when will i finish my task if (1)the time i need to pick up an apple and throw it in the second basket remains constant throughout the duration of my task? and (2)the time taken decreases by half for each successive apple i choose?
i have given here two conditions. the second case seems to me to be somewhat equivalent to the xeno's problem. the question is whether it is possible to complete the said task in a finite time and if so how is that physically possible?

As stated neither of your problems has an answer. This is because it is possible to remove an infinite number of apples from the basket and still have an infinite number remaining. In order to get a solution we need to assume that the apples are ordered, say by each haveing a label with a number on it, and that you take them out in order.

Given that revision, and given the assumption that it is possible for you to finish the tasks at all, the answers are:

1) When eternity has passed.
2) In twice the time that it took for you to move the first apple.

Now, let me add a further twist. Suppose you place a coin on a table, and each time you move an apple you turn the coin over. The coin starts out showing heads, and so shows tails after the first apple is moved, heads after the second, and so on. When you finish your task, is the coin showing heads or tails?

[dschou]

zeno will finish the race quite quickly since the calculator he uses to determine the length of each step has limited precision and will quickly underflow (after about 330 steps for a TI-83). thus an age-old paradox is unraveled with the advent of the semiconductor.

:)

[HallsofIvy]

I'm glad we've got THAT settled!