Infinity Question

Moving on, it is, to me at least, an interesting paradox to note that the following problem is ill-posed.
[ul]Suppose that at time t = 0 a dish is empty, and that for each non-negative integer N, at time t = 2 - (1/2)[sup]N[/sup], God takes two chips from a cache of identical chips, and simultaneously places them into the dish and removes one of the chips in the dish. How many chips are in the dish at time t = 2?[/ul]Somehow, in what order the chips are removed matters, even though they are all identical. The universe, as it were, knows in what order the chips are being placed into the dish, and, dispite their identicalness, keeps track of when each is removed. For, after all, something has to happen at the two-hour mark, and the only way for the universe to know what to do is to keep track of that information.

In other words, the problem is ill-posed because it doesn’t mention the additional presence of a sort of “tagging” operation that would be occuring, were the experiment actually performed. Even though the chips are, in all respects, indistinguishable, the universe is saying “for the duration of this experiment, you’re chip 1, you’re chip 2, …” and so on. But there are degrees of freedom to this hidden tagging operation that make different outcomes possible, depending on how the tagging operation is performed in a particular case.

Perhaps this is a clue to a solution to the paradox with the lightbulb being turned on and off at Zeno-type intervals. Is the light on or off at the 2 hour mark? This phrasing of the question seems not to determine an answer, but perhaps, in each particular case where this procedure is performed, a similar unmentioned “tagging” operation is performed in one of several possible ways, and the particular way in which it is performed does suffice to determine the outcome.

Here’s another classic example to think about:

Imagine a square, each side one unit long. Consider two opposite corners of the square, say the bottom left corner and the top right corner. Imagine there’s a “stairway” from this bottom corner to the top corner. Say it has n steps, and the steps are evenly spaced, as staircases usually are. What is the length of this staircase? Well, all the horizontal segments add up to one unit, as do the vertical segments, for a total of two units.

What happens as we let the number of steps increase, without bound? Well, the length of the staircase as still two units. Additionaly, the staircase begins to approximate the diagonal connecting the two corners, with greater and greater precision as the number of steps increase.

By the logic being used to show there are infinitely many chips in the box, we can conclude that the length of the diagonal is equal to the limit of the length of the staircase, as the number of steps increases.

And we’ve just proven that 2 = sqrt(2).

The lesson to be learned–limits don’t preserve everything. In particular, limits don’t preserve the number of chips in the bowl in the example we’ve been considering.

Naturally, at any stage in the staircase process, it is true that the distance between two points that are along the diagonal line between two steps is sqrt(dxdx+dydy) where dx is the x delta, and dy is the y delta. Since dx = dy, this is sqrt(2dx) of course, or sqrt(2)sqrt(dx), and if you sum all those up over the width of the cube, you get sqrt(2). This must remain true even if you have stairs that approach being smooth.

While the staircase approximates the diagonal, at any particular finite stage dx+dy (or 2dx since dx = dy) never really represents the distance along the diagonal even though the staircase approaches a smooth diagonal.

This is an excellent demonstration of the fallibility of limits – one has to be terribly careful.

oops, I meant since dx=dy, this is sqrt(2dx*dx) which ends up being dx * sqrt(2), but the rest of it applies.

At each step, the thickness of the stack increases by 1/(2k(2*k-1). For any epsilon > 0, I can give you a step N beyond which the sum is always within epsilon of S (= log(2)). Isn’t this (more or less) the definition of convergence?

Does not this argument apply to any infinite sum? How can I ever say any infinite sum, e.g. Sum [k = 1 to infinity] {(-1)[sup]k+1[/sup] / k }, has a value? My book says this is equal to log(2), and I bet you’ve got more than one book which agrees with this. How would you “demonstrate that it still has any meaning when we take it to the infinite level”?

If the chips are not numbered, does the height of the stack converge to a value greater than zero? If instead of summing chip thicknesses, I sum line lengths in the same fashion, certainly you agree that converges, don’t you?

ZenBeam

Yes, but what’s missing is whether or not this is an appropriate way to describe the height of the chips after the full two hours is up.

It’s important to remember that we’re not actually adding up infinitely many numbers. The sum of the infinite series means that, as we add up the terms of the series (in that particular order), the sum gets arbitrarily close to log(2). That’s important in some cases. For example, if you wanted a decimal expression for log(2), you could sum up some initial terms in that series–the farther you go out, the closer you’ll be–that would be a nice application of an infinite series.

I should mention here yet another counter-intuitive aspect of infinity. About that infinite series just mentioned above–pick a real number. I can rearrange the terms of that series so that it will converge to the real number you picked. Again, order is important here.

In the original example, at each step the box contains certain integers; after all the steps have been taken, the box will still contain a subset of the integers. The limit of the number of chips in the intermediate steps is not the same as the number of chips in the box in its final state–it’s that simple.

If you’re talking about adding chips and removing chips along the way, then no there’s no way to talk about convergence in any way. Remember, the ordering of the chips is important, just as the ordering of the above infinite series is important. You might add and remove the exact same chips, only in a different order than before, and the height of the stack will converge to a completely different value.

Same thing with line lengths, if we go along and say something to the effect, “Then at this step I add in a line segment of length one, and take away a line segment of length 1/2, then add a line segment of length 2, take away a length of 2.3,…”, then it’s absolutely impossible to say anything about the what the length will be in the end. And if you do the same steps in a different order, you might get it converging to a different value (just as the infinite series demonstrates).

That’s right, I see what you’re saying. Thanks, ultrafilter.

No problem.

Even if a function f(x) converges to a particular value y[sub]0[/sub] as x approaches a point x[sub]0[/sub], it needn’t follow that f(x[sub]0[/sub]) = y[sub]0[/sub].

So, in the case of you height function H, given by
[ul]H(N) = Sum [k = 1 to N] {1/(2k*-1) + 1/(2k* )- 1/k },[/ul]we do indeed have that H(N) converges to some nonzero number y[sub]0[/sub] as N goes to infinity, and by a certain abuse of notation, we could write
[ul]H(infinity) = limit [N --> infinity] H(N) = y[sub]0[/sub].[/ul]Now, by doing this, we would be defining H(infinity) to be the limit of a certain sequence of partial sums. But there is no guarantee that this definition would match the more natural definition of H(infinity), which is as the height of the stack of chips in the dish at t = 2 hours. And, in fact, under this second definition, H(infinity) = 0.

Okay, I was wrong - about my last post being the last post I’ll make in this thread. Once more into the breech…

It seems to me that the underlying sticking point here is that I don’t accept that the process as described can ever stop. Because the set of +ve integers is infinite (a point on which I trust all agree), therefore there is no last step - the whole argument about what’s in the bowl after the deity of your choice has finished simply doesn’t apply.

But the limit as n aproaches infinity of 1 + n is infinity. Surely you don’t have trouble understanding this? I’m not saying that is is a particular number, but that is is larger than any number you can name, including 0.

Alright, at which step was the last chip in the bowl removed? You can’t answer this because there is no such chip. The number of chips in the bowl at each step increases. There is no last step. The bowl is never empty. So there.

And that is the last I have to say here.

I’ve been away for awhile, (at least from this thread). So, what’d ya’ll come up with?

Did y’all reach some kind of agreement or what? I don’t know if I even remeber the OP at this point, but IIRC dish #1 was the infinite set of chips and …dish #2 was the I set of chips minus 1, right?

That makes one set all even # chips and one set all odd # chips. Which is in fact a set of two which can be described as (n, n +1…to infinity).

Did I miss something or is someone attempting to say there are no chips in the first dish?

While I might be willing to ponder the idea of there only being one chip in set one…I don’t think I’ll agree with zero chips being in set one.

anyway, just thought I’d check in and see how it was going with you diehards…:smiley:

Everything you say here is true, except for the very last sentence. That conclusion simply does not follow. If such reasoning were valid, Zeno’s paradoxes would prove the impossibility of motion. If I leave my room, there may be no last point that I pass through before I reach the door, yet still I do pass through allthe points between here and the door, and in finite time to boot.

I’ll give a situation more similar to the one under discussion, to show that there being no last chip proves nothing. Consider this thought experiment. Two dishes lie before God. At the beginning, dish A contains infinitely many chips, each labeled with a unique one of the nonnegative integers, and dish B is empty. For each nonnegative integer N, at time t = 2 - (1/2)[sup]N[/sup] hours, God takes the chip labeled N from dish A and puts it in dish B.

With this setup, you’ll have to agree, every chip is taken from dish A and placed in dish B before 2 hours have passed. Therefore, after 2 hours have passed, dish A will be empty. This dispite the fact that there was no last chip removed from dish A.

If God threw the removed disks into another bowl, would it be empty too? Would it have more disks that the first bowl, or less? Why? Doesn’t it get disks at the exact same rate? So, an infinite series of steps which is stated to have ended must have an infinite number of steps, but it must also be ended, and therefore must have a maximal value, a unique number. But the definition of infinite denies that it can have such a value. Now you want to know the value.

This insistence upon naming a specific transfinite integer is silliness. OK, let’s ask God. “What is the lowest number?” God says, “Gazornipleximan.” “But wouldn’t that one have been removed after you had done gazornipleximan steps?” We ask. “Gazornipleximan is an infinite number, two times Gazornipleximan is still Gazornipleximan. Not only that, but Gazornipleximan over two, minus one is also equal to Gazornipleximan.” But you can’t have numbers that work like that! We howl. “Yes, I can, I’m God, remember? That’s why you needed me to do this silly trick in the first place, because no one else can do something infinite and be done at a specific time. When you do that, all the numbers come out Gazornipleximan.”

Infinite doesn’t have a specific value. Asking for the specific integer value of a number stated to be developed by an infinite series is just a silly question. It’s infinite. No, the arithmetic doesn’t work, because it’s infinite. This is sort of why infinite shit doesn’t happen with integers in physical phenomena.

Tris

So there?

OK I give up.

The very fact that ‘infinite doesn’t have a specific value’ immediately suggests that there isn’t a specific chip in the bowl. If, at any point of time, including, if you wish, the point at the two hour mark, you look at the bowl, you can only say ‘there are this many chips’ is if you can actually look and see that a chip is there. Any chip.

You can easily do this at any point up to but not including the two hour mark, knowing full well that any chip you can describe that you see will be gone in a finite period of time.

At the two hour mark, there is no way to enumerate, no way to describe, no way to see, no way to point to, no way to discuss any coin or coins in the dish. Whether you believe that naming a specific one is silliness or not, every actual single coin ever involved in the problem is given a unique integer by the definition of the problem, and coin i has already been removed.

If you start talking about the coin of index Gazornipleximan, and Gazornipleximan is ‘an infinite number’, and infinite numbers are not specifically an integer, then which coin are you referring to? If you’re not referring to a specific coin, then the fact that you’ve given it a name is irrelevant – it could refer to any coin. Using an index that cannot possibly refer to a specific coin to suggest that that coin is in the dish is, to me, the true silliness. There’s no justification.

Appealing to the ineffability of infinity is not very convincing. It seems perfectly reasonable that one bowl is empty and the other isn’t.

Give god an infinite number of bowls. Have him take out the Nth prime at time N and all of its multiples, and put them in one of his infinite bowls (which are marked 2, 3, 5, 7, 11, and so on). Now he has an infinite number of bowls with an infinte number of chips. (given the time formula as before etc). This is damned counter-intuitive, but this is how we do math.

That is simply what happens. Because we are dealing with infinity, the rate at which we approach it is irrelevent. It takes just as long to get there if we count by 1s, 2s, or primes, and we pass just as many numbers. Only, if we are removing some on the way, the manner in which we do so will determine if any, and how many, are left.

Can I get a hell yeah?

Works for me. Infinity is counterintuitive, and despite objections otherwise, we’ve established some of these counterintuitive ‘truths’ and ‘rules’ to help us deal with goofy things like Zeno’s Paradox, converging geometric sums, and integration near one of these infinite points.

Integrating under the curve f(x)=1/(x^2) is always a good example… Chaos to those who try to evaluate f(x) at point 0, but we can still compute the area under the curve in a range of x that lies across this point.

What if god decided to pursue a slightly different strategy? Instead of putting the 2 smallest remaining chips into the dish, he put the 2 smallest remaining even chips in, and if there were no even chips left, then he’d put the 2 smallest remaining odd chips in. Then he’d remove the smallest chip (perhaps with the same preference for even chips?) from the dish as before. So, he’d put in 2,4, take out 2, put in 6,8, take out 4, etc. Perhaps, at some point, all the even chips would be in the dish or already taken out, so he would put in the 1,3, etc.

If he followed the same time pattern, where would the chips be after 2 hours? Case 1: if he takes out the smallest even chip, and only removes an odd chip if there are no even chips to remove, then, assuming ultrafilter’s arguments to be correct, all the even chips would have been taken out of the bowl. But where would the odd chips be? Are they all in the bowl? But when did he put them in? Did they all go in instantaneously at the very end of the time limit? Or not yet in the bowl? But then why did he stop putting chips into the bowl if there were still some left? Or all taken out of the bowl? But then when did he remove them? Wasn’t he always in the process of removing even chips? Did they all go in and out instantaneously?

Case 2: he removed the smallest chip from the bowl, whether it was even or odd. Similar questions apply for the odd chips, and the state of the even chips is also not clear.

Is there an answer in these situations? Are the answers different? Does the question even make sense?

In that last case, no odd chip will ever be put in the bowl, because there will always be more even chips left to put in.

All of these problems are good examples of why we don’t talk about completing infinite processes. There are just too many contradictions.

I agree with ultrafilter… ‘Enter At Your Own Risk’, and do so treading lightly.

knock knock points out an interesting problem. Strictly speaking, there is no time period before two hours when all the even coins are gone. After the two hour mark it cannot reasonably be said that even coins remain, since no distinct coin can have been left behind. However, since this condition only ‘exists’ instantly following the two hour mark, there has been no time to begin putting in or taking out the odd chips. Since the two hour mark ends the experiment, the even chips have been put in and taken out, but the odd chips remain untouched, never entering the dish.

That’s my take on it anyway.

sorry, I meant ‘since no distinct even coin can have been left behind’, but I’m sure you parsed that.