owl

What about the survival or welfare of other species?

Extra nitpicking: "Natural numbers" is sometimes used to refer to all positive integers, and sometimes to all nonnegative integers. I.e., sometimes "0" counts, sometimes not, but it's more common not to include "0". :) See, e.g., natural number definition - Dictionary - MSN Encarta Definition of natural number - Merriam-Webster Online Dictionary Natural Number -- from Wolfram MathWorld

Here's a simpler version of the last paradox discussed: There's a device for counting natural numbers. It displays the natural numbers on a screen, in sequence: "1", "2", "3", and so on. It displays "1" for one second. Then it displays "2" for half a second. Then it displays "3" for a quarter second. And so on. Anything on the display remains there until it is changed by another number being displayed. At the end of 2 seconds, what is on the display? Let me add this: the device has no such symbol as "[math]\infty[/math]". It only has the characters "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". If this were a finite sequence, it would be displaying the last number in the sequence at the end. But there is no last member of the sequence of natural numbers. The only available answer seems to be, "This series is impossible."

Great question. Probably if you did that, the jar would explode, annihilating you and all the balls. The scenario, I take it, goes like this: You start with balls labeled with all the natural numbers except numbers of the form 5n. In the first round, ball #1 gets relabeled "5". Then ball #2 gets relabeled "10". And so on. In general, in round n, ball #n gets relabeled 5n. So the original ball #1 will get relabeled like this: Round 1: "1" [math]\rightarrow [/math] "5" Round 5: "5" [math] \rightarrow [/math] "25" . . . Round [math]5^n[/math]: "[math]5^n[/math]" [math] \rightarrow [/math] "[math]5^{n+1}[/math]" . . . Your question, then, is what is the end result of carrying out this process, according to this pattern, infinitely many times? "Undefined" seems to be the only answer. The succeeding changes don't converge on any determinate result. It would be tempting to answer that ball #1 will end up labeled with the symbol that refers to [math]5^{\infty}[/math]. But there is no such symbol, or at least, there need be none as far as the scenario is concerned. Ball #1 only gets relabeled with natural-number symbols. We could suppose that the symbol "[math]\infty[/math]" doesn't even exist.

No, you know there is no last stage. The stages are infinite (endless). I meant to say the set of natural numbers (integers greater than 0) is well-ordered. Anyone in set theory will tell you that, besides that it's intuitively obvious. This means that any subset of it has a least member. I don't mean any finite subset; I mean any subset. Hyperreals are generated in the relabeling case because each ball gets a "0" added to its label, infinitely many times. They're not good in the original case, because no ball ever gets its label changed in any way. An easier way to think about the problem, in these two cases, is to think about the balls outside the jar. Erasmus put forward the principle that the end result should be a function of what's going through the neck of the jar during each stage. But here's an equally plausible principle: The set of balls that end up in the jar at the end should be the complement of the set of balls that end up outside the jar. This must be true because no balls are ever created or destroyed, no ball is ever in more than one place, and the scenario only has two locations: in the jar, and outside the jar. Now, in the original problem, just look at the pile of balls outside the jar. In the succeeding stages, it looks like this: 1 1, 2 1, 2, 3 ... There's an ever-growing pile, with labels 1, 2, 3, and so on, including all the natural numbers with no gaps. But in the relabeling version, the balls outside the jar look like this: {} {} {} ... That is, a person looking outside the jar repeatedly sees nothing there, ever. So in the original version, the set of balls inside the jar is the complement of {1, 2, 3, ...}, which is the empty set. In the relabeling version, the set of balls inside the jar is the complement of {}, which is the set of all balls (although they've been relabeled).

Basically, yes. That's a similar objection. However, it is not as compelling. The universe may be infinite in extent. If it is, there may actually be infinitely many balls scattered throughout the universe. However, even if this were the case, the ball-and-jar process still couldn't be performed, as doing so would require the ability to move objects arbitrarily fast. I think an infinite object, or collection of objects, can exist spread throughout space. So you might say, analogously, it should be possible for an infinite event to exist spread over time. I would say, yes, there can be an infinite event; however, it cannot be completed by the nature of the case. The infinite event would take an infinite amount of time, just as an infinite object would take up an infinite amount of space. But an event that takes an infinite amount of time literally "takes forever", so it is never complete.

The above two quotations ask pretty much the same question. The answer is that the hyperreal numbers are not generated at any particular stage of the series. Rather, each of the hyperreal numbers is generated gradually, and takes the entire infinite series to be finished. One zero gets added at a time; only after infinitely many stages of the series have been completed do you have the hyperreal number with the infinite number of zeros. The reason these hyperreals don't appear in the original version of the problem is that, in the original version of the problem, no ball ever gets any zeros added to its label, so all balls have their original (natural number) labels at the end. But in the relabeling version, each ball gets infinitely many zeros added over the course of the series. Not exactly. It's true that the end result can't be determined by looking at the qualitative character of the balls entering and leaving the jar. Instead, it depends on the numerical identity of the balls entering and leaving the jar. In other words, when ball "10n" goes into the jar, it matters whether this is a new ball that hasn't previously been seen, or if it is the same ball as one that was previously in the jar, merely disguised to look like a new ball. This doesn't matter at any finite step of the series, of course; it only matters, at the end, which alternative has been the case over the entire series.

Right. In the relabeling version, the jar will be filled with balls each of which has a hyperreal number on it, viz., a natural number followed by infinitely many 0's. How about this: after the first minute, you turn the dial up to twice the speed. Then after another 30 seconds, you turn it up to twice that speed. Etc. So you also finish this process in 2 minutes. And you don't find a jarred integer. Good point. It tells us that the assumption is false. The integers are well-ordered. There is no set of integers that doesn't have a smallest member--except the empty set.

But there's another way of extracting an answer, which works in some cases: in some cases, it's possible to reason that the same thing happens (at some stage) to each item in a certain infinite class of objects, so at the end of the infinite series, that thing has happened to all of them. When the "limit" method of reasoning fails us, this other method can still work. Well, the procedure is not quite the same at any step: in the relabeling procedure, no ball leaves the jar; in the removal procedure, a ball leaves the jar at each stage. In the latter but not the former procedure, there is an ever growing pile of balls outside the jar. Weird, isn't it? The pair of cases violates the very plausible principle that the end-result of the infinite series should be a function (solely) of the qualitative state of the jar at all the individual stages of the series. In the two versions of the story, the jar has qualitatively indistinguishable states at every point in the series--i.e., it contains the same number of balls, with the same numbers written on them. And yet the end result is different. Although this is very weird, I think it's true, because of the reasoning we've been discussing here. Note that I said the state of the jar was "qualitatively" the same at each point in both versions of the series. It's not absolutely the same: In one version, the ball labeled "10" is identical to the ball previously labeled "1"; in the other version, it is a new ball. A similar weirdness afflicts this other version of the scenario that I previously mentioned: what if, instead of taking out the lowest-numbered marble at each stage, you take out the highest-numbered marble? Then you wind up with infinitely many marbles at the end. This is weird, because the number of marbles in the jar is the same at every stage as in the original version of the story; all that differs is which marbles are in the jar. For any finite series, this wouldn't make any difference. But this is just one more of the weird things about infinity: it does make a difference with infinite series. I think Ross' original point was that the series is logically impossible; a supertask cannot be completed. If so, then perhaps we do not need to (or perhaps it does not even make sense to) answer what would result if this impossible scenario occurred. In this case, I think the series is, indeed, impossible: no one could move marbles fast enough to complete the series. But there are other infinite series that can be completed (Zeno). So the interesting question is: when is an infinite series completable, and when is it not?

Again, that is not what I'm saying. My argument is not that because infinitely many balls are removed, there must be none left. that because you can group numbers as above, you can make the infinite sum converge to 0. Both of those arguments are fallacious, for reasons that have been pointed out in this thread, at length. (b) is wrong because it's illegitimate to rearrange the terms of an infinite series, for the reason I pointed out in post #34. Also, (a) is bad because infinitely many balls are added as well. And both (a) and (b) are bad because parallel reasoning could be used to support any answer to the problem, from -infinity to +infinity, or anything in between. Again, what my reasoning is is just this: For every ball, it is removed at some stage and never subsequently returned. Specifically, ball #n is removed at stage n of the series. Therefore, every ball will be outside the jar when the series is completed. No one has addressed this reasoning, as opposed to the clearly fallacious reasoning of (a) and (b) (which have been responded to repeatedly). You're right about this: the sequence whose terms are the numbers of balls at the succeeding stages of the series diverges. That sequence has no infinite sum; the infinite sum is undefined. What you're wrong about is thinking that that's relevant to the problem. The problem is not an infinite sum problem in mathematics. The question is not, "What is the limit of the number of balls in the jar, as the number of stages in the series approaches infinity?" The question is, instead, "How many balls are in the jar at the end of the series?" As I pointed out above, those questions are logically independent. The answer to the first question is just a matter of the trend that you can observe while you're at any finite stage in the series. The second question, on the other hand, asks you to suppose that infinitely many stages have been completed. Again, the crucial point here is that the definition of "[math]\sum_{i=1}^{\infty}a_i[/math]" in mathematics is not that it's the number you get after adding together all of the infinitely many terms of the series. The expression is defined in a certain way specifically to avoid the assumption that you can ever complete the infinite series. That's why the balls-and-jar problem is not a problem about infinite sums in the mathematical sense: because this problem does ask you to assume that the infinite series gets completed. Everyone agrees that the limit of the series is +infinity. But that's just not the question. That is valid. That example is in the Wikipedia "Supertask" article that started this discussion, where the correct answer to that problem is also given. The answer can be arrived at by the same kind of reasoning, namely, by considering what happens to each ball in the series. In the original version, for every n, ball #n is removed at stage n and never returned. So the jar is empty at the end. In the relabeling version (by Allis and Koetsier), for every n, ball #n stays in the jar, but has a "0" added to its number at stage #n, 10n, 100n, and so on. So every ball remains in the jar but has infinitely many zeros added to it. So at the end, there will be infinitely many (specifically, [math]\aleph_0[/math]) balls in the jar, each of which will have a natural number followed by infinitely many 0's written on it. No, in the original version of the story (Ross' version), there is no such ball. Again, all the balls are labeled with natural numbers. And by the way, the expression "[math]\lim_{n\to \infty}10^n[/math]" is undefined, since the series is divergent.

This seems to be intended to suggest that, because the argument can be used to "win" a debate with someone, it is therefore not really, objectively sound, or it is somehow less cogent than a purely "mathematical" argument. This is mistaken: the reason why the argument could be used to win a debate with someone is precisely that it's valid. (It's not as if it would help you "win" a debate by means of rhetoric or emotional appeals; it's a purely logical appeal.) It's also mistaken because, as I mentioned in posts 45 and 50, the balls-and-vase reasoning is perfectly analogous to arguments that are used all the time in mathematics. So there's no relevant contrast between this reasoning and "mathematical" reasoning. First, I don't know why you didn't say the [math]\aleph_0[/math]th ball is in the jar, because that ball never gets removed either. Only natural-numbered balls get removed. Second, either answer would be mistaken, because no ball labeled "[math]\aleph_0[/math]" or "[math]\aleph_1[/math]" is ever in the jar in the first place. The problem description says that the balls are labeled with natural numbers. There is never any ball, anywhere, labeled "[math]\aleph_1[/math]" or anything else that's not a natural number. Among people who study this sort of thing, I'm virtually certain that almost 100% of them, whether mathematicians, logicians, or philosophers, would agree with the argument that the jar will be empty. There's no difficult mathematics involved, and the details of the theory of transfinite cardinals are irrelevant since, as noted, no transfinite cardinals are on any of the balls in the first place.

No, you can't label infinitely many objects with finitely many numbers. But yes, you can label infinitely many objects, each with finite numbers. There are infinitely many finite numbers, so they are enough to label infinitely many objects. There are infinitely many real numbers between 0 and 1. But none of those numbers is infinite: they're all between 0 and 1. Likewise, there are infinitely many natural numbers. None of those numbers is infinite: they're all natural numbers.

Yes, of course. If the reasoning works for the "10 and 1" case, it must work equally for "x and 1" for any other x. Right. The limit of the number of balls as x approaches infinity is +infinity, but the actual number of balls after completing infinitely many steps is 0. As I mentioned earlier, this isn't inconsistent, because the limit claim is just a claim about what is happening as you go to larger and larger finite-numbered steps of the sequence. It says nothing about what happens at "stage infinity," if you will. This is not all that strange. Compare a somewhat analogous case. Consider the function, f(x) = 2x, for all x not equal to 3 = 1, for x=3. This is a legitimate function, though a discontinuous one. The limit of f(x) as x approaches 3 is 6. But the actual value of f(3) is 1, not 6. This should drive home the point that "Lim f(x) as x approaches a" is not the same as "f(a)". I'm not sure what you mean by "we are not dealing only with integers." Are you saying that at the end of the series, there will be some balls labeled "infinity" in the jar? In one sense, we are dealing only with integers: in the description of the problem, every ball has an integer written on it. No ball has "infinity" written on it. Thus, once all the integers have been accounted for, all the balls have indeed been accounted for. Where you might be getting confused is in the distinction between the number of steps in the series, and the numbers on the balls in the series. The number of steps is infinite. But the numbers on the balls are all finite. This is perfectly consistent, though it can easily confuse one: there are infinitely many, finite numbers.

It is very tempting to think the paradoxes of infinity should be addressed by saying that it's impossible to complete an infinite series. However, the case of Zeno's paradox seems to refute this. For Zeno's argument is based exactly on the claim that you can't complete an infinite series. It seems that the answer to Zeno is that the infinite series can be completed. (The answer is not to deny that a line segment is infinitely divisible.) Thus, if you say the infinite series in the other paradoxes of infinity are not completable, then we need an explanation for why that is.

No, you can't determine the answer from looking at the number of balls present in the jar at each stage. Again, the argument is not "infinitely many balls are removed, so the jar will be empty." The argument is that every ball is removed, so the jar will be empty. Notice that if you remove a different ball at each stage, you can get a different result. Suppose that in each stage, you remove the highest instead of the lowest numbered ball. In that case, the jar will end up with infinitely many balls. This is because all and only the balls with numbers of the form 10n are removed. In the original version of the problem, you always remove the lowest numbered ball, so the jar will be missing the balls with numbers of the form 1n (which is all of them). The reasoning that you posted above overlooks these crucial facts: it overlooks the fact of which ball is removed at each stage, and only pays attention to how many balls are added and removed at each stage. Although that would be fine for any finite series, it's not fine for this infinite series. CraigD's equivalence principle is mistaken for the same reason. The problem as originally posed does require that the balls be numbered, or at least that the balls be ordered in some way. What determines the ultimate outcome is whether there's an ordering of the balls that corresponds to the order in which they're removed, such that for every ball, there is a time at which that ball is slated to be removed. If so, then the jar will be empty; if not, then not. My question to you, Craig, and Q. stands: name a ball, any ball, that might be in the jar at the end of the series.