Questions about infinity

[owl]

The paradox depends on an assumption made by Zeno's hypothetical protagonist: that any infinite number of finite numbers sum to infinity.

 

As indicated in my last message, there's no need to attribute this error to Zeno. Instead, the argument can be read as relying on the claim that a supertask cannot be completed. This is closely related to Aristotle's doctrine that an "actual infinity" is impossible; only "potential infinities" are possible. This sort of view has been very widely held, historically. Before Cantor, I believe it was the standard view.

 

Their intuition didn't include the idea of infinitesimals necessary to conclude that an infinite number of finite numbers including infinitesimal numbers may be finite.

 

It's true that the standard view did not include infinitesimals. However, the standard view today does not include infinitesimals either. The point of all those "delta-epsilon" proofs in calculus was precisely to eliminate infinitesimals, so that the calculus would have a sound conceptual footing. Cauchy & Weierstrass (sp?) proved (pace Leibniz) that you don't need infinitesimals to do calculus.

 

I think there's still some confusion about this, however, because Leibniz' symbolism (the dy/dx stuff) survived, and it was designed to suggest reference to the "infinitesimal" quantities that he believed in.

 

You don't need infinitesimals to talk about convergent infinite series, either. As you know, the sum is just defined as the limit of the series as the number of terms increases.

 

(Aside: You don't need to talk about "infinity" as a number either. You don't have to say the number of terms "approaches infinity"; instead, you can just say the number of terms "gets larger without bound". The talk of approaching infinity is nonsensical anyway, if you think about it: every finite number is equally far away from infinity: namely, infinitely far. So as long as you're in the finite realm, you're never really approaching infinity. And if you're in the infinite realm, you're not

approaching

it either.)

 

The terms in the sequence get smaller and smaller, but they're always finite (not infinitesimal, not zero).

 

A study of the history of mathematics supports the idea that Zeno and his contemporaries were not unaware of, but rejected the idea of infinitesimals. Classical Greek Math – which was not as distinct from Physics and other "natural philosophies" as it is now – appears to have subscribed to atomism, implying the existence of a smallest number greater than 0.

 

I don't think this was a widely held view. For one thing, Zeno's paradox assumes the contrary: he assumes that for any distance, you can always cut it in half. Also, the Greeks had a proof that the diagonal of a square is incommensurable with the side (in our terminology: their quotient is an irrational number), which is obviously incompatible with both lengths being made up from a (natural) number of basic units.

 

Mathematical thought then (ca. 500 BC) and for the next couple of millennia, continued to be uncomfortable or actively reject infinitesimals. Not until the 20th century were infinitesimals truly formalized, by folk including Abraham Robinson.

 

Right, but the current standard analysis still rejects infinitesimals, despite Robinson. I think it's right to do so. My main reason is that I don't have any grasp of any such number. It's supposed to be a number that is "so small" that when you take infinitely many of them together, they add up to a finite number, like 4. That doesn't make any sense to me. You can say those words, and you can come up with formal axioms for it, but I still don't have any idea of what you're talking about.

 

By the way, another manifestation of the contemporary rejection of infinitesimals (besides the delta-epsilon definitions) is the standard treatment of continuous variables in probability theory. The probability of any given point is standardly said (correctly, IMO) to be 0.

[Qfwfq]

OK, yesterday I wasn't able to get what I was actually thinking into concrete terms, I tried instead to make it workable in terms of "after all the infinite steaps have been performed", which is fraught with inconclusivity. If one admits that this itself makes any sense at all, it follows of necessity that one must reason on labels with a number not finite. If you don't admit to that making sense, then abandon all hope ye who enter...

 

Anyway ;) the above troubles made me loose sight of what I originally had in mind, the essence is that the jar is inherently a problem about a sequence, while the prime number issue isn't. This holds independently of how one may choose to argue the set of primes and is basically why I deem the comparison inappropriate. Your argument 2a isn't reliant on any whatsoever sequence and is conclusive. 2b "pretends not to be" but hence can't truly represent the jar case.

 

At each step of the jar riddle, one ball is removed and 10 are added, this is simultaneous to the purpose of the riddle, so when ball #n is removed ones with number greater than n + 1 are added and this is before the one numbered n + 1 is removed. At no whatsoever step the last remaining one is removed. About your query, when saying "For each n" it can't mean an n greater than all numbers. While it is true that any specific n will eventually have it's day, the empty jar is a non sequitur. There is no mechanism by which the assert, referred to each, comes to imply the same referred to all. This is essentially the failing of all 4 of your b versions.

 

The first "2" of Zeno's paradoxes are equivalent. What you present as Zeno's argument fails regardless of not considering quantities; when one knows that a series may be convergent to a finite sum, the premise of refutation is defeated by modus tollens.

 

Zeno’s Paradox of the Arrow:

 

1. When the arrow is in a place just its own size, it’s at rest.

 

2. At every moment of its flight, the arrow is in a place just its own size.

 

3. Therefore, at every moment of its flight, the arrow is at rest.

 

1 is a very arbitrary assumption and, if 2 is granted being true and 3 false, then 1 simply is false by modus tollens. I don't see why it should be reasonable to suppose all three, and the only thing contradicted by his argument is the aut-aut between rest and flight.

 

However, the standard view today does not include infinitesimals either. The point of all those "delta-epsilon" proofs in calculus was precisely to eliminate infinitesimals, so that the calculus would have a sound conceptual footing. Cauchy & Weierstrass (sp?) proved (pace Leibniz) that you don't need infinitesimals to do calculus.

 

I think there's still some confusion about this, however, because Leibniz' symbolism (the dy/dx stuff) survived, and it was designed to suggest reference to the "infinitesimal" quantities that he believed in.

In my calculus courses the definitions of infinitesimal and infinite hold for terms and not for numbers (apart from the infinite cardinality which a set may have, a quite different matter). Leibniz notation is used simply because it is handy, it's OK as long as it is used in manners based on proper formal footing.

 

You don't have to say the number of terms "approaches infinity"; instead, you can just say the number of terms "gets larger without bound".

These two aren't equivalent...

The talk of approaching infinity is nonsensical anyway, if you think about it: every finite number is equally far away from infinity: namely, infinitely far. So as long as you're in the finite realm, you're never really approaching infinity. And if you're in the infinite realm, you're not approaching it either.)

Approaching is, again, shorthand for a statement about limits. The way to define limits involving infinity is by defining a neighborhood of infinity.

 

You can say those words, and you can come up with formal axioms for it, but I still don't have any idea of what you're talking about.

This isn't a modern mathematician's point of view.

 

By the way, another manifestation of the contemporary rejection of infinitesimals (besides the delta-epsilon definitions) is the standard treatment of continuous variables in probability theory. The probability of any given point is standardly said (correctly, IMO) to be 0.

Correct, when the distribution isn't singular. Discrete combinatorial problems may be viewed on par with continuous ones by adopting singular measures in lieu of the euclidean one. An example is the way quantum formalism is constructed for general spectra. Most of all, here be we talking examples of application. They don't necessarily say all there is to say about mathematics.

 

The real, proper grounding for limits and convergence is topology, as I have repeated already in this thread, actually very basic topology is sufficient. Are you feeling up to discussing these matters based on it?

[CraigD]

A study of the history of mathematics supports the idea that Zeno and his contemporaries were not unaware of, but rejected the idea of infinitesimals. Classical Greek Math – which was not as distinct from Physics and other "natural philosophies" as it is now – appears to have subscribed to atomism, implying the existence of a smallest number greater than 0.

I don't think this was a widely held view. For one thing, Zeno's paradox assumes the contrary: he assumes that for any distance, you can always cut it in half.

 

Another way to interpret the paradox is as a logical impeachment of the assumption “that for any distance, you can always cut it in half“, and an assertion of the alternative that some distance cannot be subdivided, just traversed in some amount of time. This is consistent with an atomist’s view of reality. The paradox then becomes a syllogism, roughly:

• If the physical universe were infinitely divisible, per Zeno’s arguments, motion would be impossible;
• motion is not impossible;
• therefore, the physical universe is not infinitely divisible.

This interpretation seems to have renewed popularity with the growth of popular awareness of quantum physics – even the wikipedia article “Zeno’s paradoxes”, arguably the most popular reference on the subject, alludes to a connection between the paradox and quantum physics.

 

Even if an atomist interpretation was what Zeno actually favored (a difficult thing to confirm, as none of his original work remains), I think it’s misguided. IMHO, the ancient and modern popularity of Zeno’s paradoxes – that they were and are considered to be paradoxical, not simply wrong – is due to a deep-rooted, intuitive human reluctance to equate time and space. Ancient and modern philosophers seem comfortable with the idea that an infinite number of ideal measuring sticks can be concatenated to make a finite distance, but uncomfortable with the idea that the same is true of the concatenation of an infinite number of durations. On a gut level, something seems to wrong with the latter idea – our everyday experience with such things as picking up spilled beans makes us skeptical of the idea that an infinite number of happenings can require less than an infinite amount of time.

 

:shrug: Intuition, IMHO, is like fire – very useful and powerful, but catastrophic if not carefully kept in check.

[ughaibu]

Qfwfq: At each time there are two actions, putting in ten marbles and removing one, the actions are peformed in that order, so the latest action is the removal of a marble, and as all marbles have numbers on them and each number corresponds to a time as in Gallileo's paradox, all marbles will be removed.

[owl]

Qf,

 

We seem to be failing to communicate somehow, so I'm going to try responding in some detail (please forgive me if this is tedious). You should be aware at the start, though, that the reasoning given in the Wikipedia article about the infinite series is standard logic. It's definitely valid. Any logic professor would tell you that, and you could formalize it using principles that you would find in any standard formal logic textbook. (I've not only taken, but taught formal logic, and while I have some unconventional views, this isn't one of them.)

 

That's not to say that everything about those examples is uncontroversial. In particular, the matter of whether the series in question are logically or metaphysically possible is controversial. But it's uncontroversial that the reasoning given about what would follow IF you suppose the series possible is correct. I know an appeal to authority is not very satisfying if you don't see how the conclusion follows, so I'll try to make it clearer presently. But I think the appeal to authority should be enough to motivate you to try to see why the reasoning is correct, rather than trying to figure out why it's not.

 

Anyway :shrug: the above troubles made me loose sight of what I originally had in mind, the essence is that the jar is inherently a problem about a sequence, while the prime number issue isn't.

 

I don't understand why one would think that would make any difference.

 

At each step of the jar riddle, one ball is removed and 10 are added, this is simultaneous to the purpose of the riddle, so when ball #n is removed ones with number greater than n + 1 are added and this is before the one numbered n + 1 is removed. At no whatsoever step the last remaining one is removed.

 

This is true, but does not show a flaw in the reasoning. It does not show that some ball will be in the jar at the end. There is no last ball, so the fact that "the last ball" doesn't get removed does not show that something will remain in the jar.

 

Analogy: At no step is a giraffe ever removed from the jar, so does this show that the jar will not be empty at the end? No, because no giraffe was ever in the jar. Similarly, a "last ball", presumably a ball numbered "infinity", is never in the jar, so doesn't have to be removed. This is related to the following point:

 

About your query, when saying "For each n" it can't mean an n greater than all numbers.

 

Good, we agree on that. There is no n greater than all the other numbers. There are just greater and greater natural numbers, without bound. That's why the reasoning is correct. Since it holds for every finite n, and the finite values of n are all the values, it holds for all n.

 

While it is true that any specific n will eventually have it's day, the empty jar is a non sequitur.

 

It isn't a non sequitur. I don't see why you say that. "Specific" n's are the only n's there are, so if all of them are removed, then all the balls are removed.

 

All I can think to do at this point is to rephrase the argument. So I'll rephrase it as a reductio. You think that the jar will be nonempty. You agree that this means that there will be at least one ball in the jar, correct? Tell me what you think about this ball: Do you think it will have a number written on it? If so, will it be a natural number? Could you give an example of one of the numbers that might be written on it?

 

Any answer you give to these questions will contradict a stipulation of the scenario.

 

There is no mechanism by which the assert, referred to each, comes to imply the same referred to all. This is essentially the failing of all 4 of your b versions.

 

Okay, I think you are saying this: "For each x, Fx" does not entail "For all x, Fx" ("Fx" stands for any statement about x). That's mistaken. "For each x, Fx", "For every x, Fx", and "For all x, Fx" are equivalent in standard predicate logic. You can look at any standard formal logic text. If you decide to do so, the rule of inference used in the argument is commonly known as "Universal Generalization (UG)" or as "A-introduction" (but put the "A" upside down to make it a "for all" symbol; I don't know how to do that here). It should be intuitive: if, for any particular man that you choose, that man is mortal, then this is the same thing as saying all men are mortal. And there's nothing funny introduced here by infinity: this holds even if there are infinitely many men. It holds just because the words "For every" and "For all" are synonymous in this context. (It also holds whether or not there exists an ordering of the men, so the introduction of sequences makes no difference.)

 

Digression: There are contexts in which "each" and "all" have different uses in English, but this isn't one of them. An example is "There's a number greater than each natural number" versus "There's a number greater than all natural numbers." The former is true (on its most natural reading), the latter false. But the difference is just one in the

scope

of the quantifiers; it's not that "each" and "all" introduce different kinds of quantifiers. What I'm calling the "quantifiers" are things like "for each", "for every", and "there exists an x such that". To explain:

• "There's a number greater than each natural number" = "For every x, there exists a y such that y is greater than x".
  
  
• "There's a number greater than all natural numbers" = "There exists a y such that, for every x, y is greater than x".
  
  

So the difference is just that you switch the order of the quantifiers when you use "for each" and "for all". But the same quantifiers are present. Now, in the context we were discussing, this isn't relevant, because the relevant statements only have one quantifier (the "for each" or "for all" quantifier), so there is no such scope ambiguity possible.

 

The first "2" of Zeno's paradoxes are equivalent. What you present as Zeno's argument fails regardless of not considering quantities; when one knows that a series may be convergent to a finite sum, the premise of refutation is defeated by modus tollens.

 

Of course, I agree that Zeno's arguments are wrong, since motion in fact occurs. But you haven't explained why the original one is wrong. To do so, you'd have to say which premise of it was false, and then explain why. The argument went like this:

 

1. An infinite series is a series with no end.
   
2. If a series has no end, then it is not possible to get to the end of it.
   
3. To complete a series, one must get to the end of it.
   
4. Therefore, it is not possible to complete an infinite series.
   
5. The series "1/2, 3/4, 7/8, ..." is an infinite series.
   
6. Therefore, it is not possible to complete the series, "1/2, 3/4, 7/8, ..."
   
7. To get to the ground, the ball must complete that series.
   
8. Therefore, it is not possible for the ball to get to the ground.
   

To respond to Zeno's paradox, you'd have to say specifically which premise, (1), (2), (3), (5), or (7) you're rejecting; or you'd have to say which inference, (4), (6), or (8), you're saying is invalid. (And then, of course, explain why. But you haven't done even the first step, so explaining why is moot.) The talk about convergent series and limits doesn't address that.

 

These two aren't equivalent...Approaching is, again, shorthand for a statement about limits. The way to define limits involving infinity is by defining a neighborhood of infinity.

 

I can't tell if you're trying to agree with me or not. I was saying that "as x gets larger without bound" is a better phrasing than "as x approaches infinity." So maybe you're agreeing with me.

 

But "As x approaches infinity" is just a misleading way of saying "as x gets larger". The two statements are equivalent in the way they are used in standard calculus--that is, the former is used as just a (misleading) way of saying the latter. So maybe you're disagreeing with me when you say "these two aren't equivalent".

 

If I understand what you mean in your last statement, I think it's mistaken. I think the idea of a neighborhood of infinity doesn't make sense, since no real number is any distance from infinity (unless you want to say it's an infinite distance, in which case they're all the same distance). Rather, here's the right way of explaining the "as n approaches infinity" talk:

 

We say "the limit of f(x) as x approaches infinity is L," when:

For any d > 0, there exists an e > 0 such that, whenever x is greater than e, f(x) is within d of L.

 

The important part is that that indented statement does not make any reference to infinity--it does not imply that infinity is a number, or that you get "closer and closer to infinity", or that infinity even exists. Now as far as I can tell, the indented statement doesn't "define a neighborhood of infinity", so I think I'd disagree with your statement.

 

[i said about infinitesimals:] You can say those words, and you can come up with formal axioms for it, but I still don't have any idea of what you're talking about.

This isn't a modern mathematician's point of view.

 

I'm a philosopher, not a mathematician. However, I think some mathematicians, namely the Platonists, would agree with me. That is, Platonists would agree that you have reason to believe a mathematical object exists if and only if you "grasp" that object.

 

The real, proper grounding for limits and convergence is topology, as I have repeated already in this thread, actually very basic topology is sufficient. Are you feeling up to discussing these matters based on it?

 

I don't understand what you mean by that, so you'd have to explain it. I think the standard treatment of limits and convergence in calculus books is perfectly adequate (it's just that many people misunderstand it). I'm not sure if you're questioning that, or if you're rather referring to that standard treatment as a basic kind of topology.

[CraigD]

I’m not sure the following, from the wikipedia article on the “balls and vase” (Ross-Littlewood) paradox, is accurate:

Since by noon every ball n that is inserted into the vase (at step n/10) is eventually removed in a subsequent step (step n), the vase is empty at noon.

 

Mathematicians generally agree that this solution is correct, given the particular conditions of the question.

As a calculation, the problem is trivial. The number of balls in the vase at step n is simply 9n. Allowing n to be the transfinite number aleph-null ([math]\aleph_0[/math]), 9n = [math]\aleph_0[/math] – both are just “countable infinity”.

 

The argument that the number of balls at noon is zero is contained, essentially, in the challenge “tell me the number of one ball in the vase at noon”. No matter what number you select, the challenger can tell you precisely when it was removed, and that time will always be before noon.

 

The argument for “zero balls at noon” is, IMHO, more one of “argumentative logic” than of mathematics. Though it’s difficult to distinguish a “mathematician” from a “philosopher” – many mathematicians are very philosophical, and many philosophers very mathematical – I find the claim that “most mathematicians” accept the “zero balls at noon” solution, or that anyone would accept this solution unless they accept a definition of “correct” like “a correct claim is that made by whoever wins an argument over it”.

 

PS: I’ve been unable to find much about either David Ross or (first name unknown) Littlewood. Publications by and about them seem to be in Philosophy journals, leading me to suspect that at least one was a professional philosopher, possibly at Oxford. Does anyone know anything of the history of the Ross-Littlewood paradox, or its namesakes? :QuestionM

[pgrmdave]

It seems to me, at least, that the jar problem is a problem of comparing infinities - hence the logical fallacy.

 

Consider that for every marble removed, there are nine more added. Consider, though, that we remove every marble. We remove infinite marbles, so infinite integers are gone. Even though we added nine times more than that. I'm not a mathematician, and I'm not nearly as good as some others here, but doesn't it seem logical to simply conclude that the infinities can't be compared, hence the apparent paradox? That [infinity] / [infinity] does not equal 1?

[owl]

Responses to 3 posts:

 

Another way to interpret the paradox is as a logical impeachment of the assumption “that for any distance, you can always cut it in half“

 

That's a conclusion someone could draw from it, but it isn't Zeno's conclusion, and I doubt whether anyone has drawn that conclusion. Zeno was really, seriously arguing that nothing moves. He was a student of Parmenides, who maintained that change is impossible.

 

Aristotle's response to Zeno also grants the infinite-divisibility assumption. He tries to solve the paradox by claiming that there is only a "potentially infinite" series, not an "actual infinity". But he definitely grants that you can always cut a line in half.

 

And again, the existence of lengths with irrational quotients, which was known to the ancient Greeks, conflicts with the discrete view of space.

 

Ancient and modern philosophers seem comfortable with the idea that an infinite number of ideal measuring sticks can be concatenated to make a finite distance, but uncomfortable with the idea that the same is true of the concatenation of an infinite number of durations.

 

I can't speak for anyone else, but I don't myself find any such intuitive resistance. Time and space seem to me, intuitively, to have the same structure (the structure of the continuum, except that space is 3D).

 

I think the explanation of the paradox's appeal is just this: It seems logically impossible to complete a series that has no end.

 

---------------------------------

 

As a calculation, the problem is trivial. The number of balls in the vase at step n is simply 9n. Allowing n to be the transfinite number aleph-null ([math]\aleph_0[/math]), 9n = [math]\aleph_0[/math] – both are just “countable infinity”.

 

That's not right, and here mathematicians would back me up. You've written the (putative) equation, "nine times infinity = infinity." This doesn't make sense in standard mathematics, because one cannot perform arithmetical operations on "infinity", since it is not a real number. "9 times infinity" doesn't make sense. One way to see this is just to see what happens if you try applying standard arithmetical operations to infinity:

 

9i = i

Subtracting i from both sides:

8i = 0.

 

So, the argument for the jar being full is defective. The argument for its being empty, on the other hand, is entirely conclusive.

 

I find the claim that “most mathematicians” accept the “zero balls at noon” solution, or that anyone would accept this solution unless they accept a definition of “correct” like “a correct claim is that made by whoever wins an argument over it”.

 

You seem to be saying that the argument isn't objectively correct, that it's a matter of opinion or of how you define "correct". It isn't. The argument is deductively valid. That's an objective fact. There is no room at all for disagreement about that. That might sound dogmatic, but it is, nevertheless, the truth, as any professor of logic would tell you. Just as there's no room for disagreement about whether there are infinitely many prime numbers. Anyone who disagrees with that claim simply doesn't understand the proof.

 

And formal logic is just as objective as (the rest of) mathematics.

 

------------------------------------

 

Consider that for every marble removed, there are nine more added. Consider, though, that we remove every marble.

 

That's correct. The latter consideration is what shows that the jar is empty.

 

We remove infinite marbles, so infinite integers are gone.

 

That is not the argument. The argument is not, "We removed infinitely many marbles, so it must be empty by now." If that were the argument, it would indeed be subject to dispute, and in fact it would be wrong. (What if the problem had stipulated that all the even-numbered marbles are removed, while the odd-numbered ones remain? Then infinitely many marbles are removed, but it would obviously be wrong to say that the jar would be empty.)

 

Instead, the argument is, "For every marble n, n is not in the jar. Therefore, the jar is empty."

 

...doesn't it seem logical to simply conclude that the infinities can't be compared, hence the apparent paradox? That [infinity] / [infinity] does not equal 1?

 

Again, that would be a fine response if the argument were as you represented it above. But it isn't. The argument does not in any way involve doing arithmetical operations on "infinity" as if infinity were an ordinary number. It doesn't involve multiplying infinity, dividing it, subtracting it, or comparing the sizes of infinities. You don't even have to use the word "infinity" or any of its cognates at all.

 

Again, the idea is very simple: if every marble is removed from the jar and never put back in, then the jar is empty.

 

Are Craig and Pgrmdave suggesting that "For every x, Fx" does not entail "For all x, Fx"?

[Qfwfq]

Qfwfq: At each time there are two actions, putting in ten marbles and removing one, the actions are peformed in that order, so the latest action is the removal of a marble, and as all marbles have numbers on them and each number corresponds to a time as in Gallileo's paradox, all marbles will be removed.

I think you aren't considering that, before the nth ball is removed, ones with larger n have been added. Perhaps it wasn't too clear in my previous post, in terms of n and n + 1 and trying to be brief, so I'll resort to a single case. At the step in which ball 10 is removed, balls are added with label greater than 11 and this, obviously, is before ball 11 is removed. How then can the jar remain empty?

 

It seems to me, at least, that the jar problem is a problem of comparing infinities - hence the logical fallacy.

In the jar problem there aren't infinities of different order. 10 times an infinity is of the same order, this is why the ratio of two infinities is indeterminate even if they aren't of different orders.

 

What's more, and relevant to this problem, the difference between infinite quantities is also an indeterminate. Specifically, 10I - I (where I is an infinite positive term) is infinity and certainly isn't zero.

 

We seem to be failing to communicate somehow

Yes, I think so too, and I can't spend nearly as much time as would be necessary, much as I like this kind of thing. :)

 

Now, don't get me wrong, I don't think the problem is of using non standard logic. I think you are still missing what I repeated to Ughaibu above. I'm not surprised of you saying that you even teach formal logic :cool: but I have found you less qualified (as you admit) and criticized you is on the mathematical aspects here.

 

I don't understand why one would think that would make any difference.

Well I do!!! :D And I think you yourself should try to see why the reasoning is correct, rather than trying to figure out why it's not. ;)

 

This is true, but does not show a flaw in the reasoning. It does not show that some ball will be in the jar at the end. There is no last ball, so the fact that "the last ball" doesn't get removed does not show that something will remain in the jar.

It depends on how you understood my words. I was trying to be brief. As I repeat in replying to Ughaibu, no ball will be removed when it is "the last" or, better, "the only one at that step" in the jar. This is the difference between saying "every ball gets removed at some step" and "the jar will be empty", making the empty jar a non sequitur.

 

Analogy: At no step is a giraffe ever removed from the jar, so does this show that the jar will not be empty at the end? No, because no giraffe was ever in the jar.

Unlike the giraffe, balls having a greater number on their label have been put in, and before the removal of the n-th.

 

It isn't a non sequitur. I don't see why you say that. "Specific" n's are the only n's there are, so if all of them are removed, then all the balls are removed.

 

All I can think to do at this point is to rephrase the argument. So I'll rephrase it as a reductio. You think that the jar will be nonempty. You agree that this means that there will be at least one ball in the jar, correct? Tell me what you think about this ball: Do you think it will have a number written on it? If so, will it be a natural number? Could you give an example of one of the numbers that might be written on it?

This gets into the matters I began my previous post with, we'd have to agree whether or not it makes any sense at all to say certain things. Loosely, "after all steps" there will be more than one ball, so it won't be "a" number. In order to give any meaning to what your asking, we'd have to admit we can say there's an infintiy of balls in the jar and none of them have finite n on there label. Where does this contradict a stipulation of the scenario? The fact that you don't contemplate a ball's label's number not being finite? Consider this and answer my question: You are talking about after all steps, consider the set of balls that have been put in. There is at least one ball in this set and, instead of asking your question, I ask you what number is on that ball's label. Does this question make more sense than yours, or less?

 

I fully inderstand what you say about predicate calculus, including the digression and that's exactly what you need to see in the problem of the jar and which I couldn't have spelt out so clearly myself especially for lack of time (except I might have used LaTeX to write [math]\norm \forall[/math]). With a bit more thought, I might have just said that the jar riddle can't be treated by arguing on a fixed set, you must apply what you call "the scope of the quantifiers" in making the distinction I made yesterday, which I have repeated here above. The set of balls in the jar changes at each step, not only with balls getting removed but with balls of greater n being added. The set does not commence as being all of N (set of naturals) and, in everyday nitwit's terms "the more balls you take out, the more ya got left to take out". The subtleties of infinity: "every ball will be removed" fails to imply "the jar will be empty".

 

To respond to Zeno's paradox, you'd have to say specifically which premise, (1), (2), (3), (5), or (7) you're rejecting; or you'd have to say which inference, (4), (6), or (8), you're saying is invalid. (And then, of course, explain why. But you haven't done even the first step, so explaining why is moot.) The talk about convergent series and limits doesn't address that.

First define the meaning of "the end" of the infinite series (sequence?) and that of "getting to" it, and that of completing the series. Then I might be able to argue with Zeno. ;)

 

We say "the limit of f(x) as x approaches infinity is L," when:

For any d > 0, there exists an e > 0 such that, whenever x is greater than e, f(x) is within d of L.

 

The important part is that that indented statement does not make any reference to infinity--it does not imply that infinity is a number, or that you get "closer and closer to infinity", or that infinity even exists. Now as far as I can tell, the indented statement doesn't "define a neighborhood of infinity", so I think I'd disagree with your statement.

That is a high-school definition, not one from a proper treatment of calculus based on topology. See what I said upstream. The neighborhood of infinity can be defined without a metric and this is what is done --implicitly-- in the definition you quote.

 

Don't be in a hurry now because, unfortunately, I can't afford much more time on this and I've hardly read your exchange with Craig. I hope I'll be able to further discuss your points by Wedensday or so. Sorry for any typos or worse, cheers...

[ughaibu]

Qfwfq: During the enactment of the task, if the contents of the jar are examined, there is always a time at which such an examination is made and, accordingly, there is a finite number of marbles in the jar defined by the time and the number of actions that have occured. So, at any point while the task is underway, all numbers are finite it's only when the task is completed that there is an infinite number of marbles, and as finite numbers have different properties from infinite numbers, the contents of the jar, when the task is complete, don't follow in a direct sequence with the preceding numbers.

[owl]

What's more, and relevant to this problem, the difference between infinite quantities is also an indeterminate. Specifically, 10I - I (where I is an infinite positive term) is infinity and certainly isn't zero.

 

Sorry to keep harping on this, but: Expressions like "10I - I" don't make sense. :naughty: It's not that 10I - I = infinity, and it's not that it's unequal to infinity; it's that it's undefined.

 

We might agree, since you said it was "indeterminate". But then you proceeded to putatively evaluate the expression anyway.

 

In some contexts, if you treat "10I - I" as if it made sense, you can conclude that it equals 0; in other contexts, you can conclude that it equals I; and in other contexts, you can conclude that it equals -I. This is just because an infinite set can be mapped:

1. One-to-one onto itself;
   
2. One-to-ten onto itself (meaning you could pair integers with groups of ten integers, and have none left over in either sequence); or
   
3. One-to-eleven onto itself.
   

As I repeat in replying to Ughaibu, no ball will be removed when it is "the last" or, better, "the only one at that step" in the jar.

 

Sorry, I misunderstood what you said earlier. The above is true. But it doesn't imply that the jar will be nonempty. It's true that there is no particular stage in the sequence when the jar is empty. At every stage, the jar gets more and more full. We can even say that the limit of the number of marbles in the jar, as the sequence progresses, is infinity. But all that is consistent with the fact that, after the sequence is completed, the jar is empty.

 

How is that consistent? Note two things:

• Saying what the limit of the number of marbles is as the number of steps "approaches infinity" does not logically entail anything about what happens when you've actually completed infinitely many steps. Rather, it only says something about what happens when you're in the sequence, i.e., it says something about the finite sub-parts of the series.
  
• Similarly, the fact that at no finite-numbered stage of the sequence is the jar emptied does not imply that the jar isn't eventually emptied, because again, we're talking, not about what happens at any finite stage, but about what things are like after completing infinitely many stages. "After any finite number of stages, the jar is full" is consistent with "After all the infinitely many stages, the jar is empty", just as it would be consistent to say "F is true of stage 1" but "F is not true of stage 2".
  

Unlike the giraffe, balls having a greater number on their label have been put in, and before the removal of the n-th.

 

I think you're saying: For every n, before ball #n is removed from the jar, a ball with a number higher than n is put in. True. So what? Those balls are removed later. Why should all the balls have to be removed at the same stage? This seems to be the same argument as above.

 

Loosely, "after all steps" there will be more than one ball, so it won't be "a" number.

 

It's because I thought you might say that, that I said "Can you give me an example of a number that might be written on it?" rather than "Tell me the unique number that must be written on it." If you think there's more than one ball in the jar, just pick any of them.

 

In order to give any meaning to what your asking, we'd have to admit we can say there's an infintiy of balls in the jar and none of them have finite n on there label.

 

So your answer is that there will be infinitely many balls in the jar, and they'll all have "infinity" written on them? Or they won't have anything written on them? Or a mix of the two?

 

You are talking about after all steps, consider the set of balls that have been put in. There is at least one ball in this set and, instead of asking your question, I ask you what number is on that ball's label. Does this question make more sense than yours, or less?

 

If you ask, as I did, "Give an example of a number that might be written on it," then your question makes as much sense as mine, but your question (unlike mine) is easy to answer. Here's an answer: "17". That is, a ball numbered 17 was put in the jar at some stage.

 

That is a high-school definition, not one from a proper treatment of calculus based on topology.

 

Don't insult my definition! If you think the definition is not adequate, you'll have to explain why. You could also offer another definition, but first I'd like to hear why this one doesn't work.

[Qfwfq]

Well, I said a downright dumb, useless thing didn't I? :doh: Obviously it wasn't exactly what I had in mind initially (I was basically trying to show that your challenge needn't be answered) but I was running short of time and realized it involved tricky things that we disagree upon, I should have postponed addressing the matter. Instead, thinking of your points as well as mine, in a hurry to pack up and leave, I ended up in such a state of confusion that I came up with a simpler but perfectly useless thing.

 

My shortage of time is aggravated by the debate being unnecessarily inflated with points about mathematics. You say you're a philosopher and not a mathematician, I say I'm a physicist and not a mathematician. Actually, not quite one, but I can say I have enough grounding and yet I avoid lecturing you with details without knowing if you're lacking or missing them. Basically, I try to say what I'm referring to, instead of details that one might either know or look up. Giving me detailed lectures only adds to the trouble where you misunderstand me and take it for my shortcoming, as in:

 

Sorry to keep harping on this, but: Expressions like "10I - I" don't make sense. :naughty: It's not that 10I - I = infinity, and it's not that it's unequal to infinity; it's that it's undefined.

 

We might agree, since you said it was "indeterminate". But then you proceeded to putatively evaluate the expression anyway.

Note that I had not said "where I = infinity" but instead "where I is an infinite term". What you follow with, about sets of infinite cardinality, is correct and I agree with it except the part of point b in parenthesis but anyway I don't see the point of it. Further, I didn't insult your definition of limit nor did I say it is not adequate or that it doesn't work.

 

I'd be glad to keep the debate more to the point, without having to swim through honey instead of ordinary water; it's the extra time and effort that I really can't afford. I'll be back when I can, hoping there can be less misunderstanding.

 

and as finite numbers have different properties from infinite numbers, the contents of the jar, when the task is complete, don't follow in a direct sequence with the preceding numbers.

What you are saying is that a function having infinite limit for x = a cannot be continuous (unless one adopts a realm such as hyperreal numbers, in which f(a) can actually be infinty).

 

Fine, but this doesn't prove the empty jar argument; to the contrary, how could that argument be any more valid than any other one, for determinig f(a) based the definition of f(x) for x other than a? I'll be unhurriedly polishing the details of an agument no less valid than the one on removing balls, and then I'll let anyone find what's wrong with it, from which one should conclude the likes of:

 

[math]\frac{1}{0} = 0[/math]

 

Do I hear hoots of mirth, cries of indignation and even shouts of protest from the audience?

[CraigD]

Do I hear hoots of mirth, cries of indignation and even shouts of protest from the audience?

From this audience member (from a 1982 BS in Math, licensed actuary, former college instructor, and computer professional since puberty ;)), you hear fait murmurs of bewilderment. :confused: Beyond the usual treatment of infinite sequences and series, my exposure to transfinite numbers (that stuff usually blamed on Cantor) is physicist and novelist Rudy Rucker’s “Infinity and the Mind” (I think – I no longer have the book, and might misremember in which of his several math/science popularization I read about them). So while I’ve heard of these numbers, I’ve never really worked their formalism.

 

I’m convinced that, under any applicable formalism involving sets, the answer to the balls and vase problem is “the vase contains an infinite number of balls at noon”.

 

My reason is similar to Allis and Koetsier’s argument. Simply put, I believe that any sequence that can be algebraically proven to be term-for-term identical for defined number system A (such as the naturals) is term-for-term identical for a number system that covers A (such as the transfinites).

 

Since the partial sums of a series generate a sequence, this applies to the balls and vase problem. We can show algebraically that the sequence described in this problem ((10–1), (20-2), (30-3) …) is identical to (9, 18, 27 …). Even with my limited understanding of transfinite numbers, the term of this sequence at any point of infinity ([math]s(n) = 9n[/math], where [math]n=\infty[/math]) is infinite.

 

In ordinary language, I’m arguing a sort of “equivalence principle”. Since the balls and vase problem doesn’t require the balls be numbered or places is special niches in the vase, no “experiment” could distinguish at any step a vase with balls in which, at each step, 10 balls were added and 1 removed, from a vase in which, at each step, 9 were added. It’s unreasonable to conclude that adding an infinite number of balls to the vase in increments of 9 will ever cause it to be empty.

 

The valuable of this “paradox problem”, is, I think, to illustrate how the natural language argument equivalent of rearranging terms in an infinite series can lead to absurd results. Making more of it than this instructive purpose is, IMHO, unhelpful.

[pgrmdave]

It seems like there should be infinite balls in the jar - removing infinite balls doesn't stop you from adding infinite balls, and each step is adding some. Each step fits a general equation of CX - X where C is some constant and X is the number of marbles. Would there be infinite balls if we added eleven marbles and then removed ten? Would you say that there will be zero marbles so long as we remove any?

[Pyrotex]

I love this thread! I don't feel qualified to contribute heavily, as my math background did not include detailed analysis of the various infinities*. I know they exist, and I know that infinity is not a "number" or quantity by any stretch of the imagination. You guys are doing a great job of educating me.

 

Perhaps a small observation: infinity is a concept rather than a number, and it must always be approached and never reached. Any argument that involves "reaching infinity" is inherently flawed and will yield marvelous paradoxes, such as the fascinating ones in this thread.

 

Notice, if you will, the use of the word 'paradox'--usually in ordinary street English, we mean a puzzle that is difficult or even impossible to understand. But "infinity as a number" yields REAL paradoxes: puzzles where two mutually contradictory conclusions can BOTH be "proven". This should tell you up front that the "proofs", though apparently solid and logical, contain flaws deep within their very conceptual foundations.

 

And they do. But isn't it fascinating?!?!

 

*my degree only qualifies me to discuss quantities less than or equal to 2^1023 - 1.

[owl]

It seems like there should be infinite balls in the jar - removing infinite balls doesn't stop you from adding infinite balls, and each step is adding some. Each step fits a general equation of CX - X where C is some constant and X is the number of marbles. Would there be infinite balls if we added eleven marbles and then removed ten? Would you say that there will be zero marbles so long as we remove any?

 

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.

[owl]

Any argument that involves "reaching infinity" is inherently flawed and will yield marvelous paradoxes, such as the fascinating ones in this thread.

 

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.