Questions about infinity

[owl]

Then there is a problem that reduces to the following:

[math]\displaystyle(\frac{1}{3})\sum_{x=0}^{\infty}[\frac{1}{x+1}-\frac{1}{x+4}][/math]

The answer in the book claims that all that is left from this is 1 + 1/2 + 1/3 because it basically says (1 + 1/2 + 1/3 + 1/4 ...) - (1/4 + 1/5 + ...)

 

There's something I find dissatisfying with the book's reported explanation, which I don't think has been mentioned. To explain what it is, first consider the following paradox. What is the value of the infinite series,

1 - 1 + 2 - 2 + 3 - 3 + 4 - . . . ?

 

Consider three proposed answers.

 

Answer #1: The sum is 0, because you can rewrite the sum as

(1-1) + (2-2) + (3-3) + . . .

= 0 + 0 + 0 + . . .

= 0

 

Answer #2: The sum is positive infinity, because you can rewrite the series as

1 + (-1+2) + (-2+3) + (-3+4) + . . .

= 1 + 1 + 1 + 1 + . . .

= infinity

 

Answer #3: The sum is negative infinity, because you can rewrite the series as

-1 + (1-2) + (2-3) + (3-4) + . . .

= -1 - 1 - 1 - 1 - . . .

= -infinity

 

In each case, I'm keeping all the original terms, just rearranging them, so it seems like it should be okay. The standard response to this is, well, regrouping the terms is not okay; my alleged rewritings of the series are actually three different series. (That seems counter-intuitive. My view is that this reflects the fact that an "infinite sum" is not really a sum, not in the same sense as an ordinary, finite sum; that is, it's not what you think of intuitively as a sum. And I think that shows something interesting about infinity as well. But leave that aside for now.)

 

It seems to me that the math book's explanation assumes that you can rearrange terms in a series, in the way that the above paradox shows you can't. Of course, you can add further conditions such that rearrangement becomes permissible, but my bet is that the book doesn't mention that. And so I suspect that the book encourages a conceptual confusion that's already very tempting for most people. To me, that's much worse than getting the answer to the problem wrong.

[CraigD]

If you were confused about what the brackets meant maybe you should have read what you copy and pasted before you opened your mouth. If your not familiar with the symbol the /ceil code in the latex might have given it away.

Although I failed to immediately recognize the [math]\lceil \rceil[/math] symbols as rendered, I am familiar with the floor and ceiling functions, as is nearly anyone with a science degree.

 

In the interest of accuracy (not nit-picking criticism), it’s important to note that the expression Kriminal99 gave in post #21,

 

[math]\sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil}[/math]

 

, which uses the ceiling function ([math]\lceil \rceil[/math]), does not give the desired terms

1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8....

, when interpreted in the usual way, but instead gives

1 + 1/2 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16....

 

It should have used the floor function ([math]\lfloor \rfloor[/math]),

 

[math]\sum_{k=1}^\infty 2^{-\lfloor \log_2 k \rfloor}[/math]

 

The floor function returns the greatest integer less than or equal to its argument. (eg: [math]\lfloor 1.5 \rfloor = 1[/math]).

The ceiling function return the least integer greater than or equal to its argument. (eg: [math]\lceil 1.5 \rceil = 2[/math]).

 

Although recognizing notational conventions in Math is important, particularly in publicly published forums like this, one shouldn’t forget that they are just conventions – as long as everyone involved in a conversation understands the notation, it has served its purpose. Providing explanations of one’s notation, before or after any confusion is noted, is, IMHO, more important than strict adherence to convention – especially as convention varies from place to place and time to time.

 

:naughty: Kriminal, some unsolicited advice on pedantic style: being scathingly critical (eg: “you should have read what you copy and pasted before you opened your mouth”) can, on rare occasion, be effective. However, when doing so, you’d best have certain knowledge, or do a little research beforehand, to assure your understanding of the subject at hand is correct.

 

PS: I’m wrong about the choice of the floor vs. the ceiling function in the series

[math]\sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil}[/math]

. It does give the desired terms: 1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8....

 

Hopefully, my friendly demeanor will cause readers (including Kriminal99) to forgive me my mistake :)

[Erasmus00]

There's something I find dissatisfying with the book's reported explanation, which I don't think has been mentioned. To explain what it is, first consider the following paradox. What is the value of the infinite series,

1 - 1 + 2 - 2 + 3 - 3 + 4 - . . . ?

 

Consider three proposed answers.

 

Answer #1: The sum is 0, because you can rewrite the sum as

(1-1) + (2-2) + (3-3) + . . .

= 0 + 0 + 0 + . . .

= 0

 

Answer #2: The sum is positive infinity, because you can rewrite the series as

1 + (-1+2) + (-2+3) + (-3+4) + . . .

= 1 + 1 + 1 + 1 + . . .

= infinity

 

Answer #3: The sum is negative infinity, because you can rewrite the series as

-1 + (1-2) + (2-3) + (3-4) + . . .

= -1 - 1 - 1 - 1 - . . .

= -infinity

 

In each case, I'm keeping all the original terms, just rearranging them, so it seems like it should be okay. The standard response to this is, well, regrouping the terms is not okay; my alleged rewritings of the series are actually three different series. (That seems counter-intuitive. My view is that this reflects the fact that an "infinite sum" is not really a sum, not in the same sense as an ordinary, finite sum; that is, it's not what you think of intuitively as a sum. And I think that shows something interesting about infinity as well. But leave that aside for now.)

 

I thought I would add a bit to clarify for people reading. What owl means (or at least what I think he means, and what I think is the right idea) is that inifinity can only be handled as a limit, not as an actual number. What we are looking at is a specific procedure: do the sum, then take the limit. I agree that introductory books are often terrible at hammering home how to treat infinity. Treating infinity as a limit solves our paradox because

[math] \lim_{N\to \infty} \sum_{x=1}^{N} (x-x) [/math]

[math] \lim_{N\to \infty} \sum_{x=0}^{N} ((x+1)-x) [/math]

 

Are different limits.

 

Edit: Another way to wrap your head around the idea of infinity as a limit vs infinity as a number is to ask yourself the question "is infinity even or odd."

-Will

[owl]

I thought I would add a bit to clarify for people reading. What owl means (or at least what I think he means, and what I think is the right idea) is that inifinity can only be handled as a limit, not as an actual number.

 

Yes, that's what I mean. Expressions like "1 - 1 + 2 - 2 + 3 - . . ." have a definition in math, but it's a special definition, just for the infinite case; it's defined as the limit of the sequence of finite sums, as the number of terms increases. It's not defined as "the result after you get to step # infinity". Mathematicians did some very clever work to excise both infinity and infintesimals (treated as numbers) from mathematics, so that everything in calculus could be defined just in terms of real numbers, but this brilliant work is often lost on students, who don't know (and aren't told) why things are defined in the sometimes convoluted-seeming way they are.

 

Edit: Another way to wrap your head around the idea of infinity as a limit vs infinity as a number is to ask yourself the question "is infinity even or odd."

Oh yes, here's another paradox that illustrates that point. (Everyone loves paradoxes of infinity.:shrug:) Suppose there is a light bulb with a switch on it. The light bulb starts out on. After 1/2 second, it gets switched off. After 3/4 seconds, it gets switched back on. After 7/8 seconds, it gets switched off again. And so on, infinitely many times. At the end of 1 second, is the light bulb on or off?

 

If you think of infinity as a number, then the answer to the question turns on whether "infinity" is even or odd.

 

Here's a simpler puzzle: If infinity is a number, what is infinity + 1? Most people will agree that it is also infinity. So we have

infinity = infinity + 1

Subtracting infinity from both sides of the equation,

0 = 1

[ughaibu]

My favourite paradox of infinity: Supertask - Wikipedia, the free encyclopedia

[Kriminal99]

Whilst having had nothing to do with the original post, you have to admit that was rather funny. Krim, did pgrmdave threaten you with violence? I he did, I didn't see it. Or are you referring to that because you don't seem to be holding your own as far as our rules are concerned?

 

Like it was pointed out: This is a private 'club', if you will, where newcomers are welcomed with open arms provided they stick to the rules. And the rules aren't the US Constitution where 'Freedom of Speech' is guaranteed; the rules are the Rules of Hypography. Grin and bear it. If we don't do it, we'll quickly degenerate into a pile of muck like the myriads of unmoderated forums out there. You will NOT lay down the rules to us that we've been successfully applying over the years.

 

And that's about it.

 

All I can say further, is 'Like it or Leave it'. 'Shape up or Ship out' also comes to mind.

 

Apparently I am not allowed to respond to that here, although I feel issues like this become relevant to the discussion, and obviously you are not refraining from discussing it here... anywyas here is a link to my response:

 

http://hypography.com/forums/user-feedback/10336-tribal-morality-lack-objectivity.html#post161855

 

There's something I find dissatisfying with the book's reported explanation, which I don't think has been mentioned. To explain what it is, first consider the following paradox. What is the value of the infinite series,

1 - 1 + 2 - 2 + 3 - 3 + 4 - . . . ?

 

Consider three proposed answers.

 

Answer #1: The sum is 0, because you can rewrite the sum as

(1-1) + (2-2) + (3-3) + . . .

= 0 + 0 + 0 + . . .

= 0

 

Answer #2: The sum is positive infinity, because you can rewrite the series as

1 + (-1+2) + (-2+3) + (-3+4) + . . .

= 1 + 1 + 1 + 1 + . . .

= infinity

 

Answer #3: The sum is negative infinity, because you can rewrite the series as

-1 + (1-2) + (2-3) + (3-4) + . . .

= -1 - 1 - 1 - 1 - . . .

= -infinity

 

 

Woah. That is exactly what I meant about there being logical contradictions in math if you look too closely.

 

That is the exact same method used to reduce

 

1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8.... to

1/2 + 1/2 + 1/2...

 

which is a proof that the harmonic series diverges. If that method is off limits than so is this.

 

I think we must look to the algorithm used to create infinity in order to deal with such situations. Note that if for any number, if its opposite is included in the series then the sum will always be zero. You only created the issue that it could reach + or - infinity by excluding the last opposite which would exactly cancel out the sum to that point. Why should we stop at a number without including its opposite, when the defining characteristic of the sum is that each number and its opposite is included? If we can concieve of any number than we can simultaneously concieve of its opposite as that function has already been defined given an integer. But I believe it is more important that (integer x, opposite x) is the defining characteristic of the series. This is just agreeing with Eramus since he basically says instead of guessing the defining characteristic of the series just specify it.

 

So I think the method itself is ok... and in many examples leaving a term off the end cannot appear to cause the series to go in a different direction. For that matter the series in your example seems similar to Limit X-> Infinity, Sin(X). I believe this is said not to have a limit correct?

 

Hopefully, my friendly demeanor will cause readers (including Kriminal99) to forgive me my mistake ;) :)

 

Your friendly demeanor? You know, perhaps my radar is a bit off. In person I feel I am very effective at reading body language and behavior to gauge someone's personality, motives, intentions etc. Over the internet I guess it is never a sure thing.

 

When I posted that, I specifically coded in the ceiling function with a clear understanding of the function. I then left for a few days to finish studying for my SOA exam (on which this topic was fairly trivial review that I just decided to look deeper into out of boredom, and which I passed with a high enough score that they told me right away instead of waiting to decide the curve). When I came back I saw your responses and completely forgot that it was the ceiling function and followed what you said. Then pyrotex pointed it out again which caused me to fall out of my chair laughing. I posted that it did so and why, and you respond that Pyrotex caught what all of us never understood. Of course it was sharp of pyrotex to catch that, but I obviously had to know it in order to code it in originally PLUS I posted that I had known what it was and explained how it worked (note pyrotex didn't explain anything before that). And still you post as if refuse to recognize this, instead giving pyrotex full credit for it.

 

That is an example of the kind of thing I think is slightly deceptive and gives me the impression that you are out to get me. Noone is perfect, I know I might end up doing things like that if someone uses it against me, but then I think if I were a moderator like you I would be even more careful because if I know if I wasn't I might get tempted to just squelch people who disagree with me too strongly. The other thing was I started out the thread with some slightly abnormal and slightly abstract questions about a fairly simple subject in math and you guys seemed to be bending over backwards to characterize me as inexperienced at math (in the posts here and in pm's etc). I did not let on how much I really knew because I didn't think that was signifigant. Rather I think someone brand spanking new to the subject should still be permitted to present interesting arguments and have them considered objectively rather than straw manned to death with the assumption that anything that comes from the inexperienced guy must be a mistake. But I think an inexperienced person would be better off than I would... I think you might want to charactize me as such because you want to believe that all philosophers are bad at math and thats why they argue for things like determinism.

 

Forcing the counter example to adhere to your beliefs

 

Pretty common human behavior. Just the other day my leasing office agent tried to talk to me like I was stupid after bringing up an in depth argument because I was in my workout clothes and am fairly muscular. She wanted to believe I was stupid because she thinks that only stupid people care about body building. When a counter example is presented, she tries to force the counter example to conform to her beliefs instead of vice versa! Happens all the time.

 

My favourite paradox of infinity: Supertask - Wikipedia, the free encyclopedia

 

One response to this brand of paradox is to simply say that the scenario is impossible because noone could move marbles infinitely fast. It seems like a copout at first but not when you consider the underlying argument that

 

A) perhaps the univers is truly finite/discrete once you reach some arbitrary point of precision

 

:) It is known that past a certain point of precision, the experiment cannot continue as described

 

C) perhaps even if the universe is infinite and continuous there is also an infinite number of changes in the nature of things caused by passing certain thresholds of precision that would prevent any one experiment from testing the nature of infinity.

[Qfwfq]

Double-ahem <cough>

 

[math]a^{-(\log_a b)} = \frac1b[/math]

 

so

 

[math]\sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil} = \frac11 + \frac12 + \frac13 + . . . [/math]

 

It is, the harmonic series.

I think the notation for ceiling (the next integer up) escaped your notice. Your argument shows why Krim's series is term by term greater than the harmonic one.

 

[math]\lceil 3.000000000001 \rceil = \lceil 3.5 \rceil = \lceil 3.9999999998 \rceil = 4[/math]

[Kriminal99]

Q but its the inverse... so the terms are smaller right?

[Qfwfq]

Sorry! :) Of course! Posting in snips of time, as usual... but I guess you got my meaning anyway... I meant to put the table on the bowl, and not vice versa!

 

Uhm... :rant: or... was it... :)

 

Woah. That is exactly what I meant about there being logical contradictions in math if you look too closely.

So that's what you meant? If you get a proper calculus course, and understand it thouroughly, you'll realize that, like most paradoxes, these are resolvable. Zeno's paradox baffled the ancients, but it's a trivial example of a convergent geometric series.

 

I think the first thing I told you in this thread was that the first thing to do is prove convergence. Without that, of course the associative and commutative properties aren't conclusive. As for the argument of Ross, it isn't conclusive. For a given n, sure the n-th ball will be missing, but for any n there will be an infinity of balls with greater n. It's fallacy to conclude that the jar would be empty.

 

The Allis and Koetsier alternative is clearly equivalent: instead of removing a ball you bump it up to a vacant number. The Ross agument simply becomes one with "has been removed" replaced by "no longer has that number" but it's just as inconclusive for the same reason.

 

One response to this brand of paradox is to simply say that the scenario is impossible because noone could move marbles infinitely fast.

This is certainly not a mathematical answer to the matter. The "experiment" with marble balls is certainly not practicable, it's a choice between doing things infinitely faster or taking an infinite time. So, let's consider the realistic thing: finite time intervals. We'll never finish the experiment, but will the content of the jar increase in number at each step or not?

[Qfwfq]

While this proves that the harmonic series diverges to infinity, if doesn’t quite meet Pyro’s challengebecause it only shows that the harmonic series diverges, not that it is the slowest growing divergent series.

I'd say it's got to do with logarithmic order...

 

Golly, I don't know how I had missed that and so many other posts yesterday! :rant:

[owl]

I said:

What is the value of the infinite series,

1 - 1 + 2 - 2 + 3 - 3 + 4 - . . .

Answer #1: The sum is 0, because you can rewrite the sum as

(1-1) + (2-2) + (3-3) + . . .

= 0 + 0 + 0 + . . .

= 0

Answer #2: The sum is positive infinity, because you can rewrite the series as

1 + (-1+2) + (-2+3) + (-3+4) + . . .

= 1 + 1 + 1 + 1 + . . .

= infinity

Answer #3: The sum is negative infinity, because you can rewrite the series as

-1 + (1-2) + (2-3) + (3-4) + . . .

= -1 - 1 - 1 - 1 - . . .

= -infinity

 

Woah. That is exactly what I meant about there being logical contradictions in math if you look too closely.

 

There's a standard understanding of these things which is non-contradictory (though it may not be philosophically correct). Your math book just has a mistake.

 

Note that if for any number, if its opposite is included in the series then the sum will always be zero. You only created the issue that it could reach + or - infinity by excluding the last opposite which would exactly cancel out the sum to that point. Why should we stop at a number without including its opposite, when the defining characteristic of the sum is that each number and its opposite is included?

 

A couple of comments. One, I didn't really stop at any number--the "last opposite" that you're saying got excluded in answer #2 above just gets included later. Answers 2 and 3 include everything, just in a different order. Now, you say the defining characteristic of the sum is that each number and its opposite is included. Why couldn't one also say that the defining characteristic of the sum is that "The number 1 is included, and every number > 1 and its-opposite-minus-one is also included"? Doesn't that define the series equally adequately (if in a more complex way)?

 

Second comment. What you say conflicts with the standard analysis (you can decide whether you care about that). Consider the infinite sum:

 

1+2-1+3+4-2+5+...

 

This has the series of partial sums:

 

1, 3, 2, 5, 9, 7, 12, ...

 

So the series diverges; you say either that it has no sum or that "the sum is infinity." It does not have a sum of 0. But it does have the property you named: namely, every integer and its opposite is included. It seems to me that the standard analysis is more intuitive than what you say: the sum is getting larger and larger as you proceed through the series, and it's clear that that continues forever. So it's counterintuitive to say the sum is 0.

 

Btw, in my original example, the standard analysis is that the series has no sum. This is because it has no limit: as you go further along in the series, the sum keeps jumping back and forth between zero and ever larger numbers, so there is no number that it approaches. The sum isn't 0, since the series only spends half its time at 0, so to speak.

 

For that matter the series in your example seems similar to Limit X-> Infinity, Sin(X). I believe this is said not to have a limit correct?

 

Exactly.

 

One response to this brand of paradox is to simply say that the scenario is impossible because noone could move marbles infinitely fast.

 

I think that's right. But most philosophers would disagree. They would say: "It doesn't matter if the scenario is physically possible. All that matters is whether it is logically possible (roughly, it can be coherently conceived). As long as it is logically possible, you should be able to reason consistently about what would happen if it were the case. The fact that when we try to think about it, we wind up in contradictory or highly counter-intuitive conclusions is puzzling."

[owl]

...you'll realize that, like most paradoxes, these are resolvable. Zeno's paradox baffled the ancients, but it's a trivial example of a convergent geometric series.

 

I think all paradoxes are resolvable, simply because reality is consistent. But I don't think that the resolution of any of these paradoxes is trivial. In particular, I don't see how the observation that

 

1/2 + 1/4 + 1/8 + . . .

 

is convergent is supposed to solve Zeno's paradox. Zeno's claim was that you never get to the end. I don't see how saying the series has a limit bears on that. :confused: Side note: Zeno's version was actually this:

 

Before the ball gets to the ground, it has to get halfway to the ground. But

before it can do that

, it has to get one quarter of the way. And before it can do that . . .

 

So a better representation of Zeno's series would be this:

 

. . . 1/8, 1/4, 1/2, 1.

 

So he claims that the ball can never get started.

 

As for the argument of Ross, it isn't conclusive. For a given n, sure the n-th ball will be missing, but for any n there will be an infinity of balls with greater n. It's fallacy to conclude that the jar would be empty.

 

If for every n, the nth ball is not in the jar, then it logically follows that none of the balls are in the jar, unless you're suggesting that some ball that's not a member of the series of numbered balls might be in the jar. This isn't a fallacy; this reasoning is used all the time in mathematics. For example, the proof that there is no largest prime number works by showing that, for an arbitrarily chosen number n, n cannot be the largest prime number. It's no good to say, "But there are numbers larger than n, and one of them might be the largest prime number," because the same reasoning applies to each of them.

 

The Allis and Koetsier alternative is clearly equivalent: instead of removing a ball you bump it up to a vacant number. The Ross agument simply becomes one with "has been removed" replaced by "no longer has that number" but it's just as inconclusive for the same reason.

 

It is just as conclusive. Instead of showing that no ball is in the jar, it shows that no natural number is written on any of the balls. That's correct. Each ball would have a natural number followed by infinitely many zeros, which isn't a natural number.

[Qfwfq]

I think all paradoxes are resolvable

Ordinarily they are, yes, indeed doxa means opinion and not truth. The original meaning is simply that they easily mislead one into a false conclusion. However, there are issues such as Russel's paradox...

 

I am not of the school that draws conclusions about math based on reality, but I'd say discussing that would belong in a different thread.

 

In particular, I don't see how the observation that

 

1/2 + 1/4 + 1/8 + . . .

 

is convergent is supposed to solve Zeno's paradox.

But Zeno's paradox is the geometric series!

 

BTW, the halfway argument is an alternative representation of it, I believe the original agument was that when, the hare reaches the current position of the tortoise, he also will have moved a bit, therefore... (reiteration)

 

If for every n, the nth ball is not in the jar, then it logically follows that none of the balls are in the jar, unless you're suggesting that some ball that's not a member of the series of numbered balls might be in the jar. This isn't a fallacy; this reasoning is used all the time in mathematics.

I know it's very subtle, but there's a difference compared with other arguments, such as those by induction and the one for the cardinality of primes.

 

For example, the proof that there is no largest prime number works by showing that, for an arbitrarily chosen number n, n cannot be the largest prime number. It's no good to say, "But there are numbers larger than n, and one of them might be the largest prime number," because the same reasoning applies to each of them.

I don't find the comparison apprropriate. The most common argument isn't just that at all, it's a reductio ad absurdum. No "for each..." but instead "If there were a..." and it needn't even be "...largest prime", it may well be argued as "...finite set". Personally I prefer my own alternative argument: Given any finite set of primes ... hence it cannot be the set of all primes.

 

That said, back to the matter of the balls in the jar. First, the "For each n" implies finite n while the resolution of the paradox shows that, just as the number of balls in the jar approaches infinity, so do the numbers on their labels. Hence it's a logical leap to draw the conclusion that there will be no balls in the jar.

[ughaibu]

I agree that Owl's comparison was inappropriate but I dont think that's important, the point is that all the balls are numbered.

A question about Allis and Koetsier, why is the result that "there are infinitely many marbles each labeled with a natural number followed by an infinite number of zeros"? Why doesn't the ball numbered "2" have the number "20" written on it?

[ughaibu]

Okay, I see. Forget about the Allis and Koetsier question.

[CraigD]

In particular, I don't see how the observation that

 

1/2 + 1/4 + 1/8 + . . .

 

is convergent is supposed to solve Zeno's paradox. Zeno's claim was that you never get to the end. I don't see how saying the series has a limit bears on that. :confused:

In a simple case of Zeno’s paradox, an object moves at a constant velocity (1 unit/sec) over a fixed distance (1 unit). The series is then both the distance that the object has moved, and the elapsed time (duration) it has been moving. Even though there are an infinite number of terms, the sum of both the distance and time are finite (1 unit and 1 second, respectively)

 

The paradox depends on an assumption made by Zeno’s hypothetical protagonist: that any infinite number of finite numbers sum to infinity. The paradox was compelling to Zeno, his contemporaries, and many people who came later, because this assumption agreed with their intuitive understanding of reality. 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.

 

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.

 

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. Mathematical formalism itself wasn’t widely popular until around the 1900, when if was increasingly promoted by folk such as David Hilbert.

Side note: Zeno's version was actually this:

 

Before the ball gets to the ground, it has to get halfway to the ground. But

before it can do that

, it has to get one quarter of the way. And before it can do that . . .

I was taught (but have not confirmed with any actual historical scholarship) that this particular Zeno’s paradox was originally described as a fast runner – Achilles – attempting to overtake a slow one – the tortoise. This is the version given by Aristotle in ”Physics”.

 

My personal favorite is the one given by Hofstadter in GEB. In this version, a questioner (Achilles or Tortoise again – the main characters in GEB), may ask a question of a genie that is slightly more knowledgeable than he is, but twice as fast. By phrasing the question “genie, if you don’t know the answer, ask the genie that knows slightly more than, and is twice as fast as, you” (there are infinitely many genies), the questioner can be assured of an answer to any question in no more than twice the time it takes to ask and receive an answer from the first genie.

[owl]

But Zeno's paradox is the geometric series!

 

I don't think so. I was going to add that the "convergent series" approach misunderstands what Zeno was getting at, because in the paradox as traditionally understood it doesn't matter what any of the terms in the series is. For Zeno's reasoning, it's irrelevant what the actual numbers are. He'd say the same thing if the series was

1/1, 1/2, 1/3, ...

as if the series was

1/2, 1/4, 1/8, ...

Zeno doesn't care about the magnitude of the terms in the sequence, and he wouldn't care about what the limit of the series of finite sums is. All he cares about is the number of terms in the series. That number of terms is infinite, so he would say you can't complete it. This is not because he thinks "the sum is infinity". It's because he thinks it's logically impossible to complete a supertask. I think the implicit argument is something like the following.

1. An infinite series has no end.
   
2. To complete a series, you must get to the end of it.
   
3. Therefore, you cannot complete an infinite series.
   

You can see that the impossibility asserted in the conclusion comes solely from the number of terms in the series.

 

Admittedly, I'm doing some interpolating and some guessing about what the paradox and the reasoning behind it is. But I think this is the most interesting and charitable interpretation. (Zeno apparently actually thought that motion was an illusion.)

 

BTW, the halfway argument is an alternative representation of it, I believe the original agument was that when, the hare reaches the current position of the tortoise, he also will have moved a bit, therefore...

 

Zeno had three "paradoxes". One was the one about the ball moving halfway, etc. The second was about the runner and the tortoise. The third was something about an arrow, but it wasn't very compelling and I've forgotten how it went.

 

I don't find the comparison apprropriate. The most common argument isn't just that at all, it's a reductio ad absurdum. No "for each..." but instead "If there were a..." and it needn't even be "...largest prime", it may well be argued as "...finite set". Personally I prefer my own alternative argument: Given any finite set of primes ... hence it cannot be the set of all primes.

 

The argument about prime numbers can be formulated in at least 4 ways. In each case, there's a parallel to the balls and Ross' jar:

 

1a) Take an arbitrary integer n. Now we can show that n is not the largest prime number... So, there is no largest prime number.

1b) Take an arbitrary integer n. Now we can show that ball #n is not in the jar (and all balls have an integer on them). So, there is no ball in the jar.

 

2a) (Reductio ad absurdum method) Suppose there's a largest prime number. Call it n. It could be shown that there would be a prime number larger than n, which is a contradiction. So it's false that there's a largest prime number.

2b) Suppose there were a ball in the jar. Call it ball number n. (If there's more than one ball in the jar, then take any of them at random.) It could be shown that ball #n is not in the jar, which is a contradiction. So it's false that there's a ball in the jar.

 

3a) Take any finite set of prime numbers. It must have a largest member. But it can be shown that there is a prime number larger than this largest member. So no finite set of primes is identical to the set of all prime numbers. So there are infinitely many primes.

3b) (This runs a little differently, but it's just as cogent.) Take any nonempty set of balls. It must have a smallest member (the member with the smallest number on it). But it can be shown that this smallest member is not in the jar. Therefore, no nonempty set of balls is identical to the set of balls in the jar. So the jar is empty.

 

4a) (Mathematical induction method) The set containing just the first prime number does not contain all the prime numbers. And if the set of the first n primes does not contain all the prime numbers, then the set of the first n+1 primes also does not contain all the prime numbers. Therefore, no finite set contains all the prime numbers.

4b) Ball #1 is outside the jar. And if ball #n is outside the jar, then ball #n+1 is also outside the jar (since it was removed in the following round). Therefore, all the balls are outside the jar.

 

That said, back to the matter of the balls in the jar. First, the "For each n" implies finite n while the resolution of the paradox shows that, just as the number of balls in the jar approaches infinity, so do the numbers on their labels. Hence it's a logical leap to draw the conclusion that there will be no balls in the jar.

 

I don't understand the "implies finite n" statement. Perhaps you mean the reasoning is implying that each n is a finite number. This is correct. That's a stipulation of the example. It was stipulated that the balls were labeled with all the natural numbers. No ball is labeled "infinity", or any other non-natural-number.

 

You can say that the numbers "approach infinity", but again, that does not mean that any of them actually is infinity. They're all finite.