Questions about infinity

[Pyrotex]

Okay, a shorter tidbit. Assume that the conclusion that the jar is infinitely full is correct. Now, ask yourself: what is the smallest integer printed on the balls? There isn't any "smallest integer" in that set.

 

We have an infinite, countable set of integers, and it contains NO smallest integer!! :) What does this tell us?

[owl]

There is no ball labeled any natural number, they've all been renumbered. Again, name me a natural numbered ball that is still in the jar (isn't this the same as the original argument?)

 

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.

 

Let's assume that we can traverse that number line as fast as we wish up to any arbitrary finite speed. We are looking for integers that are represented on balls in the jar.

Starting at 1, I dial the throttle for 10^10 integers per second. Minutes go by and I do not find a "jarred" integer. ...

 

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.

 

Assume that the conclusion that the jar is infinitely full is correct. Now, ask yourself: what is the smallest integer printed on the balls? There isn't any "smallest integer" in that set.

... What does this tell us?

 

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.

[pgrmdave]

So then, why, if we're simply relabeling the balls to the number that would've been the highest in the original question, are those hyperreal numbers not included in the original question? We're not adding numbers that wouldn't have been added in the original.

[Erasmus00]

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.

 

But when did these hyperreal balls go in? Didn't we say we always relabel the balls with finite numbers? If hyperreal balls sneak into the jar in the relabel case why don't any hyperreal balls go in for the balls in, ball out version? You asserted for the first case that our balls are always labeled with a finite number, why does this assertion break down for the case where we relabel?

 

Also,do you agree that if we except the standard answers to this paradox we are forced to conclude that the number of balls in the jar cannot be determined solely by looking at the balls that enter and leave through the neck of the jar? I.e. something outside the jar (the relabeling procedure) determines the final answer?

-Will

[CraigD]

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.

To me, this is the weirdest variation of the balls and vase problem yet!

 

A similar version would involve taking only even numbered balls. At noon, you could argue (unassailably IMHO) that the vase isn’t empty, because it contains ball #1, or any odd numbered ball. At the same time you could arguing that, as it’s had the same (infinite) number of balls added and removed as in the original version, it contains the same number of balls, which by the reasoning that you can name no ball still in the original version’s vase, is zero!

 

My conviction that the vase is not empty at noon, and that the “can’t name a ball still in it” argument is invalid, remains. The argument must contain a fallacy – the “being unable to name an member of an infinite set implies that the set is empty” fallacy.

[CraigD]

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?

As the problem is defined - each step [math]n[/math] is performed at [math]2^{-(n-1)}[/math] minute before noon – the only argument I can see for the supertask not being completable is something like “nothing physical could move that fast”, a direct rejection of the statement of the problem. A simpler objection on the same grounds would be to reject an earlier part of the statement – that you have an infinite supply of balls – with the objection “you can’t physically have an infinite supply of balls”.

 

That the first objection – a supertask cannot be completed because nothing physical is fast enough – seems to come to mind (and not be dismissed) before the second reveals to me an intuitive bias for space – the domain of finite or infinite supplies of balls – over time – the domain of steps of adding and removing balls from vases.

 

Outside of the scope of balls and vase problem the question “when is a task (super or otherwise) completable” seems to me similar to a special case of the halting problem. Supertasks that are decidable are completable, those that are undecidable, are not.

 

My spider senses tell me there’s a terrible paradoxical dragon lurking in this inocent seeming, too, that likely involves infinity. :(

[owl]

So then, why, if we're simply relabeling the balls to the number that would've been the highest in the original question, are those hyperreal numbers not included in the original question? We're not adding numbers that wouldn't have been added in the original.

 

But when did these hyperreal balls go in? Didn't we say we always relabel the balls with finite numbers? If hyperreal balls sneak into the jar in the relabel case why don't any hyperreal balls go in for the balls in, ball out version? You asserted for the first case that our balls are always labeled with a finite number, why does this assertion break down for the case where we relabel?

 

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.

 

Also,do you agree that if we except the standard answers to this paradox we are forced to conclude that the number of balls in the jar cannot be determined solely by looking at the balls that enter and leave through the neck of the jar? I.e. something outside the jar (the relabeling procedure) determines the final answer?

 

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.

[owl]

the only argument I can see for the supertask not being completable is something like “nothing physical could move that fast”, a direct rejection of the statement of the problem.

 

Basically, yes.

 

A simpler objection on the same grounds would be to reject an earlier part of the statement – that you have an infinite supply of balls – with the objection “you can’t physically have an infinite supply of balls”.

 

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.

 

That the first objection – a supertask cannot be completed because nothing physical is fast enough – seems to come to mind (and not be dismissed) before the second reveals to me an intuitive bias for space – the domain of finite or infinite supplies of balls – over time – the domain of steps of adding and removing balls from vases.

 

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.

[pgrmdave]

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.

 

But they aren't simply having zeros added, they are being relabeled to whatever number would have been put in the jar in the original problem. No ball is relabeled to a number that wasn't going into the jar in the original.

[ughaibu]

Before the task is completed, there is always a specific time corresponding to a specific number of zeros on any ball, the numbers have only become infinitely long when the task is completed, just as the balls have only all left the jar when the other task is completed.

[Qfwfq]

the numbers have only become infinitely long when the task is completed,

...and this goes whether balls are relabeled or removed. The numbers are exactly the same at any step (finite n value). It's a logical sleight of hand to accept hyperreal values only in the relabelling case and not in the removing one, in answer to last week's challenge. Even in the case in which the balls aren't removed, the numbers are. Just consider the set of numbers rather than one of balls.

 

just as the balls have only all left the jar when the other task is completed.

All those having a finite n on the label?

 

:( All of them are put in with a finite n anyway! At any finite-n-th step, that is...

 

If, at a given time, we consider the task to have been completed (supposing this to make sense), Does it make sense to ask which ball was the last one to be put in? (Or the last new label number, to be equivalent for both cases?)

 

BTW, my friend replied that the word congruence isn't wrong in that wiki, this makes the Banach-Tarski result certainly more surprising than the other things I mentioned but it remains that it's the magic of fractal objects. I could say more, despite not being an expert, if it were more on topic. ;)

 

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.

No finite set of finite integers. Why are hyperreals good only in the relabelling case?

 

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?

Good point, and one I've been hoping to address when possible. The keen observer could find material to the purpose in my breakdown of the Ross argument. Zeno doesn't (or, more strictly, needn't) involve anything (apart from the values of n) that lacks a finite limit at x = a, unlike the vase problem in which it is inescapable. Using the same countable sequence with accumulation point at a doesn't make the vase problem physical (realistic), only pseudo-physical.

 

Now, it is certainly a matter of saying whether it makes sense to talk about when the task has been completed. I have mentioned this upstream and said that, if one only allows discussion of finite numbers, then "lasciate ogne speranza voi ch'intrate"...

[Qfwfq]

the numbers have only become infinitely long when the task is completed,

...and this goes whether balls are relabeled or removed. The numbers are exactly the same at any step (finite n value). It's a logical sleight of hand to accept hyperreal values only in the relabelling case and not in the removing one, in answer to last week's challenge. Even in the case in which the balls aren't removed, the numbers are. Just consider the set of numbers rather than one of balls.

 

just as the balls have only all left the jar when the other task is completed.

All those having a finite n on the label?

 

:( All of them are put in with a finite n anyway! At any finite-n-th step, that is...

 

If, at a given time, we consider the task to have been completed (supposing this to make sense), Does it make sense to ask which ball was the last one to be put in? (Or the last new label number, to be equivalent for both cases?)

 

BTW, my friend replied that the word congruence isn't wrong in that wiki, this makes the Banach-Tarski result certainly more surprising than the other things I mentioned but it remains that it's the magic of fractal objects. I could say more, despite not being an expert, if it were more on topic. ;)

 

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.

No finite set of finite integers. Why are hyperreals good only in the relabelling case?

 

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?

Good point, and one I've been hoping to address when possible. The keen observer could find material to the purpose in my breakdown of the Ross argument (yesterday). Zeno doesn't (or, more strictly, needn't) involve anything (apart from the values of n) that lacks a finite limit at x = a, unlike the vase problem in which it is inescapable. Using the same countable sequence with accumulation point at a doesn't make the vase problem physical (realistic), only pseudo-physical.

 

Now, it is certainly a matter of saying whether it makes sense to talk about when the task has been completed. I have mentioned this upstream and said that, if one only allows discussion of finite numbers, then...

 

Lasciate ogne speranza voi ch'intrate.

/forums/images/smilies/devilsign.gif

[Pyrotex]

I still say the paradox is, at its root, created by our natural tendency to view infinity as some kind of "number". And to view infinite processes and infinite executions as "doable" under the rubric of some variation on Xeno's Process. You guys...read your posts again. You gloss over infinity as if, "sure, I understand infinity! what's not to understand? piece of cake!"

 

Your spider sense is correct! I humbly suggest that a piece of that adamantine cake is harder to slice than you imagine. :(

[owl]

If, at a given time, we consider the task to have been completed (supposing this to make sense), Does it make sense to ask which ball was the last one to be put in?

 

No, you know there is no last stage. The stages are infinite (endless).

 

No finite set of finite integers. Why are hyperreals good only in the relabelling case?

 

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).

[Erasmus00]

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.

 

I was hoping someone would suggest this, and I agree that it seems easier to think about. However, I again assert that the jar has no way of knowing about balls outside of it.

 

A reasonable person could suggest putting a sensor on the ball's neck that can read the numbers in, and numbers out. In this case, we are saying that the data set he gets is simply incapable of determining the number of balls in the jar. That is, again, why I suggest the question is broken- the balls you put into and take out of the jar do not influence the answer, only the balls you leave on the table influence the answer.

 

I assert that the jar has no way of "knowing" the condition of the table, and that the question should be answerable entirely by looking at the jar. It is not, so the question must be broken. Two equivalent situations, two equally valid arguments lead to different results. Hence, I believe the answer should be that the number of balls is simply undefinable.

-Will

[pgrmdave]

Owl, what if we changed the constants in the relabeling scenario? Let's say we put in five and take out one to relabel as the next highest. Then we wouldn't simply be 'adding zeros', we'd be relabeling to the next number highest.

[ughaibu]

At any stage, while the task is running, the highest numbered ball will be labeled with a number that ends with a zero. If we examine the jar while the task is running, one time in ten we can expect to find that the highest number has a terminal of two consecutive zeros, one time in a hundered, three consecutive zeros, etc. We can always know how many zeros terminate the number on the highest value ball by the time at which we make the examination of the jar's contents. These are finite numbers and behave as such.

Once the task is completed, there are no finitie numbers, and there is no highest value number, all one needs, to conceive of the results, is expressed in Gallileo's paradox and Cantor's diagonalisation. If the Ross-Littlewood result is incorrect, the concept of infinity needs to be revised.