Questions about infinity

[pgrmdave]

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.

 

So if I understand this correctly, it does not matter what the ratio is - so long as you occasionally remove a ball (in order), all the balls are gone. I could add, say, 10,000,000 balls at a time, and remove simply the lowest one, and I would arrive at zero balls, right? I could even only remove one ball every thousand times I add those ten million balls, and so long as I remove that lowest ball, I arrive at zero balls, right? In the face of infinity, finite numbers don't matter. But the trends don't seem to hold up:

Step:Added_Removed_Remaining_Total

1.___1-10_____1_______2-10_____9

2.___11-20____2_______3-20_____18

3.___21-30____3_______4-30_____27

4.___31-40____4_______5-40_____36

5.___41-50____5_______6-50_____45

6.___51-60____6_______7-60_____54

7.___61-70____7_______8-70_____63

...

 

Now, the argument that for every number n, that ball was removed is valid. However, the total balls (ignoring the numbers) is clearly growing at a rate of 9x. The limit of 9x as x approaches infinity is infinity. The number of total balls will always grow faster than you remove them, and you will end up with balls left over. The fact that no named integer will remain is logical, but since we are dealing with infinity, we are not dealing only with integers. Infinity is not an integer, so it cannot be simply accounted for with the original argument.

[owl]

I could add, say, 10,000,000 balls at a time, and remove simply the lowest one, and I would arrive at zero balls, right?

 

Yes, of course. If the reasoning works for the "10 and 1" case, it must work equally for "x and 1" for any other x.

 

But the trends don't seem to hold up:

... However, the total balls (ignoring the numbers) is clearly growing at a rate of 9x. The limit of 9x as x approaches infinity is infinity.

 

Right. The limit of the number of balls as x approaches infinity is +infinity, but the actual number of balls after completing infinitely many steps is 0. As I mentioned earlier, this isn't inconsistent, because the limit claim is just a claim about what is happening as you go to larger and larger finite-numbered steps of the sequence. It says nothing about what happens at "stage infinity," if you will.

 

This is not all that strange. Compare a somewhat analogous case. Consider the function,

 

f(x) = 2x, for all x not equal to 3

= 1, for x=3.

 

This is a legitimate function, though a discontinuous one. The limit of f(x) as x approaches 3 is 6. But the actual value of f(3) is 1, not 6. This should drive home the point that "Lim f(x) as x approaches a" is not the same as "f(a)".

 

The fact that no named integer will remain is logical, but since we are dealing with infinity, we are not dealing only with integers. Infinity is not an integer, so it cannot be simply accounted for with the original argument.

 

I'm not sure what you mean by "we are not dealing only with integers." Are you saying that at the end of the series, there will be some balls labeled "infinity" in the jar?

 

In one sense, we are dealing only with integers: in the description of the problem, every ball has an integer written on it. No ball has "infinity" written on it. Thus, once all the integers have been accounted for, all the balls have indeed been accounted for.

 

Where you might be getting confused is in the distinction between the number of steps in the series, and the numbers on the balls in the series. The number of steps is infinite. But the numbers on the balls are all finite. This is perfectly consistent, though it can easily confuse one: there are infinitely many, finite numbers.

[pgrmdave]

The number of steps are infinite. Thus the number of balls is infinite. Thus the numbers on the balls must go to infinity. You cannot label infinite objects with finite numbers.

[Pyrotex]

... You cannot label infinite objects with finite numbers.

Maybe so, but finite numbers are all we got. ;) ;) ;)

[owl]

The number of steps are infinite. Thus the number of balls is infinite. Thus the numbers on the balls must go to infinity. You cannot label infinite objects with finite numbers.

 

No, you can't label infinitely many objects with finitely many numbers. But yes, you can label infinitely many objects, each with finite numbers. There are infinitely many finite numbers, so they are enough to label infinitely many objects.

 

There are infinitely many real numbers between 0 and 1. But none of those numbers is infinite: they're all between 0 and 1. Likewise, there are infinitely many natural numbers. None of those numbers is infinite: they're all natural numbers.

[CraigD]

My question to you, Craig, and Q. stands: name a ball, any ball, that might be in the jar [in the balls and vase problem] at the end of the series.

As I noted in post #56, this challenge question appears to be the crux of the “the number of balls at noon is zero” argument. It’s a kind of argument I described as “argumentative logic”, rather than mathematics – that is, its formal representation is about a natural-language conversation (“you can’t give a winning answer”), rather than about numbers (the number of balls in the vase).

 

Nonetheless, there is, I think, an answer: “the aleph-1th ([math]\aleph_1[/math]) ball is in the jar at the end of the series. Because [math]\aleph_1[/math] is “generated” by a well-defined subset of the power series of the largest number you can name of a ball that has been removed from the jar, [math]\aleph_0[/math], and [math]\aleph_1 \gt \aleph_0[/math], it has not been removed from the jar.

 

As I noted in post #63, I’m personally uncomfortable talking about transfinite numbers, because I’ve not studied them enough to be adequately conversant in and about their formalism. I should quit my complaining and study them, but until I do, the best I can say is “some mathematicians understand them, and agree with me” – an argument to authority, and thus not very satisfying.

[Qfwfq]

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. :hihi:

I'm tired of polishing for today and can't do too much goofing anyway, but I'm only needing a it of proof-reading and I expect to find the right situation tomorrow. :) Remember, an argument no less valid than the Ross one. ;)

 

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.

I don't see it as a matter of natural language, just that the series can't be an algebric sum but only a topologic one. I've been failing to mention this during the whole thread, due to the honey. Ask a mathematician whether it's easier to give a linear space a topologic basis or an algebric one. If it has a finite basis this is, of course, algebric as well as topologic.

 

You cannot label infinite objects with finite numbers.

An infinite cardinality may be countable, for example.

[owl]

As I noted in post #56, this challenge question appears to be the crux of the “the number of balls at noon is zero” argument. It’s a kind of argument I described as “argumentative logic”, rather than mathematics – that is, its formal representation is about a natural-language conversation (“you can’t give a winning answer”), rather than about numbers (the number of balls in the vase).

 

This seems to be intended to suggest that, because the argument can be used to "win" a debate with someone, it is therefore not really, objectively sound, or it is somehow less cogent than a purely "mathematical" argument. This is mistaken: the reason why the argument could be used to win a debate with someone is precisely that it's valid. (It's not as if it would help you "win" a debate by means of rhetoric or emotional appeals; it's a purely logical appeal.) It's also mistaken because, as I mentioned in posts 45 and 50, the balls-and-vase reasoning is perfectly analogous to arguments that are used all the time in mathematics. So there's no relevant contrast between this reasoning and "mathematical" reasoning.

 

Nonetheless, there is, I think, an answer: “the aleph-1th ([math]\aleph_1[/math]) ball is in the jar at the end of the series.

 

First, I don't know why you didn't say the [math]\aleph_0[/math]th ball is in the jar, because that ball never gets removed either. Only natural-numbered balls get removed.

 

Second, either answer would be mistaken, because no ball labeled "[math]\aleph_0[/math]" or "[math]\aleph_1[/math]" is ever in the jar in the first place. The problem description says that the balls are labeled with natural numbers. There is never any ball, anywhere, labeled "[math]\aleph_1[/math]" or anything else that's not a natural number.

 

...I should quit my complaining and study them, but until I do, the best I can say is “some mathematicians understand them, and agree with me” – an argument to authority, and thus not very satisfying.

 

Among people who study this sort of thing, I'm virtually certain that almost 100% of them, whether mathematicians, logicians, or philosophers, would agree with the argument that the jar will be empty. There's no difficult mathematics involved, and the details of the theory of transfinite cardinals are irrelevant since, as noted, no transfinite cardinals are on any of the balls in the first place.

[Erasmus00]

Among people who study this sort of thing, I'm virtually certain that almost 100% of them, whether mathematicians, logicians, or philosophers, would agree with the argument that the jar will be empty. There's no difficult mathematics involved, and the details of the theory of transfinite cardinals are irrelevant since, as noted, no transfinite cardinals are on any of the balls in the first place.

 

Why is this logic not equivalent to the following sums

 

Number of balls in the jar:

[math]\sum_{1}^{\infty}(10-1)[/math]

 

Saying there are 0 marbles in the jar is saying we can group the terms as

 

[math](10-1-1-1-1-1-1-1-1-1-1)+(10-1-1-1-1-1-1-1-1-1-1)+...[/math]

 

Saying there are infinitely many seems to be saying we can group the terms as follows

 

[math](10-1)+(10-1)+...[/math]

 

But we aren't free to rearrange the terms as we please, as noted earlier in this thread. Why do we believe this question is well defined? As an infinite series, it seems to diverge, i.e. no limit exists on the number of marbles, its undefined. Why am I wrong?

 

Consider this: instead of putting 10 marbles in, and taking one out, we put 9 marbles in and relabel the marble we were supposed to take out. i.e., at step 2 we put in marbles 11-19, and then relabel the 1 marble with the 20. This task seems completely equivalent except no marble ever leaves the jar. Hence, at the end infinitely many marbles have to be in the jar. Why is this argument not valid?

 

Edit: Also, in the balls in, balls out version, is the marble labeled [math]\lim_{n\to \infty}1*10^n[/math] i.e. the ball with a 1 and infinitely many zeros, is this ball in the jar? When did this ball come out of the jar?

-Will

[pgrmdave]

Hmmm...I hadn't thought about the idea of relabeling the marbles. That would make it so that every number was erased and made to another number, but would you really argue, owl, that because every marble was relabled, that there are no marbles, or no numbers? Name a number that wouldn't have been relabled something higher.

[owl]

Saying there are 0 marbles in the jar is saying we can group the terms as

 

[math](10-1-1-1-1-1-1-1-1-1-1)+(10-1-1-1-1-1-1-1-1-1-1)+...[/math]

 

Again, that is not what I'm saying. My argument is not

1. that because infinitely many balls are removed, there must be none left.
   
2. that because you can group numbers as above, you can make the infinite sum converge to 0.
    
   Both of those arguments are fallacious, for reasons that have been pointed out in this thread, at length. (b) is wrong because it's illegitimate to rearrange the terms of an infinite series, for the reason I pointed out in post #34. Also, (a) is bad because infinitely many balls are added as well. And both (a) and (b) are bad because parallel reasoning could be used to support any answer to the problem, from -infinity to +infinity, or anything in between.
    
   Again, what my reasoning is is just this:
    
    
   
3. For every ball, it is removed at some stage and never subsequently returned. Specifically, ball #n is removed at stage n of the series. Therefore, every ball will be outside the jar when the series is completed.
   

No one has addressed this reasoning, as opposed to the clearly fallacious reasoning of (a) and (b) (which have been responded to repeatedly).

 

Why do we believe this question is well defined? As an infinite series, it seems to diverge, i.e. no limit exists on the number of marbles, its undefined. Why am I wrong?

 

You're right about this: the sequence whose terms are the numbers of balls at the succeeding stages of the series diverges. That sequence has no infinite sum; the infinite sum is undefined.

 

What you're wrong about is thinking that that's relevant to the problem. The problem is not an infinite sum problem in mathematics. The question is not, "What is the limit of the number of balls in the jar, as the number of stages in the series approaches infinity?" The question is, instead, "How many balls are in the jar at the end of the series?" As I pointed out above, those questions are logically independent. The answer to the first question is just a matter of the trend that you can observe while you're at any finite stage in the series. The second question, on the other hand, asks you to suppose that infinitely many stages have been completed.

 

Again, the crucial point here is that the definition of "[math]\sum_{i=1}^{\infty}a_i[/math]" in mathematics is not that it's the number you get after adding together all of the infinitely many terms of the series. The expression is defined in a certain way specifically to avoid the assumption that you can ever complete the infinite series. That's why the balls-and-jar problem is not a problem about infinite sums in the mathematical sense: because this problem does ask you to assume that the infinite series gets completed.

 

Everyone agrees that the limit of the series is +infinity. But that's just not the question.

 

Consider this: instead of putting 10 marbles in, and taking one out, we put 9 marbles in and relabel the marble we were supposed to take out. i.e., at step 2 we put in marbles 11-19, and then relabel the 1 marble with the 20. This task seems completely equivalent except no marble ever leaves the jar. Hence, at the end infinitely many marbles have to be in the jar. Why is this argument not valid?

 

That is valid. That example is in the Wikipedia "Supertask" article that started this discussion, where the correct answer to that problem is also given. The answer can be arrived at by the same kind of reasoning, namely, by considering what happens to each ball in the series.

 

In the original version, for every n, ball #n is removed at stage n and never returned. So the jar is empty at the end.

 

In the relabeling version (by Allis and Koetsier), for every n, ball #n stays in the jar, but has a "0" added to its number at stage #n, 10n, 100n, and so on. So every ball remains in the jar but has infinitely many zeros added to it. So at the end, there will be infinitely many (specifically, [math]\aleph_0[/math]) balls in the jar, each of which will have a natural number followed by infinitely many 0's written on it.

 

Edit: Also, in the balls in, balls out version, is the marble labeled [math]\lim_{n\to \infty}1*10^n[/math] i.e. the ball with a 1 and infinitely many zeros, is this ball in the jar? When did this ball come out of the jar?

 

No, in the original version of the story (Ross' version), there is no such ball. Again, all the balls are labeled with natural numbers.

 

And by the way, the expression "[math]\lim_{n\to \infty}10^n[/math]" is undefined, since the series is divergent.

[Erasmus00]

I had a long response typed up, but I lost it when my browser hung. I'll try to summarize:

 

first, I agree that the question of "to what number does the sum converge" and the question of "what results after infinitely many steps are actually completed" are different. HOWEVER, I think that the only way to extract meaningful information about such a thing is the limit prodedure. If the limit is undefined, the question has no certain answer.

 

If the limit converges, then we can say what would happen IF we completed an infinite number of these steps. (though this does make an assumption about continuity)

 

Second, the relabling procedure is identical to the take a ball out procedure at every step in the series: so saying these produce different results is saying that the state of the jar of the end does NOT depend on the series, but something outside the series. This, to me, seems absurd.

 

The reason this seems absurd is that the jar has no mechanism for knowing what goes on OUTSIDE the jar. The end stage of the jar can only depend on things the jar "knows" about i.e. the marbles passing into and out of its neck.

 

In the relabeling case, we pull ball 1 out of the jar, then relabel it, and place 11-20 into the jar. As far as the jar "knows" ball 1 went out its neck, balls 11-20 went into its neck.

 

In the "original" version, we pull 1 ball out of the jar, leave it out, and put 11-20 in. BUT as far as the neck of the jar knows ball 1 goes out, 11-20 go in.

 

Since the marbles that pass through its neck are what determine the balls in the jar, if the question is to make any reasonable sense the two situations need to give the same answer. They do not (as you have reasoned). Hence, I assert the question is broken- equally good logic results in a contradiction. The number of balls left in the jar is undefinable.

-Will

[ughaibu]

I'm surprised by the continued resistance to this result, as it never struck me as either particularly controversial or threatening. I'd be interested to know what the mathematically proficient members make of Banach and Tarski's paradox: Banach–Tarski paradox - Wikipedia, the free encyclopedia This strikes me as more disturbing, but seems generally acceptable to mathematicians.

[Qfwfq]

Consider a function defined, for each x < a, by:

 

[math]l(x) = \frac{a}{a-x}[/math],

 

where a > 0. For any value at wich f may be defined, we also write:

 

[math]u(x) = 10f(x);\; F(x) = u(x) - l(x) = \frac{9a}{a-x}[/math]

 

and procede to deduce whatever may be consequent about F(a), from this definition. One first remark, although it doesn't determine F(a), is that:

 

[math]\lim_{x\rightarrow a}F(x) = \lim_{x\rightarrow a}9l(x) = \infty[/math].

 

It is helpful to consider the following construct, which is logically equivalent to the well-known balls and vase problem and reduces to it by appropriate choice of measure, so the Ross argument about it may be applied. For x < a:

 

[math]J(x)\equiv[l(x), u(x)][/math]

 

is an interval of the real field. The ordinary (Euclidean) measure is given by:

 

[math]\mu(J(x))=u(x)-l(x)=F(x)=\frac{9a}{a-x}[/math]

 

whereas for an appropriate sequence of increasing values [math]\norm x_n=a\frac{n-1}{n};\; \forall n \in \mathbb{N}, n>0[/math] commencing with 0 and having an accumulation point at a, the appropriate singular measure [math]\norm\nu(J(x))[/math] for [math]\norm x_n\lt\norm x\lt\norm x_{n+1}[/math] gives the cardinality of the set of balls in the vase at the n-th step (the count of integer numbers in the interval J). The vase problem is represented in an obvious manner, with x increasing linearly in time and equal to a when all steps are complete. Given [math]\norm x_i\lt\norm x_j\lt a[/math], considering an increment of x from the first to the second value, the interval J may be viewed to vary (in the corresponding time) from [math]\norm J(x_i)=[l(x_i), u(x_i)][/math] to [math]\norm J(x_j)=[l(x_j), u(x_j)][/math] and all numbers in the interval [math]\norm[u(x_i), u(x_j)][/math] are those which are added, while those in [math]\norm[l(x_i), l(x_j)][/math] are removed. Clearly, the Ross argument in terms of balls being removed can be applied with equal validity, either to the interval J or to the set of integers it contains (equivalent to saying with the measure [math]\norm\mu[/math] or [math]\norm\nu[/math]). It is worthwhile examining it in detail:

 

The interval of numbers removed in an increment is the image of [math]\norm[x_1, x_2][/math] by the application:

 

[math]\cal{R}:\;\cal{I}^c_{[0, a)}\longrightarrow\cal{I}^c_{\mathbb{R}_1}[/math]

 

where the domain is the space of all closed intervals [math]I\subset[0, a)[/math] and the codomain is the set those of real numbers not less than 1.

 

Given [math]\norm x_i\lt\norm x_j\lt\norm x_k\lt a[/math], the numbers removed in the two consecutive increments are those in the set:

 

[math][l(x_i), l(x_j)]\cup[l(x_j), l(x_k)]\equiv[l(x_i), l(x_k)][/math]

 

that is to say the same as those removed in the single increment. It follows trivially that

 

[math]\forall x \lt a;\; \Lambda(x)\equiv\cal{R}([0, x])\equiv[1, l(x)][/math]

 

is the set of all numbers removed at the corresponding time, since the start. Now, since [math]l[/math] is monotonic increasing,

 

[math]\forall y \geq 1;\; l(x)>y \Rightarrow y \in \Lambda(x)[/math],

 

[math]\exists \bar{x}:\; \bar{x}\lt a,\; f(\bar{x})\geq y[/math]...

 

[math]\Rightarrow y \in \Lambda(\bar{x})[/math]

 

Essentially this proof is that any y will be included in some [math]\norm\Lambda(x)[/math] with x < a. It is noteworthy that this is a strict disequality, so far we have been reasoning on the constructs as defined for arguments less than and not equal to a, yet by arguing on the arguments of [math]\norm\cal{R}[/math] and their respective images it is easy to see (Ross argument) that:

 

Proposition 1) [math]\Lambda(a)\equiv\mathbb{R}_1[/math], the set of all real numbers [math]y \geq 1[/math] which leaves: [math]J(a)\equiv\emptyset[/math].

 

Given this conclusion at a, wholly on the basis of the constructs, as defined for x < a, it is no less questionable than to assert:

 

Proposition 2) [math]J(a)\equiv[l(a), u(a)], \mu(J(a))=\mu([l(a), u(a)])=F(a)=\frac{9a}{a-a}=\frac{9}{0}[/math].

 

The same argument may be completed with the measure [math]\norm\nu[/math] for the vase problem. By the fundamental properties of a measure, [math]\norm\mu(\emptyset)=0[/math] and, combining proposition 1 and 2, we can immediately conclude as a corollary that [math]\norm F(a)=0[/math] and recalling that [math]\norm\frac{0}{9a}=0[/math], this implies [math]\norm\frac{1}{0}=0[/math]. It is suprising that not only we are able to draw such conlusions as propositions 1 and 2 wholly based on the properties of the constructs for x < a (strictly less than) but also one about the value of functions being 0 for an argument at which they have an infinite limit (as of the initial remark), something hitherto considered no more than one out of many possible choices, contrary to traditional opinion that it is in no manner determined by the algebric exprssion having a pole in that value.

 

Any questions?

 

I'd be interested to know what the mathematically proficient members make of Banach and Tarski's paradox: Banach–Tarski paradox - Wikipedia, the free encyclopedia This strikes me as more disturbing, but seems generally acceptable to mathematicians.

I think the word congruent at the end of the first sentence is a mistake. I'm no expert on the topic but I think a different type of mapping is called for and I've queried a friend who can probably answer more certainly. This said, it strikes me no stranger than bijective mappings of the segment onto the square etc.

[owl]

first, I agree that the question of "to what number does the sum converge" and the question of "what results after infinitely many steps are actually completed" are different. HOWEVER, I think that the only way to extract meaningful information about such a thing is the limit prodedure. If the limit is undefined, the question has no certain answer.

 

But there's another way of extracting an answer, which works in some cases: in some cases, it's possible to reason that the same thing happens (at some stage) to each item in a certain infinite class of objects, so at the end of the infinite series, that thing has happened to all of them. When the "limit" method of reasoning fails us, this other method can still work.

 

Second, the relabling procedure is identical to the take a ball out procedure at every step in the series: so saying these produce different results is saying that the state of the jar of the end does NOT depend on the series, but something outside the series.

 

Well, the procedure is not quite the same at any step: in the relabeling procedure, no ball leaves the jar; in the removal procedure, a ball leaves the jar at each stage. In the latter but not the former procedure, there is an ever growing pile of balls outside the jar.

 

The reason this seems absurd is that the jar has no mechanism for knowing what goes on OUTSIDE the jar. The end stage of the jar can only depend on things the jar "knows" about i.e. the marbles passing into and out of its neck.

In the relabeling case, we pull ball 1 out of the jar, then relabel it, and place [2-10] into the jar. As far as the jar "knows" ball 1 went out its neck, balls [2-10] went into its neck. [My edits in brackets. --o]

 

Weird, isn't it? The pair of cases violates the very plausible principle that the end-result of the infinite series should be a function (solely) of the qualitative state of the jar at all the individual stages of the series. In the two versions of the story, the jar has qualitatively indistinguishable states at every point in the series--i.e., it contains the same number of balls, with the same numbers written on them. And yet the end result is different.

 

Although this is very weird, I think it's true, because of the reasoning we've been discussing here. Note that I said the state of the jar was "qualitatively" the same at each point in both versions of the series. It's not absolutely the same: In one version, the ball labeled "10" is identical to the ball previously labeled "1"; in the other version, it is a new ball.

 

A similar weirdness afflicts this other version of the scenario that I previously mentioned: what if, instead of taking out the lowest-numbered marble at each stage, you take out the highest-numbered marble? Then you wind up with infinitely many marbles at the end. This is weird, because the number of marbles in the jar is the same at every stage as in the original version of the story; all that differs is which marbles are in the jar. For any finite series, this wouldn't make any difference. But this is just one more of the weird things about infinity: it does make a difference with infinite series.

 

Hence, I assert the question is broken- equally good logic results in a contradiction. The number of balls left in the jar is undefinable.

 

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?

[pgrmdave]

Well, the procedure is not quite the same at any step: in the relabeling procedure, no ball leaves the jar; in the removal procedure, a ball leaves the jar at each stage. In the latter but not the former procedure, there is an ever growing pile of balls outside the jar.

 

No ball leaves the jar, but every number does. 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?)

[Pyrotex]

Pardon me for exposing my infinite-ignorance, but may I contribute something?

 

Let's assume the 10 balls in, one ball out scenario of the original problem. Now execute the scenario. If we assume that the first step takes a minute, and each additional step takes half as long as its predecessor, then we are done in two minutes.

 

It doesn't matter how--just assume we have somehow executed the infinite number of steps and we are done.

 

Rather than look at the jar to see if we have numbered balls, let's look at the Integer Number Line. 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. I crank the speed up to 10^10^10 integers per second. Hours go by and no jarred integers.

 

Now, I "KNOW" that somewhere up ahead of me, there are an infinite number of jarred integers, because that damn jar was infinitely full!:) :shrug: :shrug: But no matter how fast I traverse the number line, I cannot find even one. I set the speed for 10^10^...a...^10 integers per second, where the number of terms, a, increases by one every nanosecond. Years go by. No jarred numbers.

 

And I never will see any. But it's an infinite number line. There is room on that line for an infinite number of countable sets, with each set being of infinite size!!!!!!!

 

Those jarred numbers ARE up there, somewhere, but I can never, never, never, never reach them.