An actual infinite number of marbles

[arman]

arman, you are not using the proper notion of infinity. We are talking about ordering and counting objects, so we should use the numbers that are specifically used to order and count, the ordinal numbers and the cardinal numbers. There are indeed infinite (or transfinite) ordinal and cardinal numbers. The transfinite number $\omega + 1 = \{1, 2, \dots , \omega\}$ contains infinitely many elements, and contains an infinitieth element, or rather, $\omega ^{th}$ element, namely $\omega$.

This comes down to semantics rather than mathematics. I am not saying that an infinite series can't exist, just that the last element of an infinite series is simply a representation, there won't actually be an element that is the last.

Keep in mind that the initial question was, Will there be a marble labeled "infinity"? My answer is no, since infinity implies just that; 'not finite', not defined (0/1 is infinite/undefined). you can't label something without, well, defining it.

 

[Tisthammerw]

The numbers in the arrangement approach infinity.

True, as do the marbles themselves consequently (each marble is labeled).

Actually, that's not technically appropriate because you have not arranged the numbers in any sort of sequence. (Of course, any way you do so, that sequence would then approach infinity)

Let's call arrangement #1 the classic 1, 2, 3, 4... sequencing.

However, as I've emphasized above, this is referring to the numbers in the arrangement, not the quantity of marbles.

Perhaps so. The odd thing is that (ordinarily) the quantity of the marbles in a #1 sort of arrangement is the largest value. In this case the values approach infinity but never actually get there. The difference between cardinality and ordinaliy easily makes things confusing.

However, talking about the quantity of marbles "approaching" infinity is completely wrong.

So is there a marble labeled "infinity" in arrangement #1?

Until you give an explicit definition of "potential infinity" and "actual infinity", this question has no answer.

Please read my first post.

I spoke to a friend of mine who was first a philosopher, then got his Ph.D in mathematics, and asked him if the terms "potential infinity" and "actual infinity" made any sense to him. The only sense he could make out of it was Aristotle's introduction of the term (I think it was Aristotle), which more or less coincides with the definition we use today. Saying that a thing X is potentially Y means exactly that: X has the potential to be Y. In other words, X could become Y, or it may be possible for X to be Y, or something along those lines.

Not really. A potential infinite never really gets to infinity, even though the collection grows without limit. Consider the story of Count Int. Count Int is an immortal who is attempting to write down all the natural numbers and reach infinity with his trusty pen and never-ending supply of paper, taking him exactly one second to write down each number. He starts with one and successively adds one each second (1, 2, 3, 4…). Will he ever reach a point in time where he can honestly say, “I'm done, I've reached infinity”? No, the number will just get progressively larger and larger without limit. He can never reach infinity anymore than he can reach the greatest possible number. There will always be a bigger yet finite number in the next second. The Count will never lay down his pen because there is an infinite quantity of those numbers. This is a potential infinite, but not an actual one.

Of course, the difference with the marbles is that the numbers are all here simultaneously. But ask yourself this. If we add 1 to itself infinitely many times, would we get a transfinite value? An actual infinite of arrangement #1 would do just that, and yet there apparently is no marble labeled infinity.

Arrangement #1 almost seems like a hybrid of an actual infinite and a potential one. The cardinality is aleph-null, yet the marbles themselves appear to be only a potential infinite. An aleph-null type infinity could easily have all natural numbers plus omega. Is this the only way to have an actual infinite? Tough call.

 

[Hurkyl]

The difference between cardinality and ordinaliy easily makes things confusing.

Well, let me try to explain it: Cardinals "measure" sets, whereas ordinals "measure" well-ordered sets. More generally, an order type "measures" ordered sets. The ordinals are just well-order types.

To put it another way, two sets have the same cardinality if and only if there is a way to rename the elements of the first set so that it becomes the second set.

However, when we look at ordered sets, we can say the same about the order types. However, we cannot use just any old renaming of the first set's elements: we have to rename the first set in such a way that we not only have the same elements as the second set, but they have to be in the right order too!

Of course, any ordered set is simply a set, so if we choose to "forget" the ordering of our sets, we can turn any order type into a cardinal number.

So is there a marble labeled "infinity" in arrangement #1?

No. You said the arrangement consists exactly of the natural numbers in their usual order. (At least, if a mathematician had written what you did, that is what he would have said) Since there is no natural number called "infinity", there cannot be a marble with the label "infinity".

The odd thing is that (ordinarily) the quantity of the marbles in a #1 sort of arrangement is the largest value.

Yes. This is one of the classical examples that demonstrate that you cannot just go and pretend that infinite things act like finite things.

In this case the values approach infinity but never actually get there. The difference between cardinality and ordinaliy easily makes things confusing.

I'm pretty sure that this difference isn't what matters here. "Ordinality" is just as static of a thing as "cardinality".

The Count will never lay down his pen because there is an infinite quantity of those numbers. This is a potential infinite

WHAT is a potential infinite? I see several things to which you could be referring, some of which are infinite, and none of which is potential.


My philosopher & mathematician friend brought up a scenario which is close to yours, but with an important difference, and it meshes with my interpretation of what he said Aristotle said:

Suppose you have stumbled across a collection of marbles, and you start counting them. After counting for a while with no end in sight, (and if we ignore physical limitations) you would be justified in saying that the collection of marbles is potentially infinite, and that the amount of counting you would do is potentially infinite. This is because we don't have that knowledge.

But your scenario is entirely different: we're told from the beginning that the collection of marbles is infinite, so it's no longer a question of potentiality.


Your original definition of "potential infinite" doesn't make sense, since the collection of marbles isn't growing. It's like the collection of natural numbers: it's not growing either. When looking at individual natural numbers, inspected one at a time in some sequence, this sequence can be said to grow. But the collection itself is just one, unchanging thing.

 

[Hurkyl]

If we add 1 to itself infinitely many times, would we get a transfinite value?

Literally speaking, I would say that's nonsense. Of course, When a mathematician says that, what they usually mean is the notion of an infinite sum given by calculus. In that case, the right answer is either "that doesn't exist", or "we get +∞", depending on if we were using the extended reals or not.

In a more obscure sense, we could also appeal to nonstandard analysis, in which there is a collection of numbers, the hypernaturals which behave exactly like natural numbers. (As long as we don't peek behind the curtain) Of course, we have to peek behind the curtain to even be able to state the question of whether a given hypernatural is transfinite or not. And, of course, we're limited to these special hyperfinite sums.

An actual infinite of arrangement #1 would do just that, and yet there apparently is no marble labeled infinity.

No, it would not. The closest thing to that that we could say is that we have infinitely often repeated the "experiment" of adding 1 to itself finitely many times.

 

[Tisthammerw]

So is there a marble labeled "infinity" in arrangement #1?

No. You said the arrangement consists exactly of the natural numbers in their usual order.

But then, would it not seem to be a potential infinite rather than an actual one?

(At least, if a mathematician had written what you did, that is what he would have said) Since there is no natural number called "infinity", there cannot be a marble with the label "infinity".

Ah, but I did not say there were only natural numbers, only that the arrangement started with one.

In this case the values approach infinity but never actually get there. The difference between cardinality and ordinaliy easily makes things confusing.

I'm pretty sure that this difference isn't what matters here. "Ordinality" is just as static of a thing as "cardinality".

Really? Consider this. The cardinality is aleph-null, and thus would seem to be an actual infinite. But as far as the actual ordinals go, the collection approaches infinity but never actually gets there; thus seeming more like a potential infinite.

WHAT is a potential infinite?

I keep telling you to read my first post, but since you seem unwilling or unable to click the link that moves you to the first page of the thread, I guess I'll have to repeat myself.

First I should distinguish between an actual infinite and a potential one. Aristotle once suggested the terms potential infinite and actual infinite. Roughly speaking, a potential infinite is a collection that grows towards infinity without limit, but never actually gets there. Take for instance a finite past starting from a beginning point. The universe gets older and older (1 billion years, 2 billion years...15 billion years) but no matter how far you go into the future, you’ll never actually reach a point where the universe is infinitely old. You can always add one more year. In contrast, an actual infinite is a collection that really is infinite.

The thing about the marbles is that although the collection of ordinals in arrangement #1 grows towards infinity (i.e. as we move left to right, the ordinals get larger; 1, 2, 3, 4...), it never actually gets there. On the other hand, the cardinality is aleph-null. So is this a potential infinite or an actual one? It would unambiguously be an actual infinite if there was a marble labeled omega in addition to all the natural numbers. Arrangement #2 has the set of all natural numbers in addition to the ordinal omega, and this arrangement has the exact same infinity as arrangement #1, aleph-null (both arrangement #1 and arrangement #2 are "equal"). So far, I suspect that arrangement #2 would be an actual infinite whereas arrangement #1 might not be. But as I said, it's a tough call.


In response to post #75

If we add 1 to itself infinitely many times, would we get a transfinite value?

Literally speaking, I would say that's nonsense.

It might not be possible in the real, physical world. But in our thought experiment that is exactly what's happening.

An actual infinite of arrangement #1 would do just that, and yet there apparently is no marble labeled infinity.

No, it would not.

If so, then it seems more like a potential infinite than an actual one. Remember, as we go left to right the number 1 is added to itself each time; 1, 2, 3, 4...if there were an actual infinite of such marbles, would not the number 1 be added to itself an infinite number of times?

 

[AKG]

This comes down to semantics rather than mathematics. I am not saying that an infinite series can't exist, just that the last element of an infinite series is simply a representation, there won't actually be an element that is the last.

That's false. If we take the set of elements:

{1/n : n is natural} U {0}

and order them from greatest to least, then we have an infinite list with last element being 0.

Keep in mind that the initial question was, Will there be a marble labeled "infinity"? My answer is no, since infinity implies just that; 'not finite', not defined (0/1 is infinite/undefined)./quote]That's absurd. "not finite" and "not defined" are not the same. 0/1 is not defined, but we're not suggesting that we have 0/1 elements, we're suggesting that we have infinity elements. If we can have a set like $\omega + 1$ with an infiniteth element, i.e. with an element labelled infinity, then why can't we have the same for marbles?

you can't label something without, well, defining it.

Yes, and transifinite ordinals and cardinals are well-defined.

 

[arman]

I don't understand what you're saying. Just because we have a function that approaches a value, doesn't mean that it will actually get there. You can't say that an infinite series has a number labelled infinity because an infinite series that approaches zero has a number labelled 0. the 'last' value of an infinite series doesn't physically exist. if it did it wouldn't be infinite, because it would have been defined and we could give it a number. I don't see why this argument involves so many abstract mathematical concepts, which do not actually exist to exist in real linear physical problems such as this one.

 

[arman]

Besides, if the infinity value of the series you gave did in fact have a zero value, couldn't you logically conclude that 0/1 is in fact infinity?

 

[Tisthammerw]

That's false. If we take the set of elements:

{1/n : n is natural} U {0}

and order them from greatest to least, then we have an infinite list with last element being 0.

You're assuming that ordering them from greatest to least {1/1, 1/2, 1/3...} can be done, which I'm not sure is possible in the case of {1/1, 1/2...0}. After all, what's the list item just to the left of 0? If you keep going from left to right it seems you'll never get to the point where you can put zero at the far end of the array, because after all there is no greatest natural number for n. Ordering the list this way, {0, 1/1, 1/2, 1/3...} is more plausible.

 

[Hurkyl]

Is the collection of natural numbers finite? The answer is a clear, unambiguous no. Therefore, by the definition of the word infinite, the collection of natural numbers is infinite.

There is absolutely no wiggle room in this matter whatsoever -- the collection of natural numbers is clearly and unambiguously infinite.

So, unless you are allowing the phrase "potentially X" to describe something that is clearly and unambiguously X, it is clear that it is not appropriate to say that the collection of natural numbers is potentially infinite.

But in our thought experiment [adding 1 to itself infinitely many times] is exactly what's happening.

What thought experiment?? Although you've never said it, I imagine you're referring to the scenario I said that my philosopher friend had mentioned as a possibly appropriate use of the word "potential infinity":

You're faced with a collection of marbles, and you start counting them. Unless you arrive at a final marble, you will always have to consider the collection as being "potentially infinite", because you do not have enough information to determine if it is finite or not.


Or, you could be faced with this experiment: you are faced with a collection of marbles labelled with ordinal numbers, and for some reason, you know that if some ordinal number is used, then so is every ordinal number before it. So, you begin sifting through the marbles to see what labels you can find.

In this experiment, until you have either looked at every marble, or have come across a marble labelled with an infinite ordinal, you will also have to consider the collection as being "potentially infinite", because you do not have enough information to determine if it is finite or not.


But if you are given the information that every natural number appears on some marble, that is enough information to determine that the collection is infinite, so it would not be appropriate to call it "potentially infinite".

Remember, as we go left to right the number 1 is added to itself each time; 1, 2, 3, 4...

That "process" will not get you to ω. To "get" a limit ordinal, you have to do something radically different.

To get ω you have to appeal to the axiom of infinity.

Without the axiom of infinity, you have no way of constructing ω, because you have no way of getting all the natural numbers in one fell swoop.

To get 2ω (= ω + ω), you have to appeal to the axiom of replacement, with the key step being that you "replace" each natural number n in ω with ω + n.

Without the axiom of replacement, you have no way of constructing 2ω, because you have no way of getting all of the ω + n in one fell swoop.

You're assuming that ordering them from greatest to least {1/1, 1/2, 1/3...} can be done, which I'm not sure is possible in the case of {1/1, 1/2...0}. After all, what's the list item just to the left of 0?

He just said how to do it. He defined his ordering (which I'll write as {) by:

a { b if and only if a > b

Your mistake is assuming that each element in an ordering (aside from the least, if there is one) must have an element before it.

 

[Lifter0569]

"In contrast, an actual infinite is a collection that really is infinite." ---- I do not see how one could say that a collection of things are infinite in nature. But perhaps, I do not know the term for actual infinite well enough. But you say that an actual infinite of something or other "is a collection that really is infinite." This can not be in my opinion, a collection meaning a specific type of something and nothing else in its own definition means bounded. Infinite ANYTHING can not be bounded. How can actual infinity be defined? Seems illogical.

 

[AKG]

I don't understand what you're saying. Just because we have a function that approaches a value, doesn't mean that it will actually get there.

I don't know why you're talking about functions approaching values, or what it has to do with anything

You can't say that an infinite series has a number labelled infinity because an infinite series that approaches zero has a number labelled 0.

Huh? I'm not saying anything of the sort. I'm saying that we can have a set with an element labelled "infinity."

the 'last' value of an infinite series doesn't physically exist. if it did it wouldn't be infinite, because it would have been defined and we could give it a number.

This is false. Suppose we have a line of length 1m. Then mark off a point at 0m, 0.5m, 0.75m, 0.875m, ... and finally at 1m. Your first point will be 0m, your last will be 1m, but you will have no element directly preceeding the 1m mark. You will have infinite marks, and you could label the n^th element from the left "n" and you could label 1m with $\omega$ or "infinity". Why does this give you problems?

 

[AKG]

You're assuming that ordering them from greatest to least {1/1, 1/2, 1/3...} can be done, which I'm not sure is possible in the case of {1/1, 1/2...0}.

Of course it's possible.

After all, what's the list item just to the left of 0?

Why must there be one?

If you keep going from left to right it seems you'll never get to the point where you can put zero at the far end of the array, because after all there is no greatest natural number for n.

That's correct.

Ordering the list this way, {0, 1/1, 1/2, 1/3...} is more plausible.

"More plausible"? The only difference is that ordering it that way makes it more natural to label none of the elements as "infinity" whereas ordering it my way makes it natural to do so. But there's no reason why my ordering is impossible, it's a perfectly valid ordering. By the way, are you familiar with transfinite ordinals? The first one, $\omega$ is just:

$$\{0, 1, ...\}$$

i.e. it's basically just the natural numbers. The next transfinite ordinal is:

$$\{0, 1, ..., \omega \}$$

so it has infinity elements, and then it's last element is $\omega$. If you are still convinced that you're right and I'm wrong, tell that to Cantor.

 

[arman]

This is false. Suppose we have a line of length 1m. Then mark off a point at 0m, 0.5m, 0.75m, 0.875m, ... and finally at 1m. Your first point will be 0m, your last will be 1m, but you will have no element directly preceeding the 1m mark. You will have infinite marks, and you could label the n^th element from the left "n" and you could label 1m with $\omega$ or "infinity". Why does this give you problems?

You're using an irrelevant analogy. You're giving the example of an infinite series describing a finite length, which has very little to do with the marble example where we have an infinite series describing an infinite number of marbles.

The fact is that you cannot draw a mark at 0, 0.5, 0.75...1, because that's an infinite number of marks. You can't count an infinite number of marbles, if only by definition. If you count something it is finite.

If there was an infinite number of marbles, and you had an infinite amount of time, you would count an infinite number of marbles. But there would be no infinity marble because to define a marble you need to take a place in time, and at any point in time there would be a finite number of counted marbles.

Asking whether you could see how many marbles one has after an infinity years of counting is irrelevant because it's applying a physical and finite concept such as counting to a non-physical and conceptual situation that can't exist in a world where finite principles such as counting exist.

Besides, the question was whether there would be a marble labelled infinity if you count them. If I count a marble every second and live for a hundred years, I would count 3153600000 marbles, and that's without sleep. Unless I invent some kind of counting machine which never ends, in which case I would be counting a number of marbles that approaches infinity WITH TIME.

 

[arman]

I don't know why you're talking about functions approaching values, or what it has to do with anything

The marbles we count are approaching infinity.

 

[arman]

Huh? I'm not saying anything of the sort. I'm saying that we can have a set with an element labelled "infinity."

Well you gave this as an example:

That's false. If we take the set of elements:

{1/n : n is natural} U {0}

and order them from greatest to least, then we have an infinite list with last element being 0.

You're implying that this series, which approaches 0, somehow proves that infinity physically exists. As I said before, this has little relevance to this problem. Just because an infinite series approaches a finite value, and because we can stick a name tag on that value, doesn't really apply to the marble scenario and I think you're missing something in your analysis.

1/n doesn't converge anyway. The only way you could parallel your 1/n series to the marble thing is by taking a series of n = {0, 1, 2...}, in which case 1/n is only 0 for the 'last' value of the n series, which lands us at the same problem as with the marbles, if I count the values of n, will I reach the last one? It's the exact same problem and doesn't really shed any light on the issue.

I should say this again, the question wasn't, "if I count marbles will the number I count approach infinity". The obvious answer to that is yes. The question was "If I count marbles will there be a value labelled infinity". Saying n={1, 2, 3... infinity}, therefore the last marble is infinity, is misconcieved nonesense.

 

[AKG]

You obviously have no idea what you're talking about. Do you even know what a series is? I'm not talking about any series. In the example with the meter stick, I wasn't trying to "describe" the length, the length had nothing to do with it. Sometimes concrete examples like that help people understand concepts that they don't yet get, it just doesn't seem to work with you. I don't know why you're getting hung up on whether it is physically possible to make infinitely many marks on a ruler, that's obviously not the question at hand. The question is if an infinite set has an "infiniteth" element, and the answer is "It depends; it depends on how you order them." The set $\omega$ is an infinite set with no infinitieth element, no last element. The set $\omega + 1$ is such a set. If we have infinitely many marbles, and we can put them into correspondence with $\omega$, which is just {1, 2, 3, ...} then we can certainly put them into correspondence with $\omega + 1$. And if we can put them into such a correspondence, then we can label them correspondingly.

The marbles we count aren't approaching infinity. We are saying there are infinitely many marbles. We're then asking if such a set of marbles can be labelled in a reasonable way (and not just meaninglessly labelling them all "infinity", for example) that makes it so that one of the marbles is labelled as the infinity marble. The actual answer is yes, it's up to you decide whether you're willing to understand this.

And although "1/n converges" is meaningless, strictly speaking, the sequence <1/1, 1/2, 1/3, ...> does indeed converge. But that's irrelevant. We're not talking about the number of marbles "approaching" anything. The number of marbles you have counted to date may approach something, but if there are X marbles now, then there are X marbles, the number of marbles is what it is, it is not approaching something.

You're implying that this series, which approaches 0, somehow proves that infinity physically exists.

This is positively absurd. Do you know what a series is, seriously? Nothing is approaching anything, and there is no series here. And what in the world does it mean for infinity to physically exist? Infinity is a number, not a cupcake. Cupcakes physically exist, I can open my pantry and find them. Numbers aren't physical objects. Can there be an infinite number of something in the physical world? Well assuming that things like length can take on real values, then yes, of course, for there would be infinitely many points between your face and your monitor.

Saying n={1, 2, 3... infinity}, therefore the last marble is infinity, is misconcieved nonesense.

Again, if you believe this, tell it to Cantor.

You seem to not understand a lot of things, so I don't know if this is the right place to start, but perhaps the one thing you need to understand first is that we're not asking whether, if you start at time 0 and start labelling, one by one, an infinite set of marbles, if you'll ever write down "infinity" on any of the marbles. The question is, if there are infinitely many marbles and if they've been labelled, then will there be a marble labelled infinity. The answer is "it depends on how you label them." If you label them as per the original post, with the numbers {1, 2, 3, ...} then the obvious answer is no, since infinity is not an element of that set. If you label them with the elements of $\omega + 1$ then the answer is "yes". Now we have a natural "intuition" as to how we would label infinitely many marbles using the numbers from {1, 2, 3, ...}. For those unfamiliar with transfinite ordinals, we tried to provide a natural, "intuitive" way to label those same marbles with $\omega + 1$. One way would be to label the first one infinity, and then label the rest 1, 2, 3, ... but that seems like cheating. But I think the idea of labelling the points 1/n with the label "n" and then labelling "0" with $\omega$ should be more intuitive. Or marking off 0.5m, 0.75m, 0.875m, etc. and with natural numbers then marking 1m off with $\omega$ should also be a little more intuitive. If you still don't get it, then that's tough luck.

By the way, please look at this. What you call "misconceived nonsense" is something you very obviously have no understanding of, and it is something that is a well-established area of mathematical study. Of course, you seem to have problems understanding other related things like sets, series, convergence, etc. so rather than just barking back responses in regards to a topic you don't understand, I suggest you do some study.

 

[arman]

re AKG

I was using the term series wrong. I meant a set. sorry, I've been doing Fourier series all night and the word kind of stuck.]

As I was saying, if we have an infinite set {1, 2, 3,... infinity} then the 'last' value is infinite, but if we count them, and use the set {1, 2, 3...} then the last value approaches infinity and there won't actually be a value labelled infinity.

 

[Tisthammerw]

Is the collection of natural numbers finite? The answer is a clear, unambiguous no. Therefore, by the definition of the word infinite, the collection of natural numbers is infinite.

Great, but that still doesn't answer the question of whether arrangement #1 of the marble story is a potential infinite or an actual one.

But in our thought experiment [adding 1 to itself infinitely many times] is exactly what's happening.

What thought experiment??

Arrangement #1 of the marble story, remember?

You're faced with a collection of marbles, and you start counting them. Unless you arrive at a final marble, you will always have to consider the collection as being "potentially infinite", because you do not have enough information to determine if it is finite or not.

Wrong (at least in this case) we know it's never going to end (as is generally the case with potential infinites).

But if you are given the information that every natural number appears on some marble, that is enough information to determine that the collection is infinite, so it would not be appropriate to call it "potentially infinite".

*Sigh* Let's try this again.

A potential infinite is a collection that grows towards infinity but never actually gets there. On the surface it seems that arrangement #1 is an actual infinite rather than a potential one, however:

The thing about the marbles is that although the collection of ordinals in arrangement #1 grows towards infinity (i.e. as we move left to right, the ordinals get larger; 1, 2, 3, 4...), it never actually gets there. On the other hand, the cardinality is aleph-null. So is this a potential infinite or an actual one?

I repeat, although the collection of ordinals grows towards infinity, it never actually gets there. Is the cardinality of the marbles aleph-null? Yes it is; just as it is for the set of natural numbers. Does aleph-null represent infinity? No question there. But given the circumstance of the marbles, it isn't clear that the kind of infinity here isn't a potential one (albeit perhaps a weird kind of potential infinite).

Remember, as we go left to right the number 1 is added to itself each time; 1, 2, 3, 4...

That "process" will not get you to ω.

Are you aware that I do not believe an actual infinite cannot be formed via successive addition?

Still, the question remains: what would happen if the process (adding 1 to itself) were done infinitely many times as would (apparently) be the case in this thought experiment?

You're assuming that ordering them from greatest to least {1/1, 1/2, 1/3...} can be done, which I'm not sure is possible in the case of {1/1, 1/2...0}. After all, what's the list item just to the left of 0?

He just said how to do it.

And I just said why that means of how to do it was questionable.

He defined his ordering (which I'll write as {) by:

a { b if and only if a > b

Your mistake is assuming that each element in an ordering (aside from the least, if there is one) must have an element before it.

Except that given how he did it he seemed to be making that very assumption ordering it from greatest to least {1, 1/2, 1/3, 1/4...0} which was why I suggested an alternate way to order it. Suppose for instance we instantiated that ordering into marbles. How would that work? How could there be an endpoint at 0? It seems it would never be reached.

But perhaps I'm just not being abstract enough; that the ordering is somehow valid mathematically even if it could never possibly work in the real, physical word.

 

[Tisthammerw]

You're assuming that ordering them from greatest to least {1/1, 1/2, 1/3...} can be done, which I'm not sure is possible in the case of {1/1, 1/2...0}.

Of course it's possible.

After all, what's the list item just to the left of 0?

Why must there be one?

If you keep going from left to right it seems you'll never get to the point where you can put zero at the far end of the array, because after all there is no greatest natural number for n.

That's correct.

Okay, it seems like we might agree more than we disagree. To make it short, I'll reiterate what I said in post #89:

Except that given how he did it he seemed to be making that very assumption ordering it from greatest to least {1, 1/2, 1/3, 1/4...0} which was why I suggested an alternate way to order it. Suppose for instance we instantiated that ordering into marbles. How would that work? How could there be an endpoint at 0? It seems it would never be reached.

But perhaps I'm just not being abstract enough; that the ordering is somehow valid mathematically even if it could never possibly work in the real, physical word.

And finally,

But there's no reason why my ordering is impossible

Except for the reasons I stated (i.e. in that kind of ordering, we'd never reach the end and 0 would never be included), but as I admitted, perhaps I'm just not being abstract enough; that the ordering is somehow valid mathematically even if it could never possibly work in the real, physical word.

If you are still convinced that you're right and I'm wrong, tell that to Cantor.

I've tried contacting him, but he's never returned my phone calls. ;)

 

[Hurkyl]

Great, but that still doesn't answer the question of whether arrangement #1 of the marble story is a potential infinite or an actual one.

The collection is infinite, there's no question about that. The individual elements are not infinite, there's no question about that. I see nothing in this scenario to which the word "potential" would be applicable.

Arrangement #1 of the marble story, remember?

The marble story wasn't an experiment.

You're faced with a collection of marbles, and you start counting them. Unless you arrive at a final marble, you will always have to consider the collection as being "potentially infinite", because you do not have enough information to determine if it is finite or not.

Wrong (at least in this case) we know it's never going to end (as is generally the case with potential infinites).

In this example (which is different from your problem!), you cannot know if it's going to end until you actually get to the end. Thus, it is appropriate to say it's potentially infinite. I'm convinced your parenthetical is completely backwards -- we say it's a potentially infinite precisely when we do not know if it will end or not.

*Sigh* Let's try this again.

A potential infinite is a collection that grows towards infinity but never actually gets there.

And I've stated how I understand the usage of the term. We say something has potential to be something if it may be able to actualize that potential -- we say that collections of marbles are potentially infinite, and when we come across some collection that, for whatever reason, we know is infinite, then we say that the collection has actualized its potential -- we would no longer say that it is potentially infinite, because it is actually infinite.

We know the collection of natural numbers is infinite -- it is thus not appropriate to call it potentially infinite. We know that each individual natural number is finite, and thus does not have the potential to be infinite, so it would not be appropriate to call the natural numbers themselves potentially infinite. There is nothing in any of your scenarios that would be appropriate to call potentially infinite!


I posit that because you cannot write down a rigorous proof one way or another of your original question, that your definition is too ambiguous. This is why I like mathematics and don't like philosophy -- people like to state "definitions" that don't lend themselves to any sort of solid logical analysis, (But, I'm usually faced against armchair philosophers; I presume the actual practitioners fare much better in this regard) and we have these annoying "discussions" where people just keep restating their vague arguments.


"grows to infinity" and "never actually gets there" are not rigorous things, and I have not yet been able to divine a precise definition that is consistent with the way you use the term "potential infinite".

Still, the question remains: what would happen if the process (adding 1 to itself) were done infinitely many times as would (apparently) be the case in this thought experiment?

We are not adding 1 to itself infinitely many times. We are doing infinitely many "experiments", each of which consists of adding 1 to itself finitely many times.

Except that given how he did it he seemed to be making that very assumption ordering it from greatest to least {1, 1/2, 1/3, 1/4...0} which was why I suggested an alternate way to order it.

Do you disbelieve in the "<" relation on rational numbers?! That is precisely what he is using to order his collection of numbers. (Except that he's using the order relation in the opposite direction)

Recall that the notation {1, 1/2, 1/3, 1/4, ..., 0} is simply a presentation of the ordering -- it is not the ordering itself. The ordering itself is some irreflexive, antisymmetric, transitive binary relation.

In other words, R is a (total) ordering iff:

x R x is false
Either x R y, y R x, or x = y
x R y and y R z implies x R z


AKG's ordering is simply x R y if and only if y < x, where < denotes the standard ordering we use on the rational numbers.

Intuitively speaking, an ordering is nothing more than a rule that defines when one thing comes "before" another thing.

 

[Tisthammerw]

I see nothing in this scenario to which the word "potential" would be applicable.

*Sigh* Let's try this again.

A potential infinite is a collection that grows towards infinity but never actually gets there. On the surface it seems that arrangement #1 is an actual infinite rather than a potential one, however:

The thing about the marbles is that although the collection of ordinals in arrangement #1 grows towards infinity (i.e. as we move left to right, the ordinals get larger; 1, 2, 3, 4...), it never actually gets there. On the other hand, the cardinality is aleph-null. So is this a potential infinite or an actual one?

I repeat, although the collection of ordinals grows towards infinity, it never actually gets there. Is the cardinality of the marbles aleph-null? Yes it is; just as it is for the set of natural numbers. Does aleph-null represent infinity? No question there. But given the circumstance of the marbles, it isn't clear that the kind of infinity here isn't a potential one (albeit perhaps a weird kind of potential infinite).

If I didn't know any better, I'd swear I've been repeating myself. Wait, let me check post #89...

The marble story wasn't an experiment.

I said thought experiment, remember? The marble story is a thought experiment.

I'm convinced your parenthetical is completely backwards -- we say it's a potentially infinite precisely when we do not know if it will end or not.

No, that is not what a potential infinite is. A potential infinite is what I defined it to be. If you disagree with me, tell that to Aristotle.

And I've stated how I understand the usage of the term. We say something has potential to be something if it may be able to actualize that potential

In that case, you have not understood the term. Many potential infinites cannot be actualized, e.g. the story of Count Int.

Consider the story of an immortal person named Count Int who is attempting to write down all the natural numbers and reach infinity with his trusty pen and never-ending supply of paper, taking him exactly one second to write down each number. He starts with one and successively adds one each second (1, 2, 3, 4…). Will he ever reach a point in time where he can honestly say, “I’m done, I’ve reached infinity”? No, the number will just get progressively larger and larger without limit. He can never reach infinity anymore than he can reach the greatest possible number. There will always be a bigger yet finite number in the next second.

This is an example of a potential infinite. It grows towards infinity but never actually gets there.

I posit that because you cannot write down a rigorous proof one way or another of your original question

It's a question, not a conclusion.

that your definition is too ambiguous. This is why I like mathematics and don't like philosophy -- people like to state "definitions" that don't lend themselves to any sort of solid logical analysis

Actually, some forms of philosophy are very analytical (e.g. symbolic logic).

"grows to infinity" and "never actually gets there" are not rigorous things,

It seems to me they can be stated rigorously. I'm not a mathematician, but consider one possible formulation of a potential infinite:

{x1, x2, x3, x4...}

where xj+1 >= xj, however this series does not contain an element >= omega.

It's at least a semi-rigorous way to look at it.

and I have not yet been able to divine a precise definition that is consistent with the way you use the term "potential infinite".

And I have yet to understand why you refuse to listen to my own definition, make up your own (one that is not consistent with mine), and then claim that I've been causing the inconsistency.

Still, the question remains: what would happen if the process (adding 1 to itself) were done infinitely many times as would (apparently) be the case in this thought experiment?

We are not adding 1 to itself infinitely many times. We are doing infinitely many "experiments", each of which consists of adding 1 to itself finitely many times.

I have yet to divine a precise definition of these phrases that is consistent here.

Are we adding one to itself finitely many times? No. Then by your own argument it seems (see beginning of post #80), we are doing it infinitely many times.

Do you disbelieve in the "<" relation on rational numbers?!

No. Do you disbelieve the fact that Count Int will never reach infinity?!

See post #90 regarding the {1, 1/2, 1/3, 1/4, ..., 0} series.

Recall that the notation {1, 1/2, 1/3, 1/4, ..., 0} is simply a presentation of the ordering -- it is not the ordering itself.

Ah. Please see post #90.