An actual infinite number of marbles

[Hurkyl]

Tisthammerw: suppose that all of your marbles were labelled with either a natural number or the symbol ω, and that each label were used exactly once. I come along and take away the marble with the label ω. Now, all you have left are marbles labelled with natural numbers. Would you assert that you now have finitely many marbles?

The natural numbers go towards infinity without limit but never seem to actually get there. Hence the apparent paradox.

The paradox is apparent only because you insist on associating this fact with something almost entirely unrelated. (Actually, it is more appropriate to say that the natural numbers are unbounded. To speak about divergence towards infinity, you have to talk about a sequence. Of course, all sequences of natural numbers diverge to infinity, but that's beside the point)

The magnitude of the individual numbers within the set of natural numbers has absolutely no bearing on the cardinality of the set of all natural numbers. As a more obvious example, consider the set of all real numbers in the interval [0, 1]. This is also an obviously infinite set, but the numbers are all bounded! They don't even diverge to infinity!

 

[Royce]

Hurkyl, this is a bit off topic; and, goes back quite a bit to a previous discussion we had. I have two questions that still drive me nuts when I think about them.

If infinity is indeterminate how can it be a mathematical limit?

and

If the set of whole numbers is infinite how or why is it considered a closed set by definition?

 

[Tisthammerw]

Tisthammerw: suppose that all of your marbles were labelled with either a natural number or the symbol ω, and that each label were used exactly once. I come along and take away the marble with the label ω. Now, all you have left are marbles labelled with natural numbers. Would you assert that you now have finitely many marbles?

No.

The paradox is apparent only because you insist on associating this fact with something almost entirely unrelated.

What is "this fact"? Remember, I didn't say that the natural-number-marble-labeling-scheme causes the marbles to be of finite quantity, only that it might be a potential infinite instead of an actual one. Does an actual infinite require a ω here? It might be the case given the definition of a potential infinite and an actual infinite. (Note to non-mathematicians: ω is the infinity that comes “right after” all the natural numbers.) Unfortunately, the difference between cardinality and ordinality makes matters confusing.

The magnitude of the individual numbers within the set of natural numbers has absolutely no bearing on the cardinality of the set of all natural numbers.

Maybe so (at least in the case regarding the set of all natural numbers), but is the marble set (all labeled with natural numbers) a potential infinite or an actual one? I agree that the cardinality of set N is אo, but that is not the issue here. As I said, the difference between a cardinal infinite and an ordinal infinite makes matters confusing. Perhaps אo--when describing natural numbers--is actually a potential infinite whereas ω is invariably an actual one?

Of course, both the set of all natural numbers (set N) and set N + ω would both have the same mathematical "magnitude" of infinity: אo, even if one set is a potential infinite and the other is not.

 

[Tisthammerw]

Hurkyl, this is a bit off topic; and, goes back quite a bit to a previous discussion we had. I have two questions that still drive me nuts when I think about them.

If infinity is indeterminate how can it be a mathematical limit?

In calculus, the infinity ∞ really isn't an actual infinite but rather a potential one. Note that when denoting bounds, a parenthesis is always used instead of a bracket, e.g. [1, ∞) which means it includes 1 and goes towards infinity (though never actually getting there).

If the set of whole numbers is infinite how or why is it considered a closed set by definition?

I'm sure there's some proof that explains why, but I don't recall what it is.

 

[Hurkyl]

There are two major uses of the word "infinity" in calculus:

The first is simply as part of a phrase. When one says "this sequence diverges to infinity", it's really just a phrase -- there is no object called "infinity" which is referenced by this phrase. Similar things can be said for things like "the sum from n = 1 to infinity" et cetera.

However...

It was realized that one could formalize and make practical use of actual points at infinity (which is just another phrase, BTW). There is a topological space called the extended real line which consists of the real numbers with two additional honest-to-goodness points that we call +∞ and -∞. Then, we have honest-to-goodness intervals like [1, +∞] which consists of the real number 1, all real numbers bigger than 1, and the point +∞. (Which is the same as saying all extended real numbers that are no smaller than 1 and no larger than +∞)

This allows us to translate phrases with the word infinity in them into statements about these actual objects we call ±∞. For example, we have the function:

f(x) = x²

defined on the reals. Consider:

<br /> \lim_{x \rightarrow +\infty} f(x) = +\infty<br />

Before we started using the extended real line, this was merely a formal equation which was used as a convenient and suggestive shorthand for what was really meant. However, when using the extended real number line, this is an honest-to-goodness equality -- ±∞ are just as good as any other point in the extended reals, and there is nothing special about them. (Aside from the fact they're the endpoints of the extended real line)

And, here, it makes exactly as much sense to say that f(±∞) = +∞ as it would to say that for the function g(x) = x²/x, g(0) = 0.


And, just to make it absolutely clear, these two additional points that lie on the extended real line, ±∞, have absolutely nothing to do with the concept of infinite sets.

If the set of whole numbers is infinite how or why is it considered a closed set by definition?

That can't be answered until you say what you mean by "closed". Some particular meanings might be:

The whole numbers are closed under incrementing, addition, and multiplication. (meaning that if we increment a whole number, or add or multiply two whole numbers, we get a whole number as the answer)

The whole numbers are not closed under things like subtraction or division. (Because these operations will give things that aren't whole numbers, or might not even be defined at all!)

Taking the whole numbers as a topological space (in any way you would like to do so), it would be true to say that the whole numbers are topologically closed within themselves. (Of course, such a statement is true for any topological space)

The whole numbers are closed under the "sup" operation on finite sets, becuase if we have a finite set of whole numbers, they have a least upper bound that is a whole number. (i.e. the maximum of the set)

The whole numbers are not closed under arbitrary sup's. For example, the sup of the entire set of whole numbers does not exist. However, within the ordinals, we have that the supremum (= least upper bound) of the whole numbers is ω. This is an instance where the supremum of a set exists, but the set does not have a maximum. This is quite analogous to the fact that the interval (0, 1) has a supremum, 1, although that interval has no maximum element.

emember, I didn't say that the natural-number-marble-labeling-scheme causes the marbles to be of finite quantity, only that it might be a potential infinite instead of an actual one.

Well, here are two problems you face:

(1) You have to explain how the act of removing a single member of an "actual infinity" would yield a mere "potential infinity". (Of course, you couldn't explain it without actually defining the terms)

(2) You agree that you would not be left with a quantity that is not finite. Therefore, by definition, that quantity is infinite. In the English usage of the term, it would be quite appropriate to say the quantity is actually infinite.



A lot of times when people talk about "potential infinity", they are referring to some sort of "changing" thing. For example, people like to talk about how natural numbers keep "growing" larger and larger, and I'm sure I've heard that called a "potential infinite".

The problem here is that there is nothing changing at all: each natural number is what it is, and the set of natural numbers is what it is, they are not dynamic, changing things.

What people seem to mean, I think, is that there is some sort of "process" where they "start" with the number one, then move onto the number two, then three, and so forth, and then confuse the set of identified numbers with the process, or even worse, speak about some sort of "end result". (Similar to some people like to think of 0.999~ as being some sort of process that starts with 0.9, then moves onto 0.99, and so forth, and then confuse the actual honest-to-goodness number 0.999~ with this process)

 

[BoardTracker]

Here are some variations on the theme:
(a) What is the last digit in the decimal representation of \pi
(b) A lightbulb is switched on for 1/2 minute. It is switched off for the next 1/4 minute and on again for the 1/8 minute following that and so on. At the end of one minute, is the light on or is it off?

Not really the same :)
I'll try to explain why I think that
(a) If \pi has a finite number of digits, then there would be a last digit (while in the question originally brought here there is definite infinite number of items). Even if \pi has no finite number of digits (which I am not sure if proven so... is it?) the question of "what is the last digit of an endless digit string" (which is a self contradicting question), is not the same as "is there a item numbered "infinite" in an infinite numbered items. Which as most people think here has a clear simple "yes/no" answer (which is no, btw).

(b) At the end on the minite the bulb will be off since it will be burnt ;) a bulb has a finite number of times at can be switched on/off. I bet if you tried your experiment, most bulbs would burn :D
Another difference with your (b) question is that there most certainly IS a state for the bulb at exactly 1:00 after the experiment starts. Wether it can be labeled on or off (assuming as I say it does not burn), is another question.
From a scientific pov, the bulb will actually be on :D since by the end of the minute the on/off switching will be infinitely fast that the coil will anyway be hot enough in the inbetween state to rule the bulb "on"

-

As to the orginal question..
You did hint that the numbers are integer positive (by giving the example of 1,2,3..). If so, there will be no "infinite" tagged marble, since "infinite" is either text/string OR a concept. Both not integer positive.
If you did not mean its integer positive, I would say that..
If there IS any logic in the tagging of the marbles, we need to know it in order to answer your question.
If there is NO logic and tags are totally random, there most definitely a marble tagged "infinite" since in an infinite number of random (all possible) events, one even will certainly be "infinite".
If you don't tell us what the logic behind the tagging, then the answer will be "dunno, start looking at each one and see" or... the more correct one "maybe"

 

[Johann]

(b) A lightbulb is switched on for 1/2 minute. It is switched off for the next 1/4 minute and on again for the 1/8 minute following that and so on. At the end of one minute, is the light on or is it off?

I think the answer must be that the bulb will be both on and off, or neither on nor off. And if you don't understand it, that's your problem, it doesn't mean mathematicians are capable of uttering nonsense

 

[Locrian]

No, the answer is that as you approach the one minute mark, you are flashing the bulb on and off so rapidly that it quickly burns out the filament. At the one minute mark the bulb will therefore be off.

 

[Temporarily Blah]

If there was a number labeled Infinity, you would not be able to comprehend it.

If there isn't one, there isn't.


We can't see infinity, so the question makes no sense.

 

[AKG]

Not really the same :)
I'll try to explain why I think that
(a) If \pi has a finite number of digits, then there would be a last digit (while in the question originally brought here there is definite infinite number of items). Even if \pi has no finite number of digits (which I am not sure if proven so... is it?) the question of "what is the last digit of an endless digit string" (which is a self contradicting question), is not the same as "is there a item numbered "infinite" in an infinite numbered items. Which as most people think here has a clear simple "yes/no" answer (which is no, btw).

This isn't the case. \pi does have infinitely many digits, so it indeed makes no sense to ask of the last one. However, if you have an infinite number of items, then depending on the infinity, there may or may not be an item labelled with an infinity. The infinite set \omega contains no "infinite" element, the infinite set \omega + 1 does.

(b) At the end on the minite the bulb will be off since it will be burnt ;) a bulb has a finite number of times at can be switched on/off. I bet if you tried your experiment, most bulbs would burn :D
Another difference with your (b) question is that there most certainly IS a state for the bulb at exactly 1:00 after the experiment starts. Wether it can be labeled on or off (assuming as I say it does not burn), is another question.
From a scientific pov, the bulb will actually be on :D since by the end of the minute the on/off switching will be infinitely fast that the coil will anyway be hot enough in the inbetween state to rule the bulb "on"

This really doesn't answer the question. No one is asking whether the bulb will be physically burnt out or whether it will be so hot that it will glow anyways. You're not supposed to be pedantic, you're supposed to try to see his actual point and respond to that. If we must, we can consider a function f : [0, 1] --> {0, 1} where f(x) = 1 iff there are some a and b in [0, 1] with a < b such that x is in [a, b) and such that the bulb was supposed to be on during the time interval from when a minutes had passed to before b minutes had passed. Yes, there is indeed a state for the bulb at the one minute mark, and the function f does indeed have a value f(1), but there is not enough information to determine the state or the value. If I say I have a function:

f : [0, 1] --> [0, 1]

with f(x) = x for x in [0, 1), does it then make sense to ask what f(1) is? Could you (Tide) tell me what f(1) is given only that f(x) = x for x in [0, 1)?

If there is NO logic and tags are totally random, there most definitely a marble tagged "infinite" since in an infinite number of random (all possible) events, one even will certainly be "infinite".
If you don't tell us what the logic behind the tagging, then the answer will be "dunno, start looking at each one and see" or... the more correct one "maybe"

This isn't right. If you randomly select an infinite number of labels, this doesn't mean that you will end up selecting all possible labels. You might pick all labels, and any given label can possibly be chosen, but there is absolutely no reason whatsoever to believe that every possible label will be chosen: "infinite" might never be chosen.

 

[DaveC426913]

Well, since an actual infinite number of marbles cannot be contained within our universe, the conditions are impossible, even in theory, to meet.

 

[AKG]

Why is it impossible to meet, even in theory? I am out of the loop when it comes to modern physics, but isn't it generally held that the geometry of space is flat? Given this, wouldn't the universe have infinite volume? Even if I'm wrong about the geometry, isn't it at least theoretically possible for the universe to be infinite in expanse, and thus large enough to "house" infinitely many marbles. Of course, there is the issue of where to find that much matter to build so many marbles, but it depends what we mean by "in theory." However, there is a far better way to get around all of these potential snags. Build the marbles so that each marble is half the size of the previous one. There will be infinitely many marbles, but they will only occupy a finite volume and assuming uniform density amongst all the marbles (well, all we require is that the density of the marbles doesn't grow geometrically from one marble to the next, which is a more-than-reasonable assumption), they will only require a finite amount of space. Actually, if we aren't too fussy, then we can take any slab of marble and look at it as a collection of infinitely many pieces of marble. The first piece will be the left half of the slab. The next piece will be the next quarter of the slab. The next piece will be the next eighth of the slab, etc. We don't have the precision to actually break the marble into all of these pieces (in fact, it's not even a matter of precision, we don't have the time to do it - although that brings up the question of supertasks), but whether we can cut out a piece representing the (2^{n[/sub] - 1)th (2n)th or not doesn't effect the fact that it is there. Even if a (2n)th of the slab is smaller than Planck length, it still exists. Since we can give real-number measurements to regions of space, the regions of space with such measurements exist, even if physical things cannot occupy that space. Of course, at this point, you'd be right to question whether such a region can be said to even be one of the infinite marbles any more. However, if you divide the region of space containing the slab up as prescribed, then although there may be empty ones, or ones that don't contain full atoms, there must be infinitely many that contain something, otherwise there is an upper bound to the order of the piece that contains something, and there are infinitely many pieces afterwards which would have to contain nothing, but all these contiguous pieces would have finite volume, so the total volume of the slab would be less than the total volume of the slab, and that's obviously a contradiction.}

 

[Antiphon]

I'm loosing my marbles.

Infinity can't exist except as a concept. The largest number doesn't exist
even as a concept.

Edit: If you doubt this, please name any infinite attribute of any extant
thing (mathematical constructions excluded because they are concepts.)

 

[Johann]

Infinity can't exist except as a concept.

It's interesting that some (many? most?) languages don't even have a word for "infinity". It took me quite a while to understand that the English words "infinity" and "infinite" do not refer to the same concept. And I'm still puzzled as to why people think "infinity" is not nonsense. I guess if you learn a word early in life you can never think of it as meaningless.

 

[AKG]

Well "infinity" is a noun and "infinite" is an adjective, what took you so long to understand that a noun and an adjective don't refer to the same concept? However "infinity" and "infinite" are very closely related. They don't "refer" to the same concept but they are strongly related to the same concept. A set whose cardinality is infinity is an infinite set. I'm absolutely puzzled as to why you think infinity is nonsense. Why would it be meaningless? We can speak about it meaningfully in so many ways. If you aren't able to do so, that doesn't make the word meaningless, it just means that you don't understand the meaning.

Infinity can't exist except as a concept. The largest number doesn't exist
even as a concept.

Edit: If you doubt this, please name any infinite attribute of any extant
thing (mathematical constructions excluded because they are concepts.)

What in the world is this? What's the difference between a number existing "as a concept" and just existing? Or existing as something else? When you say, "there is no largest number" I guess you're referring to something like the real numbers. Do you agree that the complex numbers are numbers? The complex numbers are not ordered, it is meaningless to talk about the largest complex number, or even to talk about whether one complex number is larger than another, but what does that have to do with anything? "Infinity" is not defined as "the largest number" so just because the largest real number doesn't exist, that has nothing at all to do with whether infinity exists. How many numbers are there? Infinity.

And naming an infinite attribute of an "extant thing" is irrelevant. I can't see how any of your sentences are relevant to the discussion, or even how anyone of your sentences is relevant to any of your other sentences.

 

[arman]

I think it'd be wise to first try and agree on a term for infinity itself. I think that arguments in this discussion are more likely to occur from semantic clashes rather than actual disagreements.

I like to think of infinity as a concept, and by that I mean that it's not actually a number. Infinity is often defined mathematically as

The limit that a function is said to approach at x = a when (x) is larger than any preassigned number for all x sufficiently near a

What I find interesting is that another common definition is

A quantity greater than any assignable quantity

Which implies that infinity is an unassignable quantity. By this definition no marble would be the infinity marble.

I don't see why people are taking this to be an argument of physics, I don't think it has anything to do with physics. Who are we to say whether an infinity of marbles would fit into the universe? To solve such a question we would firstly need to be equipped with enough data to understand the size of the universe, then we would need a definition for an infinite quantity that is properly integrated into our mathematical system. We don't have either of those things.

 

[AKG]

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.

 

[Tisthammerw]

Well, here are two problems you face:

(1) You have to explain how the act of removing a single member of an "actual infinity" would yield a mere "potential infinity".

Simple. One arrangement approaches infinity but (apparently) never actually gets there. The other actually has a transfinite value (omega).

(2) You agree that you would not be left with a quantity that is not finite. Therefore, by definition, that quantity is infinite.

But then what kind of an infinite is it? A potential or an actual one?

A lot of times when people talk about "potential infinity", they are referring to some sort of "changing" thing. For example, people like to talk about how natural numbers keep "growing" larger and larger, and I'm sure I've heard that called a "potential infinite".

The problem here is that there is nothing changing at all: each natural number is what it is, and the set of natural numbers is what it is, they are not dynamic, changing things.

It may seem that way, but don't forget how I defined the terms "actual infinite" and "potential infinite" in my first post. The collection seems to grow towards infinity but never actually gets there--or does it?

 

[Hurkyl]

One arrangement approaches infinity but (apparently) never actually gets there.

The numbers in the arrangement approach infinity. 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)

The correct term is unbounded. I've also heard it called unbounded, but finite for emphasis.


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


Something to emphase: you are talking about "approaching" infinity -- this means you are speaking about some sort of variation. This would be appropriate when talking about the labels on the marbles (at least once you've arranged them in a sequence), because you can say something like "the value of the sequence" varies as you "increase the index".


However, talking about the quantity of marbles "approaching" infinity is completely wrong. The quantity of marbles is fixed; it is a single, unchanging value. It makes absolutely no sense to speak of it "approaching" anything.

But then what kind of an infinite is it? A potential or an actual one?

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

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.

Then, saying that X actualizes Y means that X has actually attained the quality Y.

According to this notion, it is blatently obvious that the quantity of marbles is a actually infinite. (From the very statement of the problem)

The labels on the marbles would be potentially infinite, meaning that it is possible for a marble to have a label denoting an infinite ordinal. A "label on a marble" is a type of logical variable, and the things to which it could refer may or may not be infinite, because it may refer to "1", "ω", or many other things.

However, any specific label can be classified. The label "1" is clearly not infinite. The label "ω" is clearly actually infinite.


However, we can also specify that every marble is labelled with a natural number. Then, the labels on the marbles are not potentially infinite, because we know that each and every one of them is a finite number. Of course, we still have an actually infinite quantity of marbles (and of labels!), by the statement of the problem.


Saying that we have a "potentially infinite" quantity of marbles would merely mean that we do not know if the quantity has a finite or an infinite value.

 

[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 +&infin;", 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.