An actual infinite number of marbles

[Tisthammerw]

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.

Here's my question: suppose there is an actual infinite number of marbles, each one numbered (1, 2, 3...). Will there be a marble labeled "infinity"? I suspect so, but I'm uncertain. What do you guys think?

 

[TD]

No, because that would imply that you would 'reach' infinity 'all of the sudden'. After 'x' marble blocks, one would be labeled 'infinity'?
The thing with infinity is, you never do 'get' there. There are infinite natural numbers, but you can't say after x numbers, we label one 'infinity'. As long as you're labeling numbers, you can (as you said) add another one. Infinity is used to describe a quantity which isn't finite, so byond any boundary.

 

[Tide]

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?

 

[chronon]

Here's my question: suppose there is an actual infinite number of marbles, each one numbered (1, 2, 3...). Will there be a marble labeled "infinity"? I suspect so, but I'm uncertain. What do you guys think?

No, as you have stated it the collection of marbles is infinite, but each is labelled with a finite number. However, there is nothing to stop you adding another marble and labelling it 'infinity' or $$\omega$$ (just as if you have 2 marbles you can add one and label it 3). You can then go on adding marbles labelled $$\omega+1,\omega+2 ...$$ This is essentially what Cantor did in creating the ordinal numbers. See http://mathworld.wolfram.com/OrdinalNumber.html for more information.

 

[Hurkyl]

suppose there is an actual infinite number of marbles, each one numbered (1, 2, 3...). Will there be a marble labeled "infinity"?

This one is very easy. You said the marbles are labelled with positive integers. "infinity" is not a positive integer, thus there is no marble labelled "infinity".

 

[nate808]

can you actually ever say that you have an infinite amount of some kind of object? And if so, when?

 

[Tisthammerw]

This one is very easy. You said the marbles are labelled with positive integers. "infinity" is not a positive integer, thus there is no marble labelled "infinity".

Well, I didn't say the marbles were labeled with only positive integers; only that they were numbered. Given an actual infinite number of marbles, would there be a transfinite (e.g. omega) marble?

 

[Tide]

can you actually ever say that you have an infinite amount of some kind of object? And if so, when?

I think that could occur only in an infinite and unbounded Universe. I'd rather not think about what would happen if they were all in one place! :)

 

[Tisthammerw]

No, because that would imply that you would 'reach' infinity 'all of the sudden'.

I agree that an actual infinite cannot be formed via successive finite addition, but this isn't quite the same. This is an actual infinite, with no regard to its origin. Given an actual infinite number of marbles, would there be a transfinite (e.g. omega) marble?

The thing with infinity is, you never do 'get' there.

With some infinites that's true. Calculus deals with limits and in those cases it’s often potential infinites (symbolized as ∞) but there are actual infinites in mathematics symbolized by e.g. ω. (Note: there is a small albeit brilliant minority who claim that actual infinites shouldn't be used in mathematics and that only potential infinites are legitimate mathematical entities.) So if there's an actual infinite number of marbles...

 

[Tisthammerw]

No, as you have stated it the collection of marbles is infinite, but each is labelled with a finite number. However, there is nothing to stop you adding another marble and labelling it 'infinity' or $$\omega$$

But wouldn't the latter be the example of what I'm talking about? An actual infinite number of marbles (as opposed to a potential one--growing towards infinity but never actually getting there)?

 

[robert Ihnot]

Thinking of the Calculus, take a simple example, like the limit as n goes to infinity of 1-1/n. Obviously the limit is 1.

Well then what value do we substitute for "n" to reach that limit? I guess we could, formally speaking, add on to the number system the limit point called infinity.

However is that point well ordered? Not at all since we have no value one less than infinity. However, as it has been pointed out, Cantor did add things like omega +1, omega +2, etc.

 

[chronon]

But wouldn't the latter be the example of what I'm talking about? An actual infinite number of marbles (as opposed to a potential one--growing towards infinity but never actually getting there)?

If you have an actual infinity ($$\aleph_0$$)of marbles then you have a choice of how you order them. If you choose to label each with a positive integer then in won't be possible to say that anyone marble is the last in the ordering. However, if you kept one back then you could still label all of the others with the positive integers and define the extra one as the last (i.e. the $$\omega^{th}$$)

 

[Hurkyl]

Well, I didn't say the marbles were labeled with only positive integers; only that they were numbered. Given an actual infinite number of marbles, would there be a transfinite (e.g. omega) marble?

In this case, again you've said nothing about ω in your original post, so you cannot possibly argue that there is a marble labelled ω. (Nor could you argue there is no such marble)

Let me rephrase what you're trying to ask:

Suppose each positive integer is used to label a different marble in a collection of infinitely many marbles. Are there any marbles that do not have a positive integer as a label?

And the answer is that there is not enough information.

(Note that if the labels aren't placed "consecutively", it is also possible for every marble to be labelled with a positive integer, and there to be unused labels)

 

[matt grime]

Here's my question: suppose there is an actual infinite number of marbles, each one numbered (1, 2, 3...). Will there be a marble labeled "infinity"? I suspect so, but I'm uncertain. What do you guys think?

in the standard interpretation of what you have written, no: all marbles are labelled by the natural nubmers. why are you talking about marbles anyway? just take the set of natural numbers. This is an infinite set without a 'maximal element'. There infinite countable sets that do have maximal elements. and no i won't accept that you chose marbles because you wanted to make this concrete; who has an infinite number of marbles on them?

 

[Tisthammerw]

In this case, again you've said nothing about ω in your original post, so you cannot possibly argue that there is a marble labelled ω. (Nor could you argue there is no such marble)

I mentioned ω indirectly, suggesting (albeit indirectly) that there was ω quantity of marbles by saying I was talking about an actual infinite. (See below regarding a potential infinite.)

Suppose each positive integer is used to label a different marble in a collection of infinitely many marbles. Are there any marbles that do not have a positive integer as a label?

And the answer is that there is not enough information.

Well, I did say it was an actual infinite instead of a potential one. In a potential infinite, there would be no transfinite marble. The numbers would just keep getting higher and higher, growing towards infinity but never actually getting there.

 

[Tisthammerw]

If you have an actual infinity ($$\aleph_0$$)of marbles then you have a choice of how you order them. If you choose to label each with a positive integer then in won't be possible to say that anyone marble is the last in the ordering. However, if you kept one back then you could still label all of the others with the positive integers and define the extra one as the last (i.e. the $$\omega^{th}$$)

Hmm, makes sense. Methinks the last one more correctly represents an actual infinite (whereas the former seems to represent more of a potential one). What do you think?

 

[Tisthammerw]

in the standard interpretation of what you have written, no: all marbles are labelled by the natural nubmers.

If we were talking about a potential infinite (a collection that grows without limit towards infinity but never actually gets there) I would agree. However, in the example there is an actual infinite quantity of marbles--suggesting the possibility of a transfinite (e.g. ω) marble.

why are you talking about marbles anyway? just take the set of natural numbers. This is an infinite set without a 'maximal element'.

And there is also no transfinite element, hence what you’re proposing seems more like a potential infinite rather than an actual one. My example was about an actual infinite.

and no i won't accept that you chose marbles because you wanted to make this concrete

Too bad; I did.

 

[matt grime]

how can it be concrete? you have a set of marbles that is not finite? you have assigned them all labels? note that *you* need to state what your labelling is, not us. we can label an infinite countable set purely with a finite number on each marble, which is what you have implied with your notation, or we may choose not to and label them with some ordinals greater than w , depends on what you want to do with them.

as for potential v. actual infinity, well. let me put it this way (not, you understand, that this is a mathematcal opinion): the set of natural numbers i suppose to be an 'actual infinity' since it is infinite. soemthing that can be described by the natural numbers and is not bounded would be a potential infinity. consider for example the finite sets of marbles, none of these contains an infinite number of marbles, but there is no bound to the size of the sets.

in any case, i can hardly think that this is a mathematical issue since you've not given a mathematical definition. you ideas seem more based upon the idea of counting things "in the real world", if so come back to me when you've got a collection of marbles that is not finite.

 

[matt grime]

oh, you're one of those "gettign to infinity but not reaching it" people. why didnt' you say earlier? incidentally, to show you why this is nonsense, i can relabel your potentially infinite set of marbles so that it becomes an actually infinite set of marbles. take the one labelled 1, and write w on it, now take all the others and replace n with n-1. makes you think... or at least it ought to.

 

[rashoumon]

This one is very easy. You said the marbles are labelled with positive integers. "infinity" is not a positive integer, thus there is no marble labelled "infinity".

hmm then if infinity is a negative number...supposedly that it increases...i don't think they would reach a stage of infinity as said, there is always 1 more infront of another

 

[Hurkyl]

hmm then if infinity is a negative number

There is no negative integer (or real number) called "infinity" either.

 

[Hurkyl]

I want to make this important point:

Counting is not the act of labelling each object with numbers, starting with the number 1, and then looking for the largest label used.

It's a tribute to how nice the finite is that the two methodologies are equivalent. (and that counting is equivalent to answering the question of "How many?")

A great many mistakes are made by hastily generalizing to the infinite our knowledge of the finite. Here's a quote from the preface of a textbook I recently checked out of the library:

These volumes deal almost exclusively with infinite-dimensional phenomena. Much of the intuition that the reader may have developed from contact with finite-dimensional algebra and geometry must be abandoned in this study. It will mislead as often as it guides. In its place, a new intuition about infinite-dimensional constructs must be cultivated.

-- Fundamentals of the Theory of Operator Algebras, Kadison & Ringrose

It's not about counting, but the point it makes is true in general.

 

[Learning Curve]

I've just been reading along and have a question. Actual infinite is not concrete in the sense you can just add another number ( potential), but with labels such as (n-1),(n-2)?

If anyone could expand on that I would greatly appreciate it.

~peace

 

[Hurkyl]

All right, I'm going to punt this off to philosophy, since so few seem interested in discussing mathematics.

 

[chronon]

If we were talking about a potential infinite (a collection that grows without limit towards infinity but never actually gets there) I would agree. However, in the example there is an actual infinite quantity of marbles--suggesting the possibility of a transfinite (e.g. ω) marble.

You seem to be confusing an actual infinity with an infinite set with a largest element (which somehow 'completes' the set). This is wrong. Indeed, the distinction between potential and actual infinities is more philosophical than mathematical. For instance the rationals in [0,1] have a largest element, but the rationals in (0,1) don't. Nothing to do with whether you consider these to be potential or actual infinities.

 

[Vossistarts]

You seem to be confusing an actual infinity with an infinite set with a largest element (which somehow 'completes' the set). This is wrong. Indeed, the distinction between potential and actual infinities is more philosophical than mathematical. For instance the rationals in [0,1] have a largest element, but the rationals in (0,1) don't. Nothing to do with whether you consider these to be potential or actual infinities.

or is it that with a potential infinate you accept that it goes on infinately and leave it at that and go about your business. As an immortal handling the actual infinate you devote an infinate amount of time assigning numbers to the ever increasing lot of marbles youre counting.

 

[Johann]

I think the best that can be said about infinity and marbles is that, by thinking about the former one easily loses the latter.

 

[Vossistarts]

Im really not qualified to comment on mathematics so maybe I shouldn't. Still, it would SEEM to me that the answer must lay in the origins of the two concepts of potential and absolute infinites. Maybe Potential imparts the quality to whatever your talking about that it may and for all practical purposes does or can continue on infinitely and allows for a sort of placeholder into count to say so.(But perhaps it might not?) With the absolute infinite it says that it does in fact go on infinitely. That says to me that nothing can be done about the fact that it goes on infinitely. You don't have to continue numbering on and on neither do you have to create a placeholder of sorts saying that it may or does go on infinitely such as "Infinity". You already know this because you've decided for whatever reason that its an absolute infinite. There is no need to call any point infinity. I imagine you can simply leave off wherever the count stops for the moment or the equation leads you in any given situation for the moment KNOWING that there is infinitely more beyond that. Absolutely.

Infinity is rather conceptual to begin with isn't it?

Did I repeat the same things 15 times there? eek!

 

[Hurkyl]

Well, the mathematics is fairly simple once you get accustomed to it: if you ask precise questions, you can use the precise definitions to give precise answers!

IMHO, the only reason people have trouble with the infinite is because they try to reason from some nebulous, unarticulated concept they have floating around in their brain.

 

[Johann]

Well, the mathematics is fairly simple once you get accustomed to it: if you ask precise questions, you can use the precise definitions to give precise answers!

This is actually funny, because different mathematicians seem to have different, equally precise definitions of what infinity is. I don't think infinity is a well-defined term in mathematics at all. It seems to be rather a nuisance actually.

the only reason people have trouble with the infinite is because they try to reason from some nebulous, unarticulated concept they have floating around in their brain.

Because that is what infinity is: a nebulous, unarticulated concept.

When I learned that two line segments of different sizes have exactly the same amount of points, I was quite shocked. It seemed to me back then that a two feet long rod must have twice as many points as a foot-long one, no matter how you define "point". I could never get a satisfying explanation for what to me seems like an arbitrary rule, and I can't think of a way to define "point" which makes both the rule true and the existence of real rods possible. (I can think of a way to define "point" which makes the rule true but meaningless, and I suspect that is the definition of "point" mathematicians use)

Mathematics is not as clear and unambiguous as some people claim it is.

 

[selfAdjoint]

This is actually funny, because different mathematicians seem to have different, equally precise definitions of what infinity is. I don't think infinity is a well-defined term in mathematics at all. It seems to be rather a nuisance actually.



Because that is what infinity is: a nebulous, unarticulated concept.

When I learned that two line segments of different sizes have exactly the same amount of points, I was quite shocked. It seemed to me back then that a two feet long rod must have twice as many points as a foot-long one, no matter how you define "point". I could never get a satisfying explanation for what to me seems like an arbitrary rule, and I can't think of a way to define "point" which makes both the rule true and the existence of real rods possible. (I can think of a way to define "point" which makes the rule true but meaningless, and I suspect that is the definition of "point" mathematicians use)

Mathematics is not as clear and unambiguous as some people claim it is.

The things mathematicians do with infinity are perfectly well-defined and follow from consistent axioms. You mustn't confuse working on different subjects within the concept infinity with disagreements. You can't form a sufficient basis for analysis of this question from popular accounts.

 

[Johann]

The things mathematicians do with infinity are perfectly well-defined and follow from consistent axioms.

But that is beside the point. You can safely say that x - x = 0 without knowing what x is. My point was that mathematicians don't know what infinity is, not that they don't know how to apply the concept.

You mustn't confuse working on different subjects within the concept infinity with disagreements.

Contrary to what you seem to imply, there is a lot of disagreement amongst mathematicians regarding the exact meaning of some concepts, the validity of some axioms, the relationship between mathematics and reality, even the nature of mathematics itself.

You can't form a sufficient basis for analysis of this question from popular accounts.

I'm not basing my analysis on "popular accounts", I'm basing it in four years of post-secondary education on the subject.

 

[honestrosewater]

Contrary to what you seem to imply, there is a lot of disagreement amongst mathematicians regarding the exact meaning of some concepts, the validity of some axioms,

Could you give some examples of these different definitions and disagreements?

Why are they having to question the validity of axioms? I mean, how are you using validity? I can only think of it being applied to proof-related things, and every definition that I've seen makes an axiom proof of itself. I'm not very experienced with math, which may be the problem, but it seems like you're generally talking about 'real world', non-mathematical, philosophical matters.

BTW, are you intentionally saying infinity instead of infinite or infinitely many or such?

 

[Johann]

Could you give some examples of these different definitions and disagreements?

Set theory is a good example. There are several versions of it, not just one, and each version is based on different axioms. Here is a link in case you're interested: http://mathworld.wolfram.com/SetTheory.html

Why are they having to question the validity of axioms?

Because invalid axioms lead to contradictions. When you can use an axiom to prove that both a statement and its negative are true, you have an invalid axiom.

I mean, how are you using validity? I can only think of it being applied to proof-related things, and every definition that I've seen makes an axiom proof of itself.

Not always, but we don't get much exposure to axioms that are known to be invalid. But as a simplistic example consider these two axioms:

- any number can be divided by any number
- zero is a number

You can use those axioms to prove that 2 = 3 (2x0 = 3x0). Now there's nothing wrong with the two axioms taken by themselves, they are invalid simply because they are not consistent with the other axioms involved in the proof (such as, for instance, the axiom that any number multiplied by zero equals zero).

it seems like you're generally talking about 'real world', non-mathematical, philosophical matters.

I was talking about what the concept of infinity means in mathematics. It's not unlike the situation when mathematicians were faced with the square root of -1. They found a way around the problem, but they didn't know what it meant until imaginary numbers could be used to solve real problems. So we have a convenient way to deal with infinite quantities, but we haven't yet found a way to apply it to real problems. Because of that, some mathematicians believe the concept should be thrown out in favor of granular mathematics ("no infinitesimals")

BTW, are you intentionally saying infinity instead of infinite or infinitely many or such?

My mother language is not English, please forgive my misspellings. Hopefully the meaning should be clear from the context.

 

[selfAdjoint]

The fact that there are various axiomatic systems for what is vaguely called set theory does not prove that anyone of them is unsatisfactory: they are not in competition. Mathematicians who are interested in mathematical logic, or proof theory, or model theory, will learn several, and undergraduate courses teach the first few mentioned at your Wolfram link, and their interesting relations with each other. This is a perfect example of trying to find support for a preconceived idea online. It seldom works.