# An actual infinite number of marbles

## Main Question or Discussion Point

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?

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TD
Homework Helper
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
Homework Helper
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?

Tisthammerw said:
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
Staff Emeritus
Gold Member
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".

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

Hurkyl said:
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
Homework Helper
nate808 said:
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! :)

TD said:
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...

chronon said:
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)?

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.

Tisthammerw said:
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 any one 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
Staff Emeritus
Gold Member
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 &omega; in your original post, so you cannot possibly argue that there is a marble labelled &omega;. (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
Homework Helper
Tisthammerw said:
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?

Hurkyl said:
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.

chronon said:
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 any one 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?

matt grime said:
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
Homework Helper
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
Homework Helper
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.

Hurkyl said:
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
Staff Emeritus
Gold Member
hmm then if infinity is a negative number
There is no negative integer (or real number) called "infinity" either.

Hurkyl
Staff Emeritus
Gold Member
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.

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
Staff Emeritus