An actual infinite number of marbles

[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.