Re: An uncountable countable set

Virgil wrote:
In article <453d0c4e@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

David Marcus wrote:

If I read this correctly, you agree that at all times every ball that is in the vase has a natural number on it, but at noon you say that there is a ball in the vase that does not have a natural number on it. Is that correct?
No. I am saying that if only finite iterations of the ball process occur, then noon never occurs in the experiment to begin with. If noon DOES exist in the experiment, then that can only mean that some ball n exists such that 1/n=0, which would have to be greater than any finite n.

What part of the gedanken experiment statement says anything like that?

The part that says that ball n is removed at t=-1/n, combined with t=0, or t=-0. Then 1/n=0, true only for infinite n.


Now, please explain what "emptying" means.

"Empty" means not having balls. To become empty means there is a change of state in the vase ("something happens" to the vase), from having balls to not having balls.

Does "emptying" (going from a state with specific balls in the vase to a state with no balls in the vase) occupy a duration of time greater than zero?

It doesn't even appear to have that single moment to occur, in this experiment, since it can't occur before noon, nor at noon, nor thereafter. Certainly, if the vase starts with some uncountably infinite number of balls which are removed according to the Zeno schedule, it will empty, the vast, infinite majority being removed AT noon. But, if this experiment is to empty, and is an experiment in time, then you should be able to say when that occurs.

Now, when does this moment, or interval, occur?

If it is an instantaneous process, it would have to "happen" at noon.

So, you are saying that something does occur at noon. But, what causes that? Surely there are no naturally-numbered balls being added or removed at noon?


But as every ball is removed strictly before noon, it does not have to happen at all.

You mean the vase doesn't have to empty? Then what makes you think it's empty? I know, I know. You have your logic. But, it amounts to artificially creating an upper bound to a boundless set, compressing some infinity of elements into some single moment or real point, where the comparison between what is entering and what's exiting is totally clouded. The fact remains that it doesn't become empty before noon, and nothing happens at noon, so it doesn't empty.

:)
.

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