Re: infinity

Gordon Collins said:
> Martin Shobe wrote:
>
> >> Which axioms (Peano, ZF, or otherwise) allow completion of an infinite
> >> sequence?
> >
> > None. But, the argument that the vase is empty does not rely on any
> > axioms that complete infinite sequences.
>
> Thank you. I was responding to the claim that:
>
> >> All finite iterations
> >> (of which there is an infinite number) are completed
>
> So you agree that an infinite number of iterations (a term which means
> that they are done one after the other) cannot be completed.
>
>
> >> (Note that this must be strictly
> >>before noon.)
> >
> > Why? Least upper bounds do not have to be in the set that they are
> > bounding.
>
> According to the problem statement, the only things that are done to
> change the state of the vase are done at times of the form -1/n where n
> is a natural number. Nothing is done at noon (t=0) to change the state
> of the vase. Nothing "outside" the process can affect the vase. If
> the vase becomes empty it does so as a consequence of the process
> itself. The process is nothing more than the aggregation of its parts,
> the individual steps, so everything that happens happens as a result of
> one of those steps. The vase does not become empty "after" the process
> "completes" (as if that were meaningful), because nothing happens to it
> then. For the contents of the vase to be affected at all, t must =
> -1/n for some n, but we know that at that time the contents of the vase
> are increased. Every single operation on the vase has the effect of
> adding balls to it. No other kind of operation is ever performed on
> the vase.
>
> There is no step that starts with a positive number of balls and ends
> with 0 balls. Why would doing something infinitely many times make the
> nonexistent step suddenly happen?
>
> If you want to talk about the vase /being/ empty without /becoming/
> empty, then its existence cannot be continuous. Otherwise there is a
> contradiction between the facts that the vase cannot become empty and
> that it cannot retain any particular ball.
>
>
> > The pigeon=hole principle is what lets you ignore the labels on the
> > balls when the number of steps is finite.
>
> Ok, thanks.
>
>
> >>Here are a couple [more] variations for people to chew on:
> >>A) Start with an empty vase. At 1/n minute before noon put ball n in
> >>the vase. At noon, what is in the vase? (Careful!)
> >
> > Well, assuming that by n, you are referring to a natrual number, then
> > the vase contains a ball with a label for every natural number n which
> > is greater than or equal to 1.
>
> So you agree with Tony Orlow that N can be generated by a sequential
> process, an infinite algorithm.
>
>
> [re scenario B)]
>
> > By the first definition, the vase is empty at noon.
> > By the second definition, the status of the vase is indeterminite.
>
> I would go with definition 2, but since the actions on the labels are
> the same as in the original problem, and the balls merely carry the
> labels, how is it different? Why is it relevant where we get a ball
> from when we are ready to stick a label on it and put it in the vase?
>
> People keep looking at this problem from the standpoint of each ball,
> and I'm trying to get them to look at it from the standpoint of the
> vase - what's coming in and what's going out.
>
> (BTW, my answer to either scenario is that "What is the state of the
> vase at noon?" has no answer.)
>
> Gordon
>
>
Gordon, I think your perspective is good, and the focus on ball labels is not
helpful, but I don't quite understand your position. If you believe that the
number of balls is strictly increasing at each step, then how can there
possibly NOT be an infinite number of balls at noon? Is it because, despite the
time construction of the gendanken such that infinite iterations occur by noon,
you don't believe that infinite iterations can occur even in theory? If that is
the case, then wouldn't you at least consider the vase to have a very large
number of balls in it? Do you think it could possibly be empty?
--
Smiles,

Tony
.

Relevant Pages

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