Re: infinity

Gordon Collins <poster02@xxxxxxxxxxx> wrote:
> Stephen wrote:

> <a bunch of good stuff to Tony>

> This post of Stephen's has got to be the most ironic thing yet to
> appear on this thread. I'm sure it is entirely unintentional.

> First, a paraphrase:

>> 'After an infinite number of steps' is not very
>> precise, and does not describe how you get from a [positive]
>> number to [0]. There either is a step
>> that starts with a [positive] number of balls and ends with
>> [0] balls, or there is not a step that
>> starts with a [positive] number of balls and ends with [0]
>> balls.
> [...]

>> You are still talking about something that does not happen.
>> There is no such step. Why would doing something infinitely
>> many times make the nonexistent step suddenly happen?
>> Why does it [...] cause the nonexistent step of there
>> being 0 balls to happen?
> [...]

>> And you are adding balls a [positive] quantity at a time and claiming
>> that the quantity becomes [0]. So in order to get an [0]
>> number, there must be a step where n was [positive], and then n+9 was
>> [0].


> And then comes this gem:

>> You still have not explained
>> why you get to have results that are not produced by any
>> step, but noone else is.

> This is not quite correct - nearly EVERYONE on this thread claims such
> results, but no one has offered such an explanation.

What explanation would you accept? We are talking
about a physically absurd situation. Any answer is
also going to be physically absurd.

The vase is empty because every ball is removed at a definite
time before noon. That is why the vase is empty. There
is no step at which the vase becomes empty, but who says
there has to be such a step?

> A casual perusal of the various recent "Cantor" threads reveals the
> folly of considering the situation after an infinite process has
> "completed". The likes of Mueckenheim and Orlow are constantly
> reminded that an infinite process /does not/ terminate and that it is
> /meaningless/ to talk about the end of such a process. The only way
> one can describe and "end result" is in terms of limits. Otherwise we
> would have a smallest positive real, 0.999... would be < 1 (since we
> wouldn't need limits to evaluate the former), the Cantor "antidiagonal"
> would end at either the bottom, side, or corner of a rectangle, and we
> could find two rational numbers without another rational in between
> them.

> The very idea that each ball added is removeable at some later time
> depends on the idea that an infinite process *does* *not* *end*.

It depends on what you mean by 'end'. English, or any natural
language, is rather vague. If you require that the last step
of a process be preformed before the process has ended, then
no infinite process ends, because no infinite process has a last step.

However, if your infinite process consists of a series of steps
with completion times d_i, and there exists some time d such
that d>=d_i for all i, then it is quite reasonable to say
that the process has ended by time d. We may not be able
to say when the process ends, but we can say when it has ended.
That is essentially the paradox in a nutshell. There is
no time at which the process ends, but there are times
at which the process is ended.

Any argument that depends on the last step of an infinite
process is absurd, as there is no last step. However
there is a mathematically sensible way to talk about
an infinite process "ending".

<snip>

> The difficulty arises when one tries to think of this as a /sequential
> process/, where each value of x (or even just each n) is considered in
> turn. One never reaches noon. "But we always reach noon", you say.
> "Time waits for no [one]!" You're just being misled by your intuition
> about physical reality. Mathematically, there's no reason that we
> have to reach noon.

Of course you have to reach noon. Do you think that if I define
the sequence
1/2, 3/4, 7/8, ... 1-(1/2)^n
that the number 1 somehow ceases to exist? 1 is not a part
of that sequence, but it clearly exists even if we are considering
the infinite sequence.

Suppose you walk across a 10 meter room starting at time 0 moving
at 1 meter per second. Assuming time and space are continuous, at time 5,
you will pass point 5. At time 7.5, you will pass point 7.5. At time
time 8.75 you will pass point 8.75, etc. And at time 10 you will
pass point 10, even though there was no last point before point 10,
and even though you passed an infinite number of points along the
way. If you claim this is absurd, then you must not believe
that time and space are continuous. But it seems a little far fetched
that something so fundamental about nature could be proven
with such a simple argument.

> In some statements of the problem, the steps are to be performed one
> minute apart, and it is clear that the process runs forever, that is,
> without ending. Mapping the time line from [1,oo) to [-1,0) does not
> change this. Asking about the state of the vase at noon is no more
> meaningful than asking for a largest natural number.

If the steps are performed one minute apart, then there is no
time at which the process is ended. There is no time that is
greater than the completion time of each step of the process.
The sequence 1, 2, 3, ... does not have an upper bound.
The sequence 1/2, 3/4, 7/8, ... does have an upper bound.
That is a big difference.

Stephen
.

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