Re: infinity

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

>> What explanation would you accept?

> One that /shows/ the completion of an infinite iterated (i.e.,
> discrete) procedure would do.

I do not know what that means.


>> There
>> is no step at which the vase becomes empty, but who says
>> there has to be such a step?

> It follows from the problem statement that any change to the vase
> occurs as a result of a step.

That is not part of the problem statement. That is your
intuition about the behavior of real world finite tasks.

> But there seems to be some question as to whether the problem as posed
> really consists of specific /steps/ at all, as many are willing to
> consider each ball independently with only coincidental (and entirely
> dispensable) synchrony. If one interprets the problem in this way,
> there is nothing strange left of the problem but there is little point
> to it.

That is a mathematically sensible way to model the problem.
There is no last step, so any attempt to talk about what
happens at the last step is mathematically absurd.


>> However
>> there is a mathematically sensible way to talk about
>> an infinite process "ending".

> I would like to see a mathematical description of an infinite discrete
> process ending. All I have seen is statements to the effect of, "well,
> it /must/ have finished by now", with the details of how it did so lost
> in a haze of implied epsilons and deltas.

As I said, there is not time at which the process ends. There
are times at which it is ended.


>> Of course you have to reach noon.

> Why? As long as we're ignoring physical reality, we can ignore the
> usual passage of time. Time can pass from -1 toward 0 and not get
> there just as easily at it can pass from 1 toward +oo and not get
> there. [-1,0) and [1,+oo) have the same cardinality, topology, etc.
> (This isn't what stops you, of course - it's the discrete iteration
> that does that.)

Just because you define a sequence of times that all occur
before noon it does not mean noon ceases to exist. Just
as the sequence 1/2, 3/4, 7/8, etc. does not cause 1 to cease
to exist. Claiming that you cannot reach noon sounds like
you are claiming that noon somehow no longer exists.

<snip>

>> 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.

> I do not pass any "points". When I move from one place to another 10
> meters away, I do not take a 5-meter step, then a 2.5-meter step, then
> a 1.25-meter step, etc. Nor is there such thing as a fractional step
> to correspond with your (and Zeno's) arbitrary, artificial divisions.

Of course you pass points. In order to move from 0 to 10
you must pass point 5. I did not say you were taking such steps.
However there is an infinite sequence of events, all of which occur in
a 10 second interval. All of these events happen, and they
have all occurred by time 10, but there is no last event that
occurs before time 10.

>> If you claim this is absurd, then you must not believe
>> that time and space are continuous.

> Quite the opposite - I take both motion and the passage of time as
> seamlessly continuous. I do not mean epsilon-delta-continuous here -
> /nonpunctiform/ is the correct term, I believe. Since I do not pass an
> infinite number of points (an idea that is indeed absurd), I have no
> problem getting around.

I do not see how you can claim that you do not pass an infinite
number of points when travelling from 0 to 10 and also
claim that space and motion are continuous. To travel from
0 to 10 you must pass 5 and every other point in between.
How can you not pass point 5? There will be a moment when
you are at point 5.

> Time may be discrete or continuous, but if the latter it must be a
> proper continuum, free of the crippling need to move from point to
> point in succession.

I said nothing about moving from point to point in succession.
If space is continuous, there are no consecutive points. However
you still have to pass an infinite number of points to move
from one location to another, and I can easily identify an
infinite set of points you will pass, and the times you will
pass them, and you will be able to do this in a finite amount of time.

Stephen
.

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