Re: infinity

Kirby Cook wrote:
> William Hughes wrote:
>
> > Kirby Cook wrote:
> >
> >>William Hughes wrote:
> >>
> >><snip>
> >>
> >>>
> >>>I is the union of a bunch of sets.
> >>>
> >>>Define I_n to be the set of balls added at step n.
> >>>
> >>>I is the set defined by:
> >>>
> >>> e is an element of I if and only if e is an element of I_n for
> >>> at least one n.
> >>>
> >>>Since we know know how to determine if any given thing is an
> >>>element of I, we have defined the set I. (This is the standard
> >>>definition for union. There is no change whether or not there
> >>>are an infinite number of sets I_n.)
> >>>
> >>>Simlarly, we define set 0. We define set I\O to be the set
> >>>of all elements of I that are not elements of O. Since this
> >>>is the empty set, by definition the vase is empty at state E.
> >>
> >>But this is not correct for any step n.
> >
> >
> > No, but the set of balls in the vase is not I\0 for any step
> > n. By definition, the set of balls in the vase at state E
> > is I\O. The vase is never empty at any step n. The vase is
> > empty at state E. Counterintuitive, Yes. Follows from
> > the definitions, Yes.
> >
> Whoa, I have to back up for a minute. You wrote,
> "Definition 1:
>
> Let I be the union of the sets of balls added to
> the vase at any finite step.
>
> Let O be the union of the sets of balls removed from
> the vase at any finite step.
>
> Then 0 is a subset of I and we define the set of balls
> that is in the vase at state E is the set difference I\O."
>
> So, in any one finite case, I/O is defined by the "Let I be" and "Let O
> be" statements above.


No, there is only one I and only one O corresponding to the
state E. I guess the definitions are a bit ambiguous

If at fist you don't succeed...

Let I_n be set set of balls added at step n.
Let I be the union of all the I_n

Let O_n be the set of balls removed at step n
Let O be the union of all the I_n

Then 0 is a subset of I and we define the set of balls
that is in the vase at state E is the set difference I\O."

Note in particular that neither I nor O is defined by a limit
process.

> And, according to the definition, I/O is not empty
> there. And, in fact, at each and every nth step, there will be 9n more
> balls in I than in O, different balls. Now, the two sets maintain that
> relationship as they increase, in lockstep, without limit.
> I've been imputed, elsewhere, with claiming a largest natural number. I
> don't think I've done that, but you appear to be doing so. At least,
> you seem to be proposing a point, some state, in which the built-in
> non-zero difference between I and O disappears, at which point I/O
> becomes empty, and you call that state E.
>

State E, noon. comes after all the steps. Since there are an infinite
number of things happening betweem any step at which there are balls
in the vase and state E, I don't have any problem with the vase
being empty at sate E. There is certainly no point at which the
vase suddenly becomes empty (it takes an infinite number of steps
to reach noon, so even times very close to noon are an infinite number
of steps away), and no last ball that is removed.

Yes this is a bit wierd. But look at the definition of the set
of balls in the vase at state E, noon. This is very straighforward.
You either have to accept the counterintuitive result or
claim that the problem does not have an answer.

> >
> >
> >>>
> >>>As to state E, what else could the state of the vase be at noon be,
> >>>but the state after all finite steps?
> >>>
> >>> - William Hughes
> >>>
> >>
> >> Hi again, William. A couple of things above trouble me. Maybe three.
> >> When you say "There is no change whether or not there
> >>are an infinite number of sets I_n", I'm unconvinced.
> >
> >
> > The definition for the union of a bunch of sets:
> >
> > e is a member of the union of a bunch of sets if and only
> > if e is a member of at least one of the sets.
> >
> > Note this definition does not change whether the bunch of sets
> > consists of a finite number of sets or an infinite number of sets.
> > This is all that was meant by "There is no change whether or not there
> > are an infinite number of sets I_n".
> >
> > The advantage to using the concept of an infinite union is that
> > it sticks very close to the concept of a finite union and
> > our intuition is valid. On the other hand if we try to use
> > counting arguments we get things like
> >
> >
> >>When I add one to
> >>one, I get two. When I add one to infinity, I get infinity.
> >
> >
> > which are counterintuitive. (Indeed they are only a short form
> > for more exact statements about cardinality, which are themselves
> > counterintuitive).
> >
> >
> >> I get
> >>suspicious at attempts to relate the finite to the infinite, or at
> >>casual transits from one to the other, as appears to happen when time
> >>passes from that path being followed by the defined sequence (which
> >>continues indefinitely before noon) to noon, and the number of balls is
> >>said to pass from a number without limit to a perfectly finite and
> >>ordinary zero. And that is supposed to be accomplished by subtracting
> >>only one ball at a time from that limitless number, albeit a limitless
> >>number of times, and somehow bypassing the sequence 3,2,1,0 in reaching
> >>zero.
> >
> >
> > Most of this make no sense. One have to be very careful with
> > definitions when talking about infinite sets. In particular, most
> > discussions of counting make no sense.
> >
> >
> >> And the phrase, "after all finite steps" is murky to me. Each step is
> >>finite, but the number of finite steps has no limit, so "after" the
> >>limitless steps is... murky to me.
> >
> >
> > Indeed, but what else can we call noon, but "after all finite steps"?
> > If a ball is added to the vase before noon and not removed, don't we
> > have to say the ball is in the vase at noon? If a ball is never added
> > to the vase don't we have to say the ball is not in the vase at noon?
> > If a ball is removed from the vase before noon, don't we have to say
> > the
> > ball is not in the vase at noon? Put this together and we get the
> > set of balls in the vase is I\O.
> >
> >
> >> And, finally, the arguement seems to be that commutivity in an infinite
> >>sum makes it (order) ignorable.
> >
> >
> >
> > Note:
> >
> > 1. The number of balls in the vase at state E does not
> > depend on the number of balls in the vase at finite steps
> >
> Would you mind exploring, just once more, why you say this, and how that
> can be?

Why do I say this:

I and O do not depend on the number of balls in the vase
at finite steps (check the definitions), so the set I\O and the
number of elements in this set do not depend on the number of
balls in the vase at finite steps.

How can this be:

The number of balls in the vase at state E depends on which
balls are added and which balls are removed. It does not
depend on which step the balls are added or which step
they are removed.

The number of balls in the vase at a finite step depends
critically on which step the balls are added and which
step they are removed.

The two things are different. The number of balls in the vase
at state E depends on which balls are added/removed, it does not
depend on the number of balls added/removed. The number of balls
in the vase at a finite step does not depend on which balls
are added/removed but does depend on the the number of balls
added/removed.


> That may be all I need. All I see (dimly) right now is that it
> has something to do with reconciling the two observations that the
> number of balls in the vase increases at each step, without limit, and
> that each and every ball added to the vase will be, according to the
> established procedure, removed from the vase, but later, after the
> number of balls in the vase has grown.
>

Right. and as the anwer does depend on which balls are removed
and does not depend on how many balls are in the vase at any
step, the fact that the number of balls in the vase increase
at each step does not affect the answer.

> > 2. The sum of an infinite series does depend on the number
> > of balls in the vase at finite steps.
> >
> > So The number of balls in the vase at state E is not
> > given by the sum of an infinite series.
> >
> > Yes the problem is set up so intuitively the answer is the sum
> > of an infinite series. But the answer to the problem is
> > counterintuitive.
> >
> >
> >
> >> Fact is, I believe the commutivity
> >>used in this particular infinite sum, by which 9+9+9... is rewritten as
> >>0+0+0... is fallacious. I haven't articulated *exactly* why yet, even
> >>to myself, but I'm slow.
> >> Hmm, maybe I did, and forgot. The sum describing the addition and
> >>subtraction of balls is (10-1)+(10-1)+... It is proposed that the sum
> >>be reordered as (10-1-1-1-1-1-1-1-1-1-1)+... The only proper reordering
> >>would be, it seems to me,
> >>(10-1-1-1-1-1-1-1-1-1-1+10+10+10+10+10+10+10+10+10)+... I mean, the
> >>first sum reflects what happens at each step. The proposed reordering,
> >>0+0+0... reflects something that never happens at any known or proposed
> >>step. Therefore, it could only occur, in some indefinable way, "after
> >>all finite steps". I don't believe it.
> >>
> >
> >
> > The infinite series
> >
> > 10 -1 + 10 - 1 + 10....
> >
> > can be used to model the problem, but only sort of. Yes, you can
> > rearrange the series to get a sum of zero (sort of) and yes this
> > infinite rearrangement is analogous to what is happening in the problem
> > (sort of), but it doesn't really help.
> >
> I finally pinned my objection down. Saying the sum can be rearranged
> as only 0+0+0... =0 is in error, as this neglects mentioning that for
> each and every 0 that appears in the reordered sum there is a
> corresponding 90 that appears somewhere else in the same sum.
>

The asnwer to the problem is not given by the sum of the
any infinite series, in particular it is not the sum of

10 - 1 + 10 -1 + 10 ....


So any comments about what this sum should or should not be are
irrelevant.


-William Hughes

.

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