Re: infinity

Kirby -

I just want to say that you're doing very well with this. As far as I can see
your objections all make sense, and you are experiencing the same issue with
this "largest natural" nonsense that Cantorians seem to spew whenever they get
cornered. Keep up the good work. Oh, and if your interested in alternatives
that actually make sense, don't hesitate to ask. ;)

Tony

Kirby Cook said:
> 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. 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.
>
> >
> >
> >>>
> >>>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? 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.
>
> > 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.
>
> >
> >>Thanks, by the way, for taking the extra time.
> >
> >
> >
> > You're welcome
> >
> > -William Hughes
> >
>

--
Smiles,

Tony
.

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