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In article <45477121@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
> David Marcus wrote:
> > Tony Orlow wrote:
> >> David Marcus wrote:
> >>> Tony Orlow wrote:
> >>>> David Marcus wrote:
> >>>>> Tony Orlow wrote:
> >>>>>> I am beginning to realize just how much trouble the axiom of
> >>>>>> extensionality is causing here. That is what you're using, here, no?
> >>>>>> The
> >>>>>> sets are "equal" because they contain the same elements. That gives no
> >>>>>> measure of how the sets compare at any given point in their
> >>>>>> production.
> >>>>>> Sets as sets are considered static and complete. However, when talking
> >>>>>> about processes of adding and removing elements, the sets are not
> >>>>>> static, but changing with each event. When speaking about what is in
> >>>>>> the
> >>>>>> set at time t, use a function for that sum on t, assume t is
> >>>>>> continuous,
> >>>>>> and check the limit as t->0. Then you won't run into silly paradoxes
> >>>>>> and
> >>>>>> unicorns.
> >>>>> There is a lot of stuff in there. Let's go one step at a time. I
> >>>>> believe
> >>>>> that one thing you are saying is this:
> >>>>>
> >>>>> |IN\OUT| = 0, but defining IN and OUT and looking at |IN\OUT| is not
> >>>>> |the
> >>>>> correct translation of the balls and vase problem into Mathematics.
> >>>>>
> >>>>> Do you agree with this statement?
> >>>> Yes.
> >>> OK. Since you don't like the |IN\OUT| translation, let's see if we can
> >>> take what you wrote, translate it into Mathematics, and get a
> >>> translation that you like.
> >>>
> >>> You say, "When speaking about what is in the set at time t, use a
> >>> function for that sum on t, assume t is continuous, and check the limit
> >>> as t->0."
> >>>
> >>> Taking this one step at a time, first we have "use a function for that
> >>> sum on t". How about we use the function V defined as follows?
> >>>
> >>> For n = 1,2,..., let
> >>>
> >>> A_n = -1/floor((n+9)/10),
> >>> R_n = -1/n.
> >>>
> >>> For n = 1,2,..., define a function B_n by
> >>>
> >>> B_n(t) = 1 if A_n <= t < R_n,
> >>> 0 if t < A_n or t >= R_n.
> >>>
> >>> Let V(t) = sum_n B_n(t).
> >>>
> >>> Next you say, "assume t is continuous". Not sure what you mean. Maybe
> >>> you mean assume the function is continuous? However, it seems that
> >>> either the function we defined (e.g., V) is continuous or it isn't,
> >>> i.e., it should be something we deduce, not assume. Let's skip this for
> >>> now. I don't think we actually need it.
> >>>
> >>> Finally, you write, "check the limit as t->0". I would interpret this as
> >>> saying that we should evaluate the limit of V(t) as t approaches zero
> >>> from the left, i.e.,
> >>>
> >>> lim_{t -> 0-} V(t).
> >>>
> >>> Do you agree that you are saying that the number of balls in the vase at
> >>> noon is lim_{t -> 0-} V(t)?
> >>>
> >> Find limits of formulas on numbers, not limits of sets.
> >
> > I have no clue what you mean. There are no "limits of sets" in what I
> > wrote.
> >
> >> Here's what I said to Stephen:
> >>
> >> out(n) is the number of balls removed upon completion of iteration n,
> >> and is equal to n.
> >>
> >> in(n) is the number of balls inserted upon completion of iteration n,
> >> and is equal to 10n.
> >>
> >> contains(n) is the number of balls in the vase upon completion of
> >> iteration n, and is equal to in(n)-out(n)=9n.
> >>
> >> n(t) is the number of iterations completed at time t, equal to
> >> floor(-1/t).
> >>
> >> contains(t) is the number of balls in the vase at time t, and is equal
> >> to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t).
> >>
> >> Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit.
> >
> > You seem to be agreeing with what I wrote, i.e., that you say that the
> > number of balls in the vase at noon is lim_{t -> 0-} V(t). Care to
> > confirm this?
> >
>
> No that's a bad formulation. I gave you the correct formulation, which
> states the number of balls in the vase as a function of t.
TO did no such thing, as his formulation requires infinitely many
unnumbered balls to come into existence out of nowhere after all
numbered balls are removed from the vase.
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