Virgil
Posts:
4,479
Registered:
1/6/11
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Re: Matheology � 062
Posted:
Jul 9, 2012 8:58 PM
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In article <jtftlb$ala$1@speranza.aioe.org>,
"LudovicoVan" <julio@diegidio.name> wrote:
> "LudovicoVan" <julio@diegidio.name> wrote in message
> news:jtftcf$a23$1@speranza.aioe.org...
> > "William Hughes" <wpihughes@gmail.com> wrote in message
> > news:abd1cfc8-e1de-4504-bd7f-bac8fa5efc7a@h9g2000yqi.googlegroups.com...
> > <snip>
> >
> >> Balls only enter the vase at a step
> >> from the statement of the problem
> >>
> >> The only steps are n in N
> >> from the statement of the problem
> >>
> >> At every step n in N, the only balls that
> >> enter the vase are labeled 1+10(n-1) to 10n
> >> from the statement of the problem
> >
> > The statement of the problem talks about an "infinite supply of balls",
> > which justifies a broader approach in terms of ordinals (and the
> > distinction balls vs. finite/non-finite labels). Anyway, I am not any
> > good at arithmetic with ordinals, so here it is over N, i.e. your setting.
> > Of course, the "structure" we get is not as reach as that given by
> > ordinals and we can only conclude that the limit set is countably
> > infinite. (My apologies for any trivial mistakes.)
> >
> > Let N be the set of balls (w.l.o.g. as each ball is uniquely labeled by a
> > natural number here).
> >
> > Let n, m in N.
> >
> > Let V(n) be the vase at step n, defined by the rules of the game as:
> >
> > V(0) := { }
> > V(n) := V(n-1) U { m | 10n-9 <= m <= 10n } \ { m | m = n }
> >
> > Let V(w) be the limit set (the vase at time 0), defined as: V(w) :=
> > U_{n<w}.
>
> Should read: V(w) := U_{n<w} V(n).
Still wrong, as it does not allow any removals at all,
but every n in |N has a time before 0 at which it is removed..
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