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Re: Matheology § 062
Posted:
Jul 14, 2012 12:12 PM
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On Jul 12, 11:16 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "LudovicoVan" <ju...@diegidio.name> wrote in message
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> news:jtftlb$ala$1@speranza.aioe.org...
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> > "LudovicoVan" <ju...@diegidio.name> wrote in message
> >news:jtftcf$a23$1@speranza.aioe.org...
> >> "William Hughes" <wpihug...@gmail.com> wrote in message
> >>news:abd1cfc8-e1de-4504-bd7f-bac8fa5efc7a@h9g2000yqi.googlegroups.com...
> >> <snip>
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> >>> Balls only enter the vase at a step
> >>> from the statement of the problem
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> >>> The only steps are n in N
> >>> from the statement of the problem
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> >>> 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
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> >> 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.)
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> >> Let N be the set of balls (w.l.o.g. as each ball is uniquely labeled by a
> >> natural number here).
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> >> Let n, m in N.
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> >> Let V(n) be the vase at step n, defined by the rules of the game as:
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> >> V(0) := { }
> >> V(n) := V(n-1) U { m | 10n-9 <= m <= 10n } \ { m | m = n }
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> >> Let V(w) be the limit set (the vase at time 0), defined as: V(w) :=
> >> U_{n<w}.
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> > Should read: V(w) := U_{n<w} V(n).
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> Also, should be: the vase at time t->0.
Where in the world do you get V(w) := U_{n<w} V(n)?
This is an extremely silly definition of a limit.
[Also note that trivially U_{n<w} V(n) = N\{1}]
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