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Topic: Matheology § 074
Replies: 114   Last Post: Jul 22, 2012 4:15 PM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: WM's WMatheology � 074
Posted: Jul 16, 2012 5:35 PM

In article <ju1tnr\$nip\$1@speranza.aioe.org>,
"LudovicoVan" <julio@diegidio.name> wrote:

> "Virgil" <virgil@ligriv.com> wrote in message
> news:virgil-220027.14190216072012@bignews.usenetmonster.com...
>

> > Except that, as the limit of a sequence of sets, S_n, the only valid
> > limit would be the common value of lim_sup S_n and lim_inf S_n, if there
> > is one.

>
> Right (more or less).
>

> > And {} is provably that common value.
>
> Wrong.
>

> > When one looks for a limit value to a sequence of sets, one must use an
> > appropriate method for sequences of sets, which WM does not do.

>
> You even less.
>

> > The question being about the set of balls in the vase at t = 0, only set
> > methods are relevant.

>
> Right: indeed, by your set-based approach, you can only look in the vase at
> t=0, not at t->0.

Actually, the original statement of the vase problem allows one to look
at the set of balls in the vase at any time up to and including t = 0,
but does not say what happens for t > 0.

With t_n = -1/2^(n-1), for n = 1,2,3,..., one has
S_initial = {} for t < -1
S_n = { n+1, n+2, ..., 10*n} for n = 1,2,3,...,
Thus the set limit, if it exists, must be lim_inf S_n = lim_sup S_n

>
> > So any attempt to solve the problem which gives a result different from
> > the one set methods gives is irrelevant, and an example of incompetent
> > analysis of the issues.

>
> Agreed: and the correct result, by your set-based method, is { w }

Nope! lim_inf S_n = lim_sup S_n = {}.

, which is
> also consistent with all results from all other methods, not to mention
> consistent with the informal logic of the problem, so that no paradox of any
> kind is ever in sight.
>

> > And in the case of WM, arrogant incompetence.
>
> In your case, unrepentant lying and patent spamming.

Given the sequence of sets S = {S_n, n in N},
with S_n ={ n+1, n+2, ... ,10*n}

Lim_Inf S= union_(n=1...oo) [intersection(m= n...oo) S_n]

http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

But since for every k in N, k is NOT a member of any S_m with m > k,
every [intersection(m= n...oo) S_n] excludes every k, so equals {},
and the union becomes

Union_(n=1...oo) {} = {}

and thus

Lim_Inf S = {} Union_(n=1...oo) {} = {}

AND

For a sequence of sets S = {S_n, n in N},

Lim_Sup = intersection_(n=.1..oo) [union_(m=n...oo) S_n]

http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

But union_(m=n...oo) S_n = {n, n+1, n+2, ...}

so intersection_(n=.1..oo) {n, n+1, n+2, ...} = {}

and Lim_Sup S = {}

and

lim S_n = Lim_Sup S_n = Lim_Inf S_n = {}
--

Date Subject Author
7/13/12 mueckenh@rz.fh-augsburg.de
7/13/12 MoeBlee
7/13/12 Virgil
7/14/12 dilettante
7/14/12 LudovicoVan
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