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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: Matheology � 074
Posted: Jul 14, 2012 9:45 PM

In article <jtt1cq\$pii\$1@speranza.aioe.org>,
"LudovicoVan" <julio@diegidio.name> wrote:

> "WM" <mueckenh@rz.fh-augsburg.de> wrote in message

> > On 14 Jul., 20:01, "dilettante" <n...@nonono.no> wrote:
> >> "WM" <mueck...@rz.fh-augsburg.de> wrote in message

> >> > On 14 Jul., 18:38, "dilettante" <n...@nonono.no> wrote:
> >>
> >> >> "About that of which we cannot speak, we must remain silent."
> >>
> >> > That's why you are silent about mathematics, in this special case?
> >>
> >> > Mathematics, as I teach it gives the improper limit
> >> > ((((((10^0)/10)+10^1)/10)+10^2)/10)+... = oo
> >> > of the sequence

> >>
> >> > 1
> >> > 0,1
> >> > 10,1
> >> > 1,01
> >> > 101,01
> >> > 10,101
> >> > 1010,101
> >> > 101,0101
> >> > ...

> >>
> >> > Set theory gives 0.
> >> > What is correct?

> >>
> >> The limit is infinity. Set theory says nothing different,

> >
> > Set theory says the limit is less than 1, because for every digit
> > occuring left of the comma (the comma is taken from a German text,
> > here representing a decimal point) the step can be determined, when
> > this digit disappears right of the comma

>
> That remains invalid reasoning, a paralogism that simply does not model the
> problem: a sensible statement would be that for every digit that goes to the
> right 2 more are added on the left, so that never all digits are on the
> right.
>
> Then, I have checked the definitions of limit inferior and superior: I can
> see a problem when said limits are defined in terms of unions of
> intersections and vice versa (entirely due to my limited understanding),
> although it is apparent that the two limits must diverge, period.
>
> -LV
>

WRONG!

Given the sequence of sets S = {S_n, n in N},
with S_n ={ n+1, n+2, ... ,10*n} being the set of elememts in the vase
from t_n = -1/n up to but not inclusing t_(n+1) = -1/(n+1), to find the
limit set after all t_n transfers. have taken place

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 is {},
and the union becomes
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, ...} = {}

THUS

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
7/14/12 dilettante
7/14/12 mueckenh@rz.fh-augsburg.de
7/14/12 LudovicoVan
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