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Topic: Matheology � 300
Replies: 27   Last Post: Jul 9, 2013 2:50 PM

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LudovicoVan

Posts: 3,201
From: London
Registered: 2/8/08
Re: Matheology § 300
Posted: Jul 7, 2013 9:52 AM
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"Michael Klemm" <m_f_klemm@t-online.de> wrote in message
news:krbng1$i4i$1@solani.org...
> Julio Di Egidio wrote:
>

>> Pardon the basic question, but I do not understand your result. Isn't:
>>
>> lim_{n->oo} 1/Card(N\Fison(n)) =
>> = 1/Card(N\(lim_{n->oo} Fison(n))) =
>> = 1/Card(N\N)=
>> = 1/Card({}) =
>> = 1/0
>> = oo

>
>> What am I doing wrong?
>
> The first lim_{n->oo} refers to a sequence of naturals


But that is not a sequence of naturals: N is a set, Fison(n) is a set,
N\Fison(n) is a set, Card(N\Fison(n)) is a cardinal (in fact, equal to
aleph_0 for all n in N), and 1/Card(N\Fison(n)) is again not a natural (in
fact, it is equal to 1/aleph_0, or just to 1/oo, for all n in N).

> and the second to a sequence of sets giving a set as
> its result. Without a check it is not allowed to draw the limit under
> changement of its meaning inside the expression.


I'd rather not see what other meaning we could give to that limit...

In other words, relative to what you originally wrote:

lim_{n-->oo} 1/Card(|N\FIS(n)) = lim_{n-->oo} 1/oo

note that FIS(n) depends on n, and I'd rather do not see a justification for
your step there. More specifically, I'm thinking that:

for all n in N: 1/Card(|N\FIS(n)) = 1/oo = 0

does *not* entail:

lim_{n-->oo} 1/Card(|N\FIS(n)) = lim_{n-->oo} 0

Julio





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