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

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Michael Klemm

Posts: 125
Registered: 11/13/12
Re: Matheology § 300
Posted: Jul 7, 2013 11:21 AM
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Julio Di Egidio wrote:

>>>> 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:


The expression
lim_{n->oo} 1/Card(N\Fison(n))
means:
Limes of the sequence
(1/Card(N\Fison(1)), 1/Card(N\Fison(2)),1/Card(N\Fison(3)),....)
= (0,0,0,....),
the value n = oo excluded.
This limes is 0.

The expression
Card(N\(lim_{n->oo} Fison(n)))
means:
Limes of the sequence
({2,3,4,...},{3,4,5,...},{4,5,6,...},...),
{oo} not contained in the sequence and
oo not contained in any of its members.
This limit is by definition the set of all
naturals contained in infinitively many members
of the sequence. Thus, the limit is the empty set {}.
Hence one obtains
1/Card(N\lim_{n->oo} Fison(n)) = oo.

Therefore the error in the calculation is the first equation
lim_{n->oo} 1/Card(N\Fison(n)) =
= 1/Card(N\(lim_{n->oo} Fison(n))).

Regards
Michael




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