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

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Michael Klemm

Posts: 133
Registered: 11/13/12
Re: Matheology § 300
Posted: Jul 8, 2013 4:59 AM
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Julio Di Egidio wrote:

>> If you accept that 1/oo is a natural not
>> depending on the natural n, then the limit n->oo of
>> the constant sequence (1/oo, 1/oo, ...) is 1/oo.
>> The way how the number 1/oo is obtained
>> doesn't matter.

> Well, I do not see how that can be acceptable. The original expression
> has a sub-expression dependent on n under the limit, not just a constant
> value.

The original expression is lim_{n->oo} 1/Card(N\Fison(n)),
and n = oo ist not evaluated. Therefore all members of the considered
equal c = 1/oo = 0.

> Again, that the sequence is constant for all n in N does not entail that
> the limit for n->oo must be equal to that constant value.

Why not? You may state the lemma:
Let c be a real. Then the the sequence (c, c, c, ...) has the limit c,
in short lim_{n->oo} c = c.

In a second step you apply this lemma with c = 1/Card(N\Fison(n)), n < oo.

> Indeed, by your token:
> lim_{n->oo} Card(N\F(n)) = oo (1)


> yet:
> lim_{n->oo} N\F(n) = {} (2)


> or is that supposed to be N, too??


> At least, I suppose we'd agree that:
> lim_{n->oo} F(n) = N (3)


> Of course, I am assuming that the limit of the cardinality of our
> well-founded sequence is equal to the cardinality of the limit of the
> sequence.


> I still do not see how we could define the limit otherwise (I may be
> missing formal details), but even less I can see (1) compatible with (2).

This is not only a question of defining the limit but also of the
The general set convergence is explained in
In the present case the outer limit equals the inner.

Concerning the cardinality one has:
Card(A) <= Card(B) iff there exists an injective function A -> B
Card(A) = Card(B) iff there exists a bijective function A -> B
and Card({}) = 0,

> Julio

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