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Re: Mathematics in brief
Posted:
Dec 10, 2012 1:27 PM
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On 10 Dez., 18:31, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <virgil-F01676.16024408122...@BIGNEWS.USENETMONSTER.COM>, on
> 12/08/2012
> at 04:02 PM, Virgil <vir...@ligriv.com> said:
>
> >In article
> ><47934f05-f8e1-4687-b590-af47fc4be...@8g2000yqp.googlegroups.com>,
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> >> On the contrary, ZFC is a deeply unlogical theory. It requires
> >>the belief that uncountably many elements can be distinguished
> >>whereas everybody knows that this is impossible even in ideal
> >>mathematics.
>
> That's one of the dumbest claims that I've ever read. Don't confuse
> your delusions with what everybody believes.
I am not interested in everybody's beliefs but in mathematics. That is
completely free of beliefs - apart from some conjectures.
>
> >> Further in ZFC the sequence
> >> 21., 2.1, 432.1, 43.21, 6543.21, 654.321, ...
> >> has the limit < 1.
>
> No; it has no limit.
It has an improper limit in analysis, namely oo. It has a limit in set
theory. Every natural number that appears left of the decimal point
disappears right of it.
>
> >> In analysis the very same sequence has the (improper)imit oo.
> >Right, for once!
>
> No. For the sequence to have the limit oo, both lim sup and lin inf
> would have to be oo, not just one of them.
Please do not mistake analysis and set theory. In analysis we have the
simple criterion: If lim 1/a_n = 0, then lim a_n = oo. That is here
the case. Obviously.
Or do you need a proof? Take any epsilon = 1/k > 0. Then
for all j > k: |1/a_j - 0| < epsilon.
Regards, WM
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