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Topic: Cantor's first proof in DETAILS
Replies: 34   Last Post: Dec 1, 2012 10:56 AM

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Uirgil

Posts: 183
Registered: 4/18/12
Re: Cantor's first proof in DETAILS
Posted: Nov 13, 2012 3:16 PM
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In article <k7u09t$tjd$1@dont-email.me>,
"LudovicoVan" <julio@diegidio.name> wrote:

> "Zuhair" <zaljohar@gmail.com> wrote in message
> news:86a85cce-2a84-4c9f-b860-527958274b50@o8g2000yqh.googlegroups.com...
> <snip>
>

> > Theorem 4. for all i. x_i =/= L
> >
> > <...>
> >
> > Let J=L
> >
> > QED

>
> Same argument, same objection: as easily proven, the limit interval here
> must be degenerate, that is it is the singleton (in interval notation)
> [L;L]. So what you claim amounts to saying that the limit value L is not in
> (x_n), but that is just incorrect in that if you consider the limit value,
> then of course it does belong to (x_n), in the limit! More formally L =
> lim_{n->oo} (a_n) = lim_{n->oo} (b_n) , then just consider an injection from
> N* instead of N and you can even talk meaningfully about that "last value".


The limit of a strictly increasing sequence (the a_i) is NOT EVER a
member of the sequence.

The limit of a strictly decreasing sequence (The b_i) is NOT EVER a
member of the sequence.

No values which are bounded below by a strictly increasing sequence and
bounded above by a strictly decreasing sequence are members of either
seequence.

Thus proving that, given any sequence of values in R, there must be
values in R not appearing in that sequence.


>
> You should rather try and show the mistake in my objection instead of
> proposing the same argument again and again.


Until you can falsify the original argument, which you have not done,
there is no need to falsify your false argument against it.

As I had put it there:
>
> << an omega-th end-point, a_oo, would necessarily be drawn from an omega-th
> entry of the sequence! Formally, we have the following property:
>
> A m : a_m e (x_n) & b_m e (x_n)
>
> That works not only for n and m in N, but also for n and m in N*. >>



That assumes that one is forced to work with N*, whereas sensible people
work with N and have no problems.
>
> Objection to Cantor's First Proof
> <https://groups.google.com/d/msg/sci.math/T2V4Jh7zzD8/wDM_wsyQZ0QJ>
>
> (Note that mine is against Cantor's First Proof of which yours remains a
> paraphrase, not a faithful reproduction.)


Your alleged argument against the Cantor proof does not work against
either Cantor's proof, nor Zuhair's proof, nor my proof for that matter,
since your N* is irrelevant for all of them.


Date Subject Author
11/12/12
Read Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/12/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/12/12
Read Re: Cantor's first proof in DETAILS
Charlie-Boo
11/12/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/13/12
Read Re: Cantor's first proof in DETAILS
Charlie-Boo
11/15/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
12/1/12
Read Re: Cantor's first proof in DETAILS
Frederick Williams
11/12/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/12/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/12/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/12/12
Read Re: Cantor's first proof in DETAILS
Shmuel (Seymour J.) Metz
11/12/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/13/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/13/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/13/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/13/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/13/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/13/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/13/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/13/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/13/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/13/12
Read Re: Cantor's first proof in DETAILS
Shmuel (Seymour J.) Metz
11/13/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/13/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/13/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/14/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/14/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/14/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/14/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/14/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/16/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/16/12
Read Re: Cantor's first proof in DETAILS
Uirgil
11/16/12
Read Re: Cantor's first proof in DETAILS
Zaljohar@gmail.com
11/16/12
Read Re: Cantor's first proof in DETAILS
LudovicoVan
11/13/12
Read Re: Cantor's first proof in DETAILS
Shmuel (Seymour J.) Metz

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