```Date: Nov 13, 2012 4:28 PM
Author: Zaljohar@gmail.com
Subject: Re: Cantor's first proof in DETAILS

On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:> In article <k7u09t\$tj...@dont-email.me>,>>>>>>>>>>  "LudovicoVan" <ju...@diegidio.name> wrote:> > "Zuhair" <zaljo...@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.I showed in the Corollary that even if he use N* as the domain of(x_n), still we can prove there is a missing real from the range of(x_n). So Cantor's argument or my rephrasing of it both can easily beshown to be applicable to N* (any set having a bijection with N) aswell as N.Zuhair
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