Date: Nov 14, 2012 3:28 PM Author: Uirgil Subject: Re: Cantor's first proof in DETAILS In article

<192db05a-9b98-4df0-ac24-d2b607a1a099@n5g2000vbk.googlegroups.com>,

Zuhair <zaljohar@gmail.com> wrote:

> On Nov 14, 10:18 am, Uirgil <uir...@uirgil.ur> wrote:

> > In article

> > <6a63fbfd-f7e7-458f-af65-fae2c805c...@d17g2000vbv.googlegroups.com>,

> >

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> >

> >

> >

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> > Zuhair <zaljo...@gmail.com> wrote:

> > > On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote:

> > > > "Zuhair" <zaljo...@gmail.com> wrote in message

> >

> > > >news:3929e6b6-2932-401d-ba0a-0a440bb18277@y6g2000vbb.googlegroups.com...>

> > > > On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:

> >

> > > > <snip>

> >

> > > > >> 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 be

> > > > > shown to be applicable to N* (any set having a bijection with N) as

> > > > > well as N.

> >

> > > > You are simply missing the point there: we don't need N* to disprove

> > > > Cantor,

> > > > we need N* to go beyond it and the standard notion of countability. In

> > > > fact, that there is a bijection between N* and N is a bogus argument

> > > > too,

> > > > as

> > > > the matter is rather about different order types.

> >

> > > > -LV

> >

> > > Now I think I'm beginning to somewhat perhaps understand your

> > > argument. I think (I'm not sure though) that what you want to say is

> > > that when we are having arguments with "LIMITS" then we must design

> > > the whole argument such that the Limit comes from the sequence, and if

> > > this design was not made then the argument is inherently deficient as

> > > far as the truth of inferences derived from it is concerned. So what

> > > you are trying to say is that Cantor's argument began with incomplete

> > > arsenal so it ended up with misleading inferences. You are making an

> > > argument at TRUTH level of the matter, and yet it is concerned with

> > > formal technicality as well, which is an argument beyond the strict

> > > formal technicality.

> >

> > > Anyhow if I'm correct, this form of reasoning for it to stand the

> > > quest, then there must be a clear line of justification for it. For

> > > instance the argument about whether the reals are countable actually

> > > means literally whether there is a bijection between the reals and N,

> > > so N is at the heart of the subject. Now to go and say that

> > > countability of the reals (which means bijectivity of reals to N) can

> > > only be reached about by circumventing N and using another countable

> > > infinite set N* as the domain for any sequence in an argument using

> > > limits is really strange somehow.

> >

> > It is worse, mathematically speaking, than merely strange, it is

> > nonsense.

> >

> >

> >

> > > What you are having is the following:

> >

> > > [1]When we use N as the domain of injections (x_n), (a_n) and (b_n),

> > > then Cantors argument PROVES and SHOWS that there is a real that is

> > > not in the range of those functions.

> >

> > > [2]When we use N* as the domain of injections (x_n), (a_n) and (b_n),

> > > then Cantor's argument will seize from working in the same way to show

> > > the missing real.

> >

> > ?"Cease"?

> >

> >

> Yes, Cease, i.e. stop, of course I'm speaking about stopping in the

> sense of running the exact particulars of the argument per se, that's

> why I said "...in the same way" for example when you use some N* which

> has an omega_th position as the domain then for example Result 7

> cannot be proven in exactly the same straightforwards way as it is

> proved with N, to prove it you need to define it indirectly in terms

> of bijections from N* to N ...., which is a long way. But ultimately

> you will also succeed in finding a missing real as I pointed out. That

> is merely a temporary conundrum with the argument that has no

> significance to the reality of the matter, and has no philosophical

> value whatsoever.

>

> Zuhair

You wrote "seize". I was merely asking if you meant "cease".