Date: Nov 14, 2012 2:18 AM
Author: Uirgil
Subject: Re: Cantor's first proof in DETAILS

In article 
<6a63fbfd-f7e7-458f-af65-fae2c805c951@d17g2000vbv.googlegroups.com>,
Zuhair <zaljohar@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"?
>
> [3]However we also have the corollary that even when we use N* as the
> domain of those functions, still we can by a single common well
> defined way define another sequence with exactly the same range of
> those functions but from domain N, and we can apply Cantor's argument
> and SHOW a missing real in the rang of those functions!
>
> Now you call [1] deficient, [2] apt to reality standards [3] bogus.
>
> Why? because we used N in an argument that involves a higher order
> concept that must use N* instead. (That's your reply).
>
> But again: why? what is the higher order part of the argument that you
> see it demanding circumventing the heart of the subject (which is N
> really) to some N*.
>
> Is it the definition of Limit.
>
> But limit is defined in this argument as the least upper bound, and I
> don't see in the definition of L that I wrote (which is the standard
> by the way) anything that has to do with necessarily picking it up
> from some Omega_th end point? that has no meaning at all, so why?
>
> Should I adopt this rational of yours then I'd ask you: why not say
> pick L from the -1_th starting point. i.e. choose your domain to be
> {-1,0,1,2,3,...} since this clearly also preclude Cantor's argument
> and you clearly can make L be the -1_th digit of (x_n) [Remember a_0
> is x_0, so x_{-1} lies "before" a_0].
>
> Or you'll say that {-1,0,2,3,...} is also a kind of high order
> countable set?
>
> Your argument is simply shunning one of the most important two sets in
> this argument, that is N, and using some replacement, without any
> clear justification.
>
> Zuhair


Right! The fact that one can make the proof seem false by changing it
does not make the original proof false.