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.