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Re: Cantor's first proof in DETAILS
Posted:
Nov 14, 2012 5:40 AM
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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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> Zuhair <zaljo...@gmail.com> wrote:
> > On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> > > "Zuhair" <zaljo...@gmail.com> wrote in message
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> > >news:3929e6b6-2932-401d-ba0a-0a440bb18277@y6g2000vbb.googlegroups.com...>
> > > On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:
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> > > <snip>
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> > > >> 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.
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> > > -LV
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> > 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.
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> > 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.
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> It is worse, mathematically speaking, than merely strange, it is
> nonsense.
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> > What you are having is the following:
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> > [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.
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> > [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.
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> ?"Cease"?
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> > [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*.
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> > Is it the definition of Limit.
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> > 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?
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> > 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?
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> > 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.
Actually he is changing the requirement for a proof, i.e. what
constitutes a proof, to him he thinks (apparently) that the question
involves high order principles that cannot be met by using N as a
domain, But he didn't give any line of justification for that.
Zuhair
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