Virgil
Posts:
4,674
Registered:
1/6/11
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 4:38 PM
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In article
<2466aecd-5309-4468-b90d-3da7727466f7@f19g2000vbv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 4 Jan., 13:36, gus gassmann <g...@nospam.com> wrote:
> > On 03/01/2013 5:53 PM, Virgil wrote:
> >
> >
> >
> >
> >
> > > In article
> > > <a60601d5-24a2-4501-a28b-84a7b1e53...@ci3g2000vbb.googlegroups.com>,
> > > WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > >> On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote:
> >
> > >>> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*
> > >>> reals r1 and r2, then you can establish this fact in finite time.
> > >>> However, if you are given two different descriptions of the *SAME*
> > >>> real,
> > >>> you will have problems. How do you find out that NOT exist n... in
> > >>> finite time?
> >
> > >> Does that in any respect increase the number of real numbers? And if
> > >> not, why do you mention it here?
> >
> > > It shows that WM considerably oversimplifies the issue of
> > > distinguishing between different reals, or even different names for the
> > > same reals.
> >
> > >>> Moreover, being able to distinguish two reals at a time has nothing at
> > >>> all to do with the question of how many there are, or how to
> > >>> distinguish
> > >>> more than two. Your (2) uses a _different_ concept of
> > >>> distinguishability.-
> >
> > >> Being able to distinguish a real from all other reals is crucial for
> > >> Cantor's argument. "Suppose you have a list of all real numbers ..."
> > >> How could you falsify this statement if not by creating a real number
> > >> that differs observably and provably from all entries of this list?
> >
> > > Actually, all that is needed in the diagonal argument is the ability
> > > distinguish one real from another real, one pair of reals at a time.
> >
> > Exactly. The only reals that matter to Cantor's argument are the
> > *countably* many that are assumed to have been written down. There is no
> > need (nor indeed an effective way) to distinguish the constructed
> > diagonal from *all* the potential numbers that could have been
> > constructed that are not on the list, either.
>
> Therefore they cannot interfere with Cantor's argument and cannot
> result from his procedure.
According to standard mathematics, a set is "countable" if and only if
there is surjection from N to that set. And until WM can produce such a
surjection, none of his claims that the reals are countable count.
>
> > Any *one* number not on
> > the list shows that the list is incomplete and thus establishes the
> > uncountability of the reals
>
> No, it establishes the incompleteness of infinity or the infinity of
> incompleteness.
And until WM can produce a surjection from N to R, none of his claims
that the reals are countable show that the definition of countability is
met.
>
> Cantor's list establishes the uncountability of distinguishable and
> hence constructable reals. Why should nonconstructable and hence
> nondistinguishable reals matter in Cantor's argument?
Until WM can produce a surjection from N to R, none of his claims that
the reals are countable show that the definition of countability can be
met.
>
> Cantor proves the uncountability of a countable set. For some people
> that has an effect like a drug.
And until WM has produced a surjection from N to R, he is just blowing
hot air.
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