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Topic: Matheology 203
Replies: 16   Last Post: Feb 7, 2013 8:06 AM

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mueckenh@rz.fh-augsburg.de

Posts: 15,033
Registered: 1/29/05
Re: Matheology 203
Posted: Feb 7, 2013 3:54 AM
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On 7 Feb., 09:27, Virgil <vir...@ligriv.com> wrote:
> In article
> <30e6f8dd-f487-4335-ba77-35f182b79...@e10g2000vbv.googlegroups.com>,
>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 7 Feb., 08:53, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <457d0429-33c1-46ad-9151-4f5d9dc96...@fv9g2000vbb.googlegroups.com>,

>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 7 Feb., 05:21, fom <fomJ...@nyms.net> wrote:
>
> > > > > Or as Virgil would write
>
> > > > in a very lucid moment:
>
> > > > > What Cantor proved was that no list of accessible real numbers
> > > > > (accessible because listable) can include all accessible numbers,
> > > > > because any such list itself proves the existence of numbers not listed.

>
> > > > That is, Cantor proved the countable set of accessible numbers being
> > > > uncoutable.

>
> > > That may well be WM's misunderstanding but it is not an understanding.
>
> > > A number being accessible does means that it can appear in some list,
> > > but does not at all mean that all accessible numbers can appear together
> > > in a single list.

>
> > All elements of countable sets can be counted by definition, i.e.,
> > they can appear in a list.

>
> But some subsets of a set may be countable even though the set itself is
> not.


Of course, the algebraic real numbers for instance, or the definable
real numbers.

> Certainly SOME sets of reals can be counted but not every set of
> reals can be counted.


My question has been this: Why are there countable sets that are
uncountable?

Regards, WM




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