Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Matheology � 210
Posted:
Feb 6, 2013 4:56 PM
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In article
<6eb88947-23cc-4a6d-8a0c-3ae873d37ff8@g8g2000vbf.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 6 Feb., 00:31, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <3ffb012f-81a5-410d-8257-f8eee410a...@e10g2000vbv.googlegroups.com>,
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > But how to pick this dark matter of numbers? Only accessible
> > > numbers can get picked. Unpickable numbers cannot appear
> > > anywhere, neither in mathematics nor in Cantor's lists. Therefore
> > > Cantor "proves" that the pickable numbers, for instance numbers
> > > that can appear as an antidiagonal of a defined list, i.e., the
> > > countable numbers, are uncountable.
> >
> > Nonsense!
> >
> > 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 accessible
> > numbers not originally listed.
Thus every accessible number can appear in SOME list of accessible
numbers, but for every such list there is at least one accessible number
not in it.
Thus no surjection from |N to the set of all accessible reals numbers.
>
> Correct. And König proved (or at least knew the fact) that the set of
> accessible numbers is countable.
If countable implies completely listable, then he did no such thing.
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