Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Matheology � 210
Posted:
Feb 7, 2013 7:20 PM
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In article
<f5d18304-0606-4ecc-b446-ba95199ed728@z4g2000vbz.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 7 Feb., 15:13, William Hughes <wpihug...@gmail.com> wrote:
> > On Feb 7, 2:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 7 Feb., 09:10, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > On Feb 7, 9:00 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > <snip>
> >
> > > > > What does that mean for the set of accessible numbers?
> >
> > > > That this potentially infinite set is not listable.
> >
> > > Here we stand firm on the grounds of set theory.
> >
> > > Once upon a time there used to be a logocal identity: The expression
> > > "Set X is countable" used to be equivalent to "Set X can be listed".
> >
> > You say, X is countable means there are not more X than any
> > natural number.
>
> No. X is countable means there is a bijection of X with N.
That is being countably infinite, but countable includes being countably
finite, certainly a set like {1,2,3} is countable, so all that is
required is a surjection from |N to X
>
> > The standard term for this is "sub-countable".
> > Standard terminology is that X is countable iff X is listable.
>
> X is countable if and only if there is a bijection of X with N.
X is COUNTABLY INFINITE if and only if there is a bijection of X with N.
X is COUNTABLE if and only if there is a surjection from N to X.
At least everywhere outside of Wolkenmuekenheim .
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