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Re: Matheology § 210
Posted:
Feb 7, 2013 9:25 AM
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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.
> 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 can be unlistable and sub-countable, so X can be uncountable
> and sub-countable.
>
> There is nothing contradictory about saying the accessible numbers
> are uncountable and sub-countable.
There is a contradiction (in mathematics, of course there is no
contradiction in matheology because there are no contradictions - it
is surprising that even this word con... can be used in matheology)
since a subset S of the countable set F of finite words bijects with
the set D of definable numbers. F is listable, D is unlistable, i.e.,
uncountable.
So we have |D| = |S| =< |F| < |D|.
Regards, WM
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