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Topic: Matheology § 210
Replies: 80   Last Post: Feb 8, 2013 5:45 PM

 Messages: [ Previous | Next ]
 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Matheology § 210
Posted: Feb 7, 2013 9:25 AM

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.
>
> are uncountable and sub-countable.

There is a contradiction (in mathematics, of course there is no
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

Date Subject Author
2/5/13 mueckenh@rz.fh-augsburg.de
2/5/13 fom
2/5/13 mueckenh@rz.fh-augsburg.de
2/5/13 William Hughes
2/5/13 mueckenh@rz.fh-augsburg.de
2/5/13 William Hughes
2/5/13 Virgil
2/5/13 fom
2/5/13 Virgil
2/5/13 fom
2/6/13 mueckenh@rz.fh-augsburg.de
2/6/13 fom
2/6/13 mueckenh@rz.fh-augsburg.de
2/6/13 Virgil
2/6/13 fom
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 Virgil
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 Virgil
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 William Hughes
2/8/13 mueckenh@rz.fh-augsburg.de
2/8/13 Virgil
2/8/13 fom
2/8/13 mueckenh@rz.fh-augsburg.de
2/8/13 Michael Stemper
2/8/13 mueckenh@rz.fh-augsburg.de
2/8/13 Virgil
2/8/13 Virgil
2/7/13 Virgil
2/7/13 Virgil
2/7/13 fom
2/8/13 mueckenh@rz.fh-augsburg.de
2/8/13 Virgil
2/7/13 Virgil
2/7/13 fom
2/7/13 Virgil
2/7/13 fom
2/7/13 Virgil
2/8/13 mueckenh@rz.fh-augsburg.de
2/8/13 Virgil
2/7/13 Virgil
2/7/13 fom
2/7/13 Virgil
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 fom
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 fom
2/7/13 Virgil
2/8/13 mueckenh@rz.fh-augsburg.de
2/7/13 Virgil
2/7/13 fom
2/7/13 fom
2/7/13 fom
2/7/13 fom
2/7/13 fom
2/7/13 fom
2/6/13 Virgil
2/5/13 Virgil
2/6/13 mueckenh@rz.fh-augsburg.de
2/6/13 Virgil
2/7/13 mueckenh@rz.fh-augsburg.de
2/7/13 Virgil
2/8/13 Scott Berg