Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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 11:57 AM

On 7 Feb., 15:56, William Hughes <wpihug...@gmail.com> wrote:
> On Feb 7, 3:25 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> <snip>
>

> >... a subset S of the countable set F of finite words bijects with
> > the set D of definable numbers

by definition.
>
> Nope.   Every D corresponds to some finite word.

No, D is a set or at least a collection. A definable number is an
element of D.

>  However, S,
> the collection of all the correspondences, may not be a subset
> of F  (subsets must be computable).

Need not be a subset. It is sufficient to know that there are not more
than countably many correspondences, because the upper limit would be
"all elements of F".

>  Hence we know D is
> sub-countable, but we do not know D is countable.

You intermingle elements D and sets D in a really refreshing way.
Nevertheless, we know that all correspondences together cannot
surpass aleph_0.

According to this definition the real numbers are subcountable. All
paths of the Binary Tree cross every finite level in a finite number
of nodes. Aleph_0 is not reached at any finite level - and there is
nothing, in particular no path, beyond every level. Nevertheless there
is no list of all paths.

A typical case of subcountability.

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