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 2:21 PM

On 7 Feb., 20:17, William Hughes <wpihug...@gmail.com> wrote:
> On Feb 7, 7:59 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>

> > On 7 Feb., 19:50, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Feb 7, 7:28 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > On 7 Feb., 19:14, William Hughes <wpihug...@gmail.com> wrote:
>
> > > > > On Feb 7, 5:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > > 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,

>
> > > > > There is no set of correspondences thus there is no number
> > > > > of correspondences.  You cannot know anything about
> > > > > the number of correspondences.-

>
> > > > You are in error again. There is the axiom of power set. For any F,
> > > > there is P such that D e P if and only if D c F. According to it every
> > > > subset of the countable set F exists. Will you dispute that the finite
> > > > definitions of numbers are a subset of F?

>
> > > Yes.  A subset must be constructable.-
>
> > Sorry, we are in classical set theory. There nothing must be
> > constructable.

>
> In classical set theory the accessible numbers are listable
>
> Note from the Wikipedia quote
>

> > Constructively it is consistent to assert the
> > subcountability of some uncountable collections

Of course, the intuitionists accepted this nonsense, perhaps forced by
the matheologians. But that does not imply that the notion is
sensical. In particular, as I wrote already, the real numbers (all
real numbers) are subcountable in constructive maths. But here we are
not concerned with constructive maths, but with ordinary set theory
with its countably many definable real numbers.

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