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Re: Matheology § 210
Posted:
Feb 7, 2013 1:28 PM
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On 7 Feb., 19:14, William Hughes <wpihug...@gmail.com> wrote:
> On Feb 7, 5:57 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
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> > On 7 Feb., 15:56, William Hughes <wpihug...@gmail.com> wrote:
>
> > > On Feb 7, 3:25 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
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> > > <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? Not even the axiom of choice
is required to prove that.
Regards, WM
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