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Re: Matheology § 210
Posted:
Feb 7, 2013 11:57 AM
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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:
>
> <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
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