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Re: UNCOUNTABILITY
Posted:
Dec 22, 2012 4:06 PM
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On Dec 22, 8:55 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 22 Dez., 00:34, William Hughes <wpihug...@gmail.com> wrote:
>
> > > Yes it is definable. It has been defined. Nevertheless it does not
> > > exist, because the sets do not exist.
>
> > It is definable which means that WM can use it to prove that the
> > bijection exists, but it does not exist.
>
> The bijection of all finite words with all natural numbers has been
> defined in binary:
However, the bijection you need is all definitons with a subset
of the natural numbers. And there is no way to define this subset.
If you put restrictions on the 0/1 sequences you allow to exist you
put restrictions on the subsets you allow to exist. Note, that
subcountable
does not mean countable.
>
> 0
> 1
> 00
> 01
> 10
> 11
> 000
> and so on.
>
> From this definition the natural number belonging to any desired
> finite word can be obtained. It can easily be translated into any
> other language. Or do you need some help?
>
> Nevertheless there is no set of all natural numbers and no set of all
> infinite words.
>
> It is the same with pi. The (potentially) infinite string of digits of
> pi can be defined. In fact there are (potentially) infinitely many
> definitions. Nevertheless there is no actually infinite string
> expressing pi.
>
> Yes, I know that is not easy to understand. That's why so many
> mathematicians have gone astray.
>
> Regards, WM
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