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Re: Another AC anomaly?
Posted:
Dec 16, 2009 9:48 AM
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On 16 Dez., 04:56, "K_h" <KHol...@SX729.com> wrote:
> "WM" <mueck...@rz.fh-augsburg.de> wrote in message
>
> news:9904ae9d-de2c-4d5f-86da-be165e8e9d7e@p30g2000vbt.googlegroups.com...
> On 15 Dez., 13:32, "Dik T. Winter" <Dik.Win...@cwi.nl>
> wrote:
>
>
>
> > Look two lines above:
>
> > Let {1} U {1, 2} U {1, 2, 3} U ...={1, 2, 3, ...}
> > hence
> > {1} U {1, 2} U {1, 2, 3} U ... U {1, 2, 3, ...}={1, 2, 3,
> > ...} .
>
> All limit ordinals, X, satisfy UX=X. That means that the
> union of the members of a limit ordinal is equal to the
> ordinal itself. w=N is a limit ordinal so UN=N. So:
>
> {} U {0} U {0,1,} U {0,1 2,} U ... = {0,1,2,3, ...}
>
> is a true equation.
Therefore 0.000..., as a path in the tree, is nothing but the union of
its finite initial segments. pi is also nothing but the union of its
finite initial segments.
All finite initial segments form a countable set. Nothing else exists.
Hence all paths in the tree form a countable set.
Regards, WM
pi
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