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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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 Virgil Posts: 870 Registered: 7/27/09
Re: Another AC anomaly?
Posted: Dec 16, 2009 1:53 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 16 Dez., 04:56, "K_h" <KHol...@SX729.com> wrote:
> > "WM" <mueck...@rz.fh-augsburg.de> wrote in message
> >
> > 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.

If nothing else exists, then those unions of finite initial segments
which represent other than binary rationals do not exist, and thus more
reals do not exist.

> Hence all paths in the tree form a countable set.

Not if it is a COMPLETE infinite binary tree, which necessarily contains
infinite strings not in WM's trees.

Consider this tree:

Each n in the set of all naturals , N, is a node.
For each such node, n, in N, its left and right children are,
respectively, 2*n and 2*n+1.
A path, P, is, by definition, a subset of N such that 1 is a member of
P, and, for each n in P, exactly one of 2*n and 2*n+1 is a member of P.

The set of all such paths is easily shown, by a version of the Cantor
diagonal argument, not to allow bijection with N.

Thus the set of all such paths is demonstrably uncountable.

Date Subject Author
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11/27/09 Virgil
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11/30/09 Dik T. Winter
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11/30/09 Virgil
12/1/09 Dik T. Winter
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12/2/09 Dik T. Winter
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