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Topic: Matheology § 198
Replies: 40   Last Post: Jan 26, 2013 6:54 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Matheology § 198
Posted: Jan 24, 2013 4:34 AM

On 24 Jan., 09:32, Virgil <vir...@ligriv.com> wrote:
> In article
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > The following is copied from Mathematics StackExchange and
> > MathOverflow. Small wonder that the sources have been deleted already.

>
> > How can we distinguish between that infinite Binary Tree that contains
> > only all finite initial segments of the infinite paths and that
> > complete infinite Binary Tree that in addition also contains all
> > infinite paths?

>
> WM also distinguishes between the union of all finite initial segments
> of |N and |N itself,

Is the set of all terminating decimal expansions of real numbers of
the unit interval not a set? Is it poorly defined? Do not all its
elements exist in ZF? (Not even AC is necessary.) Is it impossible to
distinguish a non-terminating decimal like that of 1/3 from all
terminating decimals?

The infinite set T of all terminating decimals t_i exists. With
respect to ithe indexed digits of the t_i we can do everything that is
done in a Cantor-list, since we cannot distinguish, by indexed digits,
whether we work in T or in |R.

Of course there might be some messages from the God of matheology or
some clairvoyance that is not accessible to non-matheologians. But in
mathematics, there is no difference.

> which is the unary equivalent of his binary tree

Of course. That is what I have been arguing for nearly ten years now.
In order to finish actual infinity of |N you need more than the
natural numbers, which supply only potential infinity. But in the
unary tree the difference is too small (1 element versus infinitely
many) to be recognized by most mathematicians. Therefore if have
finally considered the Binary Tree. Here the difference is uncountably
many versus countably many. And that should be possible to recognize
by many mathematicians, not by all though.

Regards, WM