Date: Jan 12, 2013 5:12 PM
Author: Virgil
Subject: Re: Matheology � 191

In article 
<0b18976d-9651-413f-9acc-4fc7c9f55002@x10g2000yqx.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 12 Jan., 16:24, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
> > WM <mueck...@rz.fh-augsburg.de> writes:
> > > Matheology § 191
> >
> > > The complete infinite Binary Tree can be constructed by first
> > > constructing all aleph_0 finite paths and then appending to each path
> > > all aleph_0 finiteley definable tails from 000... to 111...

> >
> > >         0
> > >       1, 2
> > >   3, 4, 5, 6
> > > 7, ...

> >
> > > This Binary Tree contains aleph_0 * aleph_0 = aleph_0 paths.
> >
> > No, it contains aleph_0 * whatever the cardinality of the set of tails
> > is.  Talk about begging the question!

>
> A tail can be defined by a finite word *only*. Nobody can quote an
> infinite string digit by digit - although most mathematicians believe
> that instinctively when pondering about set theory (but never when
> doing analysis).


Every subset of |N defines the path that branches left at the levels
included in that set and branches right otherwise.
And any tail is identified similarly. And for heads of any finite
length, n, there are uncountably many tails that can be attached.
>
> Therefore every definition of a tail belongs to the set of finite
> words and hence to a set of cardinality less than aleph_0.


That only shows that there are more tails than definitions of them.


(In fact
> during the lifetime of the universe the set of used words will remain
> finite.)
>
> Regards, WM

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