Date: Jan 12, 2013 12:03 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 191

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).

Therefore every definition of a tail belongs to the set of finite
words and hence to a set of cardinality less than aleph_0. (In fact
during the lifetime of the universe the set of used words will remain
finite.)

Regards, WM