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